Some Estimates of Fourier Transforms Citation Kovrijkine, Oleg E. (2000) Some Estimates of Fourier Transforms. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/0p2k-ah86. https://resolver.caltech.edu/CaltechTHESIS:11212019-172159302 Abstract This work consists of two independent parts. In the first part we prove several results related to the theorem of Logvinenko and Sereda on determining sets for functions with Fourier transforms supported in a parallelepiped. We obtain a polynomial instead of exponential bound in this theorem, and we extend it to the case of functions with Fourier transforms supported in the union of a bounded number of parallelepipeds. When dimension d = 1 we also consider the case of infinitely many lacunary intervals. We generalize the Zygmund theorem for lacunary series whose Fourier coefficients are replaced with polynomials of uniformly bounded degree. We give also a necessary condition for the support of Fourier transforms for which the Logvinenko-Sereda theorem still holds. In the second part we prove that the L ²([0,1] d x SO ( d )) norm of periodizations of a function from L ¹(ℝ d ) is equivalent to the L ²(ℝ d ) norm of the function itself in higher dimensions. We generalize the statement for functions from L p (ℝ d ) where 1 ≤ p < (2 d )/( d + 2) spirit of the Stein-Tomas theorem. We also show that the following theorem due to M. Kolountzakis and T. Wolff does not hold if dimension d = 2: if periodizations of a function from L ¹(ℝ d ) are constants, then the function is continuous and bounded provided that the dimension d is at least three. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Mathematics Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Wolff, Thomas H. Thesis Committee: Wolff, Thomas H. (chair) Keel, Markus Luxemburg, W. A. J. Makarov, Nikolai G. Defense Date: 12 May 2000 Other Numbering System: Other Numbering System Name Other Numbering System ID UMI 9972616 Funders: Funding Agency Grant Number Caltech UNSPECIFIED Record Number: CaltechTHESIS:11212019-172159302 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:11212019-172159302 DOI: 10.7907/0p2k-ah86 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 13594 Collection: CaltechTHESIS Deposited By: Mel Ray Deposited On: 22 Nov 2019 01:41 Last Modified: 16 Apr 2021 16:42 Thesis Files Preview PDF - Final Version See Usage Policy. 2MB Repository Staff Only: item control page

Som e estim ates o f Fourier transform s Thesis by- Oleg E. Kovrijkine In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2000 (Submitted May 12r 2000) R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . A cknow ledgem ents I am deeply grateful to my advisor Professor Thomas Wolff for his support and supervision during my research. It has been a great pleasure to work under his guidance. I would like to thank also W.A.J. Luxemburg, N. Makarov, B. Simon and A. Soshnikov for useful discussions and suggestions. Finally, I would like to thank the Mathematics Department of the California Institute of Technology for giving me an opportunity to conduct my research and for supporting me during my study. R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . A bstract This work consists of two independent parts. In the first part we prove several re­ sults related to the theorem of Logvinenko and Sereda on determ in in g sets for func­ tions with Fourier transforms supported in a parallelepiped. VVe obtain a polyno­ mial instead of exponential bound in this theorem, and we extend it to the case of functions with Fourier transforms supported in the union of a bounded number of parallelepipeds. When dimension d = 1 we also consider the case of infinitely many Iacunary intervals. We generalize the Zygmund theorem for lacunary series whose Fourier coefficients are replaced w ith polynomials of uniformly bounded degree. We give also a necessary condition for the support of Fourier transforms for which the Logvinenko-Sereda theorem still holds. In the second part we prove that the L2 ([0, l]d x SO(d)) norm of periodizations of a function from L l(Rd) is equivalent to the L2(Rd) norm of the function itself in higher dimensions. We generalize the statement for functions from Lp(Rti) where 1 < p < - ^ 2 i*1 spirit of the Stein-Tomas theorem. We also show that the following theorem due to M . Kolountzakis and T. Wolff does not hold if dim ension d = 2: if periodizations of a function from L l (Ra) are constants, then the function is continuous and bounded provided that the d im ension d is at least three. R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . iv C onten ts A ck now ledgem ents ii A b stract iii 0.1 Introduction...................................................................................................... 1 S om e resu lts related to th e L ogvinenko-Sereda th eorem 2 O verview............................................................................................................ 2 1.2 Case of one parallelepiped........................................................................... 5 1.3 Case of finitely many p arallelep ip ed s........................................................ 16 1.4 Case of infinitely many lacunary intervals 22 1 .1 2 1 ............................. 1.4.1 Conjectured th eo rem ....................................................................... 22 1.4.2 Large E, small su p p ort.................................................................... 23 1.4.3 Periodic E ........................................................................................... 27 1.4.4 Generalized lacunary series.............................................................. 33 1.5 Necessary condition for su p p o r t.................................................................. 53 P erio d iza tio n s o f fu n ction s 59 2.1 O verview.......................................................................................................... 59 2.2 Case p = 1 ..................................................................................................... 62 2.3 Case 1 < p< ^ 73 2.4 Case d = 2and p = o o ........................... ............................................................................................ B ib liograp h y R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 84 89 0.1 Introduction This work consists of two independent parts. We use various estimations of Fourier transforms to obtain results. The first part considers the Logvinenko-Sereda theorem which is one of the basic examples of the so-called Uncertainty Principle in Fourier Analysis. This Principle states that a function and its Fourier transform can not be simultaneously supported on “small” sets. Although this formulation is rather vague, we obtain rigorous re­ sults. Intervals and compliments of “relatively dense” subsets play the role of small sets in our situation. The Logvinenko-Sereda theorem is based on properties of entire functions of exponential type. We will improve on this theorem by getting an optimal estimate and generalize it for the case of finitely many intervals. We will also inves­ tigate the case of infinitely many lacunary intervals. We will generalize a result of F. Nazarov for the Zygmund theorem on lacunary series. We will also give a necessary condition for the support of Fourier transforms for which “relatively dense" subsets are still determining sets. The second part is devoted to periodizations of functions from Ll (Rd) in higher dim ensions. Some results on the Steinhaus tiling problem due to M. Kolountzakis and T. Wolff are related to m ine since periodizations naturally appear in the problem of Steinhaus. The main idea can be formulated in this way. If we are given some information on periodizations, what can we say about the function Itself and vice versa? It is rather natural to formulate this problem in terms of various norms. Using some facts from Number Theory, we prove that the L~{\0, l}rf x SO{d)) norm of periodizations of a function from L l (Rd) is equivalent to the L2 (Rd) norm of the function itself in higher dim ensions. We generalize the statem ent for functions from Lp(Rd) where 1 < p < ^ in the spirit of the Stein-Tomas theorem. We will also show that the result due to M. Kolountzakis and T . Wolff, which holds when dimension d > 3, does not hold when d —2. R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 2 C hapter 1 Som e resu lts related to th e L ogvinenko-Sereda theorem 1.1 Overview The purpose of this work is to study the behavior of functions whose Fourier trans­ forms are supported in a parallelepiped or in a union of finitely many parallelepipeds on “thick” subsets of Rd. A main result of this type was proven by Logvinenko and Sereda. By a “thick” or “relatively dense” subset of Rd we mean a measurable set E for which there exist a parallelepiped II with sides of length ax,a<i, —.ad parallel to coordinate axes and 7 > 0 such that |£ ? n ( n + r ) |> 7 |n| (i.i) for every x 6 Rd. T h e L ogvinenko-Sereda T heorem , d = I: let J be an interval with |J | = 6 . If f € LP(R), p 6 [1,+oo{, and supp f C J and if a measurable set E satisfies (1.1), then ll/IUfi) > e * p (-c • - I/ll,. (1.2) It is a well-known fact that the condition (1.1) is also necessary for an inequality R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 3 of the form WflLP(B) > C - \\f\\p to hold. See for example ([7], p.113). We will improve the estim ate (1.2) by getting a polynomial dependence on 7 and show that our estimate is optimal except for the constant C: T heorem 1: let J be a parallelepiped with sides of length parallel to coordinate axes. If f € Lp(M.d), p 6 [1 , + 0 0 ], and supp f C J and if a measurable set E satisfies (1.1), then d f y \\f\Um > ( £ ) ° j^ j) • ll/llp - We will also generalize the Logvinenko-Sereda theorem to functions whose Fourier transforms are supported in a union of finitely many parallelepipeds: T heorem 2 : let Jk be identical parallelepipeds with sides of length 6 ^,60 , ..., 6 ^ * n parallel to coordinate axes. If f € Lp, p € [1 . + 0 0 ]. and supp f C (J«4 and if a 1 measurable set E satisfies (1.1), then WfWmB) > c(7, n, a - b , d,p) • ||/ ||p f Qd\ where c(7 , n, a - b, d,p) = ( J depends only on the number of paral­ lelepipeds but not how they are placed. The next natural step is to consider the case of infinitely m a n y parallelepipeds. We will conjecture the following theorem for infinitely many lacunary intervals when dim ension d = 1: R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 4 T heorem 3: let A = {A fc}*!^ be lacunary and let E be “relatively dense." If / 6 L2 and supp f C (J [A* —| TA* -F1], then k=—oo ll/IU=(E) > C ( 7 ,iV ,a , 6 ) - | |/ | | 2. See section 1.4.1 for the definition of lacunarity. We will prove some partial results: the above T heorem holds when 7 is large or when b is small. It holds too when E is periodic. We will also generalize Nazarov’s result for the Zygmund theorem on lacunary series ([13], Theorem 3.6): N azarov’s T heorem : let A = {Afc} ^ _ 00 be a lacunary sequence o f integer num­ bers with the maximal number of representations N . I f f € L2([0 ,27r]) with specf 6 A then [ \f\2 > C (\E \,N )\\f\\l Je with C’( |£ |, iV) = e i£i;t+* fo r every E with positive measure. We extend the above theorem to generalized lacunary series with Fourier coeffi­ cients being replaced by polynomials of uniformly bounded degree: T heorem 4: let I be an interval of length 1 and E be a subset of I o f positive meaOO sure. I f A = is lacunary in the sense of (1.29) and f(x ) = k=— oo where pfc(x) are polynomials o f degree at most m , then f l/l 2 > C (|£ |,J V ,m ) e J |/|2 r where C (\E \,N ,m ) = e-(k™ )^(j§F)CCm+I\ R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . Pk{x)eiXkX 5 Note that the constant C below is not fixed and varies appropriately from one equality or inequality to another without being mentioned. 1.2 C ase of on e parallelepiped P r o o f o f T heorem 1: First we treat the case when p 6 [l,+ o o ). Without loss of generality we can always assume that J is centered at the origin: By considering /(fj-r—i f^) instead of f , we can also assume that \E fl II| > 7 for all cubes II with sides of length 1 and parallel to coordinate axes and just say Define 6 = (6 i,...,&<*). Bernstein's inequality ([2], Theorem 11.3.3) applied to each variable separately gives that with C = Here a = (o-i, ...ra<f) is a multiindex in a! = a il *... *a j.. with nonnegative integer i=1 Divide th e whole Rd into cubes II with sides of length 1 and parallel to coordinate R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 6 axes. Choose A > 1. Call a cube II bad if 3 a 5^ 0 such that J | / (a)r > 2 dA Wp( C *6)“** • n J I/I1 . , ,p n Then. / i/r ^ (“ ) ip U n u n “*° n 14 bad = y I [ _ [I 14 bad 2dAl°lp(C • 6)qp “ ^° |/( J u 1 n IT 14 b a d - £ 2‘‘/1I«Ip ( C • t j o f a^O ^ (1.3) Choosing A = 3 and applying (1.3), we obtain J i/r<f|i/r U n Cl 14 b a d Therefore, J I/I”>5/ I/I”U Cl m goad (1-4) n Replace 3C with C. We claim that 3B > 1 such that if II is a good cube, then 3x € II with the property that If {a) (*) |p < 4* *B ,a|p(C -,6 )“? - J \ f \ p Va. n R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . Suppose towards a contradiction that this is not true. Then ■/ i/ip * £ m & n “ l c Vl e n - - Integrate both sides of (1.5) over II: **■/l/r - ^“ W c ^ F n/ l/fa’(l)|P n * E = ( i- ( W b k / w' n / l/l'’' Choose B = 3 and apply (1.6). So Z*- / I/I” < W ) d J I/I". n n This contradiction proves our claim. Replace 3C with C. We will need to prove the following local estimate: snn n for every good cube II. W ithout loss of generality we can assume that by considering a shift / ( x —n ) which has suppf(a: - n) C [ - j , j \ x ... x [ - j , R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 8 Let z = (zlr..., zd) E C* be a complex vector. Let D j(0, R) = {zj E C : \zj\ < i?} be a disk in the complex plane C. If z E D i(0, R) x — x Dd{0, R) C Di{x \ , f? + —) x ... x Dd{xdl R + —) then i/ m i < E O'! a E 4 "— 5 a Ql li/I lf ^ = 4?exp(C(fl + -)E f'i)-||/llw (n ). (1-7) “ j=l We can assume that/ n | / | p = L Then 3y E II such that \f(y)\ > 1.Following an idea of F.Nazarovwe can choose spherical coordinates centered at y and find a segment / E II, y E / and p jp > C(d) 7 : 7 < |£ n n | = f X E n n (x)d x n r(0 = JJ XEnu{y + r£)rd- ldrda{€). ( 1 .8 ) l€l=lr=0 It follows from (1.8) that 3r] E Sd~l such that r(.v) 7 < <fd-i J XEna(y + rq)rd ldr r= 0 where = |S d 1[. Let I be the longest interval in II centered at y in the direction of 77: r = y+t\I\n, 0 < £ < 1. R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . (1.9) 9 It is clear that |/| < d1^2. Therefore, \EM \ |/| > 7 > - -L C* It follows from (1-10) that \ { t £ [ot i l : y + t\i\r] e E n / } | = \Eni\ \t\ 7 Cd' (1.11) Define an analytic function of one complex variable w E C F{w) = f ( y + w\I\g). (1.12) Then |F (0)| = |/(y )| > 1. If w E D{0,R) then z —y + w \I\t} E Di(0, R 4- —) x ... x Dd(0, R + —) since \Uj\ < ? and \\I\rjj\ < 1 for j = 1 , ...,d. Then it follows from (1.7) that i |F(u/)| < 4pexp(C (i?-f 1) S > ) . i=i (1-13) Now we will give a local estimate for analytic functions of one complex variable. L em m a 1: Let (p(z) be analytic in D (0,5) and let I be aninterval of length 1 such that 0 E l and let E C I be a measurable set of positive measure.If |0(O)| > 1 and M = max [0 (2 ) 1, then |z |< 4 / c \ « sup [0(ar)I < ( r ~ r I xer Vl-fc' l / s^P \<t>(x ) txeE R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . (L14) 10 P r o o f o f Lem m a 1 : Let W]_, wo,...wn be the zeros of <p in D {0,2). If glz) = Mz) ■TT 4 ~ a k Z = <Kz) ■^ SI 1 W ; 1 1 % - : ) w ; P (z) then. |ff(0) | > 1 and max |ff(z)| < M by the property of Blaschke products. Applying kl<2 Hamack’s inequahty to the positive harmonic function IniV/ —In |ff(z)| in D( 0,2) we have: max(InM —In|ff(z)|) < 3 In M . Therefore, m in|o(z)| > M 2. W<i “ This gives us max (x) | nun (ar) | 16/ < M 3. We can give a similar estimate for Q: n max|Q(ar)| X 61 max n K - wkz\ k = l ________________ < < min n K - wkz\ M^ 1 fc=i 3”. Prom the Remez inequality for polynomials ([3], Theorem 5.1.1) it follows that: sup |P(ar) [ < ( x€/ * V I* !/ sup |P(ar) |. xeE R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 11 Therefore, m ax|P (x)| s u p |0 (a :)| re i < m a x\g {x)\ ■ m m |c j(x )| xe/ s u p |P ( x ) | < M 3 *3n - -= -] \\E \J - m i n |5 ( x ) | - ^ — — — xei m a x |Q ( x ) | x6 / - M3'( li) ' S Wx)|From Jensen's formula it follows that n < Therefore, sup 10(x) | < ( y^ r) sup 1<p(x) x€/ x€£ V l^ l/ □ Now we are in a position to proceed with the proof of our theorem. Let M = max | F(w) \. Applying Lem m a 1 to F(w), interval [0, ll and set {£ 6 [0, l i : y+ t\[\ri 6 H <4 E d 1} and using ( 1 . 1 1 ), we have that Enn 1/fc ) I > s u p |/(x )| Eni > ( i ^ L ) “ fl/lk-tm- Therefore, |{* £ n : l/W I < ( ^ ) ^ ll/H £ptn)}| < e Vc > 0. If we put e = ■—^n- then |{ * e n : I/W I < ^ ll/ll" « o }l < R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 12 Therefore, enn snn LP(U) >_ In ilf . t 2 (EQH)’" w . n Using (1.13) we get d M < 4? exp(5C ^ ^ b j). j=i Hence we obtain the desired local estimate d / / *y \ 3rf+Cp F If \ p > ( ^ ) £nn r J \f\p n (1.15) for every good cube n . Summing (1.15) over all good cubes and applying (1.4), we have J i/i” > J £rn u i/i’ n I I is good d / -y \ 3d+Cp £ bi > ( £ ) « r • / U f l is good I/I' n R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 13 d Replacing ^ i= i d with ^2 a.jbj and choosing a new C, we have: j= i / <t / \ 3d+Cp £ ‘h h r - J I/I' I/I" > ( ^ ) If p = oo then the proof is almost the same: ||/||l«>( y n) = ll/lloo- If II is good IT i s good 2d-rC 53 djbj then | | / | | L ~ ( B n n ) > ( ^ ) 1=1 - l ! / I U ° ° ( n ) - Hence / <y \ 2d+C 52 d-jbj ll/llt-w n > ( £ ) • ll/ll □ If we keep track of all the constants and do the calculations more accurately, then we can get that if d = 1 and p 6 [1 , oo): \ \ / \ \ l p {E) > ( 300) ' H/ilp> if p = 00: / "v \ 33a6-fl II/IIl-(s, > ( ^ 0) • 11/ 11=0. However, if we try to minimize the factor in front of a b r then we can get the following estimate: / >7 \ C +t)-a6+/l(e) \\f\\LP {E) > ' ll/llp Ve > 0. The following example suggests that the right behavior of the estimate in the d Logvinenko-Sereda Theorem is 7 to the power of a linear function of Let J be a parallelepiped with sides of length 61 , a.jbf i=i bd parallel to coordinate axes and let II be a cube with sides of length 2 parallel to coordinate axes. Denote d b = (&!,...,&<f). Let T = P ro jb (lt 1, —, 1) = ^ bj/\b\. Choose another system of j=i R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 14 coordinates yi, ...,yd with, axis y\ parallel to b . Define f{y) = fi(yi)iP(y2,-,yd) where Aim) = * X [ - ^ ,^ ] ) ( y x ) d with the number of convolutions equal to [|b |T /8 7 r] = fiT bj/8ir\ and ip is supported i= i in a small enough ball in R4*-1. Then / lives in a cylinder with, axis along the main diagonal of J parallel to b with small enough radius so that it is inside of J . We have /s in ( 2 7 ry i/T ) f(y) = f ^ - ) - <%2, . yd). Let E = Ei x R4*- 1 where Ei is a periodic subset of yi axis with period T: £ .n ( 4 || = [ 4 4 + | l n [ | - 2 ||. Then \e n ( n + x) | > C(d)7 |n | v* e Rd. d If bj is large enough we have: j= i d \ —1+ £ 6 j/8 ir ||/ l ||L J > ( £ t ) < J= l ll/lllf and therefore \\f\\L P (E ) = W fl\\LP {E t )\bP\\p d < (£ ) II/i IIpM , / *7 \ — H 6j/ 8ir - © - w- R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 15 <t Rem ark 1: when ^ a.jbj is sufficiently small the proof of the theorem is much j=i d i simpler: if $3 o-jbj < 1/p then \ \ / \ \ l p (E) > (c ) HI/IIp for certain p. This cam be proven j= i very easily. If p = 1 we have |n |- |/ M I [\!\- > Y n (i-i6) atO-.a,<l ft and |H| • |/(x )| < A / I + Y ft a° [ l ^ “’l a # 0:Qj< l ft where x € II, by induction on the dimension d . Hence, using (1.16), we have ini- f \nx)\dx > 7 |n| ( / \n Enn Y I/I - q^0:oij-1 n Therefore. i • f \f(x)\dx > A / I - Y Son n f l / ‘' a # 0 :Q j< l n Summing over all parallelepipeds II we have l - / i / i > / i/i- Y 1 JE J a?Q:a j<l > A /iJ = Y a^O:ctj ( 2 - I J (1 + “ jV J (? )” “” / i/i <1 ^ 2 )) / i= l > ( 2 —ei=I > c j I/I- I/I 7 ) J |/ | R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . f1-17) 16 If p is an integer then repeat the previous argument with j* instead of / and take into account that supp f* E p - J . We can also repeat the above argument when p > d where p is not necessarily an integer since if jo:| < p then f < (p6 / 2 )“ / | / p| (combine Holder’s and Bernstein’s inequalities to get it). It is even easier to prove d that if p = oo and £ ctjbj < 1 , then ||/|U «(£) > 5 II/II00 j= L Alternatively, using Holder’s inequality, we can obtain / i/i^^r/ i/r- ( E Ena “ n Va ^ 0:Q^ 1 n / Put ca = a“/ 2 (6/2)a/l2. Then we are done provided (eJ=l -( E ^ • / —l )2 < It shows that if £ cijbj < C then ||/||LP{E) > 3 r ||/||p for p e [I, 0 0 J. j= 1 In a similar way using inequalities analogous to (1.17) we can treat the case when 1—7 d -p (C + £ a jbj/2) is sufficiently small depending on ^ a^bf if p 6 [l,oo) and 1 — / < e ■>=l then \ \ f \ \ pLP{E) > k \ \ f \ \ $ ‘ When d = 1 we get better results. 1.3 C ase o f fin itely m any parallelepipeds P roof o f T h eorem 2: Let y = 1r _ 2h , 2h | X _ . X [1 - h2 , -2l . Then Jk = J + Xk where Ak 6 k = 1 , 2 ,..., n. Denote by Jfc — 2 J -F A*, fc — 1 ,2 ,..., 71. Fust we will prove a special case of T h eorem 2: R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 17 T h eo rem 2': ifJk, k = 1 , 2 , are disjoint then ||/| \ l p (E) > c ( 7 , n , a - M , p ) nJ llp where 0 (7 , n , a - b,d,p) = ( ^ ) ab^ 7 ) +n p . P r o o f o f Theorem 2': Let / ( x ) = 52 A ( x - A fc) where supp/fe C «/ and f (x) = fc=L f k(x)e'Xic'*-. The following fc=l lemma gives an estimate of ||/fc||p from above. L em m a 2: IIA II, < C l / l l , ( A r = l , 2 , . . . , n ). (1.18) P r o o f o f Lem m a 2: Let 0 be a Schwartz function such that supp <p C [—1, ljd and <£>(x) = 1 for x 6 Therefore, f t = [ - y i * . ThenA(ar) = Applying Young’s inequality we have ||/fc||p < ||/ ||p • \\<f>\\i. ...... □ We will also need the following auxiliary lemma on local estimates of generalized trigonometric polynomials of one real variable: Lemma 3: i f r ( t ) = 5Z P k ( t ) e t,tkt k=1 where pt(£) is a polynomial o f degree < m — 1 and E C / is measurable subset of an interval I with \E\ > 0 , then IM U « (/) < * llr I U “ ( £ ) - R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . (L19) 18 P r o o f o f Lem m a 3: If g is a pure trigonometric polynomial of order n, i.e., n 0(0 = ^ 2 c kei(tkt, k=l then it follows from a theorem on trigonometric polynomials by F. Nazarov ([13], Theorem 1.5) that N k “ (/) < * ll0 lU°°(E)- U-20) m—L If p(t) = a^L a polynomial of degree m — 1, then it can be approximated 1=0 uniform ly on an interval with a trigonometric polynomial of order < m because t = lim e m~ l /V/i* _ t \ 1 rn_l 1=0 \ l=Q r / uniformly on an interval. Applying (1.20) to the trigonometric polynomial of order mn rt r{t) = E Pk(t)eittkt k=1 and taking the limit we have the desired result: 1 • iw u -iE i. □ Now we are in a position to proceed with the proof of T h eorem 2'. First we assume that p € [l,o o ). Divide the whole R.d into parallelepipeds EI. R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 19 Consider one of them. Suppose |/ | attains its maximum in II at point y E II. We can find an interval / E II, y €: II and > C d7 (see argument before Lem m a 1): / = y + £|/|t7, 0 < t < 1. ( 1 .2 1 ) f > I'l 7SC1 ’ ( 1 -2 2 ) Define m f(y + t\i\v) = n = £ My + (1.23) fc= I Using the Taylor formula ' s'(O) t = I p (« )+ - sr-'d s 0 where p (t) is a polynomial of degree m — 1 , we obtain from (1-23) fc=i ^ >' k= i { = r(t)+ T {t) R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 20 where gk(s) = fk(y + s |/ | 7/). Applying (1.17) to fjf* we have that m a x \T(t)\ < £€[04! S"' V “ (m) ! f - ^ m a x \ f [ a^(x )|a ° *-*• a l x e n Uk v n fe=l |a|=m 1 ^ T^rE E ^ E irf^iiw^/ini. (m ) fc=l |a |= m 3 ,/3 i< l Denote the last quantity by M . Applying Holder’s inequality we have 'v/p'ni^ n S F E E E ( ^ Y ij^ hw 1 '*fc=l \ a \ = m 0 A < l V y (l«) Summing (1.24) over all parallelepipeds II E-vnni < E ( = ^ ) ' i l l ’ll? n I"1** t e l |a|=m/*,A<l \ “* / S h 5 ^ E E E ( ^ ) P(C6)-»imi; “ ■ '* t e 1 |a|=m/J,/3i<l ^ ' ^ (2 dm‘fn)P- 1(Ca - b)"*ec *~b ^ .. f ... E IIA B p - --- -------------------k= I < (Cmn)JP(^ brPeC," ‘, |l/li; d-25) where the last inequality follows from (1.18). Lem m a 3 applied to r, interval [0,1] and subset E\ = (t € [0,1]: y 4- £ |/|7 7 6 E n l } gives the following local estimate: ll/[U«(ii) = ^ < I|^IIl»(|P4D IMU“ ([04i) + M / s~id\ nm-l * llr IU“ (tft) + M ( ( a m —i sid \ nm~l Y J ' r 'd \ nm-l —J ' | |/ I U “ (£) + (?) (?) H -M rent—I -M . R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . (1.26) 21 An argument similar to the one after L em m a 1 shows that the following can be obtained from (1.26) pnm-(p-L) p £*•(13) < - y ) / C d\ Pn m _ (P_ 1 ) •ll/r » < En n )+ 2 P“ l ( Y ) •" ■ m Summing over all parallelepipeds II we have: / i/p < pnm-(p-l) / S~id\ pnm— (p—l) ss-td\pnm—(p—l) (^ -) •' ii/iit»,Ei l l l l £ P ( E ) +■ ( ~ ) pnm-(p-l) 'C d \P n m -(p -D ( C m n ) d p ( C a L . b )m p e C pa-b P • II/IIlp(e) + [mlJP pnm-(p-l) 4- (t J ^2 S ™ The second inequality follows from (1.25). The last inequality is due to Stirling's formula for ml and the fact that t < 2 £. Choose m such that it is a positive integer and m = 1 + [a - b ^ e.g., ] with so large C > 0 that the factor in front of ||/||p in the last inequality in (1.27) is less than Therefore, I i/p < ( £ • ) -fur E f r ' d \ Pa b( ^ ) +P*»-(P-1) s (t ) r •/l/rE The proof for p = oo is similar and even simpler. □ Now we can proceed with the proof of T h eorem 2. We will apply induction on rc. For n = 1 the theorem follows from T h eorem 2' or the usual Logvinenko-Sereda Theorem. Suppose the statement is true for n < m. Let n = m -f-1. R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 22 If Jfc, k = 1,2,..., n, axe disjoint then the result follows from T h eorem 2 \ If Jfc and J( intersect each other for some k and I, then we can replace J with 3 J reducing the number of frequencies At and replacing b with 3b. Therefore, byinduction: f C d\ -3a b(ir ) - m+£7 1 l l / I \ l p (E) > 'll/llP / n d \ - a-b( i r ) ( + 5 (-) □ The purpose of this theorem is to prove the existence of a constant 0 (7 , n, ab, d, p) > 0 depending only on the number of parallelepipeds and not how they are placed rather than to get the best possible estimate. We can conjecture that the right behavior of the constant c(7 , n, ab, d, p) is the following: \ Cna-bfn—? * Qd) ' (L28) The estimate (1.28) is suggested by an example similar to the one after T heorem 1 : choose Jk = J + k *( 1 —e)b, k = 1 ,..., n, so that neighborhoods of two comers of « 4 and Jk+u k — 1 , ...,n — 1 , intersect. Then we can choose / supported in a cylinder with axis along b of length n (l —e)|b|. The rest is the same as in the former example. 1.4 C ase o f in fin itely m any lacunary in tervals 1 .4 .1 C onjectured theorem The goal is to generalize a result of F. Nazarov for the Zygmund theorem on lacunary trigonometric series ([13], Theorem 3.6). Instead of a trigonometric series we will R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 23 OO consider the following sum f(x ) = ^ fk(x)eiXkX where /* € L2 (R) with supp k——oo is a lacunary sequence. A sequence of real numbers fk C [—5 , |] and A = {M&-CO is lacunary if there ea sts AT < 0 0 such that N = m axC a rd {(A/, V) : |Afc —A* —(A*/ —A^) | < 1 }. (1.29) Let Ik,1 = [At —A/ —5 , A*. —A; + ^J. Then N — max Card{(k',l') : Ik,i D lie,i> ^ 0}. k£t (1.30) Another equivalent definition: N = m axC a rd {(k ,l),k 7^ i : x 6 JT = max C ar d{(k, I), k ^ I : A* —Aj 6 [x — 1 + ^]}. (1-31) The conjectured theorem is the following: Theorem 3: let A = {Afc} ^ _ _ 00 be lacunary and let E be‘‘relatively dense. ” If f e £ 2 and supp f C IJ [Afc — Afc + |j then fc=—oc ll/ll j * a ) > C ( 7 , J W H I / l l 2 . We will prove some partial results. 1.4.2 Large E , sm all su pp ort P roposition 1. T h eo rem 3 is true if 1 —7 is sma/i enough depending on N , a, b: R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 24 *f 1 ^ — 16(l+a6)2\l+ I/a )iV ^ eTl J\f\ 2 > i / l / l 2E P r o o f o f P roposition 1: We can choose /* such that supp/*. G [— |], supp oo /*;(r —Afc) are disjoint and f(x ) = /t(x —A*). Then OO 00 / ( I ) = £ /»(*)«“ ** k=— oc and ll/lg = £ llAlli- fc=—oo Dividethe whole real line into intervals / of length a each. Consider one of them. We will need to prove the following local estimate: / l/i2 < d - 7 ) £ / ( I a i 2 + ° k/!)'|) / ih e c + ; s / ( l - 7 ) a ( l + a )jV f | M 2 £ f c = —OO + £ ^ V ( l- 7 ) « ( H - « ) A r [ ( | / : i V ( 6 / 2 ) + I A I V 2). £ f c = —OO We can assume that / = [—f , | ] . / /n E c W2 = £ / IM2 + £ fe=-00/riEc / (1.32) k^ t in E e The first term is bounded by (1 -7 ) OO * £ I r n 2 + M U IY \)- f c = — OO £ R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . (1-33) 25 To estimate the second term, we will rewrite it in the following way: £ [ W in s* ( A (y)Z (v) -f- / V = J y £ fk { y )f i(y )x in s ^ k —Aj) + k*l Mi y Mi_a 1 The first term in this expression is bounded by E < /(l—, W l + o.)N\My)\2. k=— oo To show this we will apply Holder's inequality to the first term in (1.34): £ fk{y)fi{y)X[nEc(^k —Ai) < Mi , / E l / M / M P ■. / V Mi E E I - A,)P < y Mi l/*to)ll k=—<x V f (Urns*!2 + [((.Y/nEp2)'!) < \* ‘L \ U 4,i Ml E IA fe )IW (l+ < * > f e s : —OO f Ii,w p = I oa E IA&r)lV^(H-a)(l-7)a. (1-35) & = —OO We used here Bernstein's inequality: < L[|^[|i if supp# C [—L,L\ and that R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . suPP(,Y/n£ c ) 2 6 [—a, a], la a similar way we caa bouad the secoad term ia (1.34) by •y *L E \ 1 E l^l2)•,/EIXlwW**- A,)|*<t < fc= -0 O f c = —OO V a 2 E (l/i / l2 /(V 2 ) + IA IV 2 ) 0 V ( 1 + a )(l - 7)0. (1.36) k= -o o Jy The third term ia (1.34) is treated aaalogously. Iategratiag (1.32) over y 6 I aad applyiag (1.33), (1.35) aad (1.36) we obtaia * f OO i/i 2 < a-?) » e ; / ( i / ‘ i2 + a K/t)'D /n £ c fc=-oo r iE + E + a x /( l- 7 W l fc=— oo fc=— oo +a)N J i/ fc|2 [ 00 y* x / ( l - 7 W l + a)iV / (|/fc|2 / ( 6 / 2 ) + |A p 6 / 2 ). (1.37) J fc=— oo Summiag (1.37) over all iatervals I aad applyiag Berasteia’s iaequality, we get J\f? < (1 + 6)(1-7 + \/(1 -7 )(1 + 1/ ) V ) / | / | 2. If 1 —7 < (1-38) c oi t^ea ^actor hi the right side of (1.38) is less thaa 5 . Therefore, j 1/ 12 > | / i / i 2- □ R em ark: 1. We caa always take a larger a aad try to minimize (1 -h ab)2( l 4 - 1 /a ). Thea we witt get C (I -ha 6 )2( l + mia(6 , 1 /a )), so 1 - 7 < ca+ab)i{1. R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 27 2. Using T heorem 4 with rrc = 0 we can prove in a similar way that T h eorem 3 holds if b is sufficiently small depending on 7 , a and N: i f K l + a ) < e - a*c ,l w ’ W C then j |r? > j |/|2. E S k etch o f th e proof: We can assume that a > C (otherwise if a < C we can put a = C and replace 7 with 7 / 2 ). Rescaling we can assume that \E fl I\ > 7 for every interval I of length 1 and replace A with aA and b with ab. J \f(x)\*dx = / I E /*(»)«“**+f m d t e ^ i x EM EM y m. OO / IE EM > OO - I IE I m d t e ^ f d x fe=_0° / £ fc= - ° ° S f |A to )p -C v ^ V f k = —oo fc=—oo £ Integrating over y 6 / and summing over all intervals / we get / 1/ 12 > .-»■«'>»*«)' f k= ^00 g | i/a* A.— —“*00 > '-hncm^zf r w t - c ^ a b f J |/|2. Thus we obtain the desired estimate if a& < 1.4.3 _ P eriodic E The next case we consider is when f? is a periodic set with period a. Let i? fl [0, a] = 7 *a. R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . □ 28 P roposition. 2: T heorem 3 holds in this periodic case J | / | 2 ><% ,<., 6 , JV) / 1/ | 2. P r o o f o f P rop o sition 2: We will start with, some results on periodizations. Define a family of periodizations of a function / € L1: OO &(ar) = f(x -bk a)e~ i2^ x+ka) (1.39) fc=—oo where t € [— ^] . Then gt(x) is periodic with period a and its Fourier coefficients are: m = i / ( - + 1). a a (w o ) Now we assume that / 6 Ll fl Lr. The next argument shows an important relation between the average of L1 norm of periodizations and the L2 norm of /: -L 2a _L 2a a [ [ \9 t(x)\2dxdt = -J- £n[o,tt] f a f / ( * + k a ) f ( x + I - a)e~l2*t(k- t)adtdx k'Len[o,aj __l / k,L /(a: -f- k a )f(x + la)dx £n[o,a| J \f( x + ka)\2dx fc=-°°(S-fca)n[0,oi I I/I2. (1-41) e In particular, it follows that 2a - / / | ^ £ ( x ) | 2d r d £ = / I/P- - iM R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . (1-42) 29 In the next lemma we extend these results to functions from l r . For simplicity we assume that a = 1 . Lem m a 4: i f f 6 L2 then there exists a family of periodic functions gt{x): x e [0, lj, t € [— wi t h period 1 such that gt{x) € L2 ([0t 1] x [—| r |J), I j J \gt (x)\2dxdt = i Bn[o,i] J |/|2 b and m = h i+ t) for almost all t. P roof o f L em m a 4: Consider a cut of /: A * ) = X [-n ,n |/(a r). Since f 1 6 Ll fl Lr and converge to f in Lr we can define corresponding fam ilies of periodizations y?(x) which form a Cauchy sequence in L2 ([0 , 1 ] x [— |J): 1 2 J J \g?{x) - grwfdxdt= J i r - n 2[041 R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 30 Let gt(x) be the limit of g"(x). Then we obtain the first statement of L em m a 4 2 2 f J \gt(x)\2dxdt = lim § snto,il J J \g?(x)\2dxdt § £n[o,iI = lim [ \r\2 n-«xsj = / I /I 2- E To obtain the second statement of Lem m a 4 we consider the following sum: I I OO 2 2 £ [ W ) - h ‘+ = £ l= -o o J t [ \m - g ? {i)+ r v + Q - h i + t)? dt Z=—oo ^ 2 2 i 0° < 2 £ 1 j | j , ( ( ) - s r ( ( ) i 2 + i r ( ( + o - / ( ( + ‘ ) i 2 <ft i=—OC “4 - i [o,iI < e where e can be arbitrarily small if n is large enough. □ Rescaling we can assume that the set f? has period 1, |f? fl [0 ,1]| = 7 , supp/* € [—y , y ] and A is lacunary with 1 being replaced by a in definitions (1.29), (1.30) and (1.31). Let gt be a family of periodizations of / as defined in (1.39). It follows from L em m a 4 that gt(i) = f ( l + 1) for almost ail t E [—y |J. Assume for sim plicity that t = 0 is among them. Let n* be the smallest integer in [At —y , Afc-f- y ] if such exists. Denote A = Then m=[a£j Spec g0 C ( J (A -t-m ). m=0 R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 31 Next lemma generalizes a result of F. Nazarov for the Zygmund theorem on lacunary Fourier series. L em m a 5 : I J ISb|2 > C ( 7 ,a , 6 ,iV) J \ g o \ \ Hnto.il o P r o o f o f Lem m a 5 : Denote M = [a&J. Let R = s\ipC ard{(k,k',m t m'),k ^ kf,Q < m ,m r < M : n = nk + m — (n^ + m ')} n€Z < (M + 1 ) sup C a rd {(k , A/ ) ,k ^ k' : n 6 [A* —X# —ab, A* —A*/ + a6 ]} 5; (iV/ + 1 ) • iV *( 1 -t- [26]). n6Z To obtain the last inequality we used (1.31) with 1 being replaced by a. We can write go in the following way: OO « ,(i) = £ Jfc= —OO where p* are trigonometric polynomials of degree M : M m=0 Therefore, J \9o\2 = Hn[o,ii f] | |Pfc|2 + $ 2 ^ " “ Bnto.i] °o J PM ^ E n f o .ii . = £ / w2 fc=_ooBn[0,ii + S S c£ )^ fejtfc' 0<m ,m ' <iVf )& ? n [ o ,i i ( r a f c + m - ( n ife, + m ' ) ) . R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . (L 4 3 ) 32 The first term in (1.43) is bounded from below by . —. / y \ 2 J lf + i f —^ E(§) /L. / *y \ 2 M r l 0 t E^i2=(§) /w2- k=— ca a.44) m— 0 The second term in (1.43) can be written in the form {Tga,g0) where T is a HilbertSchmidt operator on L~([0,1]): Tg0(nk> + rri) = E' E X E n [ o ,i\(n k + m - (n^ + m'))g0{nk + m) fcjtfc' 0<m<M and Tgo(n) = 0 for the rest of n. Its Hilbert-Schmidt norm is ^ ^ [X K n[o,i|(n* + m - («* + m' ) ) | 2 < ^ 7 ( 1 - 7 ). (1.45) We won’t proceed further since the rest of the argument is the same as in Nazarov’s proof of the Zygmund theorem ([13], Theorem 3.6). See details in the proof of the next T heorem . This leads us to the desired inequality L J \go\2> C { i , a , b , N ) J \gQ\2 £n[o,il where C(7 ,a , brN ) = e~ilnR)R2( ^ C{M+l) > e-t°((i+“6)d+26)iV)((i+a6)(i+26)iV)2(£) C (l+ a 6 ) □ In a similar way we have that 1 f sn[o,iI |r f > < * 7 A M 0 / b ( * 0 R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . (1-46) 33 for almost all t £ [— £]. Applying Lem m a 4 and (1.46) we obtain f \ f \ 2 > C (n ,a ,b ,N )J \f\\ (1.47) E □ 1.4.4 G eneralized lacunary series The next theorem generalizes Nazarov’s result for the Zygmund theorem for series whose coefficients are replaced with polynomials of uniformly bounded degree. T h eorem 4: let I be an interval of length 1 and E be a subset o f I of positive meaOO sure. I f A = {Afc}^=_oa is lacunary in the sense of (1.29) and f ( x ) = ^Z Pk(x )e‘XkJ: k = —oo where Pk(x ) ore polynomials of degree at most m, then J \ f \ 2 >C(\E\,N,m) J |/ |2 E I C{\E\, N, m) = e -( ,nCiV)iv2(i§r)C(m+I). P r o o f o f Theorem 4: Without loss of generality we can assume that I — [— £]. TO Let p(r) = bjxj be a polynomial of degree m. Denote b = (b0,...,bm). Then i=o W pW lhd ~ IN h- h fact> < j |p|2 < C I b | | i . I First we will treat the case when gaps [A* —At| are large: |A* — At| > C(1 -f-m3) for OO k j ^ l . We need this condition to prove that f \f[2 ~ [ JZ fc=—oo I f iPfc|2- N ote that we don’t need the lacunarity of A for the lemma below. R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 34 L em m a 6 : If |At —Aj| > C( 1 -t- m?) f o r k ^ l then k = —OO £ P r o o f o f L em m a 6 : We will prove only the left inequality. The right one can be proved similarly. Let p be a polynomial of degree ra. Hence, using the following inequality J \p'\ < C m 2 J \p\, (1.48) f |p |2 > c f |p|2 (1.49) we obtain that A I for any set A c I provided |/ H .4C| < Let <j>{x) = (j — Then J l/l2> J 0\1? I I = 52 f J 0PfePie,(Afc_Al)ldr. + f c = —OO [ (1.50) lc £ l Using (1.49) we can bound the first term in (1.50) from below by E /W- C 1 + m2 — J f c = —OO j R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . (1-si) 35 We claim that each summand in the second term in (1.50) is bounded from above by C (l + m4) (A* — Xi}2 J (W 2 + M 2). (1-52) / To show this we consider f p(x)<p(x)eiXxdx for A ^ 0 where p = pkPi is a polynomial i of degree 2m. If we write X p(x) = p(y) 4- p \y ){x - y ) + J p " ( t ) ( x - t)dt, then f p(x)4>(x)eiXxdx can be written in the following way: i p(y)k A ) + p'(y){x ~ y ) 0 ( a r ) ( A ) y y — 5 r J J 4>{x)etXxp',{t){x —t)dtdx + J J (p(x)ei>a:p,,(t)(x —t)dtdx. (1.53) _1 x y y The first and second terms in (1.53) are bounded from above by ^ Meanwhile the third term in (1.53) y t J J (p(x)etXxp"(t) (x —t)dxdt i - .12 '2 is bounded from above by The fourth term in (1.53) can be treated analogously. Integrating (1.53) overy e I and applying (1.48) twice, we obtain (1.52). If [A*—Aj| > R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 36 C (1 + m3) with, laxge enough C then I fc=— oo j >- T ^ E fcjci £ / * * • k=—oa i The right inequality can be proven similarly with <p(l , xc ) instead of 0 . □ l+m» Repeating an argument similar to the one used in P ro p o sitio n 1 to prove a local estimate, we have that T heorem 4 holds if E is large enough: / / | / n £ c| < then J l / l 2 > \ j I/I2- Now we consider the case when \I D Ec\ > (1.M ) The same argument shows (see also L em m a 5) that [ l/l2 = £ f N 2 + (Te f ’f) J_______ kL_ ^ J = -C O E R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 37 where (T e / . D fp tP t^ -^ d x = £ = H / Pk(y)Pi(y)xE{^k - h )d y k)U j y - / £ J ( P k P i Y ( t ) X [ - i / 2 ,i[n E (^ k ~ h ) d t d y + I Y 2 I ( P k P t Y ( t ) x [ t , i / 2 \nE(><k ~ h ) d t d y [ W y = 5 1 [ PkWPii^XE^k ~ >^i)dt k * i Jt ~ /(P*Pi)'(0(V 2~0X [-iA t]nB(Afc-Ai)tft Mi , + ^ J (PkPiY{t)( 1/2 -F 0X[t,i/2!nB(Afc —Af)d£. fe*/ _ VVe can view T# as an operator acting on the space of sequences of pairs of functions from L2(I): G = {Gi,Go,Gz,...), Gk = {gl,g%), gik e L 2{ I ),i = l,2 , k = l ,2 ,3 with a scalar product defined in the following way: <f’g> = E E / ^ fe=I «=I Then ( TBF)t=Y,T“‘F’‘ Mi R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 38 where each T& is a 2 x 2 matrix whose ( l r1 ) entry is XE^k —Aj), ( 1 , 2 ) and ( 2 , 1 ) entries are the same and equal to “ (1/2 —£)£[-i/2,t|n£r(Afc —Aj) + (1/2 + 0X[t,i/2]n£(Afc —A*) and (2,2) entry is 0. In our case Fk = (PfctPfc)* We claim that Te is a Hilbert-Schmidt operator since S"P £ £6/ E r a 2 < CN \E\ k,l i,j= l by an argument similar to the one used to prove a local estim ate in P rop osition 1 . Our proof will follow the one of F. Nazarov for the Zygmund theorem. Let a \ , oo, ~ be the eigenvalues of Te enumerated in the descending order of their absolute values: l^il > |er21 > .... Since 5 Z k ,|2 < s u p ^ j ^ |T ^ | 2 £^ £e' k,l k,[ i,j= l CN\E\ a=L < we have that |crrt+1 j2 < If Vn is the space spanned by the first n eigenvectors of Te , then the norm of Tg on is equal to |crn+l[. Recall that / l/l2 = E ** f M 2 + (T^ /, f) ■ _ J kr .= -o o E The first term in (1.55) is bounded from below by 2m-t-l oo R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . (1-55) 39 If f±Vn(E) with. 4m +l n= (1.57) N with large enough C then the second term in (1.55) is bounded from above by \ of the expression in (1.56). Hence, if f l.V nt then 2m-t-l oo 2m +l (1.58) / The last inequality follows from Lem m a 6 . Therefore, T heorem 4 holds if / is orthogonal to certain subspaces. Formula (1.57) gives that n > 1 . We can think that n = 0 if E is large enough (see (1.54)). Now we will do the general case. The main idea is to construct a set SE € I \ E such that |£f?| > d(|i?|) > 0 if \E\ < 1 and EUSE E Then we will iterate until we get a large enough set so that we can apply (1.54). In fact, we will reach / automatically at some iteration (see the end of the proof). Choose n as in (1.57) but with twice larger constant C so that it will work for sets of measure at least \E|/2 . Define n i=o We can choose r > 0 so small that \Et\ > \E\/2 for t €E [0, t \ . It is clear that r < 1. We will pick r in a special manner. Consider the continuous frmction0(or) = \Ecn (E + x )\. R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 40 0(0) = 0, 0(1) = [£[. Let n r = inf{arT0 < x < 1 : 0(ar) = |J£} then | £ " n ( f : + nr)| = M and l£l \Ecn ( E + kt) \ < — fo r k = l , „ . , n and t 6 [0,r] £fh Hence |£ t| = | p |( £ 7 - f c i) [ Ar=0 = a |£7\ | J (£7 —Art)c n £7| k=l n > |E |-^ |(£ -w r n £ ;| k=i n = l E I - ^ I F n O E + fci)! fc=l Let g be the following linear combination: 9{x) = '%2aj( t ) f { x + jt) i=o R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 41 with. i=o |ay( £ ) |2 = 1 then [ \9\2 < £ | a # ( 0 |2) £ I i= ° < [ \ f(x + j t ) f dx) j= ° i [ \ f{x + jt) \2dx l= QE - jt 5 1 f \f( x)\2dx = i=°E = (n + 1) J |/|2. E Since \Et \ > \E\/2 we can choose such a.j(t) that g±V Et. Then / « * * ( & ) “ / « • Let p(x) = J ! bix1 be a polynomial of degree m. Define t=o P(x) = '52aAt)elXjtp(x + j t) = ^ k x 1. j=o t=o Then b = Ab where A is a certain (m + l ) x ( m + 1) upper triangular matrix with I* 5^ on the diagonal. If this sum is not zero, then A is invertible. A - 1 is also i-0 upper triangular and A^ 1 = . In fact, Aki = 51 ^ ' j—o 0 < k < I < m. A trivial estim ate shows that [My*] < m! (2my/nnm)m < (Cn^2)m2. However, we can give a better estimate by noticing that is the sum of at most 2 m terms (open R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 42 determinants via rows or columns containing at most two nonzero entries) whose absolute values do not exceed ^m » ^m ^ n*=l 1 1=1 nm/2, n — 0 < ki < li < m. n W i-^ ) 5 i=L We used the fact that r < 1. Therefore, \Mji\ < 2 mrnln3 m / 2 < (C{m + l)n 3/2)r Hence ,-H, ^ {C(m + l)n 3/2)r | Y, a jffle W |m+i j=o Note that 9 i x ) = ' £ t aj (t)f(x + j t ) = pk{x)etXkX. k= - o o j=Q Therefore, / [ w 2= £ / N 2 k=-'x I IIM l fc=—oo 3 = iff t , IM R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 43 Integrating over t E [0, r] and applying Holder’s inequality, we have > E ||bk[||- f ( C W ' 2) " ^ fc=—oc > E m k= ~< X 3 = j = 0 Q l l b k l l ^ 2< m +l)( ^ ) - It is known that S = {A : p2 (A) < length j= 0 ( i [ i E %(i)ea *-»i2dt \ E 2(m+l,<ii r “ °n n C U /j where /j are some intervals of j=t (see the proof of Lemma 3.1 in ([13])). Thus we obtain a lemma analogous to ([13], Lemma 3.1). Lemma 7: There are n intervals A ,...,/n o/length ^4^ such that ^2 l|t»iclll < ((w» + l)(Cn)4rl+3/2)' e". Afc€A \ U h }=i n Remark: if r > C(m , iV, |i?|) then we are done since th e cardinality of AH (J Ij i= i is bounded by n (l + ^^^A ____) f ca]1 be approximated by pk(x)e°'kX. \ ke \n u It is also known that there are real nmnbers p i,..., fin and t E [0,r] such that i x > jW ^ -f=0 ‘i > ( ^ r n « r( A - w ) ' / T=1 n where 9r(x) = m in (l,r[x[) and A E (J Ij (see arguments after Lemma 3.1 in ([13])). i= i R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 44 Thus we obtain the following result: / n \ 2 (m+l) £ < ( ( m + l) ( C n ) <“+3'2) m+l£2. A*eAn u 1, 1=1 J=l Combining this result with L em m a 4 and the fact that 0 < dT < 1, we obtain L em m a 8 :There are n real numbers p \ , p n such that oo / n \ 2 (m + l) Ilb »cHl f f [ ° A X k- Afj) ) k = —oo Vj=L ^ (C (m + l ) n 4R+3^2) ,n+I e2 = e'2. / Let D = [ n (e'^x^ e - ‘^ r) ) = f | (e‘^ x be a differential operar a tor. Any solution of the homogenous equation D f = 0 is of the form pj(x)elft]X j= i with pj being polynomials of degree at most m. A partial solution of D f — g is of the form <p = f e ^ 1) j g. Note that max|0(ar)| < {5 ^ j T ] 7 f f |tf| for any interval J. Hence there exist n polynomials pj of degree at most m such that max i ej m V \J w ‘ - (L59) Define a set of In intervals Ik = (pk —^rPk + 7 ) and Ik — (pk —7 »Pk + 7 )* For some ^ n fixed constant C which will be chosen later, define A = |J {A E A : |A —pk\ < Cm2}. fc= 1 The cardinality of A is bounded from above by Cn because of our gap assumption (see L em m a 6 ). Note that we don't need the lacunarity of A for this. Put A! = A\A. n Split A' into n + 1 disjoint subsets in the following way. Ao = A '\ (J Afe = i= r R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 45 [k\ (J Ij, k = 1 , n. Then we can decompose j=k+l / = / + E a 3=0 where f = Pki^ ) ^ 1 and fj = £ pk(x)eiXkX. It follows from Lem m a 8 that AfcSA Afe€Aj / n e AfcSA{ u m « n \ 2 (m + l) 9'( A‘ - « ) \i= t ^ / In particular, it means that j l/ol2 < E I M j ^ ^2 (i.60) Afc€Ao since f j 6 r(A —fij) = 1 if A 6 Aoi= i If 1 < k < n then fk has only finitely many terms and therefore it is infinitely many times differentiable. If A 6 Afc then n n < w - « ) = n < M A - f t ). J=l /**€/* Correspondingly, instead of D we will define another differential operator Jra+I d*= n fh&k Let Dkfk = 9 k and let n* denote the number of fij G /<,. We have that I < n/t < n. Let p be a polynomial of degree at most m , then R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 46 Since Hp^Hj^/j < (Cm 2 )k\\p\\L2 {n we have WPWlhd ^ ( l ^ - A * l + C ,rra2 ) m + l | | p | | £,2( / ) . Actually we can give a better estim ate because IIp “ ° I I l ’ ( / ) < c V H m which follows from well-known Markov's inequality ([3], p.256): IIP l k«([ -i , i ! ) ^ ™-2 ' i ™ 2 ~ 1 ) ' ( ™ 2 ~ 2 2) •... • (m 2 - { k - l) 2) — . (2 fc 1 ) -------------------------------- l l P l U » ( [ - i , i ] ) ^ but what we already have is enough. Therefore, J t e l 2 < II ( | A i -n \ + CmYm*° J t e l 2 £ I t A,6Afc Mj6/ fc <£ n ( c i A , - f t i) 2< - ) / t e i 2 I At €Afc since |A —p.j\> Cm 2 if AE Afc C A \A . It is left to notice that |A —fij\ < |0 r(A —Pj) if A E A<, C Ik and pj E /*to conclude that T ^ /te I 2 < i £ AteU ^e/fe r£ J \ 2nfe(m+I) * (? ) '■ (1.59) shows that for each 1 < k < n and each interval J there exists £ Pj(x)elt*ix such that j I r |n fc(m-t-I) - K ^ n j i f M s‘ (:r) R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . = 47 for every x E J . We will consider only J C / . Mgn is the maximal function of g* on / . Then S/~>\ \\Mgk\\LHn—C\\9k\\LHn < y —j ^ (L61) Let | J\ = A t with A > 1 and <pJ(x) = f ( x ) -F £ Qkfc) fcLen fe=i \f(x) - < |/„ ( * ) | + £ = ft(x ) for every x E J . Note that <pJ can be written in the following way: Cn ®J( x ) = J 2 pj (x )ei0iX j=i where pj are polynomials of degree at most m. Applying (1.60) and (1.61) we can estimate the L -(I) norm of the remainder R by Now we will construct 5E E I H Ec. Divide I into intervals of length jJ[ = A t . We will choose A later so that so that ^ is an integer. Fix some 0 < 7 < \E\ < 1. Call an interval J good if \E fl J\ > 7 |J\ and correspondingly bad if \ECi J\ < 7 |«/|. R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 48 Denote SE = Ec n (J J . Then J is good \ECn ( E + t i t ) | < tit [J « /n ( £ + nr)| + |dT?| -F |f?c fl 7 is bad < nr + |dT?|-F ^ |J n (£ ? + nr)| J is bad < n r + \SE\ + ^ 2 ( |J \(J + nr)| + |(J + n r ) n ( £ + nr)|) J is bad < \SE\ + ^ ( n r + j\J \) < |d’E |+ n r j jj-+ 7 since the number of bad intervals is bounded by —1. Recall that \ E c n ( E + t i t ) | g . If we put 7 = g < \E\ then There are two possible situations: 1- nT < g - Then we can put g = [gg;] and obtain that We also have that t Ul 1 , lft»» " - t s it l “ 1^1' It is interesting to note that the condition n r < g implies that \E\ < 1 — since g = \Ec D (E + nr)\ < n r + l - \ E \ < g + l - \ E \ . 2. n r > g . T h ai we put | J\ = 1, 7 = \E\ and therefore I is a. good interval itself- In this case s 1 8 n2 r < \EY R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 49 Alternatively, we can use Lem m a 7 directly in this case since ^ and the n cardinality of A fl (J Ij is bounded by n Hj=i £ pk(x)eakX. M and we can approximate / with Afee U ij j=i Now we will explain how to proceed in each case. 1. f i/i2 < J SE i/i3 U J i s good. < 2 E / 1 * 7 + / ir i2 J is good j j J is goad EnJ 7 / /'■»\ 20n(mH-l)—I ± £ j / 2C » (m + i)—1 * ( f / i « i 2 J i s g a o d ^ 7' f C n \ 2C^ £y * EJn J + l)" 1 r Jj /< 7 „ \ / '^ y f /'*' • Recall that J \R \2 < (C -A ) 2n(m+I)e'2 i < (C *A)2n(m+1) (C(m + l)n 4n+3/2) m+I e2 < (C • .4 ) 2" 1" " -11 (C (m + l)n 4” +3' 2) '" + 1 (7! + 1 ) + j |/ | R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 50 Hence SEHE r c iv y v(ft) |E | ) < e , J m {lnCN)V( j f f ) / i/ p with. |K| > § - N iy C J 2. In a similar way we can get I i/p < r |/ |2 l E and we are done in this case. / _ \ C(nx+t) If we Iterate /V I j^ J times in case 1 we will reach. I and therefore j |/ | 2 < e d » ^ ( # ) c i" « ’ y i / i 2. r ( i.6 2 ) e Now we will drop the assumption that gaps |Afc—X[\ are large, i.e., [A*—Af[ > C (l-B n 3) for k ^ L Since A is Iacunary it has no more than C N ( l + m 3) pairs of (Afc, A() with. k 7^ I such that |Afc — A{| < C( 1 -F to3). To show this we w ill split these pairs in [(7(1 -I- to3)J groups such, that d < |Afe — Aj[ < d + 1 with, d = 0 , 1 ,2 ,..., [(7(1 +- to3)] and apply definition (1.29). Denote by A' = {Afc e A : 3Xt E A, I £ kt |Afc - A,[ < (7(1 -F m 3)}. R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 51 Then the cardinality of A' is bounded by C N (l + m 3). Let /(* ) , i \ kx = P k& 1 fc=—oo = Pk(x )elXlcI + Afce<V\A' = Pk(z)e'Xka: AfcgA' h (x )+ r (x ). Assume for a while that r(x) is a trigonometric polynomial of order n' = (l+ m )|A '| < (1 + m )C iV (l + m3). The proof goes in the same way with few changes. We will only discuss the changes. Let n x be the same as in (1.57): nx = Put ^ 4 m -rl N (m < N f TTTT I m) + [C W (l + m 3)] / r< \ 4 m + l Let 9(x ) = 3= 0 = S J= 0 % (f)h (x + j f ) -f- ^ o ,-(f)r (x -H yt) j= Q with £ |aj ( £ ) |2 = 1 such that % (£)r(x-{-j£) = 0 and a.j(t)h(x -f- j t ) = x=o /=o j=o 5 (x)±K tl(Be) which is exactly nx + n' = n linear homogenous equations with n -f- 1 R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 52 variables Then J IY ^ aj{.t)h{x + jt)\*dx = J |p|2 * - (&)“ /«■ (n+1)(lii) / l/|2=A B Therefore, there are <pJ |h(x) —<!>J(x)\ < H(x) for every x € J. Note that 4>J can be written in the following way: Cn j= t where are polynomials of degree at most m and iifiiiW )< (c -A r < '" +‘>€'. Then |/(r ) - 0 J(x) -r(ar)| < R{x). It is left to note that 2 C n (m + l)+ 2 re/ —1 J ^ * ( S / r ’X rl ) EHJ \ 2Cri(m.-)-l)—I s GfSr) - 1*+* R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 53 The rest of the proof is the same. Thus we obtain an estim ate similar to (1.62) J | / | 2 < e(iaCN)N2W Clm+l) J | / | 2 . (1 .6 3 ) E / Since any polynomial of degree m can be uniformly approximated on an interval by a trigonometric polynomial of order m + 1 we can drop our assumption that r(x) is a trigonometric polynomial and obtain the same estimate as (1.63). □ Rem ark: Unfortunately, the factor in (1.63) grows much faster than ml and we can not use this result to prove the conjectured T heorem 3 by approximating fk on each interval I with corresponding Taylor polynomials as we did in the case when / is supported on a union of finitely many intervals in T h eorem 2. 1.5 N ecessary condition for support Recall that a set E C R is called “relatively dense” if there exist a > 0 and 7 > 0 such that | £ n / | > 7 -a (1 .6 4 ) for every interval I of length a. It is a well-known fact that “relative density3’ is also necessary for an inequality o f the form ll/llu(E) > C ’(fr ,E ,p )-||/llP (1-65) to hold for every / € Lp with supp/ C S . Now we will consider an inverse problem. We wifi give a necessary condition for S D / so that (1.65) holds. T heorem 5: Suppose that for a given open set E € R.thereexist 0 < 7 < l r a > 0 andp 6 [lToo] such that for every “relatively dense" set E satisfying (1.6 4 .) we R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 54 have ll/II^ E ) > C(E,p) ■l l / l l . (1.66) for every f € Lp with suppf 6 S then for every bo > 0 E can not contain arbitrarily long arithmetic progressions with steps at least bo. P ro o f o f T h eorem 5; Note that the parameter a is fixed during the whole proof. First we will prove that there is a uniform constant C such that (1.66) holds for large enough sets E, i.e., there exists 70 6 [7 ,1) and C(p) > 0 such that WfWmE) > C{p) • 11/U p (1.67) holds for every “relatively dense” set E with density 7 0 and every f € Lp with supp/ 6 E. Suppose towards a contradiction that this is not true. Then there exists a sequence of /„ and corresponding “relatively dense” sets En with density 7n > 1 — -^r such that WfnWLPiEn) < “ l l / n l l p Th OO Let E = f) En. Then E is “relatively dense” with density 7 : OO OO > a - ^ T a ( l- 7 „ ) R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 55 for every interval I of length a. On the other hand we have l|/n|Uf(S) < ll/n||jy>(£„) < ~ll/n|fp IV which contradicts to (1.66). The next lemma plays a crucial role. L em m a 9: Let E € R be an open set for which there exists bo > 0 such that E contains arbitrarily long arithmetic progressions with steps at least bo then for every 0 < 7 < 1 (meaning arbitrarily small I —7 ) and every a > 0 (meaning arbitrarily small a) there exist a sequence of “relatively dense" sets En satisfying (L64) and a corresponding sequence o f functions f n 6 Lp with suppf C E such that [\fn\ p lim Ep ■ = 0 n-*°° I \fn\P for every 1 < p < 0 0 . R em ark: If E contains an infinite arithmetic progression then we can take only one set E instead of the sequence of En. Here is an example o f such E which has a OO finite measure: E = (J (k — k 4- A)- Compare with the Amrein-Berthier theorem fc=i (see for example [7j, pp. 97, 455). Note also that small 1 —7 and small a are typical cases of “easy” proofs of the Logvinenko-Sereda theorem. P r o o f o f Lem m a 9: Define L = 4 - 1 , a positive integer such that < a for every b > 60 . Suppose we have an arithmetic progression w ith step b > bo o i length Ln 1 : rr0, x 0 + b, xq 4- 26,..., xQ+ Lnb G E hence there exists e > 0 such that n [J [ar0 4 - kLb —e,xQ+ kLb 4 - e] C E because E is open. We can assume that e < y s o fc=0 that these intervals are disjoint. Let 0 be a Schwartz function w ith supp0 C [—1,1} such that 0(0) = 1, e.g., if 0 > 0. Define i t \ ^ 2 tx ~ xQ~kLb^ /n (*) = 2 ^<i>C -------- ) fc=0 R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 56 so that supp/re C £- Then l/.fo)l = (1.68) k= 0 = e\Dn(Lby)<t>(ey)\ (1.69) where the 27r-periodic function Dn(t) = is the Dirichlet kernel with the following easily verified property: Urn = q V 0 < di < 7r, l < p < o o . \\Dn\\LH[-*rt) This is true since ||A i|| £?([-*•,*•]) ~ n 7 for 1 < p < oo and ||Z?n||/.i([_jrjTr]) ~ Inn and |£>n(x)| < — if 5 < \x\ < tt. Now we will construct a ^-periodic set E: F n \ ~ — JLi - \-JL ^ i u U - t K ( 1 - 7 K i 1 Lb’ Lb 1 1 |£ c n / | < Lb’ LblX[ 2Lb ’ 2Lb J' Then ~ = ’aLb + D ( 1 - t K 2t Lb 2aLb (1 —j)ir 2t Lb (1 - 7 )a ( for every interval I of length a. We used here that ^ < a to get the second inequality. Let 1 < p < oo. |0(x)| < since 0 is a Schwartz function. Applying (1.69) we R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 57 have / OO IM ’ = I £ \ tD n (Lby)4>(ey)\pd y I J fe=— OOI 7 r » r _ l ± ________________ Lb Lb * OO - ] L f (l + (tk/Lb)*)p U> < — J |D-W W C 'e '( l+ “ ) ^ ||D . ||^ W ) f l l ^ n l l L P ( [ - J ,! J I C J (1 -7 0 ) where 5 = -l-~'1r)7r. We used the fact that e < to get the last inequality. Now we need to estimate ||/„ ||p. Recall that 0(0) = 1. Since 0 is a continuous function there is d > 0 such that 0 (x) > 1/2 if |x| < d. We can assume that e < Then applying (1.69) we have d /e f \f,\p > ^ J ID n{Lby)lpdy -d /e d L b /e J uw w r* -d L b /e > > fl_ L \ Lbd] im IIP 2 p Lb ex £p 1 7.6d p 2p Lb 2ex nll^ ([-M ) z/fP [ J1 We used here that e < — to get the last but one inequality. Hence dividing (1.70) by (1.71) we obtain the desired result / Bn l/« lP it D it < r \\L>n\\Lp([-sm . /i/n [ P " R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . The proof for p = oo is similar and even, easier. □ R em ark: If p = 2 we can use periodizations to prove L em m a 9 (see the proof of P ro p o sitio n 2 ). T h eorem 5 follows from Lem m a 9 since otherwise we would get a contradiction to (1.67). R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . □ 59 C hapter 2 2.1 P eriodizations o f functions O verview Let / be a function, from Ll(Rd). Define a family of its periodizations with, respect to a rotated integer lattice: 9P{x) = 5 3 uezd ~ t2-1) for all rotations p 6 SO(d). The main object of our study is G, the L2 ([0, ljd x SO(d)) norm of the family of periodizations, G2 = j J |gp(x)\2dxdp peso(d) [0,11* = j \W ld p . (2.2) peso(d) The purpose of this work is to show how G can give an estimate of the L 2 (R.d) norm of a function from Ll (Rd) in higher dimensions. Some results on the Steinhaus tiling problem are related to Theorem . 1 since periodizations naturally appear in the problem of Steinhaus. M. Kolountzakis ([9]) proves that if a function / 6 Ll(R2) and [x\af 6 L l (R2) where a > y and its periodizations are constants, then the function is continuous. Another result is obtained by M. Kolountzakis and T. Wolff ([10J, Theorem 1). It says that if periodizations of a function from LI(R<£) are constants, then the function is continuous provided that the dimension d is at least three. We will show that this result is false when dimension d = 2 . The main theorem s are the following: R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 60 T heorem 1: let d > 4 and let f 6 L l (Rd). If periodizations o f f 9pix ) = 5 3 ~ v€ Z d are in L2 ([0, l]d) for almost all rotations p € SO(d) and G2 = j \\9p\\ldp < oo, p € S O (d ) then f e L2 (Rd): ll/B2 <C(G+11/HO (2.3) where C depends only on d. We will also obtain an inverse theorem. T heorem 1 ': let d > 5 and let f 6 L l (Rd) fl Lr(Rd) and let gp be periodizations o ff 9p(x ) = 5 3 s o f a uezd then gp 6 L2 ([0,1]4*) for almost all rotations p 6 SO(d) and J IISplllip < C(\\fh + l l/ i ll ) 2 (2 .4 ) p€SO (d) where C depends only on d. We will generalize T h eorem s 1 and 1 ' in the spirit of the Stein-Tomas Theorem ([4], Chapter 6.5). R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 61 T h eorem 2: let d > 4 and let f E Lp(Rd) where 1 < p < If periodizations o ff 9p(x ) = X M ) x ~ ")) u ezd are in L2 ([0, l]d) for almost all rotations p 6 5 0 (d ) and J G2 = \\gP\\ldp < oo, pesow then f 6 Lr(Rd) : ||/|| 2 < C ( C + | | / | | P) (2.5) where C depends only on d and p. We will also obtain an inverse theorem. T heorem 2': let d > 5 and let f E Ll (Rd) H L2 (Rd) and 1 < p < ^ and let gp be periodizations o f f 9p(x ) = 5Z f W x ~ u ) ) i uezd then gp 6 L2 ([0, l]d) fo r almost all rotations p E 5 0 (d ) and j llffp -& (0 )id /> < C (||/I|2 + ||/||p)2 (2.6) p€SO(d) where C depends only on d and p. Note that the constant C below is not fixed and varies appropriately from one equality or inequality to another without being mentioned. R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 62 2.2 C ase p = 1 P r o o f o f T heorem 1 : We will denote f(x ) = / ( —x) and F(x) = f * f(x ). Then F e L l (Rd) and l|F ||i < \\f\\l (2.7) We will define the following functions h.hi, ho : R + —+ C h{t) j |/(O I2<M O = = jRd = [ (2 .8 ) f * f(x)dcrt(x)dx F(x)dat(x)dx, (2.9) MO = f F{x)dat(x)dx, J\x\<l (2.10) J r.* h-2 (0 = f F(x)dcrt (x)dx. (2.11) Clearly h = hi + /i2. L em m a 1: Let q : R. —►R. be a Schwartz Junction supported in [|,2J, and let b E [0 . 1 ). Define H i : R —►C Then fo r large enough N we have R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 63 C\\F\U (2.12) irfiQ where C depends only on q and d. P r o o f o f Lem m a 1: First we will estimate derivatives of hi(t) when, t > 1 |A<‘>(t)| < C id -l||F ||[ (2.13) where C depends only on k and d. This follows from (2.11) = [ F(x)dat(x)dx = td~l [ JM lx*l<t i< F(x) [ J e - aiUx^da{Cjda i= i J\Zk1= by differentiating the last equality k times. We can easily prove by induction that dk ( h \(\/t + 6 ) \ V t+ b J ~ i s h i\y /T + b ) (2.14) ‘• \ y / t T b ) ^ - ‘- It follows from (2.14) and (2.13) that when i ~ iV2 we have (2.15) with C depend in g only on k and d. Since 9 (1^±H) = q(y/tf -{-(/) = q(f) with H = ^ and V = ^ and q[f) is a R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 64 Schwartz function supported in tf ~ 1, then we have s ,( v r + 4 ) = A r - || r f O I < CJV-“ (2.16) with C depending only on k and q. Since ) is supported in t ~ /V2, it follows from (2.15) and (2.16) that dk f — < difc V v t + 6 C N d~2~k\\F\\i iV >) (2.17) with C depending only on k, d and q. Since Hi (t) is also supported in t ~ N 2 we have life'll, < CiVi- ‘ ||F ||1. Therefore, Itf.M I < p f l l ^ ’ lli Mk (2.18) for every u ^ 0 . Summing (2.18) over all v ^ 0 and putting k = d -\- 1 , we get our desired result c m i N where C depends only on q and d. In the next lemma we will use an approach related to ([10], Lemma 1.1). R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . (2.19) □ 65 Lem m a 2: Let q : R —* R be a Schwartz function supported in [ |, 2], and let b € [0,1). Define fli(i) = -^ L = A 2 ( x /i T 6 ) , ( ^ I ) . Then for large enough N we have f If'WI-I^WI .~an where D,y : (2.20) */ W >1 —*■C |A v(x)|<C ^ - 1X1 ± 77 (2.21) *71 < fcl < £ with C depending only on q and d. P r o o f o f Lem m a 2: We have Ho{u) = J H (t)e -i2m*dt = 2ei2™b f Nq{t)h 2 {tN )e -i2m,im)Zdt = 2 ei2irub f 2 j = 2 ei2™b f Nq{t}e - i2™ i m ) 2 f F{x)da^t(x)dxdt J \x \> l F(x) f N q (t)e-l2™{m)2 {N t)d- lfo(N tx)dtdx. J (2.22) We will use a well-known, fact that da{x) = j?e(f?(|x|)) with f?(r) = a{r)ea '!rr and R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 66 a(r) satisfying estimates l“‘MI^S r r (2-23) r 2 with. C depending only on k and d. Now we will need to estim ate the inner integral in (2.22) with B(\x\) instead of da(x) . . rf+t = / 9(0 e -i2^ (^ )2^ - la(iV |x|0(iV |x|)^ ei2,riV|x|tdt |x| 2 J - N 2 I I j q ( t ) a ( N \ x \ t ) ( N \ x \ ) ^ t d- le~i2™N2{t- ^ )2dt |x| 2 J M s = f <p(t)e-am,N2(t- ^ )2dt (2.24) where 4>(t) = <?(£)a(iV|x|£)(iV|x|)ii2 i'£d_l is a Schwartz function supported in [|,2 j whose derivatives and the function itself are bounded uniformly in t, x and iV because of (2.23). Note that we used here the fact that iV|x| > 1. We can say even more. Note that in fact <p(£) = <p(t, |x|). Let |x| = c *r where c > 2 and r > i . Then all partial derivatives of <p(t, c • r) with respect to £ and r axe also boundeduniformly in £r r, c and iV.However,we will use that <z>(£) dependsalso on x only informula (2.66) from the proof of L e m m a 4 and therefore we will keep writing just <£(£). From the m ethod of stationary phase ([8], Theorem 7.7.3) it follows that if k > 1 then [ [ 0(£)e^2^ J (t- ^ 2d £ - ^ 9 ( ^ 2 V 2) ^ - V 2i)(^ 4 F )l < cfc(M!V2)-* -5 . n 3= 0 Z l/V i where Cj axe some constants. R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . (2.25) 67 Since 0 is supported in [4,2] we conclude from (2.25) that If then there are no u in [|^ , and therefore if we sum (2.26) over all i / ^ 0 we will get | J 0(£)e-i2™ ^(t- ^ )2di| < CkN~2k~l. If *then (2.27) the number of u in [|^ , is bounded by ^ and therefore if we sum (2.26) over all u ^ 0 we will get J 2 1 f 0 ( t ) e - ^ uN2(t- ^ )2dt\ ujtQ < C ^ ( \ x \ N ) - ± + C kN - 2k- 1 < (2.28) Ni Summing (2.24) over all u ± 0 and applying (2.27) or (2.28) we conclude £ r 1 I N q(t)e-“*«m r m f q . iK ) ^ d-'B {N \x\t)dt\ < \ * if M > i " \ (2.29) Replacing in (2.22) da{x) with 5tiE0±£liEilr summing over all u ^ 0 and applying (2.29) with k > we get the desired result £ |& W I < “*> f |f ( r ) | • |flw(r)| w>i R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . (2.30) 68 where —+■C '(f f i^ |£/y(x)| < C < £ ifW >f (2.31) i f l < M < ? with C depending only on q and d. □ Now we are in a position to proceed with the proof of T h eorem 1. From (2.1) it follows that 9 P{™) = f(pm ) (2.32) for every m E Zd. Scaling we can assume that j,(m ) - / ( ^ ) (2.33) for every m E Zrf. It follows that = E = E m€Zd me zd iA t V ?)!1- t 2 -3 4 ) Let Td{n) denote the number of representations of an integer n as sums of d squares. It is a well-known fact from Number Theory that if d > 5 then rd(n) > C n M (2.35) r 4 (n) > Cn (2.36) and if d = 4 and n is odd then where C > 0 depends only on d. See for example ([5], p.30r p.155, p.160). R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 69 Integrating (2.34) with, respect to the Haar measure dp and applying (2.2) we have c2 = / £ \fQ \* * p peso(d) meZd = J £ l A ^ o i 2^ © t€I=l meZd = E E / i / ( % ) i 2* ( « VZ n>0 > £ £ / i / ( ^ j « i 2<He) n>0 |m|2=2n+l|^|=l = J Z rrf(2n + 1) [ \ f ( \ l n + ^ ) \ 2 M 0 n~° IC!=L Using (2.8) and (2.35) or (2.36) we conclude from (2.37) that £ - 4 = f t ( J « + 5 ) < C C 2. n>0 y j n 4- \ (2.38) v Let q : R —►R be a fixed non-negative Schwartz function supported in [ |, 2J such that q(x) + q(x/2) = I when x E [1,2J. It follows that £ « (§ )= 1 j >o when x > 1. R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . (2.39) 70 Applying the Poisson summation formula to we have = £ £ (* ) = ff(0 )+ £ £ , ( < / ) + £ £ , ( « / ) . t^O t^SO (2.40) Note that t " (0) i =i ^ k = / r xA ^T | A(v i + 2w T )di 2 (2.41) Substituting (2.41) into (2.40) we get that 2 J htyq^dt < E / t m + E i& m i + E i& m i < n E -^ M t^ O l / n + | ) ? ( - ^ ^ ) + ^ !P i + " i0 V " + 5 ' t/^0 [ |F(ar)Dw(*)|cte W>i where the last inequality follows horn L em m a 1 and L e m m a 2. R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . (2.42) 71 Prom the definition of D n (x ) (2.21) it follows that E u w * ) l = E P » (* )l + E V <2|x| j> 0 * E I ^ W I 2J> 2|i| ^ 23<2|i| + ' E 1 § S C (2.43) 2i>2|x| for every |x| > 1. Putting iV = '1? in (2.42), summing over all j > 0 and applying (2.39) we get by Lebesgue Monotone Convergence Theorem 2 [ h(t)dt { < ^ — ^ k J n +h + C m \i+ C [ n>Oy/n + h_ V ^ \F(x)\dx < C(G2 + ||F ||1) (2.44) where the last inequality follows from (2.38). Prom the definition of h(t) (2.8) it follows that Ml) < CH/11? (2.45) for t < 1. Therefore, we have OO J | / ( a r ) \2dc = J 1 / ( 0 1 dat{i)dt 2 0 oo = J h(t)dt o < C (G 2 + ll/ll2) (2.46) where the last inequality is obtained from (2.44), (2.45) and (2.7). Prom (2.46) it follows that f E L2 and ll/lh < C (G -f- [[/Hi) R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 72 with C depending only on d. □ If d > 5 then r d(n ) < C n d*~ (2-47) where C > 0 depends only on d. See for example ([5]. p.155. p. 160). An argument similar to the one used to get (2.37) but without scaling shows that g2 = f 11, i Peso(d) m*zd = 1/(0)|2 + ^ r rf(n) f \f{y/n£)\ 2 da ( 0 KM 1 = l/(0 )P + £ ^ n>l n 1 / l / K ) | 2^ © J . (2.48) Using (2.8) and (2.47) we conclude from (2.48) that c 2 < ll/ll? + c £ 4 = f c ( V 5 ) . (2.49) ST Vs Repeating arguments which we used to obtain (2.44) we get OO < 2 f /t(f)rff + C ||F ||, I < C'dl/lli + ll/ll?). (2.50) Hence we can formulate an inverse theorem to T h eorem 1: T h eorem 1': let d > 5 and let f E LI(Md) H L 2 (Rd) and let gp be ■periodizations o ff 9P{x) = ^ 2 f { p ( x - 1/)) uezd R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . (2.51) 73 then gp E Z/2([0, l]'*) for almost all rotations p E SO(d) and J II9,\\ldp < C ( H /lb + ll/ll>)s peso(d) where C depends only on d. C orollary: interpolating between the trivial p = 1 and p = 2, we obtain the following generalization of the previous theorem for 1 < p < 2: let d > 5 and let / E L l (Rd) fl Lp(Rd) and let gp be periodizations of f 9P(z) = X ) f(P (x ~ v )) uezd then gp E Lp([0, l]d) for almost all rotations p E SO(d) and IM?<fc<c(||/||,,+11/110'’ J Peso(d) where C depends only on d. 2.3 C ase 1 < p < We will generalize T h eorem s 1 and 1' in the spirit of the Stein-Tomas Theorem ([4], Chapter 6.5). T h eorem 2: let d > 4 and let f E Lp(Rd) where 1 < p < If periodizations o ff 9p(x ) = £ / ( p ( a : - v ) ) vezd R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 74 are in Z,2([0r l}d) for almost all rotations p 6 SO(d) and G2= J \\gP\lldp<oo peso(d) then f e L 2 (Rd): il/ila < C(G + \\f\\p) (2.52) where C depends only on d and p. It will follow from the proof (see (2.37)) that we can replace f with peso(d) f ||<7p —ff(0 )||\dp In T heorem 2. We will also obtain an inverse theorem. peso(d) T h eorem 2': let d > 5 and let f 6 LL(Ra) n L2 (Rd) and I < p < and let gp be periodizations o f f 9p(x ) = f(p (x ~ ")) i/€Z d then gp 6 L2([0, l |rf) for almost all rotations p 6 SO(d) and I lift. - S M U dP 2 C ([|/]b + ll/ll,)2 (2.53) pesow where C depends only on d and p. Since Schwartz functions are dense in Lp(Rd) fl Z,2(Rd) it follows from T h eorem 2' that we can define periodizations gp of / G Lp(Rd) fl L2 (Rd) where 1 < p < Jpj for a.e. p 6 SO(d) as elements of the quotient space of L 2 ([0r Ij^) modulo constants. f R em arks: 1. As the following example showsr we can not replace peso(d) R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . \\gp — 75 g(0)\\ldp with, f \\gP\%dp in T heorem 2' when p > 1. Let 0 : —►C be a peso(d) Schwartz function supported in 5 (0 ,1 ) such that 0(0) = 1. Put f(x ) = 0 ( f) . Then 9p = / ( 0) = 1 but 2. The next example from ([4], Chapter 6.3) shows that p can not be greater than jn T h eorem 2'. Put / ( * , ........................................................................................... (2.54) where 0 : Rd —* C is a Schwartz function supported in 5 (0 ,2 ) such that 0 = 1 in 5 (0 ,1 ). Then \\9P - m \ \ \ d p = 2d [ \ m \ 2 d a (0 •'KM > Ced~l p € S O (d ) but ii/n ? = ^ 1 4 It is an open question whether T heorem s 2 and 2' are valid when < p< We discuss this further in R em ark 2 at the end of the section. P r o o f o f T heorem 2: The proof goes quite sim ilarly to the one of T h eorem 1. We will replace L em m a 1 with L em m a 3: Let q : R —►R. be a Schwartz Junction supported in [|,2 ]r let f R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 6 76 Lp(Rd) where 1 < p < 2 and let b 6 [0,1). Define H i : R. —►C Then for large enough N we have V " ' I LT f M ^ ^ ll/H p 2 ^ \ H i l y ) \ < —j f s' u? 0 i r , CC\ (2*55) where C depends only on q and d. P r o o f o f L e m m a 3: The only difference in the proof is to obtain an inequality analogous to (2.13). Using Young’s inequality we have ||/ * f\\q < ||/||p where 1 -F ^ Therefore, \ f ( f * f)(x)w (x)dx\ < ll/IIJIMI*. Substituting derivatives of dat{x)x{\x\<i\ with respect to t instead of w, we get the desired inequality where t > 1 and C depends \ h f \ t ) \ < C t^ V W l (2.56) only on k and d. □ The main difficulty is to prove a lem m a analogous to Lem m a 2: Lem m a 4: Let q : R —►R. be a Schwartz function supported in [^,2], let f 6 Lp(Rd) where 1 < p < ^ and let b 6 [0,1). Define H-xjt: R —►C R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 77 Then we have £ | £ & irjtQ » M I < c » / I I 2, (2.57) j >0 with C depending only onp, q and d. P r o o f o f L e m m a 4: Recall from (2.22) that ^ 2,iv(^)| = 2 / ( f * f)(x )D x iU(x)dx where D t f A x ) = X{\*\>i}ei2™b f i V q ( t ) e - l2™(m ) \ N t ) d- l f o ( N t x ) d L (2.58) Denote by K-„(x) = £ (2.59) D * » . i>0 Then | £ f f 2l* M | i>0 = 2| / ’( / . f l ( x ) £ o , , ( * ) < f c | ^ = 2]J i>0 f(x ) (K u * f)(x)dx\ (2.60) R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 78 I f ] / = oo or ] / = 2 we have First we will show that I I K .I L < l l £ | 0 j - , . l ( * ) I L l>0 < C \v\-$. (2.61) It follows from (2.26) that |DW*(X)I < f - e - < w W 1 1 Cfc(|^|iV2)-fc-a . (2.62) if AT i [M M| If v > 0 then the number of diadic N 6 [j|f, ^-] is at most 3. If v < 0 then there are no N in [Jj^, ^ ). Therefore, choosing k > ^ and summing (2.62) over all diadic N , we have [> 0 with. C depending only on d and q. Now we will show that l | i t |L < l l £ |£ W f e ) [ L < C . (2.63) Z>0 Since supp <f> 6 [|> 2} we can re-write (2.25) for a stronger version of the method of R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 79 stationary phase ([8], Theorems 7.6.4, 7.6.5, 7.7.3) j =o max (1, ) where Cj are some constants. Therefore, D k, ( x ) = (2.64) I3*! 2 j —0 where |0fe(ar)| < X{\x\>i} H~ti Cfc(NAf3) * * . Choosing k > | i | - 3“ ||0fc||oo we have m a x d .g J ^ - ) < l|0 fc ||l = J \<pk\d x + |i|<8|t/|iV S J \<j>k\dx |x|>8|i/|iV f (2-65) where C depends only on d and q. We can assume that u > 0 since D^jU(x) = <f>k(x) for v < 0. 2^ We can also ignore X{|x|>i} hi front of the sum in (2.64) because if 6 [£, 2], then |x| > uN > 1. We will consider only the zero term in the sum. The other terms can be treated similarly. The Fourier transform of v ’ 2 vN at point y is equal to iV ^i (2i//V )^i (£//V2)-^ f i/,(\x\)ei2 m'N2 lx[2 e - i2*2,'lVx'ydx = JRrf C {vN 2 )^e~ i2 x v ^ 2 [ if(\x[)ei2m'N2[x- ^ d x JM* where xl?(t) = 0(£, 2vN t)t~ i *L is a Schwartz function supported in (2.66) 2J whose deriva­ tives and the function itself are bounded uniformly in £, u and N (see remark after R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 80 (2.24)). The same is true about partial derivatives of ^(|ar|). Applying the stationary phase method for f | / i>(\x\)e J*d ([8], Theorem 7.7.3) we get o xr [c{vN * )-i ifN e[f,2 \y\\ [CkiuN2) - ^ if AT £ [M , 2|y|] dx[ < { . (2.67) Therefore, the absolute value of (2.66) can be bounded from above by: < (2.68) Similar inequalities hold for Fourier transforms for the rest of the terms in the sum in (2.64). The number of diadic iV 6 [^f,2|p|] is bounded by 3. Using (2.65), choosing k > 1 in (2.68) and summing over all diadic N , we get £ lA ft,(» )l < C (2.69) (>0 with C depending only on d and q. Using (2.61) and (2.63) and interpolating between p = 1 and p = 2, we obtain II*. * /lip -< where a p = f ^ E. a p > 1 if p < C M -" 'll/llp Summing (2.60) over all u ^ 0, we get the desired inequality £ l £ f f « » M t^ O l < c | | / l g - j> 0 □ Now we are in a position to proceed w ith the proof of T h eorem 2. The proof is almost the same as the one of T h eorem 1. We need also to replace inequality (2.45) R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 81 with the following one: 1 J J h{t)dt = o \f(y)\2dy lsl<i cw/wl < c\\f\\l < where p < 2 and C depends only on d. An argument similar to the one used to get (2.42), (2.44) and (2.46) (note that the interchange of summation by u and iV is not a problem) yields the desired inequality OO I J f(x)\-dx = J |/ ( « |J<fai(9<ft 0 oo = J h{t)dt < C(G 2 + \\f\\l) with C depending on d and p. □ The proof of T heorem 2' is the same (see the argument after the proof of T he­ orem 1 ). The important thing is that we exclude pp(0) = / ( 0) in (2.48) now. F in a l rem arks: 1. We can farther generalize T h eorem 2 '. Fix some q 6 [1, jp j). Interpolating between the trivial p = 1 and p — 2, we obtain the following d > 5 and let / 6 L1(Re£)flL p(M.c£) x ~ u )) (2.70) generalization of the T h eorem 2' for 1 < p < 2: let and let gp be periodizations of f 9 P( x ) = M uezd R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 82 then gp 6 Z^([0, l]d) for almost all rotations p € SO(d) and lift. - m W U e ^ C (||/||, + ll/ll,/ / p€SO(d) where C depends only on d and q. We can restate this result in the following way: if 1 < p < / then llff,-fl,(0)||''4>< C ||/|£ , p e s o (d ) if 1 < q < ^ and jp , < p < 2 then / peso(d) 2. < c ( ii /n ,+ n / ii ,) ' / . Conditionally on the exponent pair conjecture ([12], Chapter 4, Conjecture 2) we can clarify what happens when ^ < p< In our case the conjecture says that I ^ < Ce|r |en5 (2.71) n<u<m where m < 2 n and |x| > n. Let (3{x) = m ax(l, |ar|). P ro p o sitio n . T heorem s 2 and 2' hold if we replace ||/||p with ||/3e/[ |p and if p< provided the conjecture is valid. Using the example (2.54) we can show that the P ro p o sitio n is sharp up to e in the range of p for the estimate (2.53). P r o o f o f P roposition : R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 83 The m ain issue is to improve the result of Lem m a 4. Denote by Ljix) = K ^ x )' (2-72) 23<v<2>+1 Using summation by parts we obtain from (2.64), (2.59) and (2.71) that H id )i = i y , 2><u<p+l < Ce\x\e2 ~ ^ j . We will deal with the following expression instead of (2.60) i//w(ij«/)w<fc| = iJ h x w {Li*f}{x)dx\ If pf = oo or p' = 2 we have n ( Lj * / ) „ II ^ Hoc W ^ A h ^ „ 4i<ixi - II - v ) \ - \ f ( y ) \ dv + 4 i > w \L A x - < ii^ iw im < C€2 - ^ W f \ \ i , < H W Ib < W liUlfh v)\- \ f(y)\ dy u Hoc < cn /h . Interpolating between p = 1 and p = 2, we obtain < c ,2^ m \ \ p where ^ ap > Q i£p < R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . O 84 3. Concerning the lower dimensional cases we can use the following results from the Number Theory: ^ (n ) < C n i Inn Ininn, < C n ln ln n . See for example ([2]). There is an infinite arithmetic progression, e.g., n = 8 k + 1 , such that r3(n) > Cen 2 _e. See for example ([6]). Then T h eorem 1 holds when d = 3 and T heorem V holds when d = 3 o r d = 4 ifw e replace M i = mezd with mezd V ' l & M li 4—' In \m\ Inlnlml m eZd,|m |> 3 11 11 or y 4^ m eZ d,|m |>3 2 |gp(m )l I n ln lm l correspondingly. 2.4 C ase d = 2 and p = oo Some results on the Steinhaus tiling problem are related to T heorem 1 since peri­ odizations naturally appear in the problem of Steinhaus. In particular, showing that there are no measurable Steinhaus sets in dimensions greater than two, M. Kolountzakis and T. Wolff proved the following theorem on periodizations in higher dimensions R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 85 ([10], Theorem 1) which can be viewed as some version of T heorem 1 for p = oo : K olou n tzak is and W olff’s T h eorem If f 6 Ll (Rd) and its periodizations gp are constants for almost all rotations p 6 SO{d), then f is continuous and, in fact, ll/ll- < cyi/ii, provided that the dimension d is at least 3. Obviously, this statement is false when d = 1. We will show that it doesn’t hold either if d = 2. The fact that gp are constant means that f(p(k,l)) — gp{k,l) = 0 for all (k, I) 6 Z2\( 0 ,0) and almost all p 6 SO(d) which means that / vanishes on all circles of radii \Jl2 + k2 > 0. Denote by X the Banach space of functions from Ll (R2) whose Fourier transforms vanish on all circles of radii \/l 2 + k2 > 0 X = { / 6 Ll {R2) : / ( r) = 0 if |r| = V P T P , (k , l ) € Z2\(0 ,0 )} . We will use the notation x < y meaning x < Cy, and x ~ y meaning that x < y and y < x for some constant C > 0 independent from x and y. The next lemma crucially depends on the following fact from the Number Theory ([5|, p.22): The number of integers in [n, 2n] which can be represented as sums of two squares is nen where en < 0 as n -»• oo. Lem m a 5: There exists a sequence of Schwartz functions f n € X such that ™ P r o o f o f L em m a 5: Let at < J M l-h l/«(0)| < oj < ... be the enumeration o f numbers R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 86 dm — y/l2 + k 2 in ascending order. Denote 5m = 0 ^ 1 —am.. As we already said the number of a™ in [y/n, 2y/n\ is nen. Let and ami be correspondingly the smallest and the largest such v . Then mi—I y~] Sm = Omi —V ~ s/n. m=TTlQ Let fi = with small enough constant C > 0 so that if M = {m, mQ < m < rrii : 6 m > J} then m m€M since mt —mo ~ nen. Choose coordinate axes x and y. We will construct f n supported hi 1J Rm where Rm is a largest possible rectangle inscribed between circles of radius mgAf dm, and CLm+i with sides parallel to the coordinate axes. Then Rm is of size ~ 6 m x y/Sm&m > Sm x y/fiy/n. We will split each rectangle Rm further into smaller [^p-] rectangles r of the same size y /5 y/n. The number of these rectangles r is um mGM T fim E m£M 5 y/n ~ 1 ~ = n fn since fim > fi for m E M. Enumerate these rectangles rk, k = l T...riV. Let r* be centered at (A^.O) It is clear that [A* — Aj| > 5 for k / I. Let 0 be a nonnegative Schwartz function on K. supported in [—A, |J. Define f n as the following sum: s 5 v tv n R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . p -73) 87 The Ar-th term in (2.73) is supported in t v Therefore, f n is a. Schwartz function supported in (J Rm- Hence /„ vanishes on all circles of radii a*. Taking the inverse rngM Fourier transform of (2.73), we get N / n ( f , V) - e,Afc€. 8 y/n) (2.74) jt=i Then /n(0) = S m y /6 r fi(fi)N 1 I 1 rs/nen y y/nen _ \/n (2.75) y/^ri We used here that <p(0) = f (p > 0 is some nonzero constant. Denote fc=i Since | Xk^x-L\ > £ = 1 for k /■ I we have '2 ~ N / « ■ for any interval / of length 4tt (see ([14j, Theorem 9.1)). Therefore, j\9 \ < y ffifjw < (2.76) VM for any interval I of length 47t. Since 0 is a Schwartz function, we have that I \m \ < T 2‘ + X‘ R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 88 The L 1 norm of (2.74) is iV [ \fn(£,v)\d£dn = ||0||i f |0(OI -I J J = < k= 1 OO (t+1)4,r c £ f w a i • w a i-ie fa- ~ 1i E tV * /= —00 < t2-77) Dividing (2.77) by (2.75) we obtain the desired result as n —>• 0 0 . □ C orollary: It follows immediately from the lemma that ll/lk«(D (0,l)) f€X iiTii II/Hi = °°- We claim that there exists a function / 6 X such that ||/||z,«(D(o,i)) = 00- Suppose towards a contradiction that this is not true. Then the restriction operator T : f -*• /|d(o,i) maps X to L°°(D(0r 1)). Note that if f n —►/ in L l and f n —►g in L°°(D(0 ,1 )), then f — g a.e. on D (0 ,1 ). An application of the Closed Graph theorem shows that T is a bounded operator acting from X to L°°(D(0r 1)). This contradicts to the C orollary. Thus we proved our claim. Obviously, this function / is not continuous. Therefore, it can serve as a counterexample to the T h eorem when d = 2. R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 89 B ibliography [1] P.T. Bateman, On the representation o f a number as the sum of three squares, Trans. Amer. Math. Soc., 71 (1951), 70-101. [2] R.P. Boas, Entire functions, Academic Press Inc., New York, 1954. [3] P. Borwein, T. Erdelyi, Polynomials and polynomial inequalities, SpringerVerlag, New York, 1995. [4j K.M. Davis, Y.-C. Chang, Lectures on Bochner-Riesz means, Cambridge Uni­ versity Press, Cambridge, 1987. [5] E. Grosswald, Representations of integers as sums of squares, Springer-Verlag, New York, 1985. [6J E. Grosswald, A. Calloway and J. Calloway The representation o f integers by three positive squares, Proc. Amer. Math. Soc., 10 (1959), 451-455. [7j V. P. Havin, B. Joricke, The Uncertainty Principle in Harmonic Analysis, Springer-Verlag, Berlin Heidelberg, 1994. [8] L. Hormander, The analysis of linear partial differential operators I, SpringerVerlag, Berlin, 1983. [9] M. Kolountzakis, A new estimate for a problem o f Steinhaus, International Math­ ematics Research Notes, 1996, No. 11, 547-555. [10] M. Kolountzakis, T. Wolff, On the Steinhaus tiling problem, to appear in Mathematika. [llj O. Kovrijkine, Some results related to the Logvinenko-Sereda theorem, to appear in Proceedings of AMS. R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n . 90 [12] H. Montgomery, Ten lectures on. the interface between analytic number the­ ory and harmonic analysis, CBMS Regional Conference Series in Mathematics, Amer. Math. Soc., 1994. [13] F. L. Nazarov, Local estimates o f exponential polynomials and their application to inequalities o f uncertainty principle type, St. Petersburg Math. J. 5(1994), 663-717. [14] A. Zygmund, Trigonometric series, v. I and II, Cambridge University Press, New York, 1968. R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .