Sharp Bounds for the Operator Norm of the Mehler Kernel Operator Citation Deutsch, Leslie Joseph (1981) Sharp Bounds for the Operator Norm of the Mehler Kernel Operator. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/zcm4-w140. https://resolver.caltech.edu/CaltechTHESIS:09192025-004709766 Abstract In this thesis the norms of the Mehler kernel operators are calculated. In particular, Babenko's conjecture about the norms of these operators with purely imaginary parameter is settled. The proof is inspired by Wiener's approach to Fourier theory and his proof of Plancherel 's theorem. An account is given of Wiener's approach, as well as of the important tools and theorems, particularly those of Beckner, needed to prove the main result. Applications to Kober operators and smoothing operators are given. 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): Luxemburg, W. A. J. Thesis Committee: Unknown, Unknown Defense Date: 11 June 1980 Record Number: CaltechTHESIS:09192025-004709766 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:09192025-004709766 DOI: 10.7907/zcm4-w140 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 17690 Collection: CaltechTHESIS Deposited By: Ben Maggio Deposited On: 04 Oct 2025 12:24 Last Modified: 04 Oct 2025 12:33 Thesis Files PDF - Final Version See Usage Policy. 16MB Repository Staff Only: item control page

SHARP BOUNDS FOR THE OPERATOR NORM OF THE MEHLER KERNEL OPERATOR Thesis by Leslie Joseph Deutsch In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1981 (Submitted June 11, 1980) ii To my father, Ralph Deutsch ii; ACKNOWLEDGEMENTS I am deeply grateful to my advisor, Professor W. A. J. Luxemburg, for his expert advice and guidance throughout my work at Caltech. I am also indebted to Dr. A. J. E. M. Janssen who provided me with invaluable help in the preparation of this thesis during Professor Luxemburg's absence. I thank my parents, both of whom helped with the editing of this thesis as well as providing encouragement. Lastly, I would like to thank the California Institute of Technology for providing financial support and an excellent working environment and Fran Williams for typing this manuscript in record time. iv ABSTRACT In this thesis the norms of the Mehler kernel operators are calculated. In particular, Babenko's conjecture about the norms of these operators with purely imaginary parameter is settled. The proof is in- spired by Wiener's approach to Fourier theory and his proof of Plancherel 's theorem. An account is given of Wiener's approach, as well as of the important tools and theorems, particularly those of Beckner, needed to prove the main result. operators are given. Applications to Kober operators and smoothing V TABLE OF CONTENTS Page Notat.ion .. l Introduction . . ... Chapter 1. L2 Theory and the Fourier Transform 3 8 Chapter 2. The Fourier Transfonn in LP. 24 Chapter 3. The Mehler kernel operator for Gaussian measure 37 Chapter 4. The nonn of the Mehler kernel operator . . . . . 49 Chapter 5. Applications 1. Babenko's conjecture . 67 2. Smoothing operators 71 3. The operators of Kober 73 References 78 NOTATION Throughout this thesis we will use the following definitions. (l) R is the set of real numbers. (2) C is the set of complex numbers. ( 3) If 1 ,::: p < a, then er -a:: and, if f E LP , then 1/ er II f II P = J Jf(x) jPdx) ( P . -er (4) ~ L = { Lebesgue measurable and, if f E La:,, f:R + t Jessential s~p Jf(x) I < ~} then ll f II"'= essential sup lf(x) I . X (5) Let A bealinearmappingof LPintoLq (1_:::p,q<a:,). We write II A II p,q = sup II Af II f E LP, f rO l II f II P and call this the p,q operator norm of A. Let dµ be a measure other than Lebesgue measure. (6) If 1 < p < 0) then cc LP ( dµ) = {µ-measurable f :1R -+ t and Ir jf(x) jPdµ(x) < "'} -a:, 2 Lcr(dµ) = {µ-measurable f:R ➔ 0 !essential supjf(x) j < ~} X The corresponding norms , the obvious definitions. II f II µ; p' llfllµ;a:' and II A II µ,.P' q have 3 INTRODUCTION This thesis deals with integral operators associated with the kernels N ( X ,y) = w where X 2 2 2 l exp { 4xyw - (x ;y ) (l +w ) } 2 ' 2( l -w ) ./n( l -w ) and y are real and w is a complex number with lwl ~ l and w f ± l . Our main results consist of formulae for the values of the norms of these integral operators as mappings of LP Lq(l _s p ~ 2 and 1/p + 1/q = l ). where the case that was considered. into This generalizes the work of Babenko w is purely imaginary and q is an even integer lwl = l, w f ± l , we get expressions for By taking the norms of what are called the Kober operators, and by taking 0 < w< l we get expressions for the norms of the smoothing operators which were considered by~ Bruijn. The kernel N w was introduced by F. G. Mehler [ 9] in 1866 in connection with the Laplace equation. He proved the identity C: N (x,y) = w L wncpn(x)Cfh(Y), n=O where cpn denotes the nth Hermite function. This identity, which plays a very important role in this thesis, was reestablished as an identity for the corresponding integral operators by Myller and Lebedeff in 1907 [10]. For sufficiently well behaved functions f we define a:, U\f) (x) = J Nw(x,y)f(y)dy -cc (x ER) . 4 If we take f. w= - i then If 1 ~ p ~ 2 and Nwf becomes J f, the Fourier transform of 1/ p + 1/ q = 1 then it is we 11 known that a bounded linear mapping of LP into Lq. a: is In fact, we have the classi- cal Hausdorff-Young inequality I!;;!! p,q ~ (2n) l/q-1;2 . However, of ~ J is not a compact operator. difficult. This sometimes makes the study The Mehler kernel operator Nw, for compact and it may be used to approximate J. lwl < 1, For this reason is NUJ has become an important tool in Fourier transform theory. In 1933, N. Wiener [15] used the Mehler kernel operators to give a proof of Plancherel 's theorem which states that 3' is an isometry of L2(R). Wiener's approach was used in 1961 by K. I. Babenko [2] to find the norm of the Fourier transform as a mapping of LP into Lq in the case q is an even integer. This was done by calculating the norms of N as a mapping of LP · into Lq for w even integer. w purely imaginary and q an Babenko conjectured that the formula he found for this norm (with q even) holds for all q,::: 2. One of the purposes of this thesis is to settle this conjecture. In the proof of Babenko's conjecture we shall use a result of W. Beckner [4] which was generalized by F. B. Weissler [16]. This result, which was proven using a method of E. Nelson [11] concerns the norms of Mehler-type operators defined on Lp(dµ), where 1 ~ p ~ 2 and dµ denotes the Gaussian measure on the real line. method, to find the exact va 1ue of Beckner was able, by this l!Jllp, q. We now briefly mention some areas in mathematics where operators of the Mehler type occur. 5 In a paper of 1939, H. Kober [8] considered the operators Nu; with /wl = l as roots of the Fourier operator J. At about the same time (1937), E. U. Condon [5] considered the same operators to construct a continuous group of transformations that includes the cyclic group of transfonnations generated by J. In 1961, V. Bargmann [3] constructed a Hilbert space of entire functions on which Fock's operator solution to a corrmutation relation is realized (Bargmann-Fack representation). Hilbert space L2 (1R) The transition from the usual to this space is given as follows. The mapping B defined by 0C (Bf)(x) = TT-¾ maps 2 L (1R) onto the space of all entire functions, g, of order< 2 and type ~~such that f 2 2 /g(z) 1_ e- lz 1 dz ( The latter space is a Hilbert space if we take the obvious inner product. Then B is· an isometry that maps the nth Hennite function onto zn/,.fa!. If, now, w is a complex number of modulus less than or equal to one, then (where we take (N u.,f)(x) = f(UJ<) if w = ±1). N. G. De Bruijn, in a paper of 1973 [£] presented, among other things, a method for studying generalized functions by means of smoothing 6 operators. For a.> 0, the smoothing operator Sa. is essentially the Mehler operator with w = e-a. (see proposition 5.5). He also noted that the set of smoothing operators (and so the set of Mehler operators) forms a semigroup under composition. To sketch De Bruijn's approach to generalized functions, we let S be the set of all functions of the form Sa.f with a. > 0 and f E L2 (1R). Thus S can be thought of as the test function space on which the theory is built. A generalized function is then defined as a trace in the space S, i.e. a mapping F of (Q,cx,) into S such that Sa.F(S) = F(a.+ ~) for all O<a.,S< CXl, Both De Bruijn and Bargmann note that if H is the Hermite operator (harmonic oscillator), then Nw = UJ-l/2e-(-}'log w) H . In pa rt i cul a r, if we put U(t) = eit/ 2N . for t ( R, e,t then u is the sol ut ion to the Schr~di nger equation Ut = - '}' i HU with U(O) = I. We finally give a survey of the chapters of this thesis. In chapter one of this thesis, the basic properties of Nw are developed and Plancherel 's theorem is proven by the method of Wiener. Chapter two deals with the Hausdorff-Young inequality for result and conjecture of Babenko. a:- and the The Gaussian measure form of the Mehler operator is treated in chapter three. Chapter four contains the 7 main result, i.e. the calculation of a function of a parameter II Nwll p,q· The norm is expressed as y which is given impl icit_ly i.n terms of the roots of a certain fourth degree polynomial. late y and hence ing operators. In chapter five we cal cu- NwII p,q in the special cases for Kober and smoothWe also complete the Wiener-style proof of the sharp II Hausdorff-Young inequality for the Fourier transform. 8. Chapter l L2 THEORY AND THE FOURIER TRANSFORM We begin by defining the Fourier transfonn . f E L1 (lR) we may define, for x ER, For functions (Jf)(x) (1.1) Norbert Wiener, in his book The Fourier Integral and Certain of Its Applications Ds], showed by using Hennite expansions that J is also definable for f ~ L2(1R) and that J is in fact an isometry of L2(1R). This result is usually referred to as Plancherel 's theorem. Because Wiener's methods are similar to those we will use in a more general form later in this thesis, it is instructive to study them in some detail. We begin with some elementary properties of J. PROPOSITION 1 .l(i) J is a linear operator from (ii) If A is the operator defined by Jf' = AJf and L1 to (Af) (x) = ixf(x) L~. then JAf = - ( Jf)' . {iii) If B is the operator defined by (Bf) {x) = f (X) - x2f(x) 8 then Here we assume f is sufficiently well behaved for these statements to 9 make sense. The validity of (i) is clear, (ii) follows from direct calculation, and (iii) is a consequence of (ii). From (iii) one might expect the eigenfunctions of the operator B to be of some importance in the study of the Fourier transform. we fix a complex number Hence A and consider the differential equation ( l . 2) + ... + a _ xn-1 +2nxn , 1 2 complex, and substitute g (x) = e-x / 2hn (x) 0 h were eac h ai into this equation. is This yields the formulae a k-2 = k(k-1) ak 2k + A- 3 from which we find (k = 2, 3, ... , n) , A= - (2n + 1) and where the last term in the series involves x or 1 according to 2 whether n is even or odd. It is clear that e-x 12 h (x) is the only n solution of (l .2)(aside from constant multiples) of the form 2/ e-x 2P(x), A = - ( 2n+l). where P is a polynomial of degree n, and l0 DEFINITION l .2. integer n is oaZZed the PROPOSITION l .3. PROOF. hn defined in (1 .3) for a nonnegative The poZynomiaZ nth Hermite polynomiaZ. n 2 hn(x) = (-1 )nex _d_ e-x dxn 2 n Let g(x) = (-l)nex j__ e-x dxn 2 2 • Then one may verify that g is 2 a polynomial of degree n and e- x /2 g(x) 11. = - ( 2n+l). • is a solution of (1 .2) with Furthermore, its leading coefficient is 2n whence g = hn by uniqueness. DEFINITION l .4. We define the th Hermite funotion by n (n=O, 1, 2, ... ). We notice that % E LP for each p ~ 1 as hn decays exponentially to zero at +m and -m. Some important pr~perties of the Hennite functions are contained in the following theorem. THEOREM 1 . 5. Let n be a nonnegative integer. Then the foZZowinq are true. n.I ✓TT , ( i) (ii) % isasoZution of (1.2) with 11. = - (2n+l ). (iii) Jcpn = (-i)n ~n· (iv) [Cfk };=O is an orthonormaZ system in 2 L. 11 (v) 2 e-x /2+2x"A-l = f / [2~f]} C/\(x) k=O PROOF. Using proposition l .3 and integration by parts we get the recur- sion relation Also, ho= l so ex: dx = and (i) follows. J (ii) is clearly true. m r n be nonnegative integers. 2 e-x dx = Jn, To prove (iv) let n and Then from (ii) we know We integrate this equation by parts twice to get a, J cpm ( x)cpn (x )dx = 0 , -ex: is an orthonormal system in L2 . 2 To prove (v) let gn(x) • e-x 12 hn(x). Then expanding the function exp(-x 2/2 + 2x"A-l) = exp(x 2/2)exp(-(x-"A.} 2 ) about "A.= 0 we get whence {cq,) g0 (X) = Ld:n exp(-x212 + 2xA- i)) (0) 12 so (v) follows from Taylor's theorem and (i). We may now prove (iii) by calculating cc = (iii) J;- f exp(-x 2/2 x(211.-iy) - l) dx + now follows by comparing coefficients of 11.k. We are now ready to introduce the Mehler kernel. DEFINITION l . 6. Let jw I ~ l • but w 'f ± l and Zet (,r is principal, vaZue) l Nw(x,y) = ~~~ f 2 ,/IT( l - u.; ) 13 Nw(x,y) is aaZZed the Mehler kernel of parameter THEOREM 1 .7. Let lwl < 1. w. Then C0 N (x,y) = w PROOF. ju;j < 1. Fix L wn Cft,(X)Cft,(Y) . n=O Direct computation yields -i N - x2N u.: = ~ i Nw - y2N = - 2<L - o N - NW . ~ ~~ w ~~ ~ w w It may be shown by induction that there exist polynomials Bn(x,y) x and y with coefficients dependent on u.: in such that Taylor's theorem then implies N (x,y) w where the Pn are polynomials in x and y that are independent of w and so cN a: 2 2) - 2w~- N = L - (2n+l)Pn(x,y)u.:ne-(x +y . / 2 ~ w n=O (1.5) 14 We may write N ( X ,y) w where the contour of integration lies inside jzj < l. may show that for all positive A and B, integers k and uniformly in r > l, From this one and nonnegative £ we have jxj~A, jyj ~B. Hence the sum in (1.5) converges uniformly on compact sets and hence we may differentiate it with respect to x and y freely. Doing this we obtain and, of course, a similar equation with x and y interchanged. Then (1.5) implies whence P0 satisfies the differential equations i P - 2x __~ P + 2nP =--,, i P - 2yPn + 2nPn = 0 • - oX2 n ox n n aye. n This has the same form as the differential equation for hn (compare with (1.2) and following paragraph), therefore there exist functions 15 F and G such that Now G(x)/hn(x) = F(y)/hn(y) whence F and G are constants. We have obtained for constants en. Hence N ( X ,y) UJ for some constants dn that depend only upon n. We now have and, in particular, for x = 0, From theorem 1.5 (v) we know that for n even, (~ (0))2 = n whence, expanding n! 2nJrr[(n/2)!] 2 2 -k (1-w) 2 in its power series, we find dn = 1 for 16 n even. Furthermore and a similar, though more involved calculation of ~~(O) dn = l for n odd also. DEFINITION 1 .8. N w , For with parameter implies The theorem is now proven. jwj < 1, we define the Mehler kernel operator, w by For now we will consider the domain of Nw to be L2 although later we will consider other domains. Some important properties of Nw are listed in the following proposition. PROPOSITION 1 .9. Let ( i) Jwl < 1. Then the following are true. n Nw~n = w er., for n = 0, 1 , 2, ... . (ii ) Nw is a bounded operator of L2. (iii) w is a aompaat operator of N L2. PROOF (i) follows directly from theorem 1 .7. f E L2 and calculate, To prove (ii) we let 17 r If cg l!Nwf 11~ = -co cg 2 f(y)Nw(x,y)dy 1 dx -co cg cg 2 IN ( x , y ) I dydx . w -co -co However, we also know cg [ f IN w(x,y) 12dydx = f -cc cg CD cg -co -cc J £ 2 I wncpn(x)cpn(y) 1 dydx n=O -co = CD - J n=O ~ < luPcpn(x) !2dx -ai 1 = 2 • 1 - lwl Finally, N L2 (R2 ), hence it is compact and we have (ii) and (iii) (see [17] page w is a bounded integral operator with a kernel that is in 319). We have mentioned that cp E LP for each ing theorem gives bounds on the LP norm of THEOREM 1 . 10. c(a,p) 0 Let 1 ,::: p ,::: co, 0 < a < 1 . depending only on (n = 0, 1, 2, ... ) . a and p 1 ,::: p ~ =. The follow- ~. n Then there exists a function suah that l:%I:p ~ a-"c(a,p), 18 PROOF. Let 1 ~ p ~ ~. From proposition 1 .9 (i) we have = c(a,p). THEOREM 1 . 11 . PROOF. so Let lw I < 1. Let f E: L2. Then Then 19 Also, we clearly have equality if and only if f is a constant multi- ple of C/b· THEOREM 1 . 12. PROOF. ( i) Let Let O <a< 1. f E LP;. 1 < p < ex,. Then We calculate 2 2 2a - -____,,.... 1+a ) x2 exp --,,4 2 \( 1 - a 2 ( 1 -a ) l 2 2 = J ( / 1 +a ) exp YJ(l-a2)- x)l-a4 dy ,•. J -a: §Z"_) 1 2 l ( _ 2 exp x2 . 1-a - ( ~ ) 1'2 2(1+a 2) Let 1/p + 1/q = 1. We will now prove IINa l~,p ~ ~- The case p = 1 may be easily verified so we will assume er, = )l p > 1. Then if f E LP, 20 .e. ex: ~(JZ)qf -a, ex: J jf(y) jPNa(x,y)dy dx. -ex: We use Fubini s theorem to interchange the o~der of integration and in 1 so doing we conclude We now allow the value p = 1. Let e > O and let tion with a finite number of jumps such that g be a step func- l:f-glip < e/4. Then it may be verified that there exist positive numbers M and A such that for all O < a < 1 and all real x. Also, Nag-+ g pointwise except possibly at the points where g is not continuous. from Lebesgue's dominated convergence theorem that Let a besoclosetoonethat Then and we have proven (i). llNag-g[IP<e/4. It now follows 21 Now let f E L2. l:Naf-fl:2 < €/2. Let e > 0 and let a be so close to one that Define for each positive integer m Then ~ llNa f - Pm f 1:2 = ll L an ( f, %)cpn 1:2 n=m This completes the proof of (ii). We may now prove Plancherel 's theorem by Wiener's method. THEOREM 1.13. whence J For aZZ l f E: L 2 f1 L we have ma:y be extended to an isometry of 2 L . 22 PROOF. Let O < t < 1 , and f E Ll n L2 . Then s . jf(y) I E L1 . Also • N ( e-ixy • 1,m ·t x,y ) = _1_ !'J-:"n t-+1 _, ..;c..11 It then follows from Lebesgue's dominated convergence theorem that lim (N_itf)(x) = (Jf)(x) t-+1 pointwise. Fatou's lerruna and theorem 1 .11 imply Since any function in L2 may be approximated in L2 by functions in L1 n L2, we have shown that a can be extended to a bounded linear operator of L2 . We now have, by Theorem 1 .12, for f E L2, C0 IIJf lk = Ua I; ( f, CfJi, ) q;n 112 n=O 23 =( n=O f (f , 't'n,,2)~ rn = /'fl' ! 12 • 24 Chapter 2 THE FOURIER TRANSFORM IN LP In this chapter we will examine the development of the classical Hausdorff-Young inequality. Throughout this chapter p will be a real number in the range 1 ~ p ~ 2, conjugacy relation 1/p + 1/q = 1. and q will be defined by the We will allow the value q = +~. The first inequality we will look at is due originally to Titchmarsh and its complete proof may be found in his book Introduction to the Theory of Fou.rier Integrats [14]. THEOREM 2. 1. F·o r f E LP n Ll we have liJfl~ ~ (2n)l/q-l/ 2 llfllp, so J may be extended to a aontinuous Zinear operator of LP into suah that the ahove inequaZi ty ho ids. PROOF. The case p = 1 is treated separately. 1 =- ,/2.n Notice that if we take g(x) -- e-x2 /2 then If f E L1 then Lq 25 whence II ;:, 11 1 , oc = 1/ ,fin. p > l. The proof of theorem 2.1 makes no use of the Mehler kernel oper- ator. Titchmarsh first treats the case of such q, :Jf of ;;.f In the remainder of this proof we wi 11 assume may be written as a q an e"en integer. For q-fold convolution and so the norm may be easily calculated. To do this, several lemmas are needed. LEMMA 2. 2. ( YOUNG I S INEQUALITY) 11., µ > 0 where and a- 11. + µ < 1 , If f E Ll / ( l - "' ) ana then l-11.-µ Q) 1£ fgdx I~ (Dfjl/(1-A) Jg 11/(l-µ)d1 (L Let 11.,µ,f,g µ er ( l LEMMA 2.3. g E Ll / ( l -µ) Jfjl/(1-A)dx) Jg 11/(1-µ)dx) "' be as in Zemma 2.2, and Zet J Q) c(x) = f * g (x) = f(t)g(x-t)dt. Then We now assume f to be a continuous function of compact support. . 26 LEMMA 2.4. Le~ ck(x) = k be an integer. k-fotd convolution f f(uk-1) f f(u 1)f(x-u1-u 2•.. - uk-l )du1 •.. duk-l -cc be tongs to Then the -ex: 2 L and Lemma 2.2 follows directly from Holder's inequality for three functions. Using lerrma 2.2 and c as in lemma 2.3 we see L a:, lc(x) I.'.: ( ( ( ) ) lf(t) 111 l-A) lg(x-t) 111 l-µ dt 1 -1,,-µ II f II 1/(H,) µ/ ( l - 1,,) • lb 11"'0-µ} 1/(1-µ) vJhence -a. and lerrma 2.3 follows. To prove lerm,a 2.4 we notice so upon several applications of lerrma 2.3 we have 27 and hence the desired result. Now let ck be as in lemma 2 .4. Then by writi.ng the iterated inte- gral as a multiple one and performing a change of variables, (Jf)k = ;, ((2n)-k/2+1/2ck) ' and we may use theorem l .10 to get By our usual extension procedure, the above inequality holds for all f E LP and q an even integer. Titchmarsh then attacks the problem of general q by using the following lemma of Hausdorff and Young [lA]. LEMMA 2.5. z;, • ·te se t o f rea~1 numb ers cor any f ~n~ [d lTl111 1n=-n we have The proof of this lemma is long and cumbersome;it will not be presented here. Let f be a continuous function of compact support, and define av= ( v+l )/ A f(x)dx J v/ A for integers v, and A> 0, b > 0, 28 n • / = 6 a e-,vx "v=-n v where n = [ ~ J - l . So uniformly on compact sets. Furthennore, by lemma 2.5 b ~ 2n([ jf(x) jPdx) l / ( p-1 ) Hence, by Fatou's lemma a: 0:, -a: -a: J j(Jf)(x) Jqdx = f l~+i~f Jg"-(x) lqX(-n;\,n;\)dx We again approximate a general f E LP by continuous functions of com- pact support to finally prove the theorem. 29 There is another, very quick proof of theorem 2.1 using M. Riesz's convexity theorem (see [12] chapter 5). THEOREM 2.5 (M. RIESZ) p. L 1 q. to L 1 (i=l ,2) Let T be a linear operator of norm Ki (the pi Then T has norm < K1-tKt from - 1 2 and qi from are not neaessariZy conjugate) LP to Lq where The proof of this theorem is not important to this thesis. However, if J is an isometry on L2 and that we use the already proven facts that it has norm equal to 1/..;Zn from which is exactly theorem 2.1. L1 to Lea we get immediately It is interesting to note that even this extremely powerful theorem gives no better result than theorem 2.1. In order to improve on theorem 2.1 we return to the study of Mehler kernel operators, - this time on LP. We will need a few more proper- ties of the Hennite functions and of the operator N~ in order to proceed. THEOREM 2.6. Let 1 _::: a _sex., 1 < b ~ °"• lwl < l, and f E La. Then 30 Lb. where the sum converges in Moreover, N UJ is a compact operator of (i.e. [N UJ j forms a semi group) . PROOF. From theorem 1 .7 we know that, for fixed x, C0 Nw(x,y) = L,wn C+t,(x)%(y) , n=O and from theorem 1.10 this sum converges in for a11 0 < t < 1 , 1 ,::: c ,::: co). Lb sense (as !lCfinllc = O(t-n) Hence a: ( Nwf) ( x) = J N ( x, y) f ( y) dy w 00 = I: wn ( f' qh) Cfln ( x) n=O for all x. The convergence of the latter series is surely in Lb sense, again because of theorem 1 .10. Also, C0 L, n=m lwlnl(f,%)1 11%llb er. < II f ll a L lwln n=m This last sum approaches zero as m ~er., again by theorem 1 .10. shows that This Nw can be approximated by the finite rank operators Pm, 31 where It follows that ty of N is a compact operator of w La. The semigroup proper- N follows easily from the identity for N f proven in this theorem. w w THEOREM 2.7. The Zinea:t' span of the set [%};=O is dense in La for 1 < a < o:,, PROOF. Let f E La and compact support such that one that Since II Ntf-fl!a < €/4. g E Lb, where e > 0. Let g II f-g II a < €/4. Let Pm be a continuous function of Let t < 1 be so close to be defined by 1/a + 1/b = 1, it follows from the proof of theorem 2.6 that form sufficiently large : II (Nt-Pm)g Ila < e/4. Hence and the theorem is proven. A bit more calculation gets us our first Hausdorff-Young inequality for N . w 32 THEOREM 2.8. Let lwl <l,] <p < 2, 11 Nw II p , q ~ [ n( l PROOF. 0::, Let a= (l+w2)/2(1-w2 ). w be real. -1/2 1/q ( 2n/ q ) Then • Then a is positive and we calculate 00 f f -o::, -w2 )J and let J J exp {-aq(x 2+y 2 ) 0::, (X) jN 0}x,y) lqdy dx = [n(l-w2)] -q/ 2 -cc 1 -00 -o:> ± 2 ~ xy} dy dx We now have, for any f E LP. The theorem follows. Theorem 2.8 is already good enough to give us a result for the smoothing operators of De Bruijn (see page 71 ). !IN w11 for w purely imaginary. We may also bound 33 THEOREM 2.9. lwl < 1, 1 < p ~ 2, and "let w be purely imagirzax,y. Let Then ' 2 -1/2 . 2 2 1/q IINwllp,q ~ [n(l-w )] [2n(l-w )/q(l+w )] PROOF. We again let a= (l+w2)/2(1-w2). Then a is positive and we have co JJ -co co co 2 -q/2 IN w(x,y) jqdydx = [n(l-w )] co f J exp[qa(x 2+y2)] dydx -co 2 -q/2 . = [n(l-w )] [n/qa]. We now have, for any f E LP. The theorem follows. We observe that the bounds of theorems 2.8 and 2.9 both approach infinity as lwl-+ 1. For this reason theorem 2.9, for example, is not good enough to prove a Hausdorff-Young inequality for the Fourier transform. If a bound for jjN wII is proven that remains finite as w-+ -i, then such an inequality could be proven using the method of Wiener. For special values of w and p, late the norm IINwllp,q explicitly. K.I. Babenko was able to calcu- We summarize his 34 results in the following theorem. The full proof may be found in [2 ]. THEOREM 2.10 ( BAB EN KO I S THEOREM) Let -1 < t < 1 and let q be a positive even integer. Define Then 2 2 ) l/2p-l/2q II Nit JI p q = Jy ( l + y ~ ' . rry( 1+t ) Moreover, this norm is achieved for the Gaussian function 2 k(x) = exp {-x 2 ( 1 + y2~ ) } • . py(l +t ) If we let t = -1 we get SKETCH OF PROOF. Babenko's proof is much too long to be presented here in detail. Instead we present an outline of his proof. the theorem as an extremal problem. be an We approach Let ~ = II N-it II p,q and let f LP function of LP norm one for which N-it achieves its norm. Let g = N-itf. If f exists then it can be shown that f satisfies 35 co LEMMA 2.12. ~ JNt( jg lq- 2g)eixydy ~q jf(x) I P-2f(x). = -ex: This lemma is proven by perturbation off. We now define u(x) = jf(x) jP- 2f(x). Then it can be shown that u and g are entire functions of order two and finite type. a= Let 1 + 2t2 0 2 y , E;(x) = eµX u(x). qy{l +t 2) Then the following is true. LEMMA 2.13. For all real x and y we have -a: The proof of l_errma 2.13 relies heavily on the theory of entire functions. At one point in the proof the path of integration must be changed. do this, To q must be an even integer to assure that the integrand is en- tire. As a consequence of lemna 2.13, for any h > 0, the entire function is bounded and hence is constant. Therefore 36 and, as h is arbitrary, ; is a constant. Since we have l/2q ;(x) = ( ~) • It follows that f(x) 1/2p = ( B!) 2 e -qSX /p and the theorem follows by direct computation of ~itf. 37 Chapter 3 THE MEHLER KERNEL OPERATOR FOR GAUSSIAN MEASURE Throughout this chapter we denote by µ the Gaussian measure for R, 2 dµ(x) = - 1- e-x 12 dx .,/Zn Lebesgue measure will be denoted by A or simply by dx if there is no confusion possible. The measure µ is finite and has been normalized so that µ(R) = 1. We define the modified Hermite polynomials Hn by The role played by the Hermite functions in the previous chapters will now be played by the polynomials Hn. Some of the properties of Hn are listed in the following proposition, which is a direct consequence of theorems 1 .5 and 2.7. PROPOSITION 3 .1. ( i) [Hn }~= 0 is a aompZete orthonormal system in 2 L ( dµ). (,·,·,·) eAX - /12 = ;._ t..k H ( ) '-- -kl k X k=O • ( t.. E 0:). 38 We now define the Mehler kernel operator for Gaussian measure. DEFINITION 3. 2. lw I < 1 . We denote by Let 1 .{ ( 2+ 2 ) 2 K (x,y) - - - -2 exp 2wxy - x 2Y w w - J -w 2(1-w ) 1 ' J the Mehler kernel for Gaussian measure and parameter K w and we call parameter K UJ We denote by f E LP(dµ), 1 ~ p ~ 2 , by the operator defined for (K wf)(x) = w. J f(Y) ' Kw(x,y)dµ(y), the Mehler kernel operator for Gaussian measure and w. F. Weissler, in his paper [16], presents a more general version of the following theorem. For the case we are immediately interested in, this simplified version suffices. THEOREM 3.3. Let 1 ~ p < 2, 1/p + 1/q = 1, and let w be a complex number of modulus less than one that satisfies the two relations Re 1 > 1/p 2 1- w ( 3. 1 ) and [Re 1 2 w 2 1 - -] [Re > 2J p 1- w 1 - w2 (3.2) 39 PROOF. p > 1 Let and put a = Re 1 1 1 - ,} p and b • Re w 2 . 1- w Then f J 0: 2n - co 0: IKw(x,y) JQdµ(y)dµ(x) = -co = = jl - w2 I-q/2 11 -w21 JCD JO: -a: -cc -q/2 exp { 1/2 (2rv'qa) - T (x 2+y2 ) + qbxy } dydx J CIC 2 2 exp{.9f- (ba - a) }dx -CIC which is finite by (3.1) and (3.2). (see . [17] page 319). If It follows that w is compact p C? 1 then we notice that (3.1) implies Iw2 - 1; 2 I < 1; 2 so a + b = Re _1£_ > O 1 -w and K 40 a - b = - Re 1 ~ w < O whence a2 - b2 < 0 for all fied. w and so (3.1) and (3.2) are never satis- The theorem now follows trivially. We also have the following theorem. THEOREM 3.4. Let w, Wi and wz be aomplex nwnbers of modulus less than one that satisfy (3.1) and (3.2). Then the following statements are true. (i ) K H = J"H w n n (n = 0, 1, 2, ... ). CX) (ii) PROOF. Kw(x,y) = I: wn Hn(x)Hn(y). n=O To prove (i) we let s = x/,/l, t ~ y/,/l and calculate ex: 1 =- f ex: -= 2 hn(t)e-t Kw(,/ls, $t)dt 41 2 12 = es J(hn(t)e-t 12 )Nw (s,t)dt cc 2 -cc The kernel defined in the sum in (ii) that also maps Hn to defines a kernel operator hence it must be the same operator as Kw. W. Beckner, in his paper [ 4 ], was the first person to precisely calculate the norm of Kw for and (3.2). operator J. w pure imaginary and satisfying (3.1) He was able, from this, to determine the norm of the Fourier The norm Beckner derived for J is the same as that de- rived by Babenko for the special case that q is an even integer. state and sketch a proof of Beckner's theorem below. We The main idea is to prove the result first for Mehler-type operators over a discrete probability measure and then use the central limit theorem. The main tool in the proof is the two point inequality" ( see lemma 3. 6). 11 THEOREM 3.5 Then (BECKNER'S THEOREM) Let l < p < 2, .!_p + lq = l and 42 REMARKS. Beckner states this theorem only for t = Jp'=1'"", however his proof still works if t not consider the case SKETCH OF PROOF. is in the above specified range. Also, he does P = 1 which is trivially proven as then t = 0. The idea of Beckners proof is due to Nelson [11]. will use the central limit theorem to obtain the Gaussian measure, We dµ, as a limiting probability measure of convolutions of Bernoulli measures. Let dv(x) be the discrete probability measure with weight 1/2 at the points x = ±1. The measure dv is referred to as Bernoulli measure, Let dvn be then-fold convolution of the measure dv(,.fa" x) with itself. The central limit theorem says that dvn converges to dµ in the sense of the space C0 (]R) * ( C0 {R) that vanish at is its dual). is the set of continuous functions of R with the topology induced by the L norm;c (R) * CD ±ex> 0 So, if f c C0 {R}, ex, ex, CD and CX> l im n-+o: co f f{x)d vn{x) = f f{x}dµ{x) • -0:l -co Beckner proves that a result analogous to theorem 3.5 holds with respect to each of the measures dvn. The inequality of the theorem then follows as a limit of inequalities with respect to these product measures. 43 The product measures dfln = dv(J, x1 ) •·· dv(,jnx.1) and each xi can assume only the va 1 ues on the corresponding measure space, then ± 1/ ,jn. If f are discrete is a function always has a representative f that is a polynomial of degree at most one in each of the n variables. By imitating the action of Kw nomials on the first two modified Hermite poly- (H 0(x) = 1, H1 (x) = x) we define, inductively, Mehler type operators over these discrete measure spaces. On the measure space over d v we define an operator C by C (a+bx) w = a + u;b x. The following lemma may be proven by careful calculation [4]. LEMMA 3. 6. Let - ~ _s t _s ~ . norm one of Lp ( dv ) to Then Cit is a bounded operator of Lq ( d v) . Lemma 3.6 is known as a two-point inequality. We define operators Bw,n, k on the set of functions defined on the measure space over by Bw,n, k ( a+bx k) = a + w b xk (k = 1, 2, ... , n) , where a and b are functions of the remaining n-1 variables. Let Cw, n = Bw, n , l ••• Bw, n , n From lemma 3.6 each Bit,n,k is an operator of norm one of LP(dvn) Lq(dyn). It follows that Cit,n is also an operator of norm one of to 44 Let Xn denote the space of functions symmetric in the n variables x1 , ... xn over dvn' and let Dw, n be the restriction of Cw, n to Then Dit,n may be shown to also be an operator of norm one. Xn . The sym- metric functions = i! 6 form an orthonormal basis in X m1 < • • •< m.e m1 L2(Xn). .•. X mi, (.e=O, 1, ... ,n) , We also have D w, n Let g E Xn and write n = I:; d a ..e=O .e n, .e Then we have II Dit n9 II v ·q s_ II 9 ll v ·p , ' n' n' or more explicitly, (J.·· j -a: -a, (3 . 3) By comparing the generating functions for Hn and that if each xi = ± 1/J,, then an,;t,O one may prove 45 l [i/2] +x)+-6 a n where the coefficients an n r=l .e,r are bounded with respect to n for fixed A/,r .e. It may now be shown that the inequality (3.3) as n + 00 • implies the theorem A comp1eta account of this proof may be found in [4]. It should be noted here that leITTna 3.6 is still true for w real and 1 - p ~ w < p - 1, that is Cw, 1 has norm one for these values of The rest of the theorem remains unchanged and we get the following. w. COROLLARY 3. 7. Let 1 < p ~ 2, ¼+ ½= 1 , and 1 - p ~ t ~ p - 1 , then IIKt II µ;p,q = 1. Beckner was then able to use his theorem to prove the sharp Hausdorff-Young inequality that Babenko conjectured. COROLLARY 3.8. PROOF. Let Then The case p = 1 was treated in chapter 2 (see theorem 2.1). now assume p > 1 . L a: { 1 < p < 2 and 1/p + 1/q = 1. co ~~e Theorem 3. 5 for t = ~ may be written as IL°K;Jp::r (x,y)g(y)dµ(y) lqdµ(x) }1/q :<, ¥ co jg(y) jPctµ(y) }1/p , 46 for all If we let x = Jl u and y = ~ v then this g E LP(dµ). becomes {£ .l CX) ih(v1°dv }l/ p { where 2 h(v) = g(hv)e-v 12 . F. Weissler was able to determine necessary and sufficient conKw to be bounded. In the case that Kw is bounded, Weissler showed further that its norm is usually equal to one. ditions on w THEOREM 3.8. (WEISSLER'S THEOREM) Let l < p, q < ~, vaZues for 2 < p < q < 3 and 3/2 < p ~ q < 2. but e:r:cZude the l l l Let -lp + -::::-r p = -q + -=-r q = l, lwl < 1, w F ± l satisfy and Zet 1 Re l -w2 ~ max[l/p, 1/q'} (3 .4) and ( Re l 2 - .!_) ( Re 1-w P l 2 - ~) ~ ( Re 1-w 2 w 2) 1-w (3.5) 47 Then Ii KUJ II µ,·P ' q = 1 • Moreover, K UJ is bounded if and on Zy if w satisfies ( 3 . 4 ) and ( 3. 5) . We observe that the case 1 -< p -< 2, J_p + lq ~ 1 is covered by this theorem. In this case, (3.4} implies lwl < i with equality allowed only if p = 2. For some p and q the case theorem. lwl = 1 is allowed in Weissler's By this we mean that Kw may be extended to lwl = 1, w 'f ± l and that the extension is an operator of nonn one. SKETCH OF PROOF. Weissler proves the boundedness of K UJ by relating KUJ to the Gauss-Weierstrass operator cc 2 (e ~)(x) = (4nz)- 112 J f(y)exp { - 4 (x-y) dy. 1 2 2 } -cc If w satisfies (3.4) and (3.5) then the Gauss-Weierstrass operator with z = y(l-w2)/4w (where Re(y/w) ~ O) cludes that Kw is bounded. is bounded, from which he con- It is possible, however, to prove this theorem without using the Gauss-Weierstrass operator. sati~fies Re and 1 1 - UJ2 > max { l l, 1 p' g J If w 1wi < 1 48 ( Re 1 ,)( 1 1 ) ( w ) 2 - -q, > Re 2 2 - -P Re 1- w 1- w 1- w 2 then calculations similar to those in theorem 3.3 imply the boundedness of Kw. The result that IIK wIIµ;p,q = 1 for these boundedness for the region (3.4) and (3.5). w wil 1 then imply To prove that Kw is bounded implies (3.4) and (3.5), Weissler exhibits explicitly the effect of Kw on Gaussian functions and shows that for and w satisfying (3.4) (3.5), their images are unbounded under K. w To show that IIK wII= 1 in the prescribed cases, Weissler proves a two point inequality. It is the same one as in Beckner's proof but Weissler was more careful with his estimates and so he could prove it to hold for more general p,q, and w. The remainder of the proof is then identical to the proof of Beckner's theorem. proof of this theorem is to be found in [16]. A detailed account of the 49 Chapter 4 THE NORM OF THE MEHLER KERNEL OPERATOR This chapter is devoted to proving the main result of this dissertation. The norm of N as an operator from LP to w jwj < 1, w 'f ±1. explicitly for all is calculated Since Nw is unbounded for all other w this settles the question completely. if pf 1 then Lq We will also show that N always achieves its norm for Gaussian functions of w a particular form. The main result is stated in theorem 4.1. :s 1, w f ±1 , 1 < p :S 2, ana THEOREM 4.1 (MAIN RESULT) Let p1 + q1 • 1 . Then there exists a comp Zex nwnber y such that lwl ( 4. 1 ) and (4.2) For any such y we have 22 - l l~ ( l - y w ) IiNw !'ip,q I y l/2p-l/2q rry(l-w2) • Moreover, this norm is achieved for the function l 2 2 ), /2p 2 f ( x) = ( 1 - Y w2 ex P x2 ( 1 + w 2 rry( l - w ) 2( l - w ) - l 2 ) y( 1 - w ) l. 50 We remark that the operators for N have not yet been defined on w LP [wl = 1, w "f ± 1 , and for this case the theorem should be read more carefully. What we mean to say is that Nw' defined at least on 1 2 L nL , satisfies for 1 2 f EL nL , Hence, N where C is the bounding constant in the theorem. may be extended linearly to w lI'N W IIp,q < C and the norm LP in such a way that is assumed for the Gaussian function f of the theorem. In the course of the proof it will be apparent that we can choose y = y(w) in such a way that for luial = 1, w I± 1 we have 0 IINlllo l~,q < C is thus a consequence of the corresponding in- The inequality equality for lw I < 1, for if 1 2 f EL nL , then we have 1.3 for method), C0 IINWof l~,q = J li; ti~f [(Nw rf)(x) [qdx -cc ~ 1 im inf C( llbr) r t 1 0 IJf II 'P ( see theorem 51 We also remark that the case p = 1 will be treated separately at the end of this chapter. We begin the proof of theorem 4.1 with the following lemma. LEMMA 4 . 2 . Let IwI _s l , w 'f ± l , l < p ~ 2 and there exists a complex number PROOF. l / p + l / q = l . Then y that satisfies (4.1) and (4.2). Let a and b be real numbers with a> O. Then (4.1) and (4.2) are true if and only if we can choose a and b so that the following are true: 2 2 2 l +w p- l + y w 2 ( l -w2 )- p y( l -w2 ) = ib These may be rewritten in the form l +w2 2 2(1-w) p -1 p{l-w2) l l w2 l w2 2 • Y = a, •y - p(l-w2) 2 •-y - 1-w • y = ib 1-w which are equivalent to the following equations . 2 - .P. 2 • l + w2 • 1-w P2 . l 1-w Y = a - i pb 2 l + u} _LP• 2 .......P._P l • w 2 • Y = a + bp - ib 2 ( p- l ) l - w l - w This system of equations has a solution y if and only if there 52 exist a > O and b E lR such that ( a-ipb+¥ •; :w: 2) ( ¥ l1 +tu~ tu (p-l)a-ipb) = 2 P w~ (l -w2 )2 (4.3) Hence there exists such a y if and only if there exist real numbers a and b with a> 0 that satisfy (4.3). We let l + 2 _ _Ul.,..2 = X + iy 1 - tu where x and y are real. It follows that x > 0. Also whence (4.3) becomes [(a+x) + i(y-pb)][(x-a(p-1 )) + i(y-pb)] = (x 2-y 2-p 2/ 4) + 2ixy. We take real and imaginary parts of the above equation to get the following two equations. (4.4) (4.5) (2-p)ya - (2-p)pab - 2xpb = 0. It will now be shown that for all tu and p satisfying the hypothesis of the lerrma we may find real solutions a,b of (4.4) and (4.5) such that a > 0. This suffices to prove the lemma. - First suppose p = 2 or x = y = 0. b = 0 , .:. p • a - 2,1p-:i- • Then we have the solution 53 Also, if pf 2 and x is equal to zero then (4.4) and (4.5) have the solution b = y/ p , a = / IT ¾- 1) / + f ] / ( p-1 ) If pf 2 then from (4.5) we obtain a = (2-p) (4.6) substituting this into (4.4) gives us g(b) = - (p-1 )4 x2p2b2 + 2(2-p) 2x2pb(y-pb) + (2-p) 2~-pb) 2(2pyb - p2b2 + f2 ) = 0. Now g(b) ( 4. 6a) is a fourth degree polynomial in b. • • • h 1s • a1ways negative • as 1n g 1s - ( 2-p )2p4 wh1c g(0) = [(2-p)py/2] 2 which is nonnegative. If y = 0 then (4.5) implies pr"' 2 . Al so, b = 0 whence (4.4) becomes the fol- lowing quadratic equation in a. - (p-l)a 2 + (2-p)xa + p2/4 = 0. We then take The coefficient of b4 54 which is always positive. Finally, if y f 0, then g(O) > 0 and so g has at least two real roots, one positive and the other negative. We now notice that g(y/p) as x f 0. So g has a real root b that satisfies 0 < Jb I < jy/p I , sign(y) = sign(b), whence (4.6) yields a positive a. The lemma is now proven. We need to analyze the root b of (4.6a) for circle in more detail. then x0 = 0. we can take Let w near the unit IWo I = l, We f ± l . If we now write Suppose Yo f 0. From what we found above we see that a0 = This gives rise to a y(Wo). Let f 1 (b,y) = (2-p) 2(2pyb - p2b2 + p21/ 4) , = 2(2-p) 2pb 2 2 = ( p-1 ) 4p b , . 55 We may then rewrite (4.6a) as By an appeal to the implicit function theorem one can show the above equation has two roots (one to the left and one to the right of which are close to Taking for positive b a y /p 0 if x > a and x + iy the root whose modulus is less than and a y(w) that satisfies is close to /·y/pj, y/p) iy . 0 we get a lwl < l y(w) + y(uu) if and w + ut)· If lLb = ± i . y O = O then If w = it, for - 1 ~ t ~ 1 , then y may be calculated explicitly (see theorem 5.1) This y is clearly continuous in t for t close to ±.1. With this rather technical lemma out of the way we may proceed with the proof of the main theorem. relation between Nw and In the following pages we will find a Kyw· From the results of chapter 3, we know a great deal about the nonn of K. w In particular, we will show that for the· y of lemma 4.2 is in fact a bounded operator with norm equal 4o one from LP(dµ) to Lq(dµ}, This. will be done by applying Weisslerls theorem to yw than w. The factor y is just smal 1 enough to force the domain specified by Weissler. Kyw )OJ rather to be within 56 LEMMA 4 . 3. Let Iu; I .:S 1 , w -f: ± 1 , 1 < p .:S 2, l / p + l / q = l , y satisfy (4.1) and (4.2). PROOF. Then We need only show that R e '(I.I) and 7,et Kyw has norm equal, to one. satisfies l > l. 2 2 - p l - y w (4. 7) and ~ l2 2 - -1)2.2: Re )Ul2 2 ( Re 1-yw P 1-yu.i J (4 .8) for then Weissler's theorem (theorem 3.8) will imply that equal to one. We let a. and Re l 22 l - y w S be as in theorem 4.5. K yw has norm Then 2 2 l = Re p - l + y w p p(1-iw2) t2 2 = £ Re p - l + w py(l -w ) e 2 =l.Re l+w > 0 8 as p > 0 and implies (4.7). (l+w2)/(l-w2) maps 1 -w2 lwl .:s l into Re w.2: 0. We now notice that (4.8) may be rewritten as This 57 2 ~ w2) Re p-1 + 2y 22 p(l-y w ) We multiply both sides by inequality as ~/2 ~ ( Re )2 yw2 2 . 1- y w (which does not change the sign of the S > O) to get 2 ( Re w2 ) , 1- w which is equivalent to (R 1+w2)2 Re w ). e 2 > ( 1 - w2 2(1-w) 2 (4.9) . We now notice that 1 + w2 Re -----,,,.-2 ( l -w2 ) Re 1 w = -1 Re ~ > O 2 2 1 +w 1- w (4.10) and Re 2 2 + Re l _w w2 2(1-w) 1+w = l Re 1 + w > O 2 1- w ' (4.11) because the transformations z1 = (l+w)/(1-w) both conformal mappings of Rez 1 ~ 0 and Rez 2 ~ 0. and z2 = (1-w)/(l+w) are lwl ~ 1 into the right half plane. Hence Multiplying (4.10) and (4.11) together then yields (4.9) and our lerrma is established. In order to state the relationship between need the following definitions. Kyw and Nw we will 58 Let DEFINITION 4. 4. a > 1, a be a real nwnber, ~ 'j: O.J let and let Re a. > 0. a. be a: aomp Zex nwnber, 0/)(x) ={(2rr)-1/2a .-x3/2af(x) f(x) if a < m if a = a;, (ii) Define the map M a. from La(dx) to La(dx) by (iii) Define the map T from to La(dx) 6 La(dx) by We have the following proposition. PROPOSITION 4.5. Let a ,2: 1. Re a.= 0 then Then the following are true. (ii) If M is an isometry of a. (iii) For real PROOF. The proof is by computation. La(dx) . a f O and f E La ( dx) we have Let f E La ( dµ). Then if a < a;, , 59 Since Icof = f, this result is trivial for Now let f c La(dx) a= co, hence (i) is true. and compute I:Maf I~ and so (ii) is true. = Ii IM af I Ila = II If I I~ = llf Ila , Lastly, if a < co, co ltr sf 1ra =f jf (ex) ladx -a:, C0 ~I = J jf(x) jadx , -a:, and if a= C0 then !;T f[!(X) = llfll(X) 6 trivially. We now state the identity that relates THEOREM 4.6. L e t fy ( 4 . 1 ) and ( 4 . 2 ) . w h e r e and This proves (iii). K'YW to Nw. lwl < l, l < p < 2, 1/p + 1/q = 1, Then a n d l e t y s a t i s - 60 a. = 1 +2 w 2(1-w2) p- 1+22 y w - py(l-w2) We remark that the identity in theorem 4.5 is also true if we remove the restriction 1/p + 1/q = l and replace the leftmost M a. with where M a, a. = 1 (q-1 )lw2 + l qy(l -w2) l + 2 w 2(1-w2) This will be evident from the proof. PROOF. Let f E LP and compute = (2n).-l/2qe-x 2/2q 1 l 2 2 j(2n)l/2pey 2/2pf(y)exp -Jy~2 } dy (X) Jn(l-y w }-m 2(1-y UJ) a, = J (2n)l/2p-1/2q z2 f(y)exp Jn(l-y w ) ~~ 1-y· 2[p-l+y2w2] 2 + xy [ 2p(l-y o; w 2] 2 1-y w _ x2 [l+(g-l~y~w2 ] 2q(l -y w ) Substituting gives us ) dy . Jex and ~y for x and y respectively in the above 61 ( TJi:i'I K I -1 f)(x) .., q yw p 7 )l/2p-l/2q 2 yn(l-w) = ~TT i [ 00 J(r 'Sf) (y )exp -y2 p-1 +\2w2j + xy [ 2w21 -a, y(l-w) AJ 1-wJ whence If 1/p + 1/q = 1 then (p-1 Y(q-1) = 1 and we have, in addition = a. and the theorem is proven. We may now prove theorem 4.1. If we let wand p be according to the hypothesis of the theorem and let y satisfy (4.1) and (4.2), 62 then it follows immediately from theorem 4.5, proposition 4.4, and lemma 4.6 that IINwllp,q ~ (2n)l/2q-l/2p hll/2 (a)l/2p-l/2q 2 2 \ 1/2p-l/2q = hll/2( l-y2 ny(l -w ) / It suffice? now to prove that for the function 2 f(x) = 1 - y2w2 )l 1 P exp ( ny( 1 -w2 ) Nw actually achieves this bound. imaginary and l( x2 l + w2 _ 2 2 ( l -w ) 1 ;y( l ) -w2 ) We use the fact that l a is purely S > 0 to compute II f II P = (S/2n) 112P £ c» ( 2 )1/ e-SX 12 dx p = 1. Al so (Nr f)(x) = (S/2n~ 1/2p " )rr( l - w ) 1 0> ·_ CX) l 2 e(a-~/2p)y 2 exp 40J<y- (x 2+~ 2)(l+w) 2 (1- w ) l dy 63 2 - X 1 2 = 8/2n l PJ(X) exp (-y2[. n(l-w) -(X) = (0/2n , /2 p exp x2 l n(l-w) ( j} l +w 2 [ 2(1-w ) dy l ] + xyl- 2w ]- x2 [. l + w~ ] ] dy 2 2 y(l-w ) ll - w 2(1-w) w2 y2 - l + w22 1- w 2(1-w) = Yl /2 ( S/2n) 1/2p expjx2(A 1-w ) l a: exp - J - u;x l {JrJ-i) -cc 1 + w~ ) 2(1-w) E) l ~ t. dy ,.. l· We now have l!Nwfllq = hl 112 (S/2n) 112 P(J(X) expfqx -ex, 1 2 1 Re[A+w~])dx l/q w 2 (1-w ) ,- 2 2 - l)J ldx 1/q f exp lqx2 Re[(p-1 )(~w Lpy(l -w ) CX) = hll/2(~/2n)l/2p ( -ex, = jy 11 /2( p/2n) 1 /2p ( j •-?;x.3/ 2. dx -ex, = IY11/2(e/2n)l/2p-l/2q = jy , /2 1 2 2 )1/2p-l/2q l - y w ( 2 ny( 1-w ) y /q 64 This completes the proof of theorem 4.1 We now treat the case THEOREM 4. 7. p = l. jw I _::: l, w j: ± l . Let Then 2 -1/2 j11(l -w ) I PROOF. lwl < l. We again assume at first that Let a = Re Then so a> 0 a 2 l + 2 ~ 2 ( 1-w ) b=Re , l - w and l + w a + b = Re 2 ( 1 -w) > 0 1 - w a - b = Re 2(1 +w) > 0 2 - b > 0. We then have sup jN (x,y) y w I = 111(1-w2 )j-1/2 sup exp { 2xyb - (x 2+y 2 )a } y 2 = ln(l-w) I Hence, if UJZ 1 f EL , - l 12 J 2 2 2 } exp\x (b -a )/a • . 65 CD II Nwf II ex:= s~p If f(y)Nw(x,y)dy i -ex: ~ sup X II f 11 1 sup INw(x,y) I y 2 -l = l!fll 1 s~p ln(l-w )I = llfll1 2 ln(l-w)I 12 exp { 2 2 2 x (b -a )/a} -1/2 and so To show we have equality, we consider, for 2 g(x) = e-ux . Let s - so we know Re(t 1 + w2 2 2(1-w) t = w 2 l - w u > 0, the function ' Re s > 0 and 2 We calculate - 5 s2 - us) = _ 1 Re [-(1+2u) - (l-2u)w~] + u (1+2u) + (1-2u)w < 0. 66 CD 2 -1 /2! { 2 2 (Nwg)(x) = (n(l-w )) exp -( s+u)y + 2txy - sx } dy and so Ii NuJJ II CD = -1/2 2 j(l-w )(s+u) (n/uf 1I2 = ln(l-w2) I 119111 -1 /2 I ju/(s+u)~ • 2 -1 /2 I Since this ratio approaches jn(l-w) the proof of the theorem for lw < 1. Now let 1 f EL . Let € r < 1 where k as u ~ ~ we have completed • > 0 and let g be a continuous function llf-g I~ < €. If jwj = ·1 depends only on w. Integration by parts yields of compact support such that for all r I2 it follows that the result as uniformly in r(i.e., N obeys a Riemann-Legesgue type theorem). w l im (Nrwf)(x) r tl uniformly for x E :R. = Hence (Nwf)(x) The usual arguments may now be used to extend the result of this theorem to jwj = l. 67 Chapter 5 APPLICATIONS 1. BABENKO S CONJECTURE 1 In chapter 3 we mentioned a theorem of K. I. Babenko which exhibits the norm of Nw for w imaginary and q equal to an even integer. Based on the results of chapter 4 we can now prove this theorem for general q. The theorem is restated in this general form for convenience. THEOREM 5.1. Let -1 < t::: l, l < p::: 2, and lp + lq = l. Then Moreover, this norm is achieved for the Gaussian function FinaZZy Define 68 PROOF. The case p = 1 is a direct consequence of theorem 4.7. assume now that We p > 1 . Let a. = ( 5. 1 ) so 2 1 = -p(l -t ) + a. (5.2) 4t 2 We now compute 2 1 p - 1 - y22 t = p - 1 - :-.::-:-z (p2( 1- t 2) + a.2 - 2p ( 1 - t 2) a.) 16t 1 Bt 24 22 = ~ ( - p t + 2p t 4 2 - p - p( 1 -t ) a.) __Q__ 2 2 = -::-7 ( - p ( 1-t ) + a.)( 1 - t ) , 8t whence 22 p-1-tt py(l +t ) which is exactly condition (4.1) for w = it in theorem 4.1. 22 = p24 2 2 4 2 t + ( 8 p-2 p ) t + p - p( 1 - t ) a. y(l +t 2 ) 2(1 +t 2 ) ( -p(l-t 2)+a.) 1+y t _ p[ ( 2-p )( l - t 2 ) + a.) ] [ - p(1 - t 2) + a.] 2 2 4 ( p- 1 )( 1+t )( -p ( l -t ) + a.) Also 69 (5.3) However, we have whence from (5.l) (5.4) CL?:'.:Pll-tl?:'.:0. If we now assume t fl for Since then by (5.3),condition (4.2) is also satisfied w = it and we apply theorem 4.1 to get llNill p,q = i~r II Nit ii p,q we get by the usual argument the cor- responding equation for t = l also. Finally 4.1 says that Nit achieves this nonn for g(x) = exp {x2 c(~ :3~ - y(l :t2))} 2 2 = exp{x (:~~)~t _ 2 = exp{-x (:~;::~) )} • The theorem is now proven. y(l:t 2))} 70 We may now let t = -1 in theorem 5. l to achieve the sharp Haus- dorff-Young inequality for the Fourier transform J, first proven by Beckner (see theorem 3.5 and corollary 3.8). COROLLARY 5.2. Let l::: p::: 2 and 1/p + 1/q = l. II :r-11 Then • 1/ -1 ( l 1/ l ;2q ) = (2n) 2q ;2p P ;2p/q p,q From this it is seen that the method used by Wiener to prove Plancherel 's theorem can be generalized to yield the Hausdorff-Young inequality directly from the norm of N . w Finally, we notice that we may replace t by i- t everywhere in the proof of theorem 5.1 and so doing we get the following theorem. THEOREM 5.3. Let -1 ::: t::: l, l < p::: 2, and 1/p + 1/q = l. Then Moreover, this norm is aahieved for the Gaussian function g(x) = exp FinaZZy, if t 'f ± l 2 -x 2( l - Y 2 { py(l -t ) then t2 )~ . Define 71 2. SMOOTHING OPERATORS Smoothing operators, as used by De Bruijn [ 6] are invaluable tools in the study of generalized functions. They are defined as follows. DEFINITION 5.4. Let a> 0 and define Sa(z,u) = (sin h a)- for aU complex pa.rameter z and u. 1I2 exp {- We define sin 1a ((z2+u 2 )cos ha- 2zu)}. Sa ,the smoothing operator with w, by f CX) {Saf ){z) = Sa(z,u)f(u )du . -= We also have the following representation of S . a PROPOSITION 5.5 (MYLLER-LEBEDEFF) Let a> 0. Then (5.5) and so (5.6) 72 PROOF. The identity in (5.5) follows from comparison with theorem l .7. To prove (5.6) we take f to be a function in LP(l ,:::: p < 2) a: (S f)(z/Fn)=fs (z/,fln,u)f(u)du a. a. -a:, f f(u)Fn n=OI:. e-(n~)a.%(z)cpn(u/,fln)du C0 = -a:, f f(,fln n=OI: e-(n~)a.%(z)cp,,(u)du C0 = Fn u) -c:c from which the result follows. We may now calculate the nonn of the smoothing operator from to Lq. THEOREM 5.6. Let l < p ~ 2, 1/p + 1/q = 1, and a.> 0. Then Moreover, the norm is achieved for the Gaussian function Define LP 73 g(x) = PROOF. r exp 1-x 2 ( , - le-20. )} py(l -e-2°') . • This is an irrmeaiate consequence of proposition 5.5 and theorem 5. 3 with w = e -a.. 3. THE OPERATORS OF KOBER In [ 8 ] and [SJ, H. Kober and Condon used yet another Meh1 er-type operator in their study of the Fourier and Hankel transforms. These operators are defined as follows. DEFINITION 5.7. Let r be reaZ, -t < r < t and define i X y Tr ( x,y ) = Cr exp { sinZnr (x2+y 2 ) ( rot2nr i ) '} , where Cr = ( 2TT I. sin 2 nr I1)-1/2 etin(¾ sign(r) - 2r) for x and y reaZ. r. For f E L1 , Tr(x,y) we define is aaUed the Kober kerneZ of parameter 74 J CX) (Trf)(x) = f(y)Tr(x,y)dy -CX) and caZZ the Kober operator of pa:rconeter Tr The operator Tr may also be defined on r. LP(l < p < 2) but the same care must be taken as in the case of the Mehler operator on the unit circle. The following lemma relates the Kober operator to the Mehler kernel operator. LEMMA 5.8. PROOF. T = N . r e2n1r We let w = eie 2 = = e real and calculate l+e 2ie 2(l-e2ie)= 1 +w2 2(1-w) for isin2e 2-2cos2e i 2 cot e , and 2w = 2 1- w = _i_ sine 4i sine 2 - 2cos 2 e • Furthermore, 1 = ( l n(l -e2i e) )1 /2 75 ie-ie )l/2 = ( 2nsine = (2njsin 0j) -l/2 }in(~ sign(r) - i). e n We now have N (x,y) w = = T8/2n(x,y) and the lemma follows by letting We now know that Tr maps 8 = 2nr. LP 4.1 to see how Tr may be defined for into Lq (compare with theorem LP, l ~ p ~ 2) and we may cal- culate its norm. THEOREM 5.9. but Let 1 < p < 2, - t < r < ~- 1/p + 1/q = 1. Let r be a real number Define sign{sin 2 nr) Y = ei$-T i 2nr • Then 76 1 12 1/2 ( 1 - y2e i4nr ) ny ( 1 - e i4nr) IYI -1 P f 2q Moreover, this norm is achieved by the Gaussian function 2 • = exp { x ( ½:ot 2 nr - f (x) 1 , nr ) } • y(l -e 4 ) FinaZZy PROOF. The cases pf 1,2 and p = 1 and p = 2 are clear so we will assume 2r is not an integer. Let w = ei 2nr . Then y = ih:J sign(sin 2 nr) w and so 2 l+w 2(1-w2) 2 2 p-l+yw - 1 + w2 2 py(l-w2) 2 ( 1 -w ) - i = z<=Ot 2 Tir • Al so 2 2 1- y w 2 y(l -w ) = 2- p sign(sin 2 nr) (5.7) 77 = = (2-p)sign(sin 2 nr) i~(-2i sin 2nr) 2 - p 2,Jp-ljsin2nrj The theorem now follows from theorem 4.1. > 0 • (5.8) 78 REFERENCES [l] Askey, R. and Wainger, S., Mean convergence of expansions ~n Laguerre and Hermite series, Arner. J. Math. 87(1965) 695-708. [2] Babenko, K. I., An inequaZity in the theory of Fourier integraZs, Izv. Akad. Nauk SSSR, Ser. Mat. 25(1961 ), 531-542; English transl., Arner. Math. Soc. Transl. (2)44, 115-128. [3] Bargmann, V., On a HiZbert space of analytic functions and an associated integraZ transform, Comm. on Pure and Applied Math. 14(1961) 187-214. [4] Beckner, W., InequaZities in Fourier anaZysis, Ann. of Math. 102 (1975), 159-182. [5] Condon, E. U., Immersion of the Fourier transform in a continuous group of funationaZ transformations, Proc. Nat. Ac. Sci. U.S.A., 23(1937) 158-164. [6] de Bruijn, N. G., A theory of generaZized functions, with qppZications to Wigner distribution and WeyZ correspondence, • [7] Nieuw Archief voor Wiskunde (3), XXI(l973), 205-280. Janssen, A. J. E.M., "AppZication of the Wigner Distribution to Harmonia AnaZysis of GeneraZized Stochastic Processes," Mathematisch Centrum, .Amsterdam, 1979. [8] Kober, H., WurzeZn aus der HankeZ-,Fourier-und aus anderen stetigen Transformationen, Quart. J. Math. Oxford Ser., 10(1939), 45-59. 79 [9] Mehler, F. G., Ueber die entuJiaklung einer funktion van beliebig vielen va:r>idblen naah Laplaaesahen funationen hBherer ordnwig, J. Reine Angewandte Math., 66(1866) 161 -176. [10] Myller-Lebedeff, W., Die theorie der integralgleiahwigen in anuJendwig auf einige reihenentuJiaklwigen, Math. Ann., 64(1907) 388-416. [11] Nelson, E., The free Mazikoff field, J. Fune. Anal. 12(1973), 211-227. [12] Stein, E. M., and Weiss, G., "Introduction to Fourier Analysis on Eualidean Spaces," Princeton University Press, 1971 . [13] Szega, G., "Orthogonal Polynomials," Colloquium Publications, Vol. 23. Amer. Math. Soc., New York, 1939. [14 J Titchmarsh, E. C., "Introduction to the Theory of Fourier Inte- grals," [15] Clarendon, Oxford, 1937. Wiener, N., "The Fourier Integral and Certain of its Appliaations," Dover Publ., New York, 1933. [16] Weissler, F. B., Two point inequalities, the Hermite semigroup, and the Gauss-Weierstrass semigroup, J. Fune. Anal. 32(1979), 102-121. [17] Zaanen, A. C., "Lineazi Analysis," Interscience Publishers Inc., New York, 1953.