Generalized Multipliers on Locally Compact Abelian Groups Citation Ford, Lawrence Charles (1974) Generalized Multipliers on Locally Compact Abelian Groups. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/NB31-1Y34. https://resolver.caltech.edu/CaltechETD:etd-10122005-082659 Abstract NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. Let G be a locally compact Abelian group with dual [...], [...], and [...] supp [...] is compact}. Then for [...], the containments are proper if G is noncompact, and [...] is a dense, translation invariant subspace of [...] for [...]. Let [...] be a complex valued function defined on [...], and [...] = [...]. Suppose [...]. Define the operator, [...] by the equation [...] for each [...]. Then [...] is a module over M(G), [...] is a module homomorphism, and [...] is (p, q) closed. We call [...] a generalized (p, q) multiplier. The main results include: (1) Suppose T is an operator satisfying: (a) The domain D(T) is a translation invariant subspace of [...], and the range R(T) [...]; (b) D(T) [...]; (c) T is (p, q) closed, linear, and commutes with all translations; (d) C X T(C) is dense in [...]. Then T = [...] for some [...]. (2) The set of all generalized (p, q) multipliers, denoted [...], is a linear space, and the set of all generalized (p, p) multipliers, denoted [...], is an algebra containing [...] and contained in [...]. (3) If [...], then [...] is locally the transform of a bounded (p, q) multiplier. Further sections include a deeper study of [...], [...], and special results obtainable for compact G. 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): De Prima, Charles R. (advisor) Luxemburg, W. A. J. (advisor) Thesis Committee: Unknown, Unknown Defense Date: 7 February 1974 Record Number: CaltechETD:etd-10122005-082659 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-10122005-082659 DOI: 10.7907/NB31-1Y34 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 4041 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 14 Oct 2005 Last Modified: 25 Jul 2024 22:34 Thesis Files Preview PDF (Ford_lc_1974.pdf) - Final Version See Usage Policy. 2MB Repository Staff Only: item control page

GENERALIZED MULTIPLIERS ON LOCALLY COMPACT ABELIAN GROUPS by Lawrence Charles Ford In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1974 (Submitted February 7, 1974) To LaJean -iii- Acknowledgement I here express my sincere appreciation to all those who have made this thesis possible. First there is the Graduate Office and the Mathematics Department of the California Institute of Technology, which supported me through my four years of residence; the Ford Foundation, which provided the funds enabling me to pursue summer research; and the National Science Foundation for giving me an NSF traineeship in the year 1972-73. I would like to thank Dr. J. T. Burnham of Oklahoma State, for his helpful comments and for copies of his papers on Segal Algebras, listed in the References. I also thank Professor Ken Ross of the University of Oregon for his helpful correspondence in answer to my several questions, and Professor Ronald Larsen of Weslyan University for part four of Theorem 5.5. I give my Sincere thanks to all the members of the Mathematics Department of Caltech, but especially to Professors C. R. DePrima and W. A. J. Luxemburg for their many helpful suggestions and continual guidance. Finally, I wish to thank my parents, Mr. and Mrs. J. E. Reiswig, and my dear wife LaJean, whose love and faith made this all possible. ~ilV- Abstract Generalized Multipliers on Locally Compact Abelian Groups Let G be a locally compact Abelian group with dual T, 1 <p,q < 2, and Cy ={f ¢ LP(G)| supp(f) is compact}. Then for I<r<s<2,C,CC,CC,C¢ C,, the containments are proper if G is noncompact, and C, is a dense, translation invariant subspace of L(G) for 1 <t< oo, Let’ be a complex valued function defined — on Tr, and J}?4 = {f ¢ LPG) |af ¢ LY(G)}. Suppose JP?4 5 C,- Define the operator, T,: a} 4. 1,2(G) by the equation Ti =f for each fe gP 4, Then ghd is a module over M(G), T, is a module homo- morphism, and T, is (p,q) closed. We call T, a generalized (p, q) multiplier. The main results include: (1) Suppose T is an operator satisfying: (a) The domain D(T) is a translation invariant subspace of LP(G), and the range R(T) iS L(G); (b) D(T) 2¢,; (c) Tis (p,q) closed, linear, and commutes with all translations; | (d) CX T(C) is dense in Cy x T(C,,)- Then T = T) for some i. (2) The set of all generalized (p,q) multipliers, denoted Xp, Pe is a linear space, and the set of all generalized (p, p) multipliers, denoted Xy is an algebra containing X, and contained in X,. -V- (3) If T, E Xp oy then 2 is locally the transform of a bounded ? (p,q) multiplier. Further sections include a deeper study of X,, X,, and special results obtainable for compact G. -vi- TABLE OF CONTENTS ACKNOWLEDGMENTS . ABSTRACT INTRODUCTION . CHAPTER I Preliminary Lemmas on Translation, Convolution, and Density . II Generalized Multipliers Til Generalized L” Multipliers IV Generalized L’ Multipliers V The Compact Abelian Case REFERENCES iii iv 13 23 30 34 40 -1- Introduction Let G be a locally compact Abelian group with dual T, and A, B be two Banach spaces of measurable functions on G for which the Fourier transform is defined. Suppose A is a measurable function on I such that af is the transform of a function in space B whenever f is a function in A. Then A determines a bounded linear operator from A into B, and if the spaces are translation invariant the operator commutes with all translations. 2 is called the transform of the operator, and such operators are called (A, B) multipliers. The problem of determining all (A, B) multipliers is called the "multiplier problem". The multiplier problem has drawn the interest of many mathe- maticians since the turn of the century. The early investigators con- sidered spaces of real 27-periodic functions, and looked for real Sequences which multiplied Fourier coefficients of these functions to produce Fourier coefficients from a Second space. Bochner was the first to use the complex Fourier series of a function. This allows one to think of the functions as being defined on the circle group T, and provides the prototype for compact Abelian groups. Others, notably Hérmander, studied the multiplier problem on R", which provides the prototype for noncompact LCA groups. Since World War II much of harmonic analysis has been done on locally compact Abelian groups. This includes the study of (p,q) multi- pliers for 1 <p,q<o. A (p,q) multiplier is a bounded, linear operator from L?(G) to L(G) which commutes with all the translation operators on G. An equivalent description is an operator defined by a function A -2- on IT which multiplies TP into cy Brainerd and Edwards [2], Gaudry [17], [18], and Larsen [24], provide excellent sources for the study of (p,q) multipliers. An interesting multiplier problem arises if one considers func- tions on T with the property that xf ¢ L4G) for f in a dense subspace of LP(G). The operator determined by A is closed, but not necessarily bounded. L(G) provides a guide for this development. A version of this problem on L appears in Ford [16], where it is shown that these operators have a minimal domain, and they form a commutative algebra containing the bounded L multipliers. We call these the generalized L multipliers on G. One area of application is the study of multipliers on Segal algebras, as in Burnham [3], since every multiplier on a Segal algebra is a generalized L multiplier. Our goal is to characterize generalized (p,q) multipliers for 1 <p,q < 2, using the L’ case as a guide. Chapter I contains the pre- liminaries on density, convolution, and translation needed elsewhere in the development; the most important results are Lemma 1.7, which relates convolution of LP functions by L’ functions to sums of translates of the LP functions, and Lemma 1.8 which shows that closed, linear operators which commute with all translations also commute with con- volution by finite regular Borel measures on G. Chapter II defines a generalized (p,q) multiplier as an operator determined by a function \ having certain properties. These operators are shown to be closed, linear, and commute with convolution by measures in M(G). It follows that they commute with all translations. ~3- We show that A is locally the transform of a bounded (p, q) multiplier, and from this obtain inclusion results similar to those which hold for bounded multipliers. Chapter III extends the results on generalized L multipliers contained in [16]. Chapter IV is a deeper study of generalized L’ multi- pliers, and Chapter V gives the additional results obtainable when G is compact. Throughout the paper, Gis an LCA group with Haar measure m. The dual group Tr has Haar measure m normalized so that the Plancherel transform is an isometry. M(G) denotes the set of finite regular Borel measures onG. Iff¢ L(G), f denotes the Fourier, Plancherel, or Hausdorff-Young transform for p=1, 2, or 1< p< 2, respectively. If £ ¢ LP anda < G, then f, is the translate of f by a, i.e., f(x) = f(x-a) for x € G. Tf is the involution: f(x) = f(-x). Iff € LP andg € L’, then {*g is the convolution of f with g. If E is a measurable subset of Gor I, Xp is the characteristic function of E. If 1<p< ©, p’ denotes the dual index: 1/p +1/p’=1. Ifp=1, thenp’ =. Iff€ L(D), then f is the inverse image of f under the Plancherel transform, i.e., F. f. For 1<p<2, J\p-= {f ¢ LP’(r) | euP(q)}. Ifx € Gandy € I, (x,y) | denotes the value y(x) ( or dually, x(y)). Finally, C o(G) are the con- tinuous functions on G which "'vanish at infinity", and C o(G) are the continuous functions on G having compact support. -4- I. Preliminary Lemmas on Translation, Convolution and Density. Definition: C = {f€ L(G) |supp(f) is compact}. Lemma 1.1: For each f €L’(G) and e>0 there is a function v €C such that | |f - f*v||,< e. For a proof, see Rudin [30], Theorem [2.6.6]. An immediate conse- quence of this lemma is that C is a dense ideal of L(G). Lemma 1.2: C = f\ {I|I is a dense ideal of L’(G)}. Proof: Let I be a dense ideal of L’, andf€C. Let K = supp(f). For each y € TF, let M,, ={f¢L'(G) lf) = 0}. Then M, is a maximal * 1 4 closed ideal of L’, sol g M,. So choose f, € 1 such that fy) ~ 0. Let gs, = fiat: Then gy el, &, 7) > 0. Since g, is continuous, there is a neighborhood Vy of y such that g, > 0 on V- Now ivy, ly €K} is an open cover of K, so there exist y,, :- Vn € K such that n Cc . Kc \J ‘y, i=1 Let Then ¢g €I and g¢>OonkK. By the Wiener-Lévy theorem there is a function h €L'(G) such that h = 1/¢ on K. This implies that f = feh on Tl, or f=f*g*h. Therefore f €I, soC c I, and the Lemma follows from Lemma 1.1. Lemma 1.3: Let K C Fr be compact and « >0. Then there is a function g €C such that a(F) c [0,1], g=1 0nK, and ||g||,< 1+. The Lemma follows from Rudin [30], Theorems [2.6.8] and [2.6.1]. Lemma 1.4: Letl1<p<o, f C L(G), and pé M(G). Then ftp € L9G), and |x|, < {I¢l [pllul This is Theorem [20.12] of Hewitt and Ross [22]. Lemma 1.5: Let1 <p< «, andf ¢ LP(G). Then there is a neighborhood V of 0 with V compact such that if u is any measurable function,on G with u 2 0, supp(u) cV, and f u(x)dx = 1, then ||f- full, < €. G For a proof of this Lemma, see, for example, Loomis [25], Theorem 31E. Call such functions u V - blips. Lemma 1.6: Let1 <p,q<o. ThenC is a dense, translation invariant subspace of LP(G), and iff ¢ LP(G) and g € L4(G) we can find a sequence co {ft })4 © C such that both i|f- ot, 11,, <1/m and ||g-g*f, ||) <1/n. Proof: C is a subspace of LP(G) since L'/f LP is a dense ideal of L (G), and C is contained in every dense ideal. C is translation invariant since f(y) = (-a, y) f(y). To show C is dense in LP(G) let f € LP(G) ande > 0. We can find g € Ln LP such that Ilf-ell, <¢/3. By Lemma 1.5 we can find V a neighborhood of 0 such that if u is any V-blip then lle - g*ul|, <€/3. Also for a fixed V-blip u, Lemma 1.1 tells us we can find v € C such that -6- \lu-v|, < €/3(1) |, +1)" Now note that g*v € C, and \|f-e*v]| <e€. | E {lp +1) p So C is dense in LP(G). To prove the second part of the Lemma, for each n let Va bea neighborhood of 0 small enough so that both ||f-f*u, ||, <1/2n and lle - g*u, | la <1/2n for any V,, blip u,. Also for fixed choices of such uy choose fn € C such that -f Ma-fall, § Set sindlell Then If- ff, 1, <1, lIg-e*f, II, <i. Lemma 1.7: Letli<p<o, f€ LPG), g¢ € L(G), ande > 0. Then there are complex numbers a,,-*-, @,, and a,,***, a,€ G such that n | fxg - » af | | < €. k=1 a P Proof: Without loss of generality, g #0. Assume for the moment that ge C,(G) with support K. Let V be asymmetric, precompact neighbor - hood of 0 in G small enough that y € V implies f-f < Ht lIp< Shit mew). Now KC LU (Vta), ackK so there exist a,,+°°, a,, € K such that n Kc \) (V + a,). k=1 Let A, = K/\(V+a,), and k-1 A, = [KA (v+a,)] - [) A,] for k = 2, +++,n. i=1 Then n K=\J A,, k=1 each A, is measurable, and the union is pairwise disjoint. Let a, = S ely) ay. Ay Then n n | 1/p llie- Y ots lly = (s Jfee(x) - )) ala Px) G k=1 n 1/p = ( Sl f t&-yeyay - Y ag, Pax G G k=l n 1/p : (/ | f t-y)e)dy- Y of, Pas) GK k-1 n 1/p = ( S1Y Sf Ley) - t@-a,)] e(y)ay [Pax G k-lA, n | p \ < ) i(f Key) ~ 0-0,)| |e) lay} dx k=1 A, where the inequality is obtained by first using Minkowski's inequality and then moving the absolute value signs inside the interior integrals. For each of those interior integrals we can use Hélder's inequality and get: ( |£(x-y) - f(x-a,.) | Jeol)” Ay < m(A,)P/P" ff |(x-y) - £x-a,.)|P |ety) [Pay A k which implies n [tee Deut, |p < k=1 n L/p! p ? 1/p y m(Ay)/P (7 f |t-y) - t-a,)/" ley) | aya) k=1 G A, n ; yy 1/p = ¥ m(A,)!/P ( f ff |tx-y) - fx-a,) |" ety) | axa} k=1 A, G Now (x-a,) - (x-y)=y- a,, andify€ A,, theny€ V+a,, or y~ a € V. Consequently the translation invariance of Haar measure gives us p f |£(x-y) - f(x-a,,) [ax = f |f(x-(y-a,,))- f(x)| dx G G < (‘Aiellemce) Therefore nN HIe- Dat, Ily < k=1 n | 1/p y m(A,)/P" : le(y) [Pay kel K [lel |,m(K) J n 1/p <> mtayve" elle le bay ae Tiel am J n n LD may” + MP fn - ED mt) siRy- ¢ The lemma now follows from the density of C,(G) in L(G). Remark: Note that if we have 1< p,q < ~, f € LP(G), h e LAG), and g € L(G), and € > 0 we can find the complex numbers Qy,*, A, and group elements a,, «°°, an such that both n n | |i*s - 2 aif | lp < € and | |hxg - 2 ach, | la < € hold. To do this we need only choose the neighborhood V small enough that translating either f or h by y € V leaves | |f - f, | lp and | |h - hy | la small. After that step the construction depends only on the function g. -]1- The Lemma is actually stronger than we need. - For example, if f ¢ LP(G) and S(f) is the closed subspace generated by the translates of f, then the Lemma implies that S(f) = £*L'(G). When p = 1 we get the Proposition below and from it Wiener's Tauberian theorems, as done in Loomis [25], Chapter VII. Proposition: Let f ¢ L(G) such that f is never 0. Then S(f) = L(G). Proof: Let g ¢ C, with supp(é) = K. f is bounded away from 0 on K, so there exists anh € L(G) such that fh = 1 on K. Then g = fhg on I, or g=f*h*g. Lete > 0. By the Lemma we can get n lle - » af Ili <e ) k=1 “k so g € S(f). The proposition follows from the density of C in L(G). Lemma 1.8: Let 1 <p,q<, and T a linear operator with the following properties: (1) the domain D(T) c LP(G), the range R(T) ¢ L(G), (2) T is closed, (3) T commutes with all the translations on G. Then D(T) is a module over L(G) and if f ¢ D(T) andg € L(G), we have T(f*g) = Tf*g. Proof: Let f¢ D(T) andg ¢€ L(G), ande>O0. As we have already observed we can find a,, ---, a, complex and a,, ---, a, € G such that -12- n | fxg - af < 2 Kay . and n | |THg- Y aT), Iq < € k=1 hold simultaneously. Note that n »; mf € D(T) k=1 since D(T) is a translation invariant subspace, and that n n r( af, |} = ), a, (Th) k=1 ia) k=1 ie since T is linear and commutes with translations. Since T has a closed graph we now have f*g € D(T) and Tf+g = T(f*g). -13- Il. Generalized Multipliers The model for generalized multipliers is provided by the L’* case. If T is a densely defined closed linear operator on L(G), such that T commutes with all translations, then Lemma 1. 8 implies the domain D(T) is a dense ideal of L'(G), and Lemma 1.2 implies that D(T) DC. If f € C with supp(f) = K, Lemma 1.3 allows us to choose g € C such that ¢=-1onK. This gives 1? = Tie) = Ton = FEL allowing us to define a function A on I such that Ti = rf for each f € D(T). In this manner we can identify the class of all densely defined, closed, linear operators which commute with all translations on L(G) with a function algebra of multiplication operators on 7 (Lr). There are some minor complications in attempting to copy the same development for operators mapping from L(G) into LUG). First. of all, for noncompact LCA groups we restrict our attention to 1 <p,q < 2 in order to have the transform algebra available. Second, there is not usually a minimal dense submodule of L?(G) over L’(G) available for 1<p <2. As an example, let G be noncompact, p= 2, andy € T. Let M,={f¢ L'G) (\ L7(G) |f¢) = 0}. Then M, isa dense submodule of L? over L’, but [{\ ™, =f. yer To avoid this difficulty we could restrict our attention to operators T with domain D(T) D C. This however would not be sufficient to guarantee that we could compose operators. For example, consider -14- L?(R), and the operator defined by Tf = 0, ii This T does not map C into C, and if we could find an operator S with D(S) = C we could not make the composition S-T. We avoid this difficulty by requiring the domain of the (p,q) - operator to contain Cy = {f ¢ L(G) |supp(f) is compact}. We will begin by making some observations on the Plancherel and Hausdorff-Young transforms. Let 1 <p <2, and 1/p+1/p’=1. Then for f€¢ L(G) MN LPG) we have f € LP’ (I) and | |?| los | Fal lo Equality holds when p = 2. We can extend the transform uniquely to all of LP(G), and the trans- form is well behaved in the sense that the following hold: Lemma 2.1: Let1< p <2, f£€ LPG), andge LYNG). Then i) f= 0 ifff=0 ii) Teg = 18 iii) If f € L4G) for 1 < q < 2, p q, then the transform of f as an LP function and the transform of f as an L® function agree a.e. on Ir. For a proof, see [22], Theorems (31.32), (31.27), and (31. 26). Definition 2.1: Let 1 <p <2. Cy = C,,(G) = {£€ L(G) |supp() is compact}. Note: We agree to make the usual identification of functions that agree a.e., and under this equivalence find a representative function with com- pact support. ‘Note also that C, is just C. We will continue to write C for -15- C,. Finally, if G is compact then C= C, = Cy = C,= {trigonometric polynomials } = n = {f € C(G) | £(x) = » a, (X,%;,), ay, complex, y, € T}. k=1 Theorem 2.1: Let1 < p< q<2. Then Cy = LP*c, C, ccC,cc cC,, q and Cy # Cy if G is noncompact. Proof: Iff € Cy with K = supp(f), choose g € C with g=1o0nK. Then f -?¢- fae, andf=f*e¢ LPC. Clearly LPacc Cy, 50 LPaC = C,, Next note that f€ C,(G). Ifp=1, thenf€ C Cc L(G) by Lemma l.6; alsog € L2(G). Ifp> 1, thengé C Cc LP'(G) by Lemma 1.6. In either case f = {*g € C,(G) by Rudin [30], Theorem (1.1.6)d.. Since C,(G)A LPG) c LUG), we have f € L4G), and Lemma 2. 1 (iii) gives f€ Cy: To show that the containment is proper for noncompact G, let h € C with h positive real, h(0) > 1, and h(x) = h(-x)V x¢€ G. LetV bea symmetric, precompact neighborhood of 0 in G with |h(x)| >1 for x € V+V. Since G is noncompact there is a sequence {a} 1 of points in G such that (V + a) (\(V + a.) = ¢forn = m. Let 4 f= » 1/nl/P XvVe+a’ n=1 n Then forl <r < oO, -~16- co f/7=2 = Yo 1t/Py, 2 3 n=1 ae and oO f |£(x) |" dx = » 1/nt/P u(V). G n=1 Hence f € L4(G) - LP(G), and co 1/p h«f = ) 1/n h*X ; V+ n=l “n foo] = » i/ni/P (h¥Xy7) 5 . =1 n Also hexy@)= f h(x-y)xy(y)dy G Vv Vv Vv We claim haf ¢ LP(G), i-e., f |hat(x) [Pax = ©, G -17- To see this, note ~ 4 Y =i Obey) a, 60 [Pax f |nxt(x) [Pax = f | G nel G >f YE [oexy), GO]? ax G n=l n 2 N Y tL mexy), @I? ax n=1 n n i) he - f [mye]? ax G WV Mo 2 |— Jf (hexy(«))P ax Vv ii} rar n N = »; * f @ hy(o)ay) ? ax V OV n i —_ N N 1 1 1 > Vos ft-uv)Pax= Y= wvy*?. n=1 V n=1 This holds for any N 2 1, so f [het (x) |P dx = 0, G -18- Definition 2.2: Let 1 <p,q < 2. x, qm {X measurable on IT laf € FA p) for each f € Cy. If p=q, bd . d x by xX. we denote xy g y X, Theorem 2.2: Letl<r<p<2, 1<q<s <2, Then X, 4 ¢ X, 5° ’ ? : EX Ec, EC fe FUY), Proof: Letrx€ &, ,, andf€ C,. Thenf €C,, and 23 FAr). But K = supp(f) is compact, and supp(af) Cc K. So rAF e Cy and by Lemma 2.1 (iii) and Theorem 2.1, Af € C. Cc Bsr). Thereforer€ X. .. = r,s Definition 2.3: Let 1 < p,q <2, andarc xX, a" > gP4- {te LPG) |afe FUL}. Again we will write JP for JP P. p,q Note that J)’ “> Cy Definition 2.4: Let 1 <p,q <2, andre & T, = TP’? is the p,q° oA operator with domain gy a range contained in L4G), defined by the equation Ti = af for each f¢ JP, = {TP a S r r Baqi tty’ la € Roa ? h 2.3: Let1l< < 2 and € . Theorem e p,q and T, xg closed, linear, and commutes with convolution by elements of M(G). Then T, is (p,q) - Proof: Iff, g € gP q and a, 8 are complex numbers, then T (at + Bg) = rA(af + Bg) = arf + Pr(Y = aT, f+ T,g, so T, is linear. If p € M(G), then T, fxp = Afp =Afxu, sofxp €J\’?* and T,fxu = T, (f+). Suppose {f,} c JP?4, £¢ LPG), and g € L4(G) such that f,—P>f and Tf, 96. Let h € C with K = supp(h). Then £, +h, -19- feh € CC IPG, and | ati - Bill, < |AHE-F)[lqr+ [BOE - Alley < [JAG ~ 2) [ge + [In] Ll ltt, - el ly. Now ?, 5 f, so there exists a subsequence f, such that?, ~f a.e. oI. Alsoh€CCC,, so Ah € Lq (Pr), and dh is finite xe. on I. Therefore Ahf,, k ~ hf a.e. on I. But Ahi, -1'5 ig, so there is a sub- . An ana k an Cand sequence fn, such that ADIng ~+hg a.e. on I. Therefore hg = Ahf a.e. ony. Since h € C was arbitrary, ¢ = rf a.e. on I, so ||g- ai| la’ = 0. Therefore f € gP a T,f=g, and T, is closed. Corollary 1: gh 4 is a translation invariant subspace of LP(G) and Ty commutes with all translations. Corollary 2: T, is bounded iff JP?4 = LP@), Proof: Closed graph theorem. Corollary 3: T, maps Cy into Cg: Theorem 2.4: LetA€é X, q andheécC. Then rh € x and Tyh is (p, q) bounded. Proof: Let f¢€ LP(G). Then h*f € Cy: Since A € X, q we know 3 Peon hc & el _7P _—a ther- Tht = ahf ¢ Cy. Therefore J) -° = L (G) 2 Cy so Ah € X, q- Further more, by Theorem 2.3 Tih is (p,q) closed, so the Closed Graph Theorem implies T, ¢ is (p, q) bounded. Definition 2.5: Let A be a function on Tr. A has property P locally on T if A x Das property P for each compact KC TI. -20- Examples of local properties we shall consider are locally measurable, locally L and locally the transform of a bounded (p, q)-multiplier. Theorem 2.5: X q is the set of all functions on T which are locally 3 the transform of a (p, q)-multiplier. Proof: Suppose that A is locally the transform of a (p, q)-multiplier on Tr. Let fe Cy with K = supp(f). Then there is a (p,q)-multiplier T such “ - s&s <3 that X = Ton K. Then Af = Tfon TI, and Tf € L*(G). Therefore, Pd 5 Jy 2 Cy andrA € X P; q" Ifx € X, andK C Tis compact, choose h € C such that h = 1 P, 4 = on K. Then dA = Ah= T,f on K, so on K 2 is the transform of a (p, q) multiplier. Corollary 1: Letl <q <p <2, andG noncompact. Then x, q = {ol | ? Proof. From Gaudry [17], Sec. 5, we know that the only (p,q) multiplier for q< pis 0 when Gis noncompact. Mary 2: Letl <p<q <2. Then X CX. Corollary Le p<q Then Xy CX Proof: This follows from the fact that an LP-multiplier is also an L¢- multiplier for 1 <p <q <2. (See Larsen [24], Corollary 4.1.3, page 97). Corollary 3: Let T be a (p,q)-multiplier. Then T € Xp q: , Proof: T is locally the transform of T, which is a (p,q) multiplier. -21- We call Xy q the set of generalized (p, q)-multipliers. Note 2 that every multiplier is also a generalized multiplier by Corollary 3 above. Theorem 2.6: Let 1 <p,q < 2, and T be an operator with the following properties: (1) the domain D(T) satisfies C, ¢ D(T) C LP (2) the range R(T) c L4 (3) T is (p,q) closed, linear, and commutes with translations by elements of G (4) C X T(C) is dense in Cy x T(C,). Then T € Xp, q’ Proof: By Lemma 1.8 D(T) is a module over L(G) and T(itg) = Ti*g for each f € D(T) andg€ L(G). Let K C T be compact and h € C such thath=1onK. LetA=Thonk. Ifg € C with ¢ = 1 on K, then OS . Th = Thg = Tgh = Tg, a.e. on Kby Lemma 2.1. So Xd is well defined ? as a function in LP (K), and we can extend a to a well defined, locally a LP function on Fr. 2 is measurable since it is locally measurable (see Hewitt and Ross I [22], Theorem (11.42)). Iff €C, andhé C with A “« —— a onn A “a h = 1 on supp(f), then Tf = T(f*h) = Thf =af. Iff € Cy then we can : P , ve find f, ¢ C such that f,—> f and Ti. —“»T#, so Ti, = xf, —> n n But “a a there is a subsequence fhe so that fn > f, a.e., so Mn ad af, a.e., or Tf= af. Therefore JM” q> Cy and T= T, on C,. Now we can use Theorem 2.4 to conclude T = Ty, since for each h € C TyF is (p, q) bounded, and Tf*h = T,-f for each f € D(T) or Jy a, -22- Theorem 2.7: Let1<p,q,r,s <2, andq <r. Leta, € X, q and 3 € X, o. rz E xy s* Then d,A, € Xp, g? 80 Tt ody p, 8 1,77 Proof: Since Cy CC,,, we have for eachf € Cy Ait € C,, so AaAit € C, or Jn, 2 Cy. Theorem 2.8: Let1 <p <2. Then Xp is a commutative algebra. then dA, = AA, € Ky So Proof: By Theorem 2.7, if A,,A, € x p p € X.. T =T 2 Airs p Ar, tata -23- Ill. Generalized L’ Multipliers This chapter contains a deeper study of X,, the generalized multipliers on L(G) . This algebra is the prototype of generalized multipliers, and because of the structure available in L(G) we can avoid many of the difficulties encountered in Xp forl<p <2. Also from Corollary 2 of Theorem 2.5 we have X, CX: One of the areas of current research on multipliers is in Segal algebras, which are subalgebras of L'(G) which are Banach algebras under their own norms and also Banach modules (i.e., f € A, g € L(G), then f*g € A and | |f*g || A< | |e | la | lz||,). These algebras are usually taken to be dense. The virtue of studying X, is that multipliers on (dense) Segal algebras are all in X,, and inherit all of its structure. We proceed forthwith. Theorem 3.1: Let T, € X,. Then A is continuous. Proof: Lety ¢€ I’. Since IF is locally compact, there is a neighborhood Vy of y such that Vy is compact. Choose g € C such that g = lon Vy . 1 “ 11 <—— . T,g¢ L (G), T,f =Af for each f € J,’ , so T,g =A. So A is con- tinuous on Vy, and A is continuous at y. Since y ¢€ Tis arbitrary, A is continuous. Theorem 3.2: Let T, ¢ X,andfe¢C. Then T, f(x) =f AUPE) (x, y)dy. T Proof: By Corollary 3 to Theorem 2.3, Tyf ¢€ C. Since A is continuous and f has compact support, rE € L (TD). The inversion theorem can be applied to give the above formula. -24- Theorem 3.3: Let T be an operator with domain D(T) a dense ideal of L(G), range R(T) Cc L(G), such that f ¢ D(T) and g € L(G) implies f*g € D(T) and T(f*g) = Tfi*g. Then T is a restriction of some element of X,. Proof: Since D(T) is dense, D(T) DC, and we can apply the con- struction in the proof of Theorem 2.6 to get the function A on I with the property that Tf =f for each f € C. Ifgé€ D(T) andK CTis compact, let h € C such that h=lonK. Thenon K, Ts = Toh = —_~~* ~a aN Thg = Ag = Tyg. Therefore Tg = T,g and T is a restriction of T,. Definition 3.1: (Generalized Strong Operator Topology). Let “Tbe the set of all operators T with domain D(T) a dense ideal of L(G), range R(T) C L(G), such that Tis closed. A net {Ty |a € A} cT converges to T eT it | ITE - Tf | F ~ 0 for eachfe C. The topology determined by this convergence is the generalized strong operator topology on L . Remark: The generalized strong operator topology extends the strong operator topology to unbounded operators. Theorem 3.4: X, is closed in’ | under the generalized strong operator topology, and Bdd(L'(G) ) f) X, is closed in Bdd(L'(G)) under the strong operator topology. Proof: Let T cL {T ot CX, Ty ~ T in the generalized strong operator topology. LetfeC, andge L(G). Note that C C D(T) and that f and f*g ¢C CD(T). Lete > 0. Then there is an a, € A such thata >a, implies that ||T,f-Tf£]|,< ¢/2 ||g||,. Also there is an a, such that @ => a, implies | |T (ee) - T(f*g) ||, < €/2. -25- Then | ITee*g) - THe | |: < ||T(*g) - T,(é*e) ||, + ||T,fe - Tie ||, —< | (TGs) - Tye) | + [Tot - TEI, Heth < €/2+e/f2=€. Therefore, T(f*g) = Tf*g. By Theorem 3.3 T is the restriction of some T, € X). Also T = T, on C, and Tis closed. Therefore by Theorem 2.6 T € X, or T= T,- If the Ty are all bounded, and T ¢€ Bdd(L'(G)) such that for each fe L(G), Tyf tote, then T commutes with convolution on all L'(G), and T = T, on all L(G), so Bdd(L'(G)) A X, is closed under the strong av operator topology in Bdd(L). Remark: It is not true that Bdd(L (G)) A X, is closed under the gen- eralized strong operator topology. An example follows the next theorem. Theorem 3.9: Lety ¢€ X,, and A analytic on a neighborhood of A(f). Then A or€ X.- Proof: We must show Jy ,2C- So let f € C with K = supp(4). Let oO “N g € C such thatg=lonK. Thend= T\8 on K, andAoA= A(,8) on K. By the Wiener-Lévy theorem (see Reiter [28], Chapter 6) there exists h € L(G) such that h = A(T, 8) on K. So hf = h*f= A(T,8)f on Fr. Therefore AoA € X,. -26- Corollary 1: T, € X is invertible iff X is never 0. In this case Corollary 2: Let A be entire and T, bounded. Then T ‘Ac x 18 bounded. Proof: Suppose co A(z) = Y az forall z k=0 Let ro @] T = », wT) k=0 k Then T ¢€ Bdd(L ), T is the norm limit (hence strong operator limit) of elements of X,, and by Theorem 3.4 Té X,. Also co co “ “—™~ T=) & T= Y aak-a0onr K=0 k=0 Remark: It might be conjectured that if T, is bounded and A analytic, with A(T) bounded away from the singularities of A, that T ‘Aon is bounded. However, it is known (see Rudin [30], Theorem 6.4.1) that for G nondiscrete, and z, any complex number, there is a p € M(G) with n(L) c [-1,1] and z, in the spectrum of yp. Therefore }L - Z can be bounded arbitrarily far away from 0, but J. Iii # L(G). (0) ~27- Definition 3.2: Let ’ be continuous on I, andf€C. F rf is the function defined on G by Fs) = J AO, nay Tr Theorem 3.6: \¢ X, iff F, , €L'(G) for each f€ C. 3 Proof: If € X,, then F, , =T,f€ L(G). If F, , €L'(G) for each feC, then F, , =Afe L (G) for each f€ C. So Jy DC, andrA€ X. 3 To close this chapter we characterize the bounded multipliers on L(G). To do this we first need Bochner's Theorem, as stated in Rudin [30], Theorem 1.9.1: Theorem 3.7: (Bochner's Theorem). The following are equivalent: (1) X% =p where pu € M(G) and [fu || <M. (2) A is continuous on I, and m m 12 My] SM ILY ey) |e k=1 k=1 for each choice of ¢c,,..., Cn complex and y,,..., Ym € T. Theorem 3.8: Let X be continuous on Fr. The following are equivalent: (1) X=, where » € M(G) and [lu|| <M. (2) T, € X,, and ||T,]| < M. (3) F, ¢ €L'(G) for each f€ C, and ||F, pls <M||t]|,. ~28- Proof: (1) => (2): IfA =u, define T:L'(G) ~ L'(G):f+ fu. Then oN oN aa “~ 1 . Tf =f*u = uf =Af for eachfe L(G), soT =T, and HT, | = HIT]| < [ful] <™. (2) => (3). If T, € X, then € X, and by Theorem 3.6 F, ,€ L(G) : rf for eachf€C. Also HEX gl h = HT yfll, < [/T 1) beth <M] lel |. (3) > (1): Using the inversion formula My) = JF, glx) (-x,y)dx G ’ for eachfeEC. Let ¢,,... »fny be complex numbers and 7,,... Yn € I. Then for eachfeéeC, mae m D} 01 Mr )E(%,) | = ly Cy J Fy 9) (-x, 14.) 4 | k=1 k=1 G m < f | > C1.(-x, %) | IF, s@) ldx G k=l m < 11D eg Ox %) Tle HFa sl hs k=1 m = (PY elem) Leo Faye! h k=1 m k=l ~29- Let € >0. By Lemma 1.3 we can choose f€ C such that £(%4,) = 1, k =1,...,m, and ||f||,< 1+ €. So m m LD MAL =< MILY eam) [lo [tlh k=1 k=1 m < M || »; C1.%, %) | loo (1 + €) k=1 Since € is arbitrary, m m 12, Cy Md] < MIL Y eyes) [leo » k=1 k=1 and by Bochner's theorem A = uw, where pu € M(G) with | | | | <M. Corollary: X, is the set of all functions which are locally the transform of a finite regular Borel measure on G. -30- Iv. Generalized L’ Multipliers This chapter contains a complete characterization of X,, the generalized multipliers on L7(G). The main result is that X, is the set of all locally L” functions on T. A function A is locally L” on Tif AX, € L” (Pr) for each compact subset K of Yr. The cornerstone of this chapter is the Plancherel Theorem, which states that the Fourier transform is an L’” isometry when restricted to L'f\ L?, and since L'/\ L’ is dense in L’, it can be extended uniquely to an isometry from L7(G) to L*(I). The extension is called the Plancherel transform, and under this transform L(G) and L?(I) are isometrically isomorphic. Lemma 4.1: Let f,g € L2(G), andh € L'G). Then (f,g) = Gg, and AA “~ f*h = fh. Proof: The first equality holds because the inner product in a Hilbert space is completely determined by the norm. The second equality holds for allf € L(G) N L(G), so it holds on L’(G). Definition 4.1: Let A be a complex valued function defined on T. A_is locally L™ on I if AX, € L” (Ir) for each K C I, K compact. Lemma 4.2: Let > be locally L”~ on I. Then2 is measurable. Proof: Since AX, € L” (I) for each compact K, A K is measurable for each compact K. Then 2 is measurable by Hewitt and Ross I, Theorem (11.42). -31- Theorem 4.1: Bdd(L’(G)) A X, = L’(P). Proof: If € L” (I) thena € X, and Jy have IT £1]. = Hesale s lal fll. = lato ltl. 50 |, || S lal]. - Let T, € Bdd(L’), X,. Suppose ¢ L (I). Then there is = L(G). Also for f € L(G) we r K CT, K compact with m(K) > 0 such that la| = IT, | on K. Let 1 wv ——r X m(k)jz | ‘Then f¢ L(G), |{£||,= 1, but oytlle= (bE = step J bxgl’ar> UI Tr a contradiction. To show | |, | leo < | IT, | |, let € > 0. Then there is a com- pact K, C I with 7 (K,) > 0 and |a| > llaAllo- €onK,. Then (LI | [oo € (Kx) < fo Do) Pay = f PROX, @) Pay K, | r Vv = Jf ID Xe, Pax = | [ty Xe, He G V < | [TIP IIx, 1h? = Hel P iy, Eh? = [ie [PF 7 05,). So Hate e< [|r 11, and since € is arbitrary, Hrllo< IIT, |. Combining with the previous result gives | la | | = | IT, | , so X, f\ Bdd(L”) and L™(P) are isometrically isomorphic as Banach spaces. ~32- Corollary: Let T, € X, Ty bounded. Then the spectrum of T, is the closure of the essential range of 2X. Theorem 4. 2: X, is the set of all locally L” functions on I. Proof: This follows from Theorems 4.1 and 2.5. Theorem 4.3: X, is a commutative, self-adjoint algebra containing X,, and the containment is proper if G is noncompact. Proof: We have already shown that X, is a commutative algebra con- taining X,. If G is noncompact then IT is nondiscrete and there are noncontinuous functions in L” (I). Since L”(r) c & and every rE x, is continuous, the containment is proper. Letr € &. ThenX € X, and Jy =Jy. Also if f, g € Jy’, on A “a ~ a ~ ~ oN then (T,f,¢) = (T,,8 = af g = Gre) = GT) = &Ty)- Therefore T,* = Ty: Corollary 1: X, is anormal algebra, i.e., T,T,* = T)*T) for each T, € &- Corollary 2: T, € X, is unitary iff AX = 1. Corollary 3: T, € X, is Hermitian iff is real. T, is skew Hermitian iff is pure imaginary. Corollary 4: Ty € & is positive definite iff’ > 0 anda = 0 only ona set of measure 0. Corollary 9: T, € & is invertible iff X is locally bounded away from 0. wenssccneowsetas -33- Proof: If 1/A is locally L” on TF then 1/r € X, and T,T1/, =I. 1/fr is locally L’ on Tiff a is locally bounded away from 0 a.e. Remark: Note that Ty € X, is invertible iff X is never zero, while T, € X, is invertible iff X is locally bounded away from zero (except r on sets of measure 0). Since X, C X,, this seems to conflict, but recall that every function in X, is continuous, so A never zero implies 2 locally bounded away from 0. ~34- V. The Compact Abelian Case Throughout this section we will take G to be a compact Abelian group. Haar measure m on G will be normalized so that m(G) = 1. Then the dual measure ™ on I is counting measure, i.e. som ({ 0} ) =1. T is discrete, soC = Cy = Cy = C, = {trigonometric polynomials} for 1 <p <q <2. Also by Lemma 5.1 below we have the Fourier transform available for all the spaces L?(G), 1 <p < ~, and we can study all generalized (p,q) multipliers for 1 <p,q <©. Lemma 5.1: Letl <p<q<o. Then LUG) C L(G) and if fe LYG), HELL = (el lg: Proof: This is a fact contained in any good real variables book. Lemma 5.2: Letf¢€C. Then f(x) = f(x,y) y T Proof: I is discrete, so compact subsets are finite subsets. There- fore the sum above has only a finite number of nonzero terms and is absolutely convergent. If y, is a character on G, then |v | ls =1 since m(G) = 1, so the sum above is in L(G). Finally -35- ¥,(y) = f (x, y,)(-x, dx = f (x, ¥, - dx G G lify=y 0 otherwise So both f and y FM6,y) yer have the same Fourier transforms, and equality holds. Remark: Note that we have the Fourier transform available for LP(G) for 1 <p < ©. Our definitions of X D; 4 p efinition XQ Jy ; and X, 4 now make sense for 1 < p,q <, and from now on we assume this extension. Theorem 5.1: X, a7 r€ - {all complex valued functions defined on I}. >? . Denote x by X for every p,q, 1 <p,q <@. 3 p,q Proof: Let A be a complex valued function defined on TI, andf€ C. Let g(x) = Yo AEY(x,y) yeT Then g € C and g =Af, so JI 3c and A€ Ky q’ Theorem 5.2: Let1 <p,q < ~, and Ty € Xn q (p,q) closed, and commutes with convolution by elements of M(G). Then T) is linear, ~36- Proof: If f,g € ged, a,8 are complex, and p € M(G), then Naf + Be) = anf + Bag and hf = rs py, so T, (af + Bg) = aT,f + BT,g and (Tf) * u = T(t * pL). Suppose {f,}-) Cav4, fe LP(@), g¢ LUG), andt, Py f, T)f, te g. Lethe C. Then | |, +h) -geh||y < ||T,(f* h) - T,(, *h) | lat | |T,£,*h - g*h| la < [[T yh] ly lifp-tlh+ Path It yf,-81 lg < [!T,b] lq Ian lp + lelh Wtyfa-8! lq - 2 asn-~. So T,(f*h) =g*hor \fh = gh for eachh €C. Therefore Mf =g, sofe ged, T,f =g, and T, is (p,q) closed. Theorem 5.3: Let 1 < p,q < ©, and T a (p,q) closed, linear operator with domain D(T) a translation invariant subspace of L(G), D(T) >C, range R(T) C L1G), such that T commutes with all translations. Then TEX ig: Proof: Let y € I’. Define A(y) = T(-,y)(y). Then T € Xp q and on C T= T,- Let f € D(T), and {ff C¢C such that fn Py f. Then for he C, {,*h—> f¥h, and Ti, *h—4> Treh, since ||Tf,+h - Ti+h| lq q . . < ||Th| la | lf,-£| lb" So T,f,*h — > Tf*g, and since T) is (p,q) closed, f*h € gy 4 and T, (f*h) = Tfth. Taking Fourier transforms we aa oN AN get Afh = Tfh for eachh€ C, or A=Tf. Sofé ead and T,f = Tf. Therefore D(T) coh and T = T, on D(T). -37- Now suppose f ¢ JP, Using Lemmas 1.1 and 1.6 we can find p qd a sequence {h,} CC such that fh, —» fand T,f*h,—» T,f. But T(f+*h, ) = T, (f*h,) = T)fth,, so f € D(T) and Tf = T)f. Therefore sped = D(T) and T =T,. Theorem 5.4: Let 1 <p,q< ©, and T,€ Xa! Then T, is (p,q) bounded iff | |F <M/iif for each f € C and some M 2 O. A,f!'q p Proof: Note that Fy (x) = Y ANE) = T,£(x) yer If T, ¢ Bdd(p,q), then Fas | la < ||T,| loa | |e | J fr eachf¢C. If T, is (p,q) bounded on the dense subspace C, then T, is (p,q) bounded since it is (p,q) closed. Theorem 5.5: LetA€E xX. (1) IgA ¢ L” (1) then T, € Bdd(p,q) for any p,q, 1 <p,q < ©. a (2) If € M(G) then T, € Bdd(p,q) for 1 <q <p <-. (3) If € L"(T) for some r, 1 <r <2, then T, € Bdd(p, q) for r <p < ~andl <q <~. (4) (Larsen) IfA€ L'(I) for some r, 2< r< © then T, ¢ Bdd(p, q) for 1 <q < or <p <o. (5) If’ € L’ (1) then T, € Bdd(p,q) for 1 <q <2 <p <<. x Proof: (1) If A ¢ L’ (X), then for each n there isa ¥, © T such that IA) | >n. Lett, =(-,¥,) (i-e-, £,(8) =(,7))- Then f, € C, and -38- IT,f, | la = | R(v,)(- »Yp) | la = AG) | >n, but | fy | lo = 1 for every p,l<p<. SoT,é Bdd(p, q). -_—~ a (2) If A € M(G), then A = w for some yp € M(G) and T,f = fy for each £€ LP(G). Then from Lemma 5.1 we get for q <p that | IT,f | la = teu ly <[lEllq Hell < Iflly [lal | 80% € Baalp,«) and (3) Letfé C. Then for each x €G, IT.2@) | = [PF @] = 1 Ate, | yer - | i/r <Y LAWiyl< ( [A(y) " [El Lee yeT yer <|[U I, Telh using Hélders inequality and the Hansdorff-Young theorem. Therefore Tye L(G) and |[Tyf| [ay < [Ally Ilfllp» andif t <a< ©, and r<p<, |[T fll < [ltyll. < [Atl tll, < [PT Ilel lp: Therefore by Theorem 5.4, T, ¢ Bdd(p,q) for 1 <q < ~ and r <p < ©, Nowfor any gé€ L(G), ||g||, =lim ell: So in particular HT eI <|/a||. llgll, for every gq, 1 <q <<, and we have T,8 [eo < ATI. lle Il. for each g € L(G). There- fore T) € Bdd(p, ~) forr <p < %. Finally, if h ¢ LP(G), forr <p<™, andl <q <~, Th l lg <= [[Tyh Too <1 Tle Uhl < LAT, HBL [,- 0 T, € Bdd(p,q) and HT Ig < /[Al|,- -39- (4) (Proof due to Larsen). Let t = 25. Then t > 2, and t 2 t ¢ 2 _ ar L(G) CLG). Letfe L(G). ThenfeL(I). Lets =“ and a=r/2+1. Thenli<s<2,1< a< ©, and . 1/ . \ 1/a’ y prey |® < e pon ° (» lf(y) en) ° yer yer yer 1/ ~ 2 \l/a' = (2 [a(y) | . (» font ne, yer yer Therefore Af ¢€ L9(T), and by the Hansdorff- Young theorem there is , a a A ge € L® (G) such that g = Af. Buts’ = or =t, and af € La), or T,f¢ L(G). Since T, is (t,t) closed and jt =1'(q), T, € Bad{t,t) = Bdd (25, or). Then by arguments similar to those in part (3) of . 2 this proof, T, € Bdd(p,q) for 1 <q <y <p<o, (5) By Theorem 4.3 T, € Bdd(2,2). Soifl<q<2<p<™, Ty € Bdd(p,q) by arguments similar to those above. [1] [2] [3] [4] [5] [6] [7] [8] [10] [11] -40- References S. Bochner, Uber Faktorenfolgen fiir Fouriersche Reihen, Acta Sci. Math. Szged 4, 125-129, (1928-29). B. Brainerd and R. E. Edwards, Linear Operators which Commute with Translations I, IIl., J. Austral. Math. Soc. 6, 289-327 and 328-350, (1966). J. T. Burnham, Multipliers of A-Segal Algebras on LCA Groups I, and Multipliers of Commutative Segal Algebras I, preprints. J. T. Burnham and R. 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