THE HARDY SPACES AND OTHER RELATED FUNCTION SPACES Thesis for the Degree of PM). MTCHTGAN STATE UNIVERSITY STEVEN JOEL LEON 1970 *5;LLJ.9 This is to certify that the thesis entitled THE HARDY SPACES AND OTHER RELATED FUNCTION SPACES presented by Steven Joel Leon has been accepted towards fulfillment of the requirements for Ph.D. Mathematics degree in________ ZZMAA) D QJN Major professél Date n'V'. :2 41 (770 0-169 ABSTRACT THE HARDY SPACES AND OTHER RELATED FUNCTION SPACES by Steven Joel Leon The Hardy spaces Hp are closely related to certain other spaces of analytic functions. For 0 < p < 1, let BP denote the class of all functions f analytic in the unit disk satisfying \lpr ‘27 I0 ll: 1r(1 r)(H1/P 21|f(re g)|d9 dr < ”- Bp with the above norm is a Banach space. For 0 < p,q < m, let Hp’q denote the space of all functions f analytic in the unit disk for which “rum = {I01 (:11? fi'rlflreig)Ipdofl/Pdrll/q < co and define HP’00 to be the Hardy space Hp. If 0 < p,q S 1 or O < p < 1, q = w and o = (%-+-§)'1, then B0 is the "containing Banach space" for Hp’q in the sense that Hp,q is a dense subset of Ba and Hp’q and B0 have the same continuous linear functionals. The relationships between the spaces H0, Hp’q, and B0 (o = (%-+-l)'l) are studied for all p and q. In particular, q if 0 < P:Q.S 1, then Hp,q is an intermediate space between H0 and BC. This relationship, HO c.Hp’q-c:B°, may be used Steven J. Leon to determine certain coefficient properties of Hp,q functions. The general properties of the corresponding spaces hp, bp, and hp’q of harmonic functions are also studied. If 0 < p < 1, it is shown that hp is a non-locally convex F-space with enough continuous linear functionals to separate points. Next, the properties of bp are discussed and its dual space is determined. Finally, the spaces hp’q are studied and the relationships between hc, hp’q, and b0 (o = (l-+-%d'l) are examined. In particular, it is shown that bp/E is the containing Banach space for hp’p, o < p g_l. The last topic to be considered is composition operators on Bp. If D is an analytic function mapping the unit disk into itself and r is in Bp, the composition operator C¢ is defined by C¢(f) = f o O. It is shown that C¢ is a bounded linear operator on Bp. Conditions are also given on ¢ in order that C¢ be a bounded operator from Bp into Hg, 0 < q.g w. Isometric and invertible com- position operators are characterized, and finally compact operators and their spectra are discussed. THE HARDY SPACES AND OTHER RELATED FUNCTION SPACES by Steven Joel Leon A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1970 ACKNOWLEDGMENTS I would like to thank Professor Gerald Taylor for his help and encouragement throughout my graduate career. ii \ ‘3’“,‘5 ' 71:1 Wei-r. i: TABLE OF CONTENTS Chapter I. INTRODUCTION 1. Fundamentals 2. Background 3. The spaces Hp and Bp, O < p < I II. MIXED NORM SPACES 1. Preliminaries p q o 2. Relationships between H ’ , H , BG 3. Coefficients III. HARMONIC FUNCTIONS l. Conjugate fuBctions 2. The spaces h , O < p < l 3. The spaces bp, 0 < p < l 4. The spaces hp,q IV. COMPOSITION OPERATORS l. Bounds on C 2. Characteriz tion 3. Operators into Hq 4. Compact operators BIBLIOGRAPHY iii Page 15 25 80 ~ew3’ ‘33- l.) CHAPTER I INTRODUCTION 1. Fundamentals. The unit disk in the complex plane will be denoted by D. We will assume throughout that z e D has the form relg and will often write r in place of [Z]. For a function u harmonic in D, the integral means of order p, O < p < w are given by 2w - r,u) {3;_I |u(relg)|pd9}l/p. m 2W 0 M p( The infinity means of u are given by Mw(r,u) = max Iu(reig)l. OSO<2Tr We denote by hp, O < p g w,the class of functions u harmonic in D such that sup M (r,u) < m, O_<_rO,a_}_O,p g Kc<1 - r>'a'(l/p)+(l/Q). where C is an absolute constant and K(p,a) depends only on p and a. Notation: C(a1,a2,...,an) will denote a constant depending only on the numbers a1,a2,...,an. Theorem 1.2 (G. H. Hardy and J. E. Littlewood [9], p. 413). If p > O, a > O and Mp(r,f') S C(l — r)‘a'1 then Mp(r,f) g K(p,a)(l - r)'a. Notation: f(r) a g(r) means f(r)/g(r) a l as r a 1. f(r) ~ g(r) means f(r)/g(r) and g(r)/f(r) are both bounded for r sufficiently close to 1. The next theorem may be proved by simple computations (see [2], p. 65). Theorem 1.3. If a > 1, then 2 W I ll - zl’ade ~ (1 - r)'a+l. 0 There are various definitions of fractional derivatives and integrals. We shall use the one given by P. Duren, B. W. Romberg, and A. Shields in [3]. This definition differs only by a factor of z8 from the one given by G. H. Hardy and J. E. Littlewood in [10]. Definition 1.1. If f(z) = ganzn, the fractional derivative of order s of f is defined as f[B](Z) = 2 ME) anzn n! and the fractional integral of order 8 is defined as n: n rial”) = '23 tom‘s—7 anz A function u harmonic in D can be written in the form rlnleinG. We define and °° IHI! ' o u[B](Z) = A}: f(ln]+l+5) cnrlnleln . We will assume many of the elementary properties of harmonic functions. For example, each u harmonic in D can be completed analytically and any two analytic completions of u differ by a constant. If f = u + iv is an analytic completion of u and v(O) = O, then we say that f is the normalized analytic completion of u or simply "the" analytic completion of u. 2. Background. If f e Hp, then we define (1 1) HfHHp = :2? Mp(r.f). If p 2_1, then (1.1) defines a norm on Hp. If p < 1, then (1.1) does not satisfy the triangle inequality and hence fails to be a norm. However, H ”pp satisfies the triangle inequality and induces a metrig on Hp. The spaces HP, 0 < p‘g m, are complete. Indeed, Hp is a closed sub- space of Lp(D), the class of complex-valued functions f satisfying: 2W 1 . HfHLp = :13 i 27 IO |flpd©}l/p < 00. Thus, if p‘Z l, Hp is a Banach space. Definition 1.2. An F space is a linear space with a complete translation invariant metric under which scalar multiplication is a continuous operation ([6], p. 51). If p < 1, then Hp is an F space. F spaces have many of the important properties of Banach spaces. In particular, the Open Mapping Theorem, the Closed Graph Theorem, and the Principle of Uniform Boundedness all hold for F spaces. Furthermore, the Hahn Banach Theorem holds for locally convex F spaces. For precise statement of these theorems, we refer the reader to [6]. Definition 1.3. We say that an F space has the Hahn Banach Extension Property, H.B.E.P., if every continuous linear functional on each closed subspace has a continuous linear extension to the whole space. In particular, a locally convex F space has the H.B.E.P. J. H. Shapiro [17] has shown that for an F space with a basis, the H.B.E.P. is equivalent to local convexity. 3. The spaces Hp and Bp, O < p < l. The properties of HP, 0 < p < l, as a linear space, were first studied by S. S. Walters in [18] and [19]. He showed in [18] that Hp has enough continuous linear functionals to separate points, in contrast to the Lp spaces (0 < p < l) which have no continuous linear functionals other than the zero functional. In [19], he conjectured that the Hp spaces were not locally convex. This was later proved by P. Duren, B. W. Romberg, and A. Shields in [3]. If A is a linear space, we will denote its dual by A*. We will also need the following definitions. Definition 1.3. If f and g are harmonic in D, and g e hl, then define 2w = lim I f(re19)g(e'ig)d9 r-0 0 provided the limit exists. Definition 1.h. Two Banach spaces A and B are equivalent if there is a one-to-one linear mapping T of A onto B such that both T and T"1 are continuous, i.e., A and B are equivalent if they are linearly homeomorphic. s. s. Walters showed in [19] that corresponding to each ¢ 6 (Hp)*, there is an analytic function g such that (1.2) ¢(f) = for each f e Hp, and conversely if g is a function such that exists for every f in Hp, then (1.2) defines a continuous linear functional on Hp. B. W. Romberg [1A] continued this study improving upon Walters' results. In the case p is not the reciprocal of an integer, Romberg gave a condition on g which is necessary and sufficient in order that (1.2) define a continuous linear functional on Hp. Romberg also gave a partial characterization of (Hp)* in the case that p is the reciprocal of an integer. Finally in [3], P. Duren, B. W. Romberg, and A. Shields gave a characterization of (Hp)* up to equivalence for all p < 1. To present these results, we will need to define certain Lipschitz classes of functions. Definition 1.5. If f is a complex-valued function defined on I2] = l, the modulus of continuity of f is given by w(h;r) = su Ir(ei“>.- f(e13)|. IG'B.Sh f is said to belong to the Lipschitz class AG (0 < a.S 1) if w(h;f) = 0(h“) as h a 0. Furthermore, f is said to belong to class A* if |r(e1(t+h)) — 2r(eit) + f(ei(t‘h))| = 0(h). The classes kc and x* are defined in a similar manner with "0" replaced by "0". For a function f analytic in D, we say f E Aa(xa,A*,x*) if f is continuous in D and f(eig) E “g(xa,A*,x*). Definition 1.6. Let A2,(n = 0,1,..., 0 < a g_l), be the space of functions f(z) analytic in D and continuous in D such that f(n) 6 A“. A: is a Banach space under the norm ’ - n i e+t n '0 “r” = urn a, + sup t a|r( )(e( )) - r( )(aL )l. H t,0 t>O Similarly, A: denotes the Banach space of functions f analytic in D and continuous on D such that f(n) e A* and urn = urnH. + gag t‘llr<“) - 2r + f(n)(ei(9't))l tic “.3 ‘ The spaces x2, A2 are defined in the same manner. Theorem 1.4. (P. Duren, B. W. Romberg, and A. Shields [3], p. 35). 1 l l p * . f ___. —-, = -- n and H then there 18 I 11+]. < p < 1'1 (X. p CO 6 ( ) 3 . n-l a unlque g 6 AG such that cp(f) = I for each f e H? Conversely, if g e A:_ , then exists for each f E Hp and defines a continuous linear functional on Hp. n-l If p = then g E An-l and conversely any function g e A* l EFT ’ defines a continuous linear functional on Hp. Theorem 1.5. (Duren, Romberg, Shields [3], p. 39). If 1 < p < l. = E—- n then (Hp)* is equivalent to an: niap ’ An-l. If p = 1 , (Hp)* is equivalent to A2_1. a n+1 We may also talk about the spaces Ad’ A*, ya, y*, A2, etc., of harmonic functions. The following theorem allows us to characterize these spaces in terms of growth conditions. Theorem 1.6. (See Zygmund [20], vol. 1, p. 263). (i) A necessary and sufficient condition for a harmonic function u to be in Ad’ O < a S_l is that :19. = 0((1 - r)a_l). (ii) A necessary and sufficient condition for u to be in A* is that Theorem 1.7. (Duren, Romberg, Shields [3], p. 44). If f is analytic in D and f(z) = 0((1 - r)'a), a > 0, then <1) r[B] = o<<1 - r>‘ii+8)>. s > o and (ii) f[B], endowed with the norm Hen p = sup {Ms (r,e[l/p])<1 - r>i. y r<1 The spaces wp are defined similarly replacing "O" by "0". 10 It has been noted ( J. H. Shapiro [17], p. 27) that P n—l . . _ 1 1 y 18 equlvalent to Ad , a — - n for HTI < p <.H and I p 1 equivalent to AE_ if p = This may be proved easily n+1 ' using Theorems 1.6 and 1.7. It was remarked earlier that Hp, O < p < l, is not locally convex. Actually the following stronger theorem was proved. Theorem 1.8. (Duren, Romberg, Shields [3], p. 51). There exists a proper closed subspace Hp(E) of Hp and a continuous linear functional m on Hp(E) which cannot be extended to all of Hp. Thus, Hp does not have the H.B.E.P. and hence is not locally convex. In [3], Duren, Romberg, and Shields found the ”containing Banach space" of Hp, that is, they found a Banach space Bp which has the same continuous linear functionals as Hp and which contains Hp as a dense subspace. Definition 1.8. For 0 < p < 1, let Bp denote the space of all functions f(z) analytic in D such that r,f)dr < m. (1.3) “ran = (:(l - r)(l/p)-2Ml( The spaces bp of harmonic functions are defined similarly. Theorem 1.9. (Duren, Romberg, Shields [3], p. 40). The space BP, 0 < p < 1, with norm (1.3) is a Banach space. Furthermore, 11 (i) |f|.s C(p)HfH p<1 - ,)-1/p B for each f e Bp’ and f(z) = ocnrn pn(1/P>‘1 B and an = o(n(l/p)'1). Conversely, if 0 < p < l and an = O(na), a <(1/p) — 3/2, then f E Bp. The (l/p) - 3/2 is best possible, in that, there exists g(z) = 2b 2 such that bn = O(n(1/p)'3/2) and g is not in RP. The next theorem shows that for O < p < q < 1, the spaces Bq and Bp are equivalent under the correspondence f a f[(1/p)-(1/q)] for each f equ. 12 Theorem 1.11. (Duren, Romberg, Shields [3], p. 43). If O < p < q < 1 and B = (l/p) - (1/q), then (i) f 5 BP implies f ] 6 Eq and [B Hf[B]HBq S,C(P,Q)HfHBp3 (ii) f E Bq implies f[B] e Bp and 11481an _<_ c\\anq. Theorem 1.12. (Duren, Romberg, Shields [3], p. 46). Theorems 1.4 and 1.5 remain true with Hp replaced by Bp. Theorems 1.9 and 1.12 imply Bp is the containing Banach space of Hp. The next theorem relates Bp to the 96* O closure of Hp in (Hp) Theorem 1.13. (Duren, Romberg, Shields [3], p. 46). For each f e Hp (O < p < l), c

nran s urn (Hp)... _<. K

uran. Hence Bp is equivalent to the closure of Hp in (Hp)**. Finally, Duren, Romberg, and Shields [3] showed that Bp is itself a conjugate space. Theorem 1.14. (Duren, Romberg, Shields [3], p. 49). If 1 < p <‘l , a = (l/p) - n, and w E (kg-l)* , then there is 11+]: 1’] a unique function f e Bp such that v(s) = 13 for each g e yg'l. Conversely for each f e Bp, determines a bounded linear functional on kngl. Furthermore, (kg-l)* and Bp are equivalent. If p = i1 , the above state- n 1 ments remain true with kn; replaced by yETl The coefficient properties of Hp and Bp functions have been studied by P. Duren and A. Shields, [4] and [5]. These properties are generally stated in terms of coefficient multipliers. Definition 1.9. Let A and B be two complex sequence spaces. We say that a sequence {An} multiplies A into B if {khan} E B whenever {an} E A. Each analytic function can be associated with its sequence of Taylor coefficients and hence Hp and Bp can be treated as sequence spaces. Theorem 1.15. (P. Duren and A. Shields [5], p. 70). If O < p.g 1, then {An} multiplies Hp into Lm if and only if 1- l/ (1.4) An = O(n ( p)). If p < 1, then {An} multiplies Bp into tm if and only if (1.4) holds. Theorem 1.16. (P. Duren and A. Shields [5], p. 70). Let O < p < 1, then (i) {An} multiplies Hp into tq (p‘g q < 00) if and only if N (1.5) 2 nq/p lxnlq = omq); n=l 14 (ii) if 1 5 q < oo, {An} multiplies Bp into tq if and only if (1.5) holds. CHAPTER II MIXED NORM SPACES In [1], Benedick and Panzone studied the spaces Lp’Cl of functions f(x such that 1’X2) 1/q Hpr’q = { IKE [ le lf(Xl)X2)]pdul]q/pdu2} < m, where Xi e Xi and “i is a measure on Xi (i = 1,2). A complex function f(z) = f(reig) may be considered as a function of r and O. In particular, we will be concerned with the Lp’q functions which are analytic in D. These classes of functions were introduced by J. H. Shapiro in [17] and are denoted Hp’q. Shapiro showed that if 0 < p,q S l, and %-=p%-+-%, then B0 is the containing Banach space of Hp’q. He also considered the relationships between these spaces in the case 0 < P.S 1, q > 1. In section 2, we continue this study. In particular, we consider the relationships between Hp,q and BC when p > 1, and the relationships between H0 and Hp,q for all p and q. In section 3, we make use of the relationships given in section 2 to study coefficients of Hp,q functions. 1. Preliminaries. We begin by defining the Hp,q spaces and stating some of their basic properties. 15 16 Definition 2.1. Let Hp’q denote the class of functions f analytic in D such that (2 1) Hf” = T [l (M (r f))qdr}l/q < w ' p,q. o p ’ ' If q = w, define Hp’oo to be Hp. The corresponding classes hp’q of harmonic functions are defined similarly. If 1.3 p,q S m, then (2.1) defines a norm and Hp’q, hpaq are Banach spaces. If m = min (p,Q) < 1: then H Hp q 9 fails to be a norm, however, induces a complete m u ”p,q metric. The next two theorems summarize some of the general properties of Hp,q determined by J. H. Shapiro in [17]. Theorem 2.1. (J. H. Shapiro [17], pp. 28-30). The spaces Hp’q (O < p,q 3.”) are complete with respect to the metric mentioned above. Furthermore, <1) lfnrup,q<1 - r>‘<<1/p>+> for each f e Hp’q3 (ii) pr ‘ pr,q a O as p a 1 for each f e Hp,q (ro = f(oz))- Theorem 2.2. (J. H. Shapiro [17], pp. 40 and 52). If m = min(p,q) < 1, then Hp,q is not locally convex. If 0 < p < 1, O < q.g m, then Hp’q does not have the H.B.E.P. To avoid repetition, we introduce the following notation. Notation: If p and q are positive numbers such that %-+-% > 1, then 0 will denote the number defined by the equation: l-= l.+.l, o P q l7 2. Relationships between Hp’q, HG, BC. We consider first the relationships between Hp,q and BC. The case 0 < p,q S 1 has been studied by J. H. Shapiro [17]. We present his results in the following two theorems. Theorem 2.3. (J. H. Shapiro [17], p. 30). If f 6 Hp’q, O < p,q S 1, then f e B0 and Hmflgcmnmmgq Theorem 2.4. (J. H. Shapiro [17], pp. 35 and 37). Let O < p,q S_1. If m is a continuous linear functional on Hp’q, then there exists a unique g 6 yo such that (2.2) u = for each f E Hp’q. Conversely, if g 6 yo, then (2.2) :q. defines a continuous linear functional on Hp Moreover, the spaces v0 and (Hp,q)* are equivalent. It follows then that (BO)* is equivalent to (Hp’q)*, O < p,q g_1, and BC is the containing Banach space of Hp’q. Furthermore, if we consider Hp,q and B0 as subspaces of their second dual, then B0 can be associated with the ** closure of Hp,q in (Hp’q) Theorem 2.5. For each f E Hp’q, O < Paq 5,1, C(paqufHBc, S [\f1\(Hp,q)** S K(p’q)”fHBo' Hence, BC is equivalent to the closure of Hp’q in (Hp,q)**. 18 Proof: By Theorem 1.12,(B0)* and v0 are equivalent under the correspondence g e m, where m(f) = for each f 6 BO. Let 5 be m restricted to Hp’q- Then O(f) = for each f e Hp’q and, by Theorem 2.4, (Hp,q)* and yo are equivalent under the correspondence g a i. It follows that (B°)* and (Hp’q)* are equivalent under the correspondence m A $ and C(p.q)H°T3H _<_ Hep”: K(p,q )HCPH If f 6 Hp’q, then f e B0 and since BC is a Banach space, we have HfHB0 = HfH B,)** = llf‘|i0_— Than)l ELLE < “ C M Manic Hen = 7—7,], will we 3 (H J ) Similarly, HfH(Hp,q)** S K(p:Q)HfH(BO-)** = K(pJQ)HfHBg Let [fn] be a sequence in Hp’q, then {fn] is Cauchy in the (Hp,q)** norm if and only if it is Cauchy in the B0 norm. Thus, each element in the closure of Hp’q in l9 (Hp,q)** can be associated with an equivalence class of *-)(- Cauchy sequences in (Hp’q) norm and hence with a unique 130 function. Let 0 < P.S 1, then B0 is the containing Banach space for Hp,q if either 0 < q'g 1 or q = m. It is natural to ask whether this is still true if 1 < q < w. Theorem 2.6. If 1 < q < w, 0 < p <~agf , then Hp,q is not contained in BC. Proof: The functions (2.3) ra,, = <1 - z>'1/“i§-1os I%Ei'1/B . a > 0 were examined by J. E. Littlewood ([12], p. 93). He showed that if y > a: then <2.i> Mx‘(1/a)+<1/*)(losf%;)'1/B- Thus, if we set a = a, then (i) f = f is in Hp’q if G:B 0:8 and only if a < q; (ii) f0 8 is in B0 if and only if B < 1. D In particular, if q > 1, we may choose a = $(q+l). Then 1 < B < q and hence f E Hp,q but f E BC. 0:8 038 Although Hp,q is not contained in B0 if q > 1, we do have the following result. Theorem 2.7. (J. H. Shapiro [17], p. 33). If f E Hp’q, O < p.g l, l < q < m then f 6 BJ6 for each t < o and ngsKmnmmm%¢ 20 We next turn our attention to the case p > 1. Theorem 2.8. If p > 1, O < q < PET , t < p, then B0 is contained in Ht’q and Hdgqsxomsmmfl for each f 6 BO. Proof: We may assume t > 1. If f 6 BO, then l n HfHBo 2 Ir Ml(o:f)(1 ‘ p)(l/C)—Cdp .2 M1<1 - r>(1/0)'1<<1/o>-1>‘1. Thus, (2.5) Ml.s cHfH ,(1 - r)1'<1/0>. B If t = 1, (2.6) M%(r,f) = M‘11(r,f) -1 scesmetI-nHL-FR BC Hence (2.7) M%(r,f) S K(p,q)ququ (1 - r)Q((1/’°)‘(1/P))‘1, B The conclusion follows by integrating (2.6) and (2.7) and then taking l/q powers. Theorem 2.9. If p'Z l, O < q < Pgi) and s < q, then B0 c Hp’S and 21 urup,s.s K(P:Q:S)HfHBo for each f 6 Bo. Proof: If f 6 Bo, then we have by (2.5) that Ml(r,f) _<_ C(p,q)]]f\\BG(1 - I‘)1-(1/a). It follows from Theorem 1.1 that Mp(r.f>.g Knru ,(1 - r>'1/q B and hence s s s _ -s/q Mp(r:f).S Kp’quHBG(1 r) - Corollary 2.10. If s < q.g 1, then Hl.q C Bq/(q+l) C HLS. If q > 1 and s < t < q, then Hl,q C Bt/(t+1) l,s. c H Proof: The first statement follows from Theorems 2.4 and 2.9. The second statement follows from Theorems 2.7 and 2.9. We next consider the relationships between Hp,q and HG. These relationships can be determined using a theorem of Hardy and Littlewood. Theorem 2.11. (G. H. Hardy and J. E. Littlewood [10], p. 411). If0nrnH, It is easy to see that the above containment is strict, since Hp,q contains all functions f satisfying f(z): O((l - r)'a), a < l/q. Thus, Hp,q contains functions which are not of bounded characteristic. 'In fact, there exist functions having radial limits a.e. which are in Hp’q, but are not in Ho. The functions f defined by (2.3) have radial limits a.e. (1:8 and J. E. Littlewood ([12], p. 96) showed that for 1 = a, x < B (2.8) Mx(r,fm ) a A(a,s)(1og_ :)B+(1/x) Thus, if we set a = o and a = %(q + 0), then 0 < B < q and it follows from (2.4) that fo 3 E Hp’q; however, - _ 3 (2.8) implies f0 8 is not in H°. ) 23 3. Coefficients. It follows from Theorem 2.3 and Corollary 2.12 that if 0 < p,q S 1, then Hp’q is an intermediate space between H0 and BC, i.e., H0 c Hp’q c BC. 5" . Thus, Hp’q functions possess properties common to H0 and BG functions. In particular, coefficient results for Hp’q functions are obtained as immediate consequences of the HO and Bc properties given in Chapter I. f Theorem 2.13. Let 0 0 such that 5 nl-(l/o) n .2 e > O, n = 1,2,..., In [4], p. 259, P. Duren and A. Shields showed that if f(z) = zanzn is in H0, then I S -5 (2.10) z n Ian < w for s > o, 6 = l + s((l/g)—1), O < 0.3 1. They then showed that B0 and HG functions differ in allowable moduli of coefficients by giving an example of a BC function whose coefficients do not satisfy (2.10). A similar example may be provided for Hp’q. Indeed, the function defined by (2.3) is in Hp’q for B = %(q + 0). However, f 098 J. E. Littlewood ([12], p. 93) has shown )n(1/O)'l( - —1 an m C(p,q log n) /B so that l -5 s _ Z n lanl .2 KCP3q93) Z BEBE—H - m CHAPTER III HARMONIC FUNCTIONS In this chapter, the spaces hp, bp, hp’q of harmonic functions are studied. These spaces are defined in the same manner as the corresponding spaces of analytic functions. Section 1 deals with the question of whether the harmonic conjugate of a function in one of the above mentioned classes is in the same class. The spaces hp, bp, and hp’q are treated in sections 2, 3 and 4 respectively. The general properties of each of the spaces as well as the relationships between the three spaces are discussed in these sections. 1. Conjugate functions. If p > 1, the spaces hp and Hp are very much alike. In fact, if f = u + iv is analytic in D, then f e Hp if and only if u 6 hp. This is a consequence of the following well-known theorem of M. Riesz (see, for example, [2], p. 54). Theorem 3.1. (M. Riesz). If f = u + iv is analytic in D and p > 1, then M r,v) S C(p)M r,u), O < r < l. p( p( _. Thus, if u 6 hp, then v 6 hp. In the case p = 1, u e h1 does not imply its harmonic conjugate v is in hl. However, we do have the following theorem (see [2], p. 57). 25 26 Theorem 3.2. (A. Kolmogorov). If u e hl, then its harmonic conjugate v is in hp for all p < l and M If p < l, the situation is much worse. G. H. Hardy and J. E. Littlewood ([9], p. 419) have shown that the function “(2) = Re T(z) = Re( § is in hp for all p < 1, however, n(z) has radial limits existing on a set of at most measure zero. If T(z) is in Hp for some p then T(z) must have radial limits a.e. But this would imply n(z) has radial limits a.e. Thus, T(z) is not in Hp for any p. We next investigate whether theorems similar to Theorem 3.1 hold for hp’q and bp. As an immediate con- sequence of Theorem 3.1, we have that if u e hp’q, p > 1, O < q S_w, then its harmonic conjugate v E hp’q and Hvlvp,q _<_ c

nuup,q. The question of whether u 6 hp,q implies v e hp’q for O < p'g l, O < q < w is still open, although it has been answered affirmatively in the case q = p. Theorem 3.3. (G. H. Hardy and J. E. Littlewood [9], p. 413). If u e hp’p, o < p < m, then its conjugate v e hp’p and 27 HVHp9p S C(p)Hqu,p' The situation for bp is much nicer. Theorem 3.4. (P. Duren and A. Shields [4], p. 256). If u 6 bp, then its harmonic conjugate v 6 bp and HVH S C(pHWH - bp ' bp We may use Theorem 3.4 to show that most of the theorems concerning Bp given in Chapter I hold also for bp. 2. The spaces hp, 0 < p < l. Recall that u 6 hp if and only if = sup M (r u) < w. Huth r<1 p ’ As was the case for Hp, p < l, H H p does not satisfy the h triangle inequality and hence is not a norm. However, H “pp does obey the triangle inequality and defines a h metric on hp. The properties of Mp(r,u), u harmonic, have been studied by Hardy and Littlewood [9]. We sum up a number of their results in the following theorem. Theorem 3.5. (Hardy and Littlewood [9], pp. 410-415). Let u be harmonic in D and f(z) = z Ynzn be its analytic com- pletion. If 0 < pug l, 3.2 O and Mp(r,u) g C(l - r)'a then <1) lynl.s Bca+‘ls (ii) If(z)| 5 B(p,a)C(l — r)'a‘(1/p); (iii) Mp.g Bc<1 - r>a‘1; (iv) if a > 0, then M (r,v) S M a. p (Inf) : B(p,a)C(l - l")" P Theorem 3.5 will be used to prove some general theorems about hp. Theorem 3.6. The spaces hp, 0 < p < m are complete, Proof: If p > 1, the result follows from Theorem 3.1 and the completeness of Hp. Assume then that p's 1. By Theorem 3.5 (ii), we have (3.1) Iul g,B(p>HUH p<1 — r>‘l/p h for each u 6 hp. It follows from (3.1) that if {um} is a Cauchy sequence in hp, then [un] converges uniformly on compact subsets of D to a harmonic function u. On the other hand, [um] is a Cauchy sequence in Lp(D) which is complete. Thus, [um] converges in Lp(D) norm to an Lp(D) function g. But then there exists a subsequence which converges a.e. to g. Thus u = g a.e. So u 6 hp and {un} converges to u in hp norm. Theorem 3.7. If 0 < p S 1, then hp has enough continuous linear functionals to separate points. 29 Proof: If u is harmonic in D, then u can be written in the form We may assume without loss of generality that u is real-valued. Let f(z) 2 § Ynzn be the analytic completion of u. Then n=O iyn ithO (3.2) 2 0 II ly ifn. M1l‘<1/p>, p b and since c 2 2 C” R - r it [u(relg)l < -£ _ , ._ n Iu Re ldt CIT IO lReJ- _relgld ( ) 3 _LL. M1(R,u)) 32 we have — 1/ g C(pmuubpu - r) ( p)- To show the "0” condition, note that for e > 0 given, the left hand side of(3.4)may'o3replaced by e if r is sufficiently close to 1. This proves (i). The proof of completeness follows in the same manner as the proof of Theorem 3.6 using part (i) in place of (3.1) and noting that op lies in the L1 space formed with respect to the measure l.(l _ r)(l/p)-2drd0. (ii) follows from Theorem 1.9 2v since if u 6 bp, then its analytic completion f 6 Bp and u-u f—f . |l pub, s n ,qu Finally, (iii) follows from (ii) since h00 c hg c bp and (ii) implies h°o is dense in op. The next theorem is an immediate consequence of (3.2), Theorem 3.4 and Theorem 1.10. CD Theorem 3.12. If u(z) = g cnr _OO InleinQ 6 bp, then 1/p)—1 n (3.6) lc l g c

\\un,bplnl< l/p)-l). Conversely, if 0 < p < l and and c = o(lnl( n cn = O(Inla), a <(1/p)-3/2, then u 6 bp. Furthermore, the (1/p)-3/2 is best possible (i.e., there exists l/p)-3/2 g = z Bnrlnlelng such that an = 0(Inl( ) and g é bp). n . Proof: Let f(z) = 3 Ynz = u(z) + lv(z). Then by (3.2), Theorem 3.4 and Theorem 1.10, 33 W = é‘vlnll SismnranlnHl/m-l g C(p)Hquplnl(l/p)-l and |Y|n|| = o(|n|(l/P)-l). If an = o(lnla), then yn = O(lnla). SO, by Theorem 1 10, f E Bp and hence u 6 bp. The (1/p)-3/2 is best possible for Bp and consequently for bp. Alternatively, (3.6) could have been derived by direct computation using (3.5) o 1.. o — Since cn — 2 (an-lbn), c_n — cn (n > 0) where l 2” i0 a = ———-f u(Re )cos n0d0 n n FR 0 and l 271' . b =-——— I u(Relg)sin n0d0. n n HR 0 Theorem 3.13. Suppose 0 < p < q < 1 and let a = l. I. p q (i) If u 6 bp, then u[B] 6 bq. (ii) If u e bq, then u[B] 6 bp. Proof: It is easily verified that if u = Re f, then u[B] = Re f[B] and u[B] = Re f[B]. The theorem then follows immediately from Theorems 3.4 and 1.11. Theorems 3.11, 3.12 and 3.13 will be used to prove the major results of this section which follow in the next two theorems. Recall that Yp denotes the space of all functions g harmonic in D such that 34 HsH p = sup{Mw(r,s[l/p])(l - r)} < w. v r<1 Theorem 3-14- Let m 6 (bp)*. Then there is a unique g E Yp such that m(u) = for each u 6 bp. Conversely, if g 6 mp, then exists for all u 6 bp and ¢(u) = defines a continuous linear functional on bp. Proof: Let m be a continuous linear functional on bp and set m(zk) for k.2 0 bk = as m. p 1 To show g 6 v , we consider first the case Bil < p < %3 Let F(z) )-(n+l) nL[2(1 - z - l] and set U(z) = Re F(§z) where g = pe:LB E D. Then and hence Now so that U(z) = ; (n+|k|)1 plkleiksrlkleiKQ -w Ikll 37 It follows that n] |s[ (s): S.HwH “Hub, .: Hm” HFpqu .: K(P)HwH HFpqu. and by Theorem 1.3 HFDHHp = o((1 - 0 Let 5 = l-- n and set h(z) = g[n](z). Then by Theorem 1.7, P (3.7) ls[l/p](a)l = lumen g c

<<1 - |e|)'l>HcpH, so that g 6 VP. 1 If p = m, define (n+1)![2(1 - z)-(n+2) - 1] '11 A N v H and set U(z) = Re F(gz), g e D. Then v(U) = s[n+l](§) = s[l/p](§) and hence 38 U) a) 09 _, l._1 \. 'o 3: /\ _ncpHIIUHbp /\ _mmnennmHp gamma-Inrhmw Thus, 2 6 WP. w Ikl iKQ p To show g is unique, suppose gl = z dkr e 6 v .00 , and = for each u 6 bp. Let uk = rlklelkg, u 6 bp for each R. Furthermore, bk = = = d k k' Hence g = g1. For the proof of the converse, assume first that 1 l m . KIT < p (.5 and g(z) = z bkrlkle1kg 6 vp is given. We '” w [R] ikQ p must show that for every u(z) = 2 ckr e 6 b , m .00 5(r) = z ckbkrIkl has a limit as r e 1 and that .4!) 11m l6(r)|.s cuuu ,5 I'd]. b where C depends only on g and p. We will prove the existence of the limit by showing 1 [ I6'(r)|dr < m. 0 Let u(z) =-§t > ([k]+n—l)1 Ikll bk rt __ a m k ikQ __a—f(z le ). _OO I t In!!- -r-—-——-u1' 39 Then h = o<<1 - r) > and so by Theorem 1.7, s[n](Z) = o<<1 - r>(l/P)‘n‘l> and hence Thus, 40 Finally, u[n-l] 6 b8 implies l/ —2 j (l - r)( B) Ml(r,u[n_l])dr < e O and l-- 2 = l-- n - 1. P _ 1 _ _ ” IKI iKQ For the case p — KIT , set U(z) — u[n_l](z) __g Akr e , 1/2 then U(z) E b by Theorem 3.13. Let G(Z) =-§; s[n'l](Z) = ; (jk|+n—l)! b r(|k|—1)eiko -°° MRI-l)! m . = z Bkrlkleiko. _W Then 2 'l 2” i0 —io r6'(r ) ='2F I U(re )G(re )d0 0 Set 00 o J(z) = U[1/2](z) = 2 [k]! Akrlklelkg. -w P(Ikl+3/2) Then by Theorem 3.13, J(z) e b‘C/3 and hence I f (l - r)'l/2Ml(r,J)dr < w. 0 Let K(Z) = G[1/2](Z) = g P(:k:+3/2) Bkrlkleik0. .00 k1 It follows from Theorem 1.7 that 1. 2" io —io io -19 '2? £3 U(re )G(re )d0 — §?-£) J(re )K(re )dQ and hence 1 1 f 6'(r2)dr < C I (1 - r) l/2Ml(r,J)dr < m. 0 —- 0 Finally, we must show that the operator m, defined by ¢(U) = for each u 6 b2, is bounded. For fixed 0 < 1, let °° Ikl ¢p(u) = _g Ckbkp . Then by Theorem 3.12 Iopl s c

nuubp§ bklkl(l/P)'1p'k', so mp 6 (bp)*. But for fixed u 6 bp, _ . °° Ikl SUP lm (u)| — 11m I E Ckbkp I o| < m. Thus, by the Principle of Uniform Boundedness w 6 (bp)*. Theorem 3.15. The Banach spaces (bp)* and Yp are equivalent. Proof: The mapping Tzw a g defined as in Theorem 3.14 is a one-to-one linear mapping of (bp)* onto vp. It follows from (3.7) and (3.8) that T is bounded. The Open Mapping Theorem implies T-1 is bounded. Theorems 3.14 and 3.15 give a characterization of the dual space of bp. These results correspond to Theorem 1.12 for Bp. In Theorem 1.14, it was observed that BP is itself a conjugate space. We will show next, by the same methods used in [3], that bp is the conjugate of mp. Given u 6 bp, we may define the operator wu on vp by (3'9) $u(g) = for each g e vp. For fixed g 6 mp, with Hg” p = 1, let mg Y be the operator on bp corresponding to g in Theorem 3.14, then |ou(g)l = lwg(u)| A. ._ Hm lHuH - 81 bp By Theorem 3.15, we have nmgn.s c

nsuvp so that Imu(g)l s_c

uunbp- Hence mu 6 (vp)* and nuun(wp)..s c

nuubp. 43 Similarly mu 6 (mp)* and < C p u . Huun(wp)* _. < >1 ubp We may now define a new norm on bp by 3.10 u = * < > IH m noun(¢p) for each u 6 bp. Lemma 3.16. The norm (3.10) is equivalent to the bp norm, i.e., Hqup S K(P)IHUHI _<_ C(PHWpr for each u 6 bpo Proof: The above remarks give [Hull] S C(p)\\uubp. It remains to show wkpsmmwm Since (bp)* is equivalent to vp, it follows that (bp)** is equivalent to (yp)*. Hence HUH = H H H S K

||cp H - bp ”“ (hp) u (rp>* Thus, it suffices to show HcpuH( p < *. ‘Y )* _ “CPUH(wP) 44 Let c > 0 be given and choose g e yp such that Hg” p = 1 Y and |l > * - e. 8 HcpuHWp) Now gp E wp and Hg H .S Hg“ = 1 p wP YP so that ll _<_ nepnwpnoun up>* < *. _ llcpullwp) But, a as p a 1. Hence * s_ u * + e. hump) Hep ll up) Theorem 3.17. If 0 < p < 1 and m 6 (mp)*, then there exists a unique u 6 bp such that ¢(g) = for all g 6 mp. Conversely, each u 6 bp determines a bounded linear functional on (p by the above formula. Finally, the Banach spaces (mp)* and bp are equivalent. Proof: In view of Lemma 3.16 and the remarks preceding it, we need only show that if m 6 (mp)*, then there exists u 6 bp such that ¢(g) = for each g 6 mp. For a given ¢ 6 (mp)*, define 45 and let Then Ickl s ncpHIIz'kJ pr c

ncpnlklwp> V\ (where Stirling's formula ([6], p. xv) has been used to estimate Hzlklu ). It follows that u is harmonic in D. l k ikO . . Let g(z) = z bkr le be ln wp and gp(z) = g(oz). Slnce go is the uniform limit of the partial sums of its power series, we have . N k k ik0 wep) = 1m cp< >2 bkp' 'r' 'e > N-o°° -N °° lkl = -020 Ckbkp . We claim that ”go-g“ p a 0 as p a 1. Indeed since 1/ -1 s[ 101(2) =o<<1-r) >. we can choose R sufficiently close to 1 so that (l - r)|g[l/p](z)| < 6/2 for r > R. 46 It follows that (l - r)lggl/p](z)| < 6/2 for r > R and hence (3.11) (l - r)|g£l/p](z) _ g[l/P](z)] < e for r < R. Choose 90 such that if p0 < p < 1 then (3.12) Isgl/p]l < e for [Z] S R. It follows from (3.11) and (3.12) that ng-gpr < e for p > po- Hence '8 09 ll lim ¢(g ) p-°l 9 oo lim 2 c b Ikl 041 —w k kp . To show u 6 bp, define * for each g 6 mp, where uR(z) = u(Rz), 0 < R < 1. mR 5 (WP) since u 6 bp. For each fixed g 6 up R lim ¢R(g) = lim Rdl R41 = lim Ral = we). 47 Therefore, by the Principle of Uniform Boundedness, {HTRH p *, O < R < 1] is uniformly bounded. It follows ‘(w ) from Lemma 3.16 that [Hu 0 < R < 1} is uniformly H , R bp bounded. However, the bp norm is an L1 norm and uR a u pointwise. So by Fatou's Lemma Hqup S 3.1—m HuRpr < 00. 4. The spaces hp’q. Recall that hp’q is the space of all functions u harmonic in D such that Hun = { fl Mq(r u)dr}1/q < w p,q. o p ’ and that hp’q is a Banach space if and only if m = min (p,q) Z 1. Theorem 3.18. If m = min (p,q) S l and u E hp’q, then u 6 bm/2 and Hqum/g s C(p,q>nuup,q. m,m Proof: If u E hp’q, then u e h It follows from m Theorem 3.3 that f = u + iv 6 Hm’ and anm,m s 21/m < C(m)| IUHm,m- 48 Furthermore, since f E Hm’m, we have by Theorem 2.3 that /2 f 6 Bm and HfHBm/Q _<_ K(m) Hf‘]m,m° /2 Thus, u E bm and Hqum/g S HfHBm/g V\ K(m)Hme,m < C(m)K(m)||uHm’m < C(m)Kl g C(p,q)llul1p,q<1 - IVE/m and 2/m)-l lcnl g Kuuup,qlnl( Proof: The result follows immediately from Theorems 3.11(i), 3.12 and 3.18. If p > 1 and u E hp’q, then by the remarks preceding Theorem 3.3, we have that f E Hp,q and 49 Ilrllp,q s c

nunp,q. It follows from Theorem 2.1 that IU(Z)| _<_ C(p,q)llullp,q(l - r)‘(1/0) 1 where —-= + o UIH QIH Theorem 3.20. The spaces hp’q are complete. Proof: If p > 1, then the result follows from the completeness of Hp’q. If p S_l, then m = min (p,q) 3'1 and the proof follows in the same manner as the proof of Theorem 3.6, treating hp,q as a subspace of Lp’q, and using Corollary 3.19 instead of (3.1). Theorem 3.21. If m = min (p,q) < 1, then hp’q is not locally convex. If 0 < p < l, 0 < q.g m, then hp’q does not have the H.B.E.P. Proof: If hp,q is locally convex, then since Hp,q is a linear subspace of hp’q, it must also be locally convex. But this contradicts Theorem 2.2. The second statement is immediate from Theorems 2.2 and 3.8. In Chapter II, we studied the relationships between Ho, Hp’q, and Bo. It is natural to ask whether the same relationships hold between hc, hp’q, and b0. It has been observed that in general h0 is not contained in b0. Hardy and Littlewood ([9], p. 416) showed that the function u(z) = Re f(z) = Re(e(l/2)KW1(1 - z)-k_l) 50 . . 1 . .. . V is in h0 for o :‘E:I (k a pOSltlve lnteger). However, f(z) is not in BG and hence u(z) is not in bc(see [4], p. 257). A similar example may be used to show h0 is not contained in hp’q for the proper choice of p and q. 1 Theorem 3.22. If p = q = i, o = 2k’ k a positive integer, then h0 is not contained in hp’q. Proof: Let A w f(z) = u(z) + iv(z) = el(Ck-l)2(l - z)-2K. . c, 1 By the above remarks, u(z) e h for g = 2k 0n the other hand, 2v 1 _f“ (Mp(r,f))q = 2'? [0 ll — 2] ‘do and thus by Theorem 1.3, (M (mm 2. T??- P Hence, f t Hp’q and since p = q = , we have by Theorem 3.3, 11¢ hp’q. WIr—J Although in general h0 is not contained in bo,'we know that h0 is contained in bt for all tIIUHG- Proof: We need only consider the cases (i) t q, s < p and (ii) t < q, s = p. To prove (i), we may assume 0 < s < p. 51 Then by Theorem 3.5 (iii), we have that u 6 ho implies -l M4nt)guww%gl-n and hence by Theorem 1.1 Ms(r,ft) _<_ K(U)Hu||ho(l - r)‘1"(1/0)+(1/S). Therefore by Theorem 1.2, r)(l/S)-(l/b). Ms(r,f) S EXO:S)”u”ho(l ‘ Let c = mlH --1-> 0, then D M§(r.q).s.(sxo.s>>quungo<1 - r>q€‘1. and hence Imhflscwemmw- To show (ii), we have as in the argument above, Mp(r,f:) g C(o)||u||ho(1 _ r).-l-(l/q) and hence Mp‘<1/q>. Thus, M;(r:f).S (K(O:P))t“uu:0(1 _ r)-(t/q)’ so that Mhfiscmmnmww. 52 We remark that if g = (l + ifl > 1, then h0 < hp’q p q and for u 6 ha, 0 . \\u\\p,q S (p,q)llullhcj The containment is strict by the remarks following Corollary 2.12. We next consider the relationships between hp,q and b0. If q > 1, we have by Theorem 2.6 that hp’q is not contained in b0 (l-= l-+ i). As a consequence of Theorem 3.4, we 0 have that i: p > 1, then Theorem 2.8 remains valid if the spaces of analytic functions are replaCed by the corresponding spaces of harmonic functions. Similarly, if p'Z 1, then Theorem 2.9 holds for harmonic functions. If m = min(p,q) S_l, then by Theorem 3.18, hp,q c bm/E. In particular, hp’p c bp/E, 0 < p.g 1. Theorem 3.24. If 0 < p 3.1, and g e wp/q) then hpip exists for each u E and defines a continuous linear functional on hp’p. Conversely, for each continuous linear p/2 functional m on hp’p, there is a unique 2 E Y such that ¢(u) = for each u e hp’p. Moreover, Hemp/2 _<_ compu- p/2 Proof: If g E v , then ¢(u) = defines a continuous p/g. Thus, in View of Theorem 3.18) linear functional on b w restricted to hp’p defines a continuous linear functional on hp’p. 53 Conversely, if c e (hP’P)*, let u e hp’p and f"be its analytic completion. Set up(z) = u(pz) and fp(z) = f(pz) for 0 < p < 1. Then I "u-uoup,p S ”f-fpllp’p and hence by Theorem 2.1, :flHW%%m=O Since m is continuous, ¢(U) = lim o(up)- pdl Let. ¢(zn) for n'z 0 ¢(21n|) for n < 0. If u(z) = z cnrlnleing, then it follows as in Theorem 3.14 that N - n inO o(up) = lim m( 2 Cn(pr)l J6 ) Nam -N .00 nn and hence m n @(u) = ti? _5 cnbnpI '- 54 Since ¢ is bounded, Ibnl.s.HmH Hz'“'np,p .s C(p)HmH|n|’(1/P). Inleing is harmonic in D. Let F(z) = 1+2 6 Hp’p l-z on So g(z) = z bnr -oo 1 and set U(z) = Re F(gz) where g = pe a, p < 1. Then as in Theorem 3.14. o(U) = 8(5) and |s(§)|.s nonnvup,p.s nonuFuH,,p so that g e H”. If B > 0, then F”N¢>=2glfimfifl<1 - Islrl. p/2 Hence g E Y and Hgpr/g S C(P)HCPH° p/2 Corollary 3.25. (hp,p)* and v are equivalent. Proof: The mapping w ~ g defined in Theorem 3.24 is con- tinuous, one—to-one, and onto. Its inverse is continuous by the Open Mapping Theorem. In the same manner Theorem 2.5 was proved, we may show: bp/2 Theorem 3.26. is equivalent to the closure of hp’p in (hpfip)**’ O < p S 1. Thus, bp/2 is the containing Banach space for hp’p. Note, that for p = 1, the spaces are identical. CHAPTER IV COMPOSITION OPERATORS Let 0 be a nonconstant analytic function mapping D into itself. If f is analytic in D, set C¢(f) = f o m where f o $(z) = f(¢(z)). C¢ defines a linear operator on Hp and Bp. It was shown by J. Ryff [15] that C¢ is a bounded operator on Hp, 0 < p < m. In [13], E. Nordgren studied the operators C¢ on H2 for 0 an inner function. Composition operators on Hp, 1.3 p S_w, were studied by H. J. Schwartz [16]. We intend to present a similar study for Bp. In section 1, it is shown that C¢ is a bounded linear operator on Bp. Upper and lower bounds on HC¢H are given, and a necessary and sufficient condition is determined in order that C¢ be an isometry. In section 2, the methods of H. J. Schwartz [16] are used to characterize those operators on Bp which are composition operators. This characterization is then used to determine which com- position operators are invertible. In section 3, conditions are given on 0 in order that C¢ be a bounded operator from Bp into Hq, 0 < q S.”- Finally in section 4, compact composition operators and their spectra are discussed. 1. Bounds on C¢. It has already been remarked that C¢ is a bounded linear operator on Hp. This was shown by J. Ryff [15] as part of the following theorem. 56 57 Theorem 4.1. (J. Ryff [15], p. 348). Let 0 < p < w. Let f be analytic in D and 0 be an analytic function mapping D into D. (i) If 0 maps Izl S.r into Izl S_R, then M (r’f 0 ¢) S_(BIIQIQIL)l/pM (R’f). P R-|¢>(O)| P (ii) If f e Hp, then f o 0 e Hp. The operator C¢, defined by C¢(g) = g o o for all g e Hp, is a bounded linear operator on Hp and 1+ 0 1/ no,” 5. (491—4) p. 1-Id>(0)l (iii) If 0(0) = 0 and for some r, 0 < r < l, Mp(r,f o 0) = Mp(r,f), then either $(z) = ez, |e| = l or f is constant. Proof: To prove (i), we will first assume 0 maps Izl S r into Izl < R. Let al"°"an be the zeros of f in Izl S R where each zero is counted according to its multiplicity. Let b (z) _ B(Z ‘ ak) k R27- z “k and n B(z) = H bk(Z)’ K=l It is easily seen that [B(z)] S l in Izl S R and IB(z)I = 1 if and only if Izl = R. If f has no zeros in Izl S R, set B(z) E 1. Note that (f/B)p is analytic in Izl S_R. If Izl S_r, then P . P f:];Z:) = 3;. 2W f Belt R2:|¢(Z)I2 dt [ ] 2F f {—L—_Ifl] IRelt-¢(Z)l2 B(¢(Z)) 0 B(Re ) and 2W . 2F 19 4 l 1‘ If(¢ rel lde <-%— lf(¢(rei )) p d0 ( ) 2w I6 ( )) —-2v IO B(¢(re 0)) Since IB(¢(reig))I 3.1 Thus, 2w . (u 2) .5; I; Ir<¢>lpde 2W 2WI [P R 2'I¢(ZlI2 dt d9 —' WC fifO IDIf IRel f-(D( z)I2 (since B(Relt) = 1). If the order of integration is changed, we may use the well-known property of the Poisson kernel that 2w R2 2 2 1L) R-|<1>(Z )12 do: R;J¢(O)l 2W |%W-©l stem? |/\ mmmf It follows that 2w . iplmew%@sfl‘ghmwrwh p( To prove (i) in the case 0 maps Izl SDr into Izl S.R: let R =R+]"R n , n > 2. So, 0 maps Izl < r into |z| < Rn and n _. ._ hence 59 Rn+l¢(0)l l/p Mp(r,f) _<_ (Rn-|¢(O)T M p(Rn,f). Letting n a w, / Munog<§$g§prwno. To prove (ii), let f e Hp and assume ¢ maps Izl S_r into Izl S_R. Then by (i) l/p (R+ O ) Mp(r,f o ¢) S. R—|¢(o)| Mp(R,f). Letting R a 1, 1+ 0 1/ Mp( ’f ¢) S'(1-Il) HfHHp Thus, f 0 ¢ 6 Hp and / Hf o ¢qu< _. (Eii91gli->l puf W” This proves (ii). To prove (iii), we note that $(O) = 0 implies by Schwarz's Lemma that ¢ maps Izl S_R into Izl S R. Thus, we may let r a R in (4.1) obtaining 1 2w (4.3) lg; I If the equality holds in (4.3), then either ¢|¢(Re Ir<¢>Ip 1£)|B nf(¢(R:: M))|p 2W19» i9>I o E R _ i9 . or B = 1. If, however, |¢(Re )l a R, then the equality case of the Schwarz Lemma implies ¢ = 62, Isl = 1. Assuming equality holds in (4.3), if ¢ is not of the form ez, then B E l and I¢(Relg)| < R, hence (4.2) becomes ' . $2.1 l'_L_ , . 2v . ‘ 2_ i9 2 (u.4> >p =-l— £ (Rel ))| g F—- [f(Re )| . . dt. cf f0 IRelt-¢(Relg)|2 Integrating both sides of (4.5) with respect to G, we get, as in the proof of (i), (4.6) Mp(R,f 0 ¢) S_Mp(R,f). If equality holds in (4.6), it follows that (4.5) must have been an equality. But then (h.4) and (4.5) imply 2w . 2 i9 2 elt p R -i¢(Re g] §%-I2"IfIp R2"¢(Reig)' dt- it i9 2 o |Re -¢(Re )l However, if in general g = u + iv is continuous and If gl = I lgl, then g = Bu, Isl = 1. So f must be of this form on Izl = R and hence f is constant. It has been shown by E. Nordgren [13] that if ¢ is an inner function, then HC¢H = (111EKQJiol/p. l-|¢(O)| On the other hand, H. Schwartz [16] has shown that there are functions ¢ mapping D into itself for which l+l¢(0)l)l/p. C H (1)” < (1-l l/2 l-|¢(O)| HC¢H.S ‘ [2(i+ $E8%:)]l/p for p.S l/2. \ _ Proof: A general form of Schwarz's Lemma gives the inequality (4.7) |Q£El:¢(0)' < Izl- |1-¢(O)¢(Z)l — It may be verified by elementary methods that if a and b are any two complex numbers such that Ial < l, Ibl < 1, then IaI-lbl a+b l-lallbl - l1+7abl This inequality applied to (4.7) yields |¢(Z)l-l¢(0)[ < (2|. 1—l¢(0)||¢(2)| _' ’ Let l+|¢(0)lr — Then |¢(relg)l S_x(r), x(r) is an increasing function of r, and A(r) « l as r a 1. Set R =-l(l + x(r)). Then 2 62 %(1 + |¢(O)|) S.R < l and ¢ maps Izl S r into Izl < R. Furthermore, R+I¢o>L< 1+|<1>(0ll R-I¢l>-Il _ 2 1+|@(o)| 1-|<1>(O)l and dr 2(1—l¢(O)I2> < 2 1+! <0 dR (l-I¢)(O)|Mr>7g — 1-|<1>(O)| (4.8) (l_r)(l/p)-2 S,(1+:$EO§: )(1/p)-2(1_R)(1/p)-2. l- o (l-I‘)(1/p)—2 < [2Il+ (D O ](O)| It follows from Theorem 4.1 (i) that M (r,f o ¢) < R+ O i M (R,f). 1 " R—|¢(O)| 1 Thus, (1(1—r)(l/p) 2Ml(r,f 0 @)dr 3 K(p,¢) (1 (1-R)(l/p)‘2Ml(R,f)dR 0 1+ 0 2 .3 K(p,¢)HfH 63 where 1-1¢: 11(LL'ELEL')WP) p >5- K ‘ l-WO)! Therefore, f 0 ® 6 Bp and Hf o (13' < K(p,¢‘)HfH ~ If p > 1- and §®(O)I is not too large, we may improve 2 the bound on HC¢H by a slight alteration to the above proof. The following is then a corollary to the proof of Theorem 4.2. Corollary 4.3. If p >-; and |$(O)l < 2(2-(l/p)) _ 1, then 2 \, I " 1 i. l/ HCoH 3.21/p<1+1¢<01 (. ”O p. HMO)! Proof: In the proof of Theorem 4.2, replace (4.8) by < <1+I¢l>t2 liiQLQLL <1-R>1(l/p)'2. " 1-l<1>(0>l In the next theorem, a lower bound for HC¢H is given in terms of $(O). Theorem 4.4. If C¢ is a composition operator on Bp, then 1 . ,. 1-ic1i2 5'“ ©H° 611 Proof: If f(z) is in Bp, then f(0) = gF-£)f(re and hence IrIneOan _<_ xxanp l where eO(z) E 1. Let g(z) = (E:$%6§2)2. Now g E Bp, so g 0 ¢ 6 Bp and lleoll p B . = e O I (l_l¢(o)l2)g H OlprIa<¢< >> 5 Hg 0 <14pr _<_ HgHBpHC¢H : Heolpr\\%HH1HC¢ll- The conclusion follows since Han = 1 . H1 1-l<1>(0)|72 Corollary 4.5. HC¢H = 1 if and only if o(o) = 0. Proof: If $(O) = 0, then Schwarz's Lemma implies ¢:|Zl 3. I‘d 'Zl S r. Thus, by Theorem 11-]- (i): M1(r:f 0 ¢) S.M1(r:f) 65 and hence 1r . 113,.s anBp p for each f e B . Thus (C | < 1. But C e = \e 5 1 $1 _. 9 H ¢ OHBp 1 oHBp’ where e0 5 1, so HC¢H = 1. Conversely, if HC¢H = 1, then Theorem 4.4 implies $(O) = O. The next theorem characterizes those composition operators which are isometries on Bp. Theorem 4.6. C¢ is an isometry if and only if ¢ is a rotation. (i.e., $(z) = €z, 161 = 1). Proof: If ¢ is a rotation, then M (r,f 0 ¢) = Ml(r,f) for l f 6 Ep and hence nrqu = nf . 11B, = nc¢1Bp- On the other hand, if C¢ is an isometry, then HC¢H :1 so that ¢(O) = 0. Let f 6 Pp, f not constant. If M (r,f 0 ¢) < Ml(r,f) for each r, O < r < 1, then 1 Hf o ¢H p < Hf” p which implies C¢ is not an isometry. B B Therefore, M r,f 0 ¢) = M r,f) for some r. But then Theorem 4.1 (iii) implies ¢ is a rotation. Results similar to the theorems given in this section have been proved for Hp, 1.3 p < w by H. J. Schwartz [l6]. ( l )l/p 2’ l-I¢(O)l He showed that C¢ is an isometry on Hp if and only if ¢ is The lower bound he obtained for HC¢H was 66 an inner function vanishing at zero, which is quite different from our case. 2. Characterization. The question of when a bounded operator is a composition operator can be answered in terms of its multiplicative properties. n _ Lemma 4.7. Let en(z) = z . Then HenHBp a C(p)n Y where l . ._ lfp +— p +1 ‘1 ifp=1aiis k a positive integer. Proof: Let n+a = (a+l)(a+2) ... (a+n) = a+n) < a > n, < n and set 8 = l-- 2. It follows from Stirling's formula that P (n;a> e rna (see [7], p. xv). Now, Henan = 1: <1-r>8rndr. 1 If p = Eil , HenHBp may be computed using integration by parts an appropriate number of times. This gives 67 k—l I . -1. HenHBp = (n+l)(n12) 1.. (n+k) H—E [(B+n)] If p + Eél" then repeated integration by parts yields: 11en11Bp=(—~1}1’-1—+T)(B_+]1:1T2) “n?” Thus, HenHBp = CW"Y l l where C(p) = ffgiij' = r(£r1) P We will use this lemma in proving our next theorem. Theorem 4.8. If A is a bounded linear operator other than zero on BP, 0 < p < 1, then A is a composition operator if and only if = (A(el))n for n = 031:2:--°: Proof: If A is a composition operator, then A = C¢ for n ' n some ¢. Hence, A(en) = C¢(en) = ¢ = (A(el)) . Conversely, suppose A(en) = (A(el))n for each n. Let ¢ = Ae then 1’ H1 1Bp_ .fl(1.-r)(1/p)‘2 _. M1(P:¢n)dr p _>. m<1-p)(1/P)‘1(-I1;- 11'1, contradicting n¢nn p 4 0. Hence ¢ E H”, ||¢>||H0° g l and B ¢ * 1. Therefore, ¢:D a D. Now, C¢(en) = ¢n = (A(el))n = A(en). Thus, C¢ and A are continuous linear operators and they agree on the polynomials (since the polynomials are linear combinations of the en's). But the polynomials are dense in Bp, hence C¢ = A. This theorem was proved for Hp, 1.5 p < w, by H. J. Schwartz ([16], p. 8). His proof is also valid for Hp, O < p < l. Schwartz then showed that the theorem could be restated in terms of the property "almost multiplicative". Definition 4.1. An operator A on a function space S is almost multiplicative if whenever f, g, fg 6 S then A(fs) = . The following corollary was proved for Hp, 1.3 p < w by H. J. Schwartz ([16], p. 10). His proof works also for Bp. Corollary 4.9. If A is a bounded linear operator other than zero on BP, 0 < p < 1, then A is a composition operator if 69 and only if A is almost multiplicative. Proof: If A = C¢ and f, g, fg 6 DP, then A(fs) = C¢(fs) fs 0 ¢ = (f o ¢)(s ° ¢) and (f . ¢)(:Z)|)‘111Lq/p. ' .. I LII— . I ! ~ . ‘ VV V'IALI- _; 71 where HC¢H denotes the norm of C¢ as an operator from Bp into Hq. 3399:: If f 6 BP, then by Theorem 1.9, lf(z)| s.c

nrqu<1-Izl>'1/P. Thus, If<¢>l.s C(p)nru p<1-I¢'1/P B so that f 0 ¢ 6 Hq and 1r . ¢an.s C(p)urquu<1-I¢I>‘11Lq/p- Corollary4.13. If (l-lcbl)‘l e L1(D), then C¢zBp a Hp and 10,1.3 c

n<1-I¢I)'1HL,. where HC¢H denotes the norm of C¢ as an operator from Bp into Hp. s Corollary 4.14. If (1-1(1)|)-1 e Lq/p(D), O < p < 1, q > p, , -1 then C¢.Hp 4 HQ and HC¢H S C(p)n(l-I¢I) “LP/q. Proof: If f e Hp, then Irl.s C(p)HrHHp<1-Izl>'1/P (see [2], p. 36). The proof follows in the same manner as the proof of Theorem 4.12. The next theorem gives a necessary condition for C¢ to be a bounded operator from Bp into Hq. "run 'o-I‘ - I-.. — —- _- I a 2:1 ' «.5. Am I-JJ . gsmfl7 I, 72 Theorem 4.15. If C¢ is a bounded operator from Bp into Hq, O < p < l, O < q < m, then [O(eit)| < 1 a.e. Proof: If C¢ is a bounded operator from Bp into Hq, then “C¢(en)HHq S-HC¢HHen”Bp' Thus, it follows from Lemma 4.7 that “C®(e )H ~ 0 as n a w. Suppose there exists a n Hq t set E c:[O,2w) of positive measure such that I¢(ei )I = 1 for t e E. Then, HC¢(en)HHq = n¢nufiq 2 . = 12-1.? fowl¢lnthluq 212151;1¢(eit)|nth}l/q {m(E)}1/q > 0 2w contradicting HC¢(en)H 4 O as n s m. If q.Z p, the condition I¢(eit)| < 1 a.e. is not sufficient. In the case q > p, choose 5 such that p < SIS q l+z nl/s and let O(z) = —§*u The function f(z) = (1-z) is in Hp and hence in Bp. However, f(¢(z)) = 2l/S’(l-z)"1/S is not in Hq. For the case q = p, again choose ¢(z) = lgi . The function fp 8 defined by (2.3) is in Bp for p < 3 < 1, ) but f(¢(z)) t Hp. Theorem 4.16. C¢ is a bounded operator from Bp into H0° if and only if “O“ m < 1. H 73 Proof: If C¢ is a bounded operator from Bp into Hm, then as in Theorem 4.15, HC¢(en)H m a O as n a m. H But, n n uwepua=wu.=>l.s c

ufu p<1-I¢l>'1/P B .: C(p)nanptl-111H.>'1/P. Hence C¢(f) 6 Hm and 110.111 5. C(p><1-H¢nH,.)‘1/P. 4. Compact operators. A bounded linear operator A on a Banach space X‘is said to be compact if the image under A of every bounded sequence has a convergent subsequence. The following theorem was proved for Hp, 1.3 p < w by H. J. Schwartz. His proof is also valid for Bp and is given below. Theorem 4.17. C¢ is a compact operator on Bp if and only if for every bounded sequence {fn} in Bp such that fn 4 f uniformly on compact subsets of D, HO f —C f” 4 O as n-O°°. 74 Proof: Assume C¢ is compact. Let [fn} be a sequence in BP such that uan p < K for each n and fn « f uniformly 'B ._ on compact subsets of D. Suppose there exists a subsequence {fn ] such that k (4.9) Hc¢fn - C¢fH p 2_a > o k B for each R. Since an ”BPS'K and C® is compact, there exists k 1 k1 then C¢1fnk ) a g uniformly on compact subsets of D. It a subsequence (fn } such that C¢(fn ) a g in BD norm. But k. follows fr6m our hypothesis that C¢1fn ) a C¢(f) uniformly k i . on compact subsets of D. Therefore g = C¢(f)-and C®(fn ) s C¢(f) in Bp norm contradicting (4.9). k. 1 Conversely, let {fn} be any bounded sequence in Bp. It follows from Theorem 1.9 (i) that {fn} is a normal family and hence there exists a subsequence [fn } such that k fn a f uniformly on compact subsets of D. By our hypothesis, k C f a C f in Bp norm, and hence C is compact. 1 nk 1 ¢ 1 Theorem 4.18. If (1-|¢(z)l)- E L1(D), then C¢ is a compact operator on Bp. Proof: Let {fn} be a bounded sequence in Bp which converges uniformly on compact subsets of D to a function f. Then - / lfn(¢(z))| < C(p)nfnan<1-I1l) 1 p s.K

<1-I¢l>'1/P 75 for each n. Letting n ~ w, lf(¢(z))| s K

<1-I¢l>‘1/P. Hence, (4.10) Irn<¢-r<¢)lp s_c

(1-I1‘1n1nqu and —l (HenHBp) e C(p)nY hence Y $11 3 II 11 11 p 5115 Il ’ As a simple example, consider the function O(z) = pZ, O < pig 1. If p < 1, then C¢ is compact by Corollary 4.19. If p = 1, then C¢ is not compact since HOnHBp = HenHBp does not satisfy (4.12). We next turn our attention to finding the spectrum c(C¢) of a compact composition operator C¢. This problem 77 was investigated for compact operators C¢ on Hp by H. J. Schwartz [16]. He relied upon the following theorems by M. Koenig (see [16], p. 72). Theorem 4.21. (M. Koenig). If ¢:D a D is analytic, O(O) = O,¢'(O) + 0 then there exists a function K(z) analytic in D such that K(¢(z)) = ($‘(O))K(z). Theorem 4.22. (M. Koenig). Let ¢:D a D and O(O) = 0. Then there exists a non—zero analytic function f, satisfying f(¢(z)) = xf(z) if and only if 1 = l or 1 = (¢'(O))n. We remark that both of these theorems may be stated for any fixed point zO of O. Furthermore, if ¢:D « D is analytic, then O can have at most one fixed point ([16], p. 74). Theorem 4.23. (H. J. Schwartz [16], p. 77). If C¢ is an operator on Hp, 1 < p < w, O(zo) = Z0 and (l-IdDI)’l e LS(D) s = max[p,p'], (l +i = 1), then P p! okc¢) = 1110)“);l u 111. We use Theorem 4.23 to prove the following: Theorem 4.24. If O(ZO) = 20, (l-IdDI)‘l e L2(D), and C¢ is an operator on Bp, then m oBp = 1(1)“1n=1 u 111. 33223: If (l—|<1)I)-l E L2(D), then C¢ is compact by Theorem 4.18. Hence c(C¢) p consists entirely of eigenvalues and by Theorem 4.22, O(C¢):p c [($‘(ZO))n} U {1]. On the 78 other hand, C is also a compact operator on H2 see [16], p. 26 . <1) By Theorem 4.23, the spectrum of C¢ as an operator on H2 is given by °1C¢)a2 = 1<¢'(zo))n1 u 111. If x 6 o(C¢) 2, then x is an eigenvalue and hence there exists H f 6 H2 such that f(¢(z)) = xf(z). But f 6 H2 implies f e Bp and hence x 6 o(C¢)Bp. Corollary 4.25. If |¢(z)1 g.r < 1 and C¢ is a composition operator on Bp, then the spectrum of C¢ is given by 00 o(c¢> = {(¢'(ZO))n} u 11} n=l for some Z0 6 D. Proof: Ozlzl S r a Izl S r. Hence O has a fixed point zO by the Brouwer fixed point theorem. The following theorem can be proved in the same manner as Theorem 4.24. Theorem 4.26. If 0 < p < 2, O(ZO) = zO, (1-1(1>l)'l 6 L2(D), and C¢ is an operator on Hp, then 00 U(C¢) = {¢'(ZO)} U {1}- n=l '. 1‘ ‘ ._ smart-‘5! .1 BIBLIOGRAPHY 1 {'1 Mug-.1; , IO. 11. BIBLIOGRAPHY Benedick, A. and Panzone, R., "The space Lp with mixed norm", Duke Math. J. 28 (1961) pp. 301-324. Duren, P. L., Theory of Hp Spaces, Academic Press, New York and London, 1970. Duren, P. L., Romberg, B. W., and Shields, A. L., "Linear functionals on Hp with 0 < p < l", J. Reine Angew. Math. 238 (1969) pp. 32-60. Duren, P. L. and Shields, A. L., "Properties of Hp (0 < p < 1) and its containing Banach space", Trans. Amer. Math. Soc. 141 (1969), pp. 255-262. Duren, P. L. and Shields, A. L., "Coefficient multipliers of Hp and Bp spaces”, Pacific J. Math. 32 (1970), pp. 69—78. Dunford, N. and Schwartz, J. T., Linear Operators, Part I, Interscience, New York, (1964). Hardy, G. H., Divergent Series, Oxford University Press, London, (1967). Hardy, G. H., "The mean value of the modulus of an analytic function", Proc. London Math. Soc. 14 (1914). pp. 269-277. Hardy, G. H. and Littlewood, J. E., "Some properties of conjugate functions", J. Reine Angew. Math. 167 (1932), pp. 405-423. Hardy, G. H. and Littlewood, J. E., "Some properties of fractional integrals II", Math. Z. 34 (1932), pp. 403-439. Hoffman, K., Banach Spaces of Analytic Functions, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, (1962). 80 _ ,1 481*;ng ~ l2. l3. l4. l5. l6. l7. 18. 19. 20. 81 Littlewood, J. E., Lectures on the Theory of Functions, Oxford University Press, London, (1944). Nordgren, E., "Composition Operators", Canadian J. Math. 20 (1968), pp. 442-449. Romberg, B. W., "The dual space of Hp’ Preliminary report", Notices American Mathematical Society 9 (1962), p. 210; Abstract no. 62T-1l2. Ryff, J. "Subordinate Hp functions", Duke Mathematical Journal 33 (1966), pp. 347-354. Schwartz, H. J., ”Composition Operators on Hp", Ph.D. thesis, University of Toledo (1970). Shapiro, J. H., ”Linear Functionals on Non-locally Convex Spaces", Ph.D. thesis, University of Michigan (1968). Walters, S. S., ”The space Hp with 0 < p < 1", Proc. Amer Math. Soc. 1 (1950), pp. 800-815. Walters, S. 8., ”Remarks on the space Hp”, Pacific J. Math. 1 (1951), pp. 455-471 Zygmund, A., Trigonometric Series, Second Edition, Vol. I, II, Cambridge University Press, Cambridge, Massachusetts, (1968). . 5 M13. I __—-_.< ___.-.. _ __ _ . rJaw n...” _ ’.~. A .. .