BOUNDEDNESS OF VOLTERRA OPERATORS ON SPACES OF ENTIRE FUNCTIONS

Transcription

1 Annales Academiæ Scientiaum Fennicæ Mathematica Volumen 43, 218, BOUNDEDNESS OF VOLTERRA OPERATORS ON SPACES OF ENTIRE FUNCTIONS Osca Blasco Univesidad de Valencia, Deatamento de Análisis Matemático 461 Bujassot, Valencia, Sain; Abstact. In this ae we find some necessay and sufficient conditions on an entie function z g fo the Voltea oeato V g (f(z = f(ξg (ξdξ to be bounded between diffeent weighted saces of entie functions Hv (C o Foc-tye saces F (C. 1. Intoduction Let Ω be the unit disc D o the comlex lane C and, as usual, denote by H(Ω the sace of holomohic functions in Ω. Given g H(Ω the Voltea oeato with symbol g, to be denoted by V g, is defined by V g (f(z = ˆ z f(ξg (ξdξ, z Ω, f H(Ω. In the case Ω = D, this oeato was fist intoduced by Pommeene [2]. He showed that it is bounded on the Hady sace H 2 (D if and only if g BMOA. A bit late the esult was extended to H (D fo any 1 < by Aleman and Sisais [1, 4]. In aticula, they showed that, fo 1 <, (1 V g (f H C g BMOA f H, f H (D, fo a constant C > deending only on. The boundedness, comactness and othe oeties of V g acting on saces of holomohic functions defined in the unit disc have been deely studied (see [5] fo weighted Begman saces, [6, 15] fo weighted saces of holomohic functions Hv (D and [17, 19] fo seveal othe saces. The eade is also efeed to [2, 3] fo diffeent esults concening the secta of the Voltea oeato in some cases. In this aticle we ae only concened with saces of entie functions. Thoughout the ae we wite P fo the sace of olynomials (with the notationu n (z = z n and H (C fo the sace of entie functions vanishing at the oigin. Fo each < <, < < and f H(C we wite M (f, = su z = f(z and M (f, = ( 2π 1/. f(e it dt 2π Given < < and a measuable function : (, R, we denote by F (C the sace of entie functions f such that C f(z e ( z dm(z < and we wite ( f F = (2π 1/ M (f,e ( d 1/. htts://doi.og/1.5186/aasfm Mathematics Subject Classification: Pimay 47G1; Seconday 3H2, 47B7, 47B37, 47B38, 3D15, 3D2, 46E15. Key wods: Voltea oeato, weighted saces, entie function, Foc-tye saces. The autho is atially suoted by the Poject MTM P(MINECO Sain.

2 9 Osca Blasco The classical Foc saces F (C coesond to (z = z 2 2. Fo the limiting case F (C we shall also use the standad notation H v (C whee v(z = e ( z, that is the sace of entie functions f such that f F = f v = sue ( M (f, <. As usual Hv (C denotes the subsace of H v (C of functions such that lim z v( z f(z =. It is well nown that we can change the values of o v in a bounded inteval [,R ] and even that we can elace fo anothe weight ϕ being continuous and inceasing so that Hw (C = H v (C and Fϕ (C = F (C with euivalent noms. Since we ae only inteested in saces containing the olynomials, that is P Hv (C o P > F (C, we shall imose the following assumtions on the weights: (2 lim m v( =, m N, o (3 m e ( d <, m N, >. Due to the above consideations we intoduce the following definition. Definition 1.1. We wite W fo the class of functions : [, R which ae continuous, inceasing in [, fo some > and fo each m N satisfy (4 su m e ( <. > Notice that conditions (2, (3 and (4 ae in fact euivalent. Examles of weights in W to have in mind ae ϕ α,β,γ ( = β α γlog fo α,β > and γ. The study of the Voltea oeato on cetain saces of entie functions was initiated by Constantin in [11]. She chaacteized continuity (and comactness of V g on the classical Foc saces. Theoem 1.1. [11, Theoem 1] Let <, < and g H (C. (i Case < : V g is bounded fom F (C into F (C if and only if g(z = az 2 +bz fo some a,b C. (ii Case < : V g is bounded fom F (C into F (C if and only if 1 1 < 1 2 and g(z = az fo some a C. Late in collaboation with Peláez [12] the esults wee extended to a class of Foc-tye saces F(C defined by cetain smooth adial weights. In [12] cetain class I of twice diffeentiable and aidly inceasing weights was intoduced. This class includes examles such as ( = α fo α > 2, ( = e β fo β > o ( = e e. Fo weights in this class they obtained the comlete chaacteization of the symbols g which oduce bounded Voltea oeatos V g acting fom F(C into F (C (see [12, Theoem 3]. In aticula fo = they showed that fo g H(C and I, the Voltea oeato V g is bounded on F (C if and only if g (z (5 su z C 1+ ( z <. Also they genealized Theoem 1.1 as follows:

3 Boundedness of Voltea oeatos on saces of entie functions 91 Theoem 1.2. [12, Coollay 25] Let <, <, g H (C and ( = α with α > 2. (i Case < and 1+(α 2( : V g is bounded fom F (C into F (C if and only if g is a olynomial with deg(g 2+(α 2( (ii Case < : V g is bounded fom F (C into F (C if and only if 1 1 < α 1 2 and g is a olynomial with deg(g < α 2( The study fo F (C = H v (C was consideed by Bonet and Tasinen [9] fo cetain classes of adial weights v. We efe also the inteested eade to [8, 11, 13] fo esults concening the secta of the Voltea oeato in this setting. In [9] cetain class of weights J (see conditions aeaing in [9, Poosition 3.2] was intoduced. This class includes examles such as ( = β α γlog δlog(log(1 +, fo some α,β >,γ,δ R, ( = (log(1 + 1+ǫ γlog δlog(log(1 +, fo some ǫ >,γ,δ R o, moe geneally twice diffeentiable weights satisfying cetain conditions (see [9, Thm 3.6, Thm 3.7]. Fo such a class, using the notation ṽ(z fo the so-called associate weight of v (see [1], they obtained (see [9, Theoem 3.4] that fo g H(C, v(z = e ( z and w(z = e ( z with J, the boundedness of V g fom Hv (C into H w (C is euivalent to the condition g (z w(z (6 su z C ( z ṽ(z <. As a conseuence they established the following theoem. Theoem 1.3. [9, Coollay 3.11] Let v( = e βα fo β > and α 1 and let g H (C. Then V g is bounded on Hv (C if and only g is a olynomial of deg(g [α], whee [a] stands fo the intege at of a >. Obseve that V g = IM g whee M g (f = fg z and I(f(z = f(ξdξ. All the evious esults ae obtained analyzing the action of M g and I on the coesonding saces indeendently, and using the euivalent definition of the nom of f in the saces Hv (C and F(C in tems of the deivative f (see [9, Poosition 3.2] o Littlewood Paley fomula (see [12, Theoem 1] esectively. In this ae we would lie to attac the boundedness of the Voltea oeato V g (and cetain modification of it diectly and not elying on the boundedness of the multilication o diffeentiation oeatos indeendently. Note that the esults in [12] do not aly to ( = α fo < α 2 and not cove diffeent weights and and the esults in [9] cove diffeent weights but only fo = =. We shall esent hee some necessay and sufficient conditions fo the boundedness of V g fom F (C into F (C fo diffeent aametes <, and diffeent weights and belonging to W, extending and oviding some altenative oofs of some esults in [9, 11, 12]. Besides the intoduction the ae is divided into fou sections. The fist section contains some esults on the class W while the second one is devoted to some eliminaies on the Voltea oeato V g and its modification Ṽg(f(z = 1 z f(ξdg(ξdξ z whee Dg(z = g(z+zg (z. The main contibutions ae in the last sections whee some necessay and sufficient conditions fo the boundedness ofv g andṽg on weighted saces of holomohic functions and Foc-tye saces and thei alications ae ovided. It will be shown (see Coollay 4.6 that the existence of a function g such that V g is bounded F(C into F (C imlies that V u is also bounded F(C into F (C fo all N such that g ( (. This foces some elationshi between,

4 92 Osca Blasco, and. In aticula we will show that thee is no entie function g H (C such that V g mas boundedly Hv 1 (C into Hv 2 (C fo v i = e ϕ α i,β i,γ i fo i = 1,2 wheneve α 1 > α 2 o α 1 = α 2 and β 1 > β 2 o α 1 = α 2, β 1 = β 2 and α 1 γ 2 +γ 1 < 1 (this actually exlains the estiction α 1 in Theoem 1.3. Moeove once such a function exists it must be a olynomial of degee less o eual than α 1 γ 2 +γ 1. In ode to ovide some sufficient conditions fo the boundedness of V g fo diffeent weights we shall intoduce a function insied by the so-called distotion function of consideed in [12]. Fo each < < and weight the authos consideed the se function, ( = (s ds,, which was cucial to descibe the nom of f (1+e ( in F(C in tems of the deivative f. We shall intoduce fo each ai (, of weights and < < the function { e ( ( 1 e (s ds 1/, < 1; H,, ( = e ((+( 1( ( 1 e (s,ds 1, 1 < < which will lay an imotant ole in finding sufficient conditions on the boudedness of V g. Namely we shall establish in Theoem 5.7 below that, fo < <,, W and g H(C, the existence of a constant A > such that (7 M (Dg, AH,, (, >. imlies thatṽg is bounded fomf (C intof (C. As a conseuence one genealizes, at least fo =, the esults in [12] to a much wide class of weights. 2. Peliminaies on weights We stat by mentioning some classical families of weights. Fo each ε,α,β > and γ R the conside the weights ρ ε and ϕ α,β,γ given by ρ ε ( = ( log(1+ 1+ε and e ϕα,β,γ( = min{(1+ γ, γ }e βα that is ϕ α,β,γ ( = β α γlog(1+ fo γ < and ϕ α,β,γ ( = β α γlog fo γ. It is easy to see that ρ ε and ϕ α,β,γ belong to W. The examles ϕ α,β,γ can be obtained fom a single one ( = using the following modifications: (8 (9 (1 β ( = (β, β >, (α ( = ( α, α >, e (γ( = min{(1+ γ, γ }e (, γ R. It is elementay to see that if belongs to W then β, (α and (γ also belong to W. Definition 2.1. Let < < and such that e (s ds < fo >. We define, fo >, (11 Φ ( = 1 ( ˆ 1 log e (s ds, o, euivalently e Φ( = 1 e (s ds. Lemma 2.1. Let < < and W. Then (i Φ W,

5 Boundedness of Voltea oeatos on saces of entie functions 93 (ii if C 1 (, and convex, then (12 sue ( Φ( <, (iii if ( = ϕ α,β,γ fo some α,β > and γ R, then (13 su α e ( Φ ( <. > Poof. (i Clealy e Φ( is deceasing and Φ ( is inceasing. Now fo each m N with m > 1 we have that m 1 m e Φ( = e (s ds s m 1 e (s ds <. This shows that Φ W. (ii Note that ( (1 = A fo 1. Hence fo 1 e Φ( = 1 e (t dt 1 A (te (t dt = 1 A e (. Since su 1 e ( Φ( < this gives (12. (iii We claim that fo any a R thee exists C a > so that (14 t a e t dt C a a e, >. Of couse the esult holds tue fo a with C a = 1. The case a N follows by induction and integation by ats. Now fo a > wite a = λ + (1 λ 1 with λ 1 and, 1 N {}, aly Hölde s ineuality and the evious case to get (14. To show (13 we conside the cases γ and γ < seaately. Case γ : Fom (14 we have e Φ( = ( 1 t γ e βtα dt 1/ = C C γ α e β α = C α e (. Case γ < : Aguing as above, ( ˆ 1 1/ ( (1+ e Φ( = (1+t γ γ e dt βtα C The oof is comlete. C(1+ γ α/ e βα α e (. ( ˆ 1 1/ s γ+1 α 1 e s dt β α β α s 1 α 1 e s dt Let us now conside a subclass of diffeentiable weights wide enough to include most of the classical weights. Definition 2.2. Let us denotew the collection of continuous functions: [, R such that C 1 ([, fo some and (15 lim ( =. Note that the classical examles ϕ α,β,γ and ρ ε belong to W fo any ǫ,α,β > and γ R. Lemma 2.2. W W. 1/

6 94 Osca Blasco Poof. Let W. Then ( > in some inteval (R, and fo each ( mlog m N, L Hosital s ule gives lim =. In aticula lim mlog (( mlog =. Hence (2 holds and then W. Poosition 2.3. Let < < and let be diffeentiable with ( > fo >. Then W Φ W lim e ( Φ( =. Poof. Diffeentiating in the fomula e Φ( = 1 e (s ds one has that Φ ( = e(φ( ( +1. Now use L Hosital s ule to obtain e lim Φ ( = lim ( +1 = lim e (s ds (. Thus both euivalences ae shown. Let us give a notation to the seuence of the noms of u in the sace F weight W and <. Definition 2.3. Let <, W and N {}. We define fo any (16 ( C (, = +1 e ( d 1/ = (2π 1/ u F, (17 C (, = su e ( = u F. << Next esult is immediate and left to the eade. Examle 2.1. Let α,β, >, γ and = ϕ α,β,γ. Then (18 C (, = (αβ +γ +γ α ( +γ α e +γ α and (19 C (2 (21 (22 +2+γ (β α (, = Γ α ( +2+γ Rema 2.1. Fo <, 1, 2 <, 1, 2, N {} and, W we have C (+, min{c 1 (,C 2 (,,C 2 (,C 1 (, }, 1 C (, 3 C (, 1 C (, 2, = 1 + 1, α. C (, 2 C (, 1 1/ 2 C (, 1 1/ 2, 1 < 2. ( Lemma 2.4. Let W and ( <. Then the seuences (C 1 (,C (, 1/ and C +1 (,/C (, ae inceasing with lim C +1 (, C (, = lim C 1/ (, =. Poof. Case = : Since e (/ e (/(+1 fo all > and N then obviously (C (, 1/ is inceasing. Let us show that ( C +1(, C (, is also inceasing. Since = 1 ( 1+ 1 ( +1, we have that 2 2 C (, = su ( 1 2 e ( ( e ( 2 C 1 (, 1/2 C +1 (, 1/2 >

7 Boundedness of Voltea oeatos on saces of entie functions 95 and then C +1 (, /C (, is inceasing. Finally, using now that C (, 1/ C +1 (, 1/(+1 we have C +1(, C C (, +1 (, 1/(+1 C. Hence lim +1 (, C = (, lim C (, 1/ =. Case < < : Alying Cauchy Schwaz we have ( 2 ( ( ++1 e d ( +2+1 e ( d +1 e ( d. This shows that C +1 (, 2 C +2 (,C (,. Thus C +1 (,/C (, is inceasing. Now conside the measue dµ ( = C (, e ( d defined in R +. Of couse, µ (R + = 1 and(c (, 1 C (, 1/ = u 1 L (R +,dµ whee u 1 ( =. This gives (C (, 1 C 1 (, 1/ 1 (C (, 1 C 2 (, 1/ 2 wheneve 1 2. In aticula (C (, 1 C (, 1/ is inceasing. Now taing into account that C +1 (, C (, (C 1 (,C +1(, 1/(+1 = u 1 L (+1 (µ C we conclude that lim +1 C is comlete. lim u 1 L (+1 (µ = u 1 L (µ =. The oof Lemma 2.5. Let W and <. Then (23 C (, C 1 (, C 2 (, 1 2 1, 1 2. In aticula C 2 (, C 2(,C (, fo all N. Poof. Let us denote M = C (, and C = C (, fo < <. We stat with the case =. Fo each, 1, 2 N such that 1 = θ θ 2, we obviously have M 1/ M θ/ 1 1 M (1 θ/ 2 2. Hence fo each 1 2, choosing θ = one 2 1 obtains M M M Fo < <, aguing as in the evious lemma we can wite fo 1 2 and 1 = θ θ 2 that u 1 L (R +,dµ u 1 θ L 1(R +,dµ u 1 1 θ L 2(R +,dµ. Now (23 follows since θ = and 1 θ = Finally selecting 1 = and 2 = 2 one gets M 2 M 2M and C 2 C 2C. Rema 2.2. The conditions aeaing in Lemmas 2.4 and 2.5 ae closely elated to the ones aeaing when defining the Denjoy Caleman classes (see fo instance [16]. 3. Peliminaies on the Voltea oeato Given g H(C we denote by M g, D and I the multilication, diffeentiation and integation oeatos esectively, i.e. fo f H(C we have M g (f(z = g(zf(z, Df(z = f (z, If(z = ˆ z f(ξdξ. Of couse I(H(C = H (C, Id H(C = DI and Id H (C = ID whee Id X stands fo the identity oeato acting on X. We denote by S and S 1 the shift and bacwads

8 96 Osca Blasco shift oeatos defined by S 1 f(z = f(z f( z = a n+1 u n, Sf(z = zf(z = n= a n 1 u n, fo eachf = n= a nu n H(C. Using the notationp m (f fo the Taylo olynomial of degee m and R m f = f P m 1 (f fo the emainde of degee m we have S m f(z = z m f(z = a m z, S m f(z = a +m z = R mf(z z. m =m This gives that S m S m f = R m f and S m S m f = f fo m N. Since P F (C we have that f F (C if and only if R mf F (C fo any m N, < and W. Note that S m f F = f (m fo each < F whee (m was defined by e (m( = m e (. = Lemma 3.1. Let m N, 1 and W. Then fo f = = a u F (C. S m f F (m (m+1 f F Poof. Fo each N {}, > and 1 we have a M 1 (f, M (f,. Thus (24 a C (, (2π 1/ f F, N {}. Theefoe P m 1 (f F m f F, R m (f F (m+1 f F and This finishes the oof. S m (f F (m = R m (f F (m+1 f F. As mentioned in the intoduction the Voltea oeato with symbol g is defined by the fomula (25 V g (f(z = IM Dg (z = z ˆ 1 n=1 f(tzg (tzdt, z C, fo each f H(C. Note that V g = fo any constant function g and that also V g (f H (C fo any f H(C. We shall conside the following modification to avoid these estictions. Fo each f, g H(C we wite (26 Ṽ g (f(z = 1 z ˆ z f(ξdg(ξdξ, z C, whee D = DS, that is Df(z = n= (n+1a nz n = zf (z+f(z. Denoting I = S 1 I, we have fo f = n= a nu n that a n If(z = (n+1 zn = 1 ˆ z f(ξdξ z n= and we obtain that Ṽg = IM Dg. In this way Ṽg is well defined fo g H(C and taes values in H(C. Moeove, fo each f,g H(C (27 Ṽ g (f = S 1 V Sg (f, V g (f = SṼS 1 g(f.

9 Boundedness of Voltea oeatos on saces of entie functions 97 Since V g is continuous (in the toology of the unifom convegence on comact sets fom H(C into H (C and the ma given by g V g is linea and continuous fom H (C into the sace of continuous linea oeatos, using (27 simila esults hold fo Ṽg. Next esult is immediate fom the definitions. Lemma 3.2. Let <,,, W and g H(C. Then V g is bounded fom F(C into F (C if and only if ṼS 1 g is bounded fom F(C into F (1 (C. Othe exessions fo the oeatos above ae given as follows: Lemma 3.3. Let f,g H(C with f = m= b mu m and g = n= a nu n. Then ( 1 (28 V g (f(z = na n b m z j, j j=1 n+m=j ( 1 (29 Ṽ g (f(z = (n+1a n b m z j. j +1 j= n+m=j Poof. The oof is staightfowad using (ˆ 1 V g (f(z = na n z n f(zss n 1 ds = = n=1 j=1 ( 1 na n b m z j. j n+m=j The othe fomula follows fom (27. ( na n z n b m n+m zm Rema 3.1. Fom (28 and (29 we obtain fo any f,g H(C and N, n=1 m= V g (u = g g(, Ṽ g (u = g, V u (f =, Ṽ u (f = If, (3 V g (u = u and (31 V u (f = u n= n=1 na n n+ u n, Ṽ g (u = u n= b n n+ u n, Ṽ u (f = ( +1u (n+1a n n+ +1 u n, n= b n n+ +1 u n. Let us efomulate the boundedness of Ṽg acting on F 2(C. Note that fo each f = m= b mu m we can wite (32 f F 2 = 1 2π ( m= b m 2 C 2 m(,2 1/2. Poosition 3.4. Let, W and g H(C with g = n= a nu n. Then Ṽg mas F 2 (C into F 2 (C if and only if the matix A = (a(m,j m,j= given by { j m+1 j+1 a(m,j = a j m C j(,2, m j; C m(,2, j < m. defines a bounded oeato on l 2 (N {}.

10 98 Osca Blasco Poof. Using (32 and (29 we obtain ( Ṽg(f F = 1 j 1 su (j m+1a j m b m C j (,2γ j 2 2π (γ j 2 =1 j +1 j= m= ( = 1 j m+1 C j (,2 su a j m 2π j +1 C m (,2 γ j C m (,2b m. Hence This gives the esult. (γ j 2 =1 m= Ṽg = su (γ j 2 =1 j=m 2 a(m,jγ j m= The analysis of V g fo g P actually deends only on the integation oeato. Let us denote by V and Ṽ the oeatos V u and Ṽu fo N {}. Hence fom (31 we obtain (33 V =, V 1 = I, V = IS 1, N, and (34 Ṽ = I, Ṽ = ( +1IS, N. In aticula V = SṼ 1 fo N. A simle conseuence of Poosition 3.4 gives the following aticula case. Coollay 3.5. Let N {} and, W. Then Ṽ mas F 2 (C into F 2 (C if and only if su m j= C m+ (,2 (m+ +1C m (,2 <. The following efomulations ae elementay and left to the eade. Lemma 3.6. Let N,, W and <,. The following statements ae euivalent: (i V : F (C F (C is bounded. (ii Ṽ 1: F (C F (1 (C is bounded. (iii I: F ( 1 (C F (C is bounded. (iv I: F ( 1 (C F (1 (C is bounded. 4. On necessay conditions fo the boundedness Taing into account that V g (u = g g( the fist condition fo V g to ma F (C into F (C is that g F (C. In aticula we have the following tivial necessay condition. Poosition 4.1. Let <,,, W and g H (C. If V g : F (C F (C is bounded, then thee exists a constant A > such that (35 M (g, AK, (, >, whee (36 K, (z = 1/2 C (, 1 z, z C. =.

11 Boundedness of Voltea oeatos on saces of entie functions 99 Poof. Using (24 fo V g (u = g(z = n=1 b nz n we obtain This shows (35. M (g, a n n (2π 1/ C n (, 1 V g (u F n n= n= (2π 1/ 1/ V g C (,( C n (, 1 n. Let us find a necessay condition fo the boundedness of V g fom F (C into F (C in the case F (C F(C. Poosition 4.2. Let <,,, W such that F (C F (C and g H (C. If V g : F (C F (C is bounded then thee exists A > such that n= (37 M (g, A(, >. Poof. Let A = max{1, u F } and C = Id F (C F. We obseve that (C V g (u = g F (C. Hence g F(C and g F C g F C V g A. Since V g (g = g2 2 we also obtain g 2 2 C g 2 F 2 C 2 V g 2 u F F (C V g A 2. This allows to iteate the ocedue to obtain gn F n! (C and gn n! F (C V g A n. Recall that F(C is a -Banach sace fo = min{,1}. Hence if n fn < F imlies that n f n F. Theefoe choosing K > C V g A we conclude that β ng n n= K n n! F (C fo any seuence of comlex numbes with su n β n 1. In aticula, choosing β n = 1 fo all n we obtain e g/k F (C. Theefoe C e (( z Rg(z K dm(z < and su z C e ( z +Rg(z K < in the cases < and = esectively. In both cases one gets R(g(z K( z +C. Selecting β n as ( 1 n,i n and ( i n one concludes that g(z A( z fo some constant A > and the oof is comlete. A simle conseuence of Poosition 4.2 is the following coollay. Coollay 4.3. Let <, ( = ϕ α,β,γ fo some α,β >, γ R and g H (C. (i Case < α < 1: V g : F (C F (C is bounded if and only if g =. (ii Case α 1: If V g : F (C F (C is bounded then g P and 1 deg(g α. Let us now show that boundedness of V g o Ṽg between saces H v (C o F (C foces cetain a ioi conditions on the weights. Poosition 4.4. Let <,,, W, g(z = n= a nz n H(C and define Λ = {n: a n }. If Ṽg: F (C F (C is bounded and Λ, then Ṽ : F (C F (C is also bounded and Ṽ Ṽg. In aticula, I: a F (C F (C is bounded wheneve g(.

12 1 Osca Blasco Poof. Let Λ. We have and theefoe ˆ z ( +1a w = ˆ 2π Ṽ f(z = 1 f(w(+1w dw = 1 1 z a z = 1 2π ( ˆ 1 z f(wdg(e a ˆ iθ wdw z Dg(e iθ we iθdθ 2π, w C, ˆ z (ˆ 2π f(w Dg(e iθ we iθdθ dw 2π e iθdθ 2π. Hence, maing the change of vaiable e iθ w = w and denoting f e iθ(z = f(e iθ z, we have Ṽ f(z = 1 2π Ṽ g (f a e iθ(e ˆ iθ ze iθdθ 2π. In aticula, Ṽf F 1 a ˆ 2π Ṽg(f e iθ(e iθ z F dθ 2π, 1, and Ṽf 1 ˆ 2π Ṽg(f F a e iθ(e iθ z dθ F 2π, < < 1. This gives, taing into account that f e iθ F = f F fo any adial weight, the estimate Ṽf F Ṽg a f F and the oof is comlete. Coollay 4.5. Let <,,, W and g H(C. IfṼg: F (C F (C is bounded then thee exists N and A > such that In aticula, C n+ (, A (n+1c n (,, n. IK, ( A S K, (, >, whee K, stands fo the enel given in (36. Poof. Since g thee exists Λ, that is a. Due to Poosition 4.4 and the fact Ṽ(u n = ( + 1I(u n+ = +1 u +1 n++1 n+ we have u n++1 n+ F Ṽ u n F. In aticula, fo all n N, u n+ F Ṽ (n+1 u n F. This shows that IK, ( = n= and the oof is comlete. n (n+1c n (, A n= n C n+ (, = A S K, (, Coollay 4.6. Let <,,, W and let g(z = n=1 a nz n H (C such that V g : F (C F (C is bounded. Then V : F (C F (C is also bounded fo each such that g ( (. Moeove, the estimate V! g ( ( V g holds. In aticula, I: F (C F (C is bounded wheneve g (.

13 Boundedness of Voltea oeatos on saces of entie functions 11 Poof. Recall that due to Lemma 3.2 we have that ṼS 1 g: F(C F (1 (C whee e (1( = e (. Theefoe invoing Poosition 4.4 and Lemma 3.6 we obtain that V : F (C F (C wheneve g( ( and the coesonding estimate in nom holds. Coollay 4.7. Let α i,β i > and γ i fo i = 1,2, v( = e ϕ α 1,β 1,γ ( 1 and w( = e ϕ α 2,β 2,γ ( 2 and g H (C. Assume that V g : Hv (C H w (C is bounded. Then eithe α 1 < α 2 o α 1 = α 2 and β 1 β 2 o α 1 = α 2, β 1 = β 2 and γ 2 γ 1 +α 1 1. Moeove, in the case α 1 = α 2, β 1 = β 2 and δ = α 1 γ 2 +γ 1 1, then g P with deg(g δ. Poof. Due to Coollay 4.6 we have thatv is bounded fomh v (C intoh w (C fo all N such that g ( (. Since V (u n = n+ u n+ we have C n+ (ϕ α2,β 2,γ 2, V n+ C n (ϕ α1,β 1,γ 1,, n N. Now tae into account Examle 2.1 to obtain fo all n N (α 2 β 2 +n+γ 2 α 2 (+n+γ 2 +n+γ 2 α 2 e +n+γ 2 α 2 Hence thee exists C > such that V n+ (α 1 β 1 n+γ 1 α 1 (n+γ 1 n+γ 1 α 1 e n+γ 1 α 1. n n( 1 α 2 1 α 1 C(α 2 β 2 e n α 2 (α 1 β 1 e n α 1 n 1 +γ 2 α 2 + γ 1 α 1, n N. This imlies that α 1 α 2. ( n β α 1 1 β 2 Cn 1 +γ 2 γ 1 α 1 fo all n N. In the case α 1 = α 2 the ineuality becomes This gives β 1 β 2. Finally in the case α 1 = α 2, β 1 = β 2 we would have n +γ 2 γ 1 α 1 1 C fo all n N. This imlies +γ 2 γ 1 α 1 1. This gives, in aticula, γ 2 γ 1 +α 1 1. To finish the oof notice that g ( ( imlies α 1 γ 2 +γ 1 which imlies that g P with deg(g α 1 γ 2 +γ On sufficient conditions fo the boundedness Let us stat esenting some sufficient conditions fo the oeatos V g and Ṽg to be bounded fom F (C into F (C fo any 1 and fo geneal weights. Poosition 5.1. Let, W and g H(C. Let us wite g (z = g(z fo > and set (38 A(, = sue ( ( g BMOA > and (39 B(, = sue ( ( (g H 1. > (i If A(, <, then both Ṽg and V g ae bounded fom F (C into F (C fo any 1 <. (ii If B(, <, then both Ṽg and V g ae bounded fom F (C into F (C.

14 12 Osca Blasco Poof. (i Let 1 < and set A(, = A. Since (g (w = g (w fo each > and w < 1, we have V g (f(w = Hence, using the estimate (1 we have ˆ w f (ξ(g (ξdξ. (4 M (V g (f, C g BMOA M (f,, >. Since Ṽg = S 1 V Sg, (Sg = S(g and Sg BMOA = g BMOA, we also have Theefoe, we conclude that M (Ṽg(f, = 1 M (V Sg (f, C g BMOA M (f,. max{ V g (f F, Ṽg(f F } 2πC 2πC A M (f, g BMOA e ( d M(f,e ( d = CA f. F (ii Let = and set B(, = B. Without loss of geneality we can assume that g H (C. Hence g(z = z 1 g (ztdt and thus M 1 (g, M 1 (g, = (g H 1. In aticula, (41 M 1 (Dg, M 1 (g,+m 1 (g, 2M 1 (g, = 2 (g H 1. Hady s ineuality (see [14] gives fo f(z = n= a nz n Theefoe, M (If, n= a n n n+1 C M 1 (f,, >. M (Ṽgf, C M 1 ((Dgf, C f F (C e( M 1 (Dg, 2BC f F (C e(. This gives the boundedness of Ṽg fom F (C into F (C. To handle the casev g we use that M 1 (D(S 1 g, = M 1 (g,. Aguing as above, we have M (V g f, = M (ṼS 1 gf, C f F (C e( (g H 1 and the esult follows with the same agument. Poosition 5.2. Let, W whee is diffeentiable with (t > fo t > and g H (C. Set (42 B 1 (, = su > e ( ( ( (g H. If B 1 (, <, then both Ṽg and V g ae bounded fom H v (C to H w (C, whee v(z = e ( z and w(z = e ( z.

15 Boundedness of Voltea oeatos on saces of entie functions 13 Poof. Let B 1 (, = B 1. Aguing as in (41 we obtain that M (Dg, 2 (g H. Now fo z = we can estimate Ṽg(f(z 1 ˆ 1 ˆ f(zt Dg(zt dt 2B 1 f v 1 ˆ 1 1 M (f,tm (Dg,tdt = 2 f v ˆ M (f,tm (Dg,tdt ˆ e (t (g t H ds t (te (t ds = 2B 1 f v (e ( e ( 2B 1 f v e ( z. This comletes the oof fo Ṽg. The case V g follows similaly using that M (D(S 1 g, = M (g,. Let us aly the evious esult to olynomials, in aticula fo V = V u. Coollay 5.3. Let v(z = e ϕ α,β,γ fo some β >, γ R and α 1 and let g H (C. Then the following statements ae euivalent: (i V g : H v (C H v (C is bounded. (ii g P and 1 deg(g []. Poof. (i = (ii This is the case = in Coollay 4.3. (ii = (i. It suffices to show that V is bounded on Hv (C fo 1 α. Now fo each 1 [α] we have ϕ α,β,γ lim ( { αβ, = α; = 1, < α. We can then aly Poosition 5.2 fo = = ϕ α,β,γ and g = u to finish the oof. Let us get now some conditions deending on fo the boundedness on F (C. We shall use the following esult. Lemma 5.4. Let < <, W. If f F Φ (C, then I(f F (C. Poof. Using that If(z = 1 Theefoe, I(f F C C f(ztdt, fo any < < we obtain ˆ 1 M (I(f, M (f,tdt 1 The oof is now comlete. (ˆ M e (f,tdt ( d C M (f,tte Φ(t dt = C f. F Φ ˆ Poosition 5.5. Let < <,, W and set M (f,tdt. (43 A 1 (,, = sue ( Ψ( M (Dg,. > M (f,t ( If A 1 (,, <, then Ṽg is bounded fom F (C into F (C. t e ( d dt

16 14 Osca Blasco Poof. Let A 1 (,, = A 1. Using Lemma 5.4 and ecalling that e Ψ( = e (s ds we have Ṽg(f F The oof is finished. = IM Dg f C M F Dg f F Ψ ( C M (f,m (Dg, CA 1 M (f,e ( d. e (s ds d We can actually weaen the condition (43 in the case > 1 using the following modification of the -distotion functions. and Definition 5.1. Let, W and < <. We define H,, ( = e Ψ( (, < 1, H,, ( = e ((+( 1(, 1 < <. e (s ds In aticula, H,, ( = e max{,1}(φ( (. Rema 5.1. Note that fo 1 we can wite (44 H,, ( = e Ψ( ( e ( 1(Ψ( ( = e (Ψ( ( e ( (. In aticula, due to (ii in Lemma 2.1 if is diffeentiable and convex, then e Ψ( ( CH,, (, > R, and fo W, fom Poosition 2.3, one has e ( ( CH,, (, > R. We shall use the following geneal fact. Lemma 5.6. Let 1 <, let U,W: (, (, be measuable functions with W L 1 ((, and let G: [, R + be a continuous function. Assume that thee exists C > such that ( ˆ 1 1 (45 G( C W(tdt U 1/ (W 1/ (, >. Then (46 ( ˆ 1 F(tG(tdt W(d C fo any continuous function F : [, R +. F (U(d Poof. Fo = 1 condition (45 becomes G(t( W(d CtU(t fo t > t and the esult follows fom Fubini s theoem. Assume > 1. Fo each R,ε > integating by ats we have ˆ R ( ˆ 1 F(tG(tdt W(d ε

17 Boundedness of Voltea oeatos on saces of entie functions 15 (ˆ ε = + ( ε + ˆ R ( F(tG(tdt ε (ˆ W(t R dt t 1 ( (ˆ 1 F(tG(tdt F(G( ( ˆ 1 ε W(tdt F(tG(tdt ε (ˆ 1 ( F(tG(tdt F(G( ( F(tG(tdt W(t dt d t 1 W(t dt d. t 1 R W(t dt t 1 Now assing to the limit as R and ε, alying (45 and Hölde s ineuality we have ( ˆ 1 F(tG(tdt W(d (ˆ 1 ( W(t F(tG(tdt F(G( dt d t 1 ( ˆ 1 1 ( F(tG(tdt F(G( W(tdt d ( ( ˆ 1 1/ C F(tG(tdt W(d ( F( G( W 1 ( ( C ( 1 ( ˆ 1 F(tG(tdt W(d This imlies (46 and the oof is then comlete. 1/ W(tdt d 1/ ( 1/ F (U(d. Theoem 5.7. Let < <,, W and g H(C. If thee exists A > such that (47 M (Dg, AH,, (, >, then Ṽg is bounded fom F (C into F (C. Poof. The case < 1 was shown in Poosition 5.5. Let us assume now that 1 < <. Witing Ṽg(f(z 1 = f(ztdg(ztdt we have fo < < and θ [,2π, Ṽg(f(e iθ ˆ 1 f(e iθ t M (Dg,tdt. Using vecto-valued Minowsi s ineuality we have (48 M (Ṽg(f, 1 ˆ M (f,tm (Dg,tdt. Let U( = e ( and W( = e ( and obseve that ( ˆ 1 1 H,, ( = W(tdt U 1/ (W 1/ (.

18 16 Osca Blasco Conside now F(t = M (f,t and G(t = M (Dg,t and notice that (47 togethe with (48 allow us to aly Lemma 5.6 to obtain This finishes the oof. M (Ṽg(f,e ( d C M (f,e ( d. We can now extend the condition in Poosition 5.2 also fo boundedness in Foc-tye saces, at least fo convex functions. Coollay 5.8. Let < < and let W be diffeentiable and convex in (,. If g H(C satisfies (49 su > e ( ( M (Dg, ( then Ṽg: F (C F (C is bounded. Poof. Fist obseve that (e Ψ( Hence assumtion (49 gives = A <, (se (s ds 1 e (, >. M (Dg, A e(1 (( Ψ( e Ψ( (. Hence, accoding to (44 we obtain the condition (47 in the case 1. On the othe hand, fo < 1 due to at (ii in Lemma 2.1 to now that su > e ( Ψ( <. Hence M (Dg, Ke Ψ( ( = KH,, (. The esult now follows fom Theoem 5.7. Coollay 5.9. Let < <, ( = ϕ α,β,γ ( fo β >, γ and α 1 and let g H (C. Then the following statements ae euivalent: (i V g : F (C F (C is bounded. (ii g P and 1 deg(g [α]. Poof. (i = (ii This was shown in Coollay 4.3. (ii = (i. Let 1 [α] and let us show that V : F (C F (C is bounded, o euivalently Ṽ 1: F (C F (1 (C is bounded. Fom Poosition 5.7 it suffices to see that (47 holds fo g(z = z 1. Recall that H 1,, ( = e (( Ψ( e ( ( fo 1 and H 1,, ( = e( Ψ( fo < < 1. Hence, in aticula fo = (1 = ϕ α,β,γ+1 we have ( ( = log(, we obtain, invoing (iii in Lemma 2.1, that (5 H 1,, ( C α+1, >. This gives The oof is now comlete. suh 1,, (M (Du 1, C su 1 1 α <. Acnowledgements. I wish to than J. Bonet fo his comments on the fist daft of the ae. I am also gateful to the efeee fo his/he caeful eading.

19 Boundedness of Voltea oeatos on saces of entie functions 17 Refeences [1] Aleman, A.: A class of integal oeatos on saces of analytic functions. - Toics in comlex analysis and oeato theoy 33, Univ. Malaga, Malaga, 27. [2] Aleman, A., and O. Constantin: Secta of integation oeatos on weighted Begman saces. - J. Anal. Math. 19, 29, [3] Aleman, A., and J.A. Peláez: Secta of integation oeatos and weighted suae functions. - Indiana Univ. Math. J. 61, 212, [4] Aleman, A., and A.G. Sisais: An integal oeato on H. - Comlex Va. Theoy Al. 28, 1995, [5] Aleman, A., and A. G. Sisais: Integation oeatos on Begman saces. - Indiana Univ. Math. J. 46, 1997, [6] Basallote, M., M. D. Conteas, C. Henández-Mancea, M. J. Matín, and P. J. Paúl: Voltea oeatos and semigous in weighted Banach saces of analytic functions. - Collect. Math. 65, 214, [7] Blasco, O., and A. Galbis: On Taylo coefficient of entie functions integable against exonential weights. - Math. Nach. 223, 21, [8] Bonet, J.: The sectum of Voltea oeatos on weighted saces of entie functions. - Quat. J. Math. 66, 215, [9] Bonet, J., and J. Tasinen: A note on Voltea oeatos on weighted Banach saces of entie functions. - Math. Nach. 288:11-12, 215, [1] Biestedt, K.D., J. Bonet, and J. Tasinen: Associated weights and saces of holomohic functions. - Studia Math. 127, 1998, [11] Constantin, O.: A Voltea-tye integation oeato on Foc saces. - Poc. Ame. Math. Soc. 14:12, 212, [12] Constantin, O., and J. A. Peláez: Integal oeatos, embedding theoems and a Littlewood Paley fomula on weighted Foc saces. - J. Geom. Anal. 26:2, 216, [13] Constantin, O., and A. M. Pesson: The sectum of Voltea-tye integation oeatos on genealized Foc saces. - Bull. London Math. Soc. 47:6, 215, [14] Duen, P.: Theoy of Hady saces. - Academic Pess, New Yo-London, 197. [15] Hu, Z.: Extended Cesao oeatos on mixed-nom saces. - Poc. Ame. Math. Soc. 131:7, 23, [16] Komatsu, H.: Ultadistibutions I, stuctue theoems and a chaacteization. - J. Fac. Sci. Toyo (IA 2, 1973, [17] Pau, J., and J. A. Peláez: Embedding theoems and integation oeatos on Begman saces with aidly deceasing weights. - J. Funct. Anal. 259, 21, [18] Pavlović, M. and J. A. Peláez: An euivalence fo weighted integals of an analytic function and its deivative. - Math. Nach. 281:11, 28, [19] Peláez, J. A., and J. Rättyä: Weighted Begman Saces induced by aidly deceasing weights. - Mem. Ame. Math. Soc. 227, 214. [2] Pommeene, Ch.: Schlichte Funtionen un analytische Functionen von beschänte mittlee Oszilation. - Comment. Math. Helv. 52, 1977, [21] Tung, J.: Foc saces. - PhD dissetation, Univesity of Michigan, 25. [22] Zhu, K.: Oeato theoy in function saces. Second Edition. - Math. Suveys Monog. 138, Ame. Math. Soc. Povidence, Rhode Island, 27. [23] Zhu, K.: Analysis on Foc saces. - Singe-Velag, New Yo, 212. Received 16 Febuay 217 Acceted 28 Ail 217