WEIGHTED INTEGRALS OF HOLOMORPHIC FUNCTIONS IN THE UNIT POLYDISC

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1 WEIGHTED INTEGRALS OF HOLOMORPHIC FUNCTIONS IN THE UNIT POLYDISC STEVO STEVIĆ Received 28 Setember 23 Let f be a measurable function defined on the unit olydisc U n in C n and let ω j z j, j = 1,...,n, be admissible weights on the unit disk U, with distortion functions ψ j z j, Un ={f f < }, where f q = [,1 n Mq f,r n ω j r j dr j, and Un = Un HU n. We rove the following result: if [1, andfor all j = 1,...,n, ψ j z j / z j z C = C,ω j,nsuchthat f,then f C f + n ψ j / z j and there is a ositive constant. 1. Introduction Let U 1 = U be the unit disk in the comlex lane, dmz = 1/πdr dθ the normalized Lebesgue measure on U, U n the unit olydisc in comlex vector sace C n and HU n the sace of all analytic functions on U n.forz,w C n we write z w = z 1 w 1,...,z n w n ; e iθ is an abbreviation for e iθ1,...,e iθn ; dt = dt 1 dt n ; dθ = dθ 1 dθ n and r, θ are vectors in C n.ifwewrite r<1, where r = r 1,...,r n it means r j < 1forj = 1,...,n. For f HU n and, we usually write M f,r = 2π n f 1/ r e iθ dθ, for r<1 1.1 [,2π] n for the integral means of f. Let ωs, s<1, be a weight function which is ositive and integrable on,1. We extend ω on U by setting ωz = ω z. We may assume that our weights are normalized so that ωsds = 1. Let = Un denotes the class of all measurable functions defined on U n such that f = f z n ω j zj dm zj <, 1.2 U n where ω j z j, j = 1,...,n, are admissible weights on the unit disk U. Coyright 25 Hindawi Publishing Cororation Journal of Inequalities and Alications 25: DOI: /JIA

2 584 Weighted integrals of holomorhic functions The weighted Bergman sace is the intersection of and HUn. For ω j z j = 1 z j 2 αj, α j > 1, j = 1,...,n, we obtain the classical Bergman sace dv α, see [1, age 33]. Let = Un, >, denotes the class of all measurable functions defined on U n such that f q = M q f,r [,1 n n ω j rj drj <, 1.3 and be the intersection of and HUn. When = q we denote by. In the case = q, these two norms are equivalent on the sace HU n, but the later one is more suitable for calculations than the first one. The result is contained in the following lemma. Lemma 1.1. The norms and are equivalent on the sace HU n. Proof. By the olar coordinates it is easy to see that f 2 n f for every f HU n, moreover f 2 n f for every f measurable on U n. Now we rove that there is a ositive constant C,whichisindeendentof f,suchthat f C f, 1.4 for every f HU n. Without loss of generality we may assume that n = 2. Let f HU n, then f = g r 1,r 2 d dr 2, 1.5 where gr 1,r 2 = M f,r 1,r 2 ω 1 r 1 ω 2 r 2. Now we estimate these four integrals, which we denote by I i, i = 1,2,3,4. Since f HU 2 then the function f is analytic in each variable searately on U and consequently M f,r 1,r 2 is nondecreasing function in r 1 and r 2,see,forexamle,[3]. Let / C ωi = ω i ri ω i ri, i = 1, Note that C ωi, i = 1,2, are well defined and finite numbers since ω i are ositive integrable functions on,1.

3 Stevo Stević 585 Using the above mentioned facts and definitions, we have Similarly I 1 M f,, = C ω1 C M f,, C ω1 C ω 1 d dr 2 ω 1 d dr 2 M f,,r 2 ω1 d dr 2 4C ω1 C M f,,r 2 ω1 r 2 dr 1 dr 2, I 2 M f,, ω 1 d ω 2 d = C ω1 C ω1 4C ω1 M f,, ω1 d dr 2 M f,,r 2 ω1 d dr 2 M f,,r 2 ω1 r 2 dr 1 dr I 3 4C M f,,r 2 ω1 r 2 dr 1 dr Finally, it is clear that I 4 4 M f,,r 2 ω1 r 2 dr 1 dr From , weobtain f C ω1 +1 C +1 f 1.11, as desired. Following [8], for a given weight ω we define the function ψr = ψ ω r def = 1 ωudu, r<1, 1.12 ωr r and we call it the distortion function of ω.weutψz = ψ z forz U. Definition 1.2 [8]. We say that a weight ω is admissible if it satisfies the following conditions: i there is a ositive constant A = Aωsuchthat ωr A ωudu, for r<1; r r

4 586 Weighted integrals of holomorhic functions ii ω is differentiable and there is a ositive constant B = Bωsuchthat ω r B ωr, for r<1; r iii for each sufficiently small ositive δ there is a ositive constant C = Cδ,ω such that su r<1 ωr ω C r + δψr Observe that i imlies Aψr 1 r thus for sufficiently small ositive δ we have r + δψr < 1 and the quantity in the denominator of the fraction in iii is well defined. For a list of examles of admissible weights, see [8, ages ]. The following theorem was roved in [8]. Theorem 1.3. Suose 1 < and ω is an admissible weight with distortion function ψ. Then U f z ωzdmz f + U f z ψz ωzdmz, 1.16 for all analytic functions f on the unit disc U,wheredmz = rdrdθ/π denotes the normalized Lebesgue area measure on U. The above means that there are finite ositive constants C and C indeendent of f such that the left- and right-hand sides L f andr f satisfy CR f L f C R f 1.17 for all analytic f. Some generalizations of Theorem 1.3 in many directions can be found in [1, 11]. In [5, 6, 9] was also investigated Bergman sace of analytic functions with weights other than classical. Closely related results for the classical weight ωr = 1 r α, α> 1, are resented in [1, 2, 3, 4, 7, 13]. Using a Bergman tye rojection B α : dv α dv α, in [1]theauthorsroved the following theorem. Theorem 1.4. Let [1,, α j > 1, j = 1,...,n and m be a fixed ositive integer and let k = k 1,...,k n Z + n.let f be a holomorhic function defined on the olydisc U n in C n. Then for α = α 1,...,α n, f dv α if and only if [ n 1 z j 2 ] k j k f z dv α z1 k1... zn kn k =m. 1.18

5 Moreover, f dv α m 1 k = k f 1 zn kn + [ n 1 z j 2 ] k j z k1 k =m z k1 Stevo Stević 587 m f 1 zn kn dv α In [11] among other things we roved the following theorem which generalizes Theorem 1.4. Theorem 1.5. Let k = k 1,...,k n Z + n, f be a holomorhic function defined on the olydisc U n in C n and ω j z j, j = 1,...,n are admissible weights on the unit disk U, with distortion functions ψ j z j.if f and >, then [ n ] ψ k f kj j zj z1 k1 zn kn z. 1.2 Moreover, let m be a fixed ositive integer. Then there is a ositive constant C = C,ω j,m,n such that f C m 1 k = k f 1 zn kn + [ n ] ψ m f kj j zj z1 k1 zn kn z k1 k =m In the same aer, we formulate and give a sketch of a roof of the following artially converse of Theorem 1.5. Theorem 1.6. Let f HU n and ω j z j, j = 1,...,n are admissible weights on the unit disk U, with distortion functions ψ j z j.if [1, and for all j = 1,...,n, ψ j z j / z j z, then f and there is a ositive constant C = C,ω j,n such that f C f + n ψ j z j In communication with other secialists in this field it has turned out that the roof is more comlicated than we have exected. Hence in this note we will resent a clear detailed roof of Theorem 1.6. Also, we rove the following generalization of Theorem 1.6. Theorem 1.7. Let f HU n and ω j z j, j = 1,...,n are admissible weights on the unit disk U, with distortion functions ψ j z j.if [1, and for all j = 1,...,n, ψ j z j / z j z, then f and there is a ositive constant C = C,ω j,n such that f C f + n ψ j z j. 1.23

6 588 Weighted integrals of holomorhic functions 2. An auxiliary results In this section, we rove an auxiliary result which we use in the roof of the main result. Lemma 2.1. Suose 1 < and f HU n. Then almost everywhere. d n dt M f,tr M 1 f,tr r i M,tr, 2.1 z i Proof. For f the result is obvious. If f, at oints where f is not zero, we have i=1 d f tr e iθ = f tr e iθ 1 d dt dt f tr e iθ f tr e iθ 1 d f tr e iθ dt = f tr e iθ 1 f tr e iθ,r e iθ f tr e iθ 1 n r i tr e iθ z. i i=1 2.2 From 2.2 and by the dominated convergence theorem we obtain d dr M f,tr 2π n n r i f tr e iθ 1 tr e iθ [,2π] n z dθ. 2.3 i i=1 If = 1 the assertion is clear. If >1, alying in the last integral Hölder s inequality with exonents / 1 and we obtain the result. Corollary 2.2. Suose [1, and f HU n. Then almost everywhere. d n dt Mq f,tr qm q 1 f,tr r i M,tr, 2.4 z i Proof. Comuting d/dtm q f,tr and then using Lemma 2.1 we rove the corollary. 3. Proofs of the theorem In this section, we rove the main results in this aer. Proof of Theorem 1.6. Without loss of generality, we may assume that n = 2, and f, =. Also we assume that f is not constant and all integrals are finite. In order to avoid some i=1

7 comlicated notations we use M f,r 1 t,r 2 tinsteadofm r 1 t,r 2 t. We have f = = + d dt M t,r 2 t M 1 t,r 2 t dt ω 1 d dr 2 Stevo Stević 589 M,r 1 t,r 2 t r i dt ω 1 d dr 2 z i M 1 t,r 2 M,r 1 t,r 2 r 1 dt ω 1 d dr 2 M 1,r 2 t M,r 1,r 2 t r 2 dt ω 1 d dr 2 M 1 s, M,s,r 2 ds ω 1 d dr 2 M 1,τ M,r 1,τ dτ ω 1 d dr 2 M 1 1 s, M,s,r 2 ω 1 d ω 2 dsd s M 1,τ 1 M,r 1,τ ω 2 d ω 1 dτ d τ M 1 s, M,s,r 2 ψ 1 sω 1 sω 2 dsd M 1,τ M,r 1,τ If >1, by Hölder inequality, we get ψ 2 τω 2 τω 1 dτ d. 3.1 M 1 s, M M,s,r 2 = f 1,s,r 2 ψ 1 sω 1 sω 2 dsd M s, ω1 sω 2 dsd 1/, ψ 1 sω 1 sω 2 dsd 1/ M,s,r 2 ψ 1/ 1 sω 1 sω 2 dsd. 3.2 Similarly, M 1,τ M f 1,r 1,τ M ψ 2 τω 2 τω 1 dτ d,r 1,τ ψ 1/ 2 τω 2 τω 1 dτ d. 3.3

8 59 Weighted integrals of holomorhic functions From we obtain the result in this case. For = 1 the result it follows from 3.3. If f is constant the result is clear. To remove the restriction of the finiteness of the integrals we consider holomorhic functions f ρ z = f ρz, ρ,1 and use the Monotone Convergence theorem, when ρ 1. Oen question 3.1. Does Theorem 1.6 hold in the case <<1? Remark 3.2. In the case when ω j z j, j = 1,...,n, are the classical weights 1 z j αj, j = 1,...,n, a ositive answer to Oen question 3.1 was given in [12]. Proof of Theorem 1.7. If f, =, is not constant and all integrals are finite, then by Corollary 2.2,as in the roof of Theorem 1.6,we obtain f q f q 1 + f q 1 M q M q,s,r 2 ψ q1 1/q sω 1 sω 2 dsd,r 1,τ ψ q 1/q 2 τω 2 τω 1 dτ d, 3.4 that is, f 2 ψ j z j. 3.5 The rest of the roof is similar to the roof of Theorem 1.6. References [1] G. Benke and D.-C. Chang, A note on weighted Bergman saces and the Cesáro oerator,nagoya Math. J , [2] D.-C. Chang and B. Q. Li, Sobolev and Lischitz estimates for weighted Bergman rojections, Nagoya Math. J , [3] P. L. Duren, Theory of H saces, Pure and Alied Mathematics, vol. 38, Academic Press, New York, 197. [4] T. M. Flett, The dual of an inequality of Hardy and Littlewood and some related inequalities, J. Math. Anal. Al , [5] T.L.KrieteIIIandB.D.MacCluer,Comosition oerators on large weighted Bergman saces, Indiana Univ. Math. J , no. 3, [6] P.LinandR.Rochberg,Hankel oerators on the weighted Bergman saces with exonential tye weights, Integral Equations Oerator Theory , no. 4, [7] J. H. Shi, Inequalities for the integral means of holomorhic functions and their derivatives in the unit ball of C n,trans.amer.math.soc , no. 2, [8] A. Siskakis, Weighted integrals of analytic functions, ActaSci. Math.Szeged 66 2, no. 3-4, [9] S. Stević, Comosition oerators on the generalized Bergman sace, J. Indian Math. Soc. N.S , no. 1-4, [1], Weighted integrals of harmonic functions, Studia Sci. Math. Hungar , no. 1-2, [11], Weighted integrals of holomorhic functions in C n, Comlex Var. Theory Al , no. 9,

9 Stevo Stević 591 [12], Weighted integrals of holomorhic functions on the olydisc, Z. Anal. Anwendungen 23 24, no. 3, [13] K. J. Wirths and J. Xiao, An image-area inequality for some lanar holomorhic mas, Results Math. 38 2, no. 1-2, Stevo Stević: Mathematical Institute of Serbian Academy of Science, Knez Mihailova 35/I, 11 Beograd, Serbia addresses: