Abstract
We give some sufficient and necessary conditions for an analytic functionfon the unit ballBwith Hadamard gaps, that is, forf(z)=∑k=1∞Pnk(z)(the homogeneous polynomial expansion off) satisfyingnk+1/nk≥λ>1for allk∈ℕ, to belong to the spaceℬpα(B)={f|sup0<r<1(1−r2)α\|ℛfr\|p<∞,f∈H(B)},p=1,2,∞as well as to the corresponding little space. A remark on analytic functions with Hadamard gaps on mixed norm space on the unit disk is also given.
Highlights
Let B = {z ∈ Cn : |z| < 1} be the open unit ball of Cn, ∂B = {z ∈ Cn : |z| = 1} its boundary, D the unit disk in C, dv the normalized Lebesgue measure of B (i.e., v(B) = 1), and dσ the normalized rotation invariant Lebesgue measure of S satisfying σ(∂B) = 1
We denote the class of all holomorphic functions on the unit ball by H(B)
The following statements are equivalent: (a) f Xp = ( B |R f (z)|p(1 − |z|2)p−1dv(z))1/p < ∞; (b). This result motivates us to find some characterizations for certain function spaces of analytic functions on the unit ball, in terms of the sequence ( Pnk p)k∈N
Summary
Let B = {z ∈ Cn : |z| < 1} be the open unit ball of Cn, ∂B = {z ∈ Cn : |z| = 1} its boundary, D the unit disk in C, dv the normalized Lebesgue measure of B (i.e., v(B) = 1), and dσ the normalized rotation invariant Lebesgue measure of S satisfying σ(∂B) = 1. We denote the class of all holomorphic functions on the unit ball by H(B). For f ∈ H(B) with the Taylor expansion f (z) = |β|≥0 aβzβ, let R f (z) = |β|≥0 |β|aβzβ be the radial derivative of f , where β = Βn) is a multi-index and zβ = z1β1 · · · znβn.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
