A Growth Behavior of Szegö Type Operators

We define new integral 
operators on the Haydy space similar to Szego projection. We show that these operators map 
from Hp to H2 for some 1 ≤ p 2, where the range of p is depending 
on a growth condition. To prove that, we generalize the Hausdorff-Young Theorem 
to multi-dimensional case.


Introduction
Let C n denote the Euclidean space of complex dimension n. The inner product on C n is given by ⟨z, w⟩ := z 1 w 1 + · · · + z n w n where z = (z 1 , . . . , z n ) and w = (w 1 , . . . , w n ), and the associated norm is |z| := √ ⟨z, z⟩. The unit ball in C n is the set and its boundary is the unit sphere S n := {z ∈ C n : |z| = 1}.
In case n = 1, denote D in place of B 1 . Let σ n be the normalized surface measure on S n .
For 0 < p < ∞, the Hardy space H p (B n ) is the space of all holomorphic function f on B n for which the "norm" it is known that f has a radial limit f * almost everywhere on S n . Here, the radial limit f * of f is defined by provided that the limit exists for ζ ∈ S n . Moreover the mapping Since H 2 (B n ) can be identified with a closed subspace of L 2 (S n , dσ n ), there exists an orthogonal projection from L 2 (S n , dσ n ) onto H 2 (B n ). By using a reproducing kernel function, which is called the Szegö kernel, we also obtain a function f from its radial function f * . More precisely, . We usually call this integral operator as the Szegö projection. It is well known that for 1 < p < ∞ the Szegö projection maps L p (S n , dσ n ) boundedly onto H p (B n ). For more details, we refer the classical text books [1,2].
In this paper we consider a class of integral operators defined by for m = 1, 2, . . . , n and a positive integer N . Compared with the Szegö projection, the growth condition in the denominator factor is better. Thus these operators are bounded on H 2 (B n ). Interestingly these operators map from H 1 (B n ) to H 2 (B n ) for any positive integer N when 1 ≤ m < n 2 . More precisely we have the following result.
For n 2 ≤ m < n, the operator T m,N maps from H p (B n ) to H 1 (B n ), but the range of p is depending on m, which determines the growth condition of the kernel function. Explicitly we have the following theorem.
For z ∈ C n , the monomial is defined as At first, we show that the Szegö type operators T m,N defined in (1.1) are actually coefficient multipliers.
Since the monomials are orthogonal on L 2 (S n , dσ n ); see ( Expanding the term inside the above integral as we obtain that To prove the main theorems, we need the Hausdorff-Young Theorem for the multi-dimensional Hardy space. For a holomorphic function f in the unit disk, we have the Taylor series expansion as For the Hardy space defined in the unit disk, a relationship between the functions in H p (D) and the growth condition of their coefficients are given by the Hausdorff-Young Theorem, see ( [3] p.76, Theorem A). H p (D)). For 1 ≤ p ≤ ∞, let q be the conjugate exponent, with 1 p + 1 q = 1.

Theorem 2.2 (Hausdorff-Young Theorem for
Before proceeding, we introduce some notation. Let N n 0 be the product set of nonnegative integers.
Define a weight function w n on N n 0 by Using the weight w n , we define a norm on N n 0 by Let l p,t be the collection of all function c defined on N n 0 with the norm ∥c∥ p,t < ∞.
For a holomorphic function f on B n , whose Taylor series is given by H p (B n )). For 1 ≤ p < ∞, let q be the conjugate exponent, with 1 p + 1 q = 1. Proof. For a multi-index α, we note that ∫ Sn |ζ α | 2 dσ n (ζ) = w n (α). (2.2) From the orthogonality of monomials on S n , we get ∫

Proposition 2.3 (Hausdorff-Young Theorem for
Thus we obtain which prove the Proposition (1). Consequently ∥f − f k ∥ H 2 goes to zero as k increase. Hence f k converges to f pointwise and by applying Fatou's lemma we finish the proof.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.