Special Toeplitz operators on strongly pseudoconvex domains

Abstract

Toeplitz operators on strongly pseudoconvex domains in C n , constructed from the Bergman projection and with symbol equal to a positive power of the distance to the boundary, are considered. The mapping properties of these operators on L p , as the power of the distance varies, are established.

Generalization of Schur’s Test and its Application to a Class of Integral Operators on the Unit Ball of $${\mathbb C^n}$$

“…The purpose of this note is to give a generalization of Schur's test for the boundedness of T from L p (X, dμ) to L q (X, dν) for 1 ≤ p ≤ q < ∞, and present an application of this test to an integral operator between different weighted L p spaces on the unit ball B n of C n . We would like to point out here that a similar generalization of Schur's test was given by Proposition 2.8 in [1] for the case of integral operators on strongly pseudoconvex domains in C n (note that the proof there actually requires 1 < p < s < ∞), but our result here is more general and more precise. Note that we also include the case p = 1.…”

“…The proof is a simple application of Hölder's inequality and Minkowski's inequality which basically follows the same way as in [1]. If f ∈ L p (X, dμ), then for almost every x ∈ X, we have…”

On Distance Estimates and Atomic Decompositions in Spaces of Analytic Functions on Strictly Pseudoconvex Domains

“…Since, by Proposition 2, we have f 1 1,p,δ ≤ C f 1 1,1,δ it suffices, in proving estimate (6), to assume p = 1. Set L = L ǫ,δ (f ).…”

“…We follow the method used to prove Theorem 2.7 from [1], which is closely related to arguments from [6] and we refer the reader to these two papers for more details. First, for every z 0 ∈ ∂D there is a neighborhood U of z 0 and a complex coordinate system φ = (φ 1 , .…”

“…Then, using results from [13], P. Ahern and R. Schneider obtained estimate contained in Lemma 2, which is of vital importance for our work. For further results in this directions see [2], [4] and recent papers [1], [6].…”