Carleson Type Measures for Fock–Sobolev Spaces

Abstract

Abstract. We describe the (p, q) Fock-Carleson measures for weighted Fock-Sobolev spaces in terms of the objects (s, t)-Berezin transforms, averaging functions, and averaging sequences on the complex space C n . The main results show that while these objects may have growth not faster than polynomials to induce the (p, q) measures for q ≥ p, they should be of L p/(p−q) integrable against a weight of polynomial growth for q < p. As an application, we characterize the bounded and compact weighted composition operators on the Fock-Sobolev spaces in terms of certain Berezin type integral transforms on C n . We also obtained estimation results for the norms and essential norms of the operators in terms of the integral transforms. The results obtained unify and extend a number of other results in the area.Mathematics Subject Classification (2010). 31B05, 39A12,31C20.

On the Spectrum of Volterra-Type Integral Operators on Fock–Sobolev Spaces

“…holds, and we call it a vanishing (p, q) Fock-Carleson measure if lim n→∞ C |f n (z)| q e − q 2 |z| 2 dµ(z) = 0 for every uniformly bounded sequence f n in F p ψm that converges to zero uniformly on compact subset of C as n → ∞. These measures have been completely identified in [12].…”

On trace ideal weighted composition operators on weighted Fock spaces

“…Questions about boundedness, compactness, and other operator theoretic properties of uC ψ expressed in terms of function theoretic conditions on u and ψ have been studied by several authors in various function spaces. The boundedness and compactness properties of uC ψ when it acts between different Fock-Sobolev spaces were described in [3]. In [4], we studied the Schatten S p class membership of the operators on F 2 m and a large class of weighted Fock spaces, where the weights defining such spaces are radial, decay at least as fast as the classical Gaussian weight, and satisfy a certain mild smoothness condition whenever p ≥ 2.…”

“…It follows that by [3,Corollary 3.2], ψ is of the form ψ(z) = Az + B where A is an n × n matrix with A < 1 and B a vector in C n . From this and substitution, we notice sup n∈N C n |e n (ψ(z))| 2 (|ψ(z)| + 1) 2m e −|ψ(z)| 2 dV (z) < ∞.…”

“…Then for any non-negative integer m, the Fock-Sobolev spaces F 2 m of order m consist of entire functions f on C n for which f (2,m) = |β|≤m ∂ β f 2 < ∞, (1.1) where |β| = β 1 + β 2 + · · · + β n . These spaces were introduced by Cho and Zhu in [2] and studied further by other authors, for example in [1,3]. One of the main results in [2] provides a Fourier characterization of the spaces as:…”

On Toeplitz Operators Between Fock Spaces

“…For finite exponents p and q such measures were described in [4]. On the other hand, the (∞, q) Fock-Carleson measures were recently identified by the author in [7]. The description of such measure µ involves its t-Berezin transform defined by…”