On boundedness of Calder\'on-Toeplitz operators

TL;DR

This paper investigates conditions under which Calderón-Toeplitz operators, associated with specific wavelets related to Laguerre polynomials, are bounded, providing criteria, examples, and counterexamples for their boundedness on wavelet subspaces.

Contribution

It offers new sufficient conditions for the boundedness of Calderón-Toeplitz operators with unbounded symbols, expanding understanding of their behavior in wavelet analysis.

Findings

Provided criteria for boundedness of Calderón-Toeplitz operators

Presented examples and counterexamples illustrating boundedness conditions

Analyzed the behavior of iterated integrals of symbols

Abstract

We study the boundedness of Toeplitz-type operators defined in the context of the Calder\'on reproducing formula considering the specific wavelets whose Fourier transforms are related to Laguerre polynomials. Some sufficient conditions for simultaneous boundedness of these Calder\'on-Toeplitz operators on each wavelet subspace for unbounded symbols are given, where investigating the behavior of certain sequence of iterated integrals of symbols is helpful. A number of examples and counterexamples is given.