An Extension of a Boundedness Result for Singular Integral Operators

TL;DR

This paper extends boundedness results for certain singular integral operators related to Littlewood-Paley theory, in a setting involving a product of Brownian motion and stable processes, and generalizes a classical multiplier theorem.

Contribution

It introduces a boundedness analysis for G-star and Area functionals in a discontinuous process setup and weakens conditions for a classical multiplier theorem.

Findings

Operators are bounded on L^p in the new setup

Generalization of a classical multiplier theorem

Extension of boundedness results for singular integrals

Abstract

In this paper, we study some operators which are originated from classical Littlewood-Paley theory. We consider their modification with respect to our discontinuous setup, where the underlying process is the product of a one dimensional Brownian motion and a d-dimensional symmetric stable process. Two operators in focus are G-star and Area functionals. Using the results obtained in our previous paper, we show that these operators are bounded on L^p. Moreover, we generalise a classical multiplier theorem by weakening its conditions on the tail of the kernel of singular integrals.