Variation and oscillation for harmonic operators in the inverse Gaussian setting

TL;DR

This paper establishes $L^p$-inequalities for variation and oscillation of fractional derivatives of semigroups and Riesz transforms in the inverse Gaussian setting, extending results to Banach spaces with UMD-property.

Contribution

It introduces new $L^p$-inequalities for variational and oscillation operators in the inverse Gaussian context, including Banach-valued extensions with UMD spaces.

Findings

Proved $L^p$-boundedness of variation and oscillation operators

Extended inequalities to Banach spaces with UMD-property

Analyzed weighted difference bounds for semigroups

Abstract

We prove variation and oscillation $L^p$-inequalities associated with fractional derivatives of certain semigroups of operators and with the family of truncations of Riesz transforms in the inverse Gaussian setting. We also study these variational $L^p$-inequalities in a Banach-valued context by considering Banach spaces with the UMD-property and whose martingale cotype is fewer than the variational exponent. We establish $L^p$-boundedness properties for weighted difference involving the semigroups under consideration.