Singular integrals on $C_{w^*}^{1,\alpha}$ regular curves in Banach duals

TL;DR

This paper extends the theory of singular integral operators to curves in Banach dual spaces, proving boundedness results for convolution operators with Calderón-Zygmund kernels on such curves.

Contribution

It establishes $L^p$ boundedness of singular integrals on $C_{w^*}^{1,eta}$ regular curves in Banach duals, generalizing previous Euclidean and Carnot group results.

Findings

Proves $L^p$ boundedness of singular integrals on Banach dual space curves.

Extends David's good lambda theorem to doubling metric spaces with upper regular measures.

Demonstrates boundedness for convolution operators with Calderón-Zygmund kernels in this setting.

Abstract

The modern study of singular integral operators on curves in the plane began in the 1970's. Since then, there has been a vast array of work done on the boundedness of singular integral operators defined on lower dimensional sets in Euclidean spaces. In recent years, mathematicians have attempted to push these results into a more general metric setting particularly in the case of singular integral operators defined on curves and graphs in Carnot groups. Suppose $X = Y^*$ for a separable Banach space $Y$. Any separable metric space can be isometrically embedded in such a Banach space via the Kuratowski embedding. Suppose $\Gamma = \gamma([a,b])$ is a curve in $X$ whose $w^*$-derivative is H\"{o}lder continuous and bounded away from 0. We prove that any convolution type singular integral operator associated with a 1-dimensional Calder\'{o}n-Zygmund kernel which is uniformly $L^2$-bounded on lines is $L^p$-bounded along $\Gamma$. We also prove a version of David's ``good lambda'' theorem for upper regular measures on doubling metric spaces.