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

Abstract

The modern study of singular integral operators on curves in the plane began in the 1970s. 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. The $$w^*$$ -derivative $$\gamma '$$ of a curve $$\gamma :[a,b] \rightarrow X$$ at $$t \in [a,b]$$ satisfies $$\tfrac{\mathrm{d}}{\mathrm{d}s} \langle \gamma (s),y \rangle |_{s=t} = \langle \gamma '(t),y\rangle$$ for any $$y \in Y$$ . Suppose $$\Gamma = \gamma ([a,b])$$ is a curve in X whose $$w^*$$ -derivative is Hölder continuous and bounded away from 0. We prove that any convolution type singular integral operator associated with a 1-dimensional Calderón–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.