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

The modern study of singular integral operators on curves in the plane began 1970s. Since then, there has been a vast array work done boundedness defined lower dimensional sets Euclidean spaces. In recent years, mathematicians have attempted to push these results into more general metric setting particularly case and graphs Carnot groups. Suppose $$X = Y^*$$ for separable Banach space Y. Any can be isometrically embedded such via Kuratowski embedding. $$w^*$$ -derivative $$\gamma '$$ curve :[a,b] \rightarrow X$$ at $$t \in [a,b]$$ satisfies $$\tfrac{\mathrm{d}}{\mathrm{d}s} \langle \gamma (s),y \rangle |_{s=t} '(t),y\rangle$$ any $$y Y$$ . $$\Gamma ([a,b])$$ is X whose Hölder continuous bounded away from 0. We prove that convolution type operator associated with 1-dimensional Calderón–Zygmund kernel which uniformly $$L^2$$ -bounded lines $$L^p$$ along $$\Gamma$$ also version David’s “good lambda” theorem upper regular measures doubling