Partial Derivatives, Singular Integrals and Sobolev Spaces in Dyadic Settings (Analysis in Theory and Applications, 2023)
In this note we show that the general theory of vector valued singular integral operators of Calderόn-Zygmund defined on general metric measure spaces, can be applied to obtain Sobolev type regularity properties for solutions of the dyadic fractional Laplacian. In doing so, we define partial derivatives in terms of Haar multipliersand dyadic homogeneous singular integral operators.