Criteria of pointwise and uniform directional Lipschitz regularities on tensor products of Schauder functions

In Ben Slimane and Ben Braiek (2012) and Ben Slimane (2012), we found characterizations of the pointwise and uniform directional regularities of a multi-parameter function in terms of decay rates of either anisotropic Triebel wavelet coefficients or continuous Calderón anisotropic wavelet transform. The purpose of this paper is twofold. We first use a result of Kamont (1996) to provide an easier criterium of uniform directional Lipschitz regularity by decay conditions on the coefficients of the function in a tensor product Schauder basis. As a consequence, we deduce the characterization of the local critical directional Lipschitz regularity. We apply our results for both multidimensional parameter fractional Wiener field in Rd and Sierpinski cascade function. We then obtain criteria of pointwise directional Lipschitz regularity by decay conditions on either two progressive difference in the given direction or the coefficients of the function in a tensor product Schauder basis.