Abstract
We consider pointwise rectangular Lipschitz regularity and pointwise level coordinate axes Lipschitz regularities for continuous functionsfon the unit cubeI2inR2. Firstly, we provide characterizations by simple estimates on the decay rate of the coefficients (resp. leaders) of the expansion offin the rectangular Schauder system, near the point considered. We deduce that pointwise rectangular Lipschitz regularity yields pointwise level coordinate axes Lipschitz regularities. As an application, we refine earlier results in Ayache et al. (Drap brownien fractionnaire. Potential Anal. 2002,17, 31-43) and Kamont (On the fractional anisotropic Wiener field. Probab. Math. Statist.1996,16, 85-98), where uniform rectangular Lipschitz regularity of the trajectories of the fractional Brownian sheet over the totalI2(or any cube) was considered. Actually, we prove that fractional Brownian sheets are pointwise rectangular and level coordinate axes monofractal. On the opposite, we construct a class of Sierpinski selfsimilar functions that are pointwise rectangular and level coordinate axes multifractal.