Anisotropic $H^{p}$ real interpolation, and fractional Riesz potentials

Abstract:

We observe that the anisotropic variants of ${H^p}$ interpolate by the real method in the usual manner. Using this fact we show that the corresponding fractional Riesz potentials and related operators perform an embedding in ${H^p},p > 0$, analogous to the one for ${L^p},p > 1$. We also state a theorem concerning the mapping properties of $f \to h \ast f$, where h is in $B_\alpha ^{1,\infty }$, which hold only for a restricted range of p.