Fatou and littlewood theorems for poisson integrals with respect to non-integrable kernels*

Abstract

Sjögren and others studied the boundary behavior of fractional Poisson integrals with respect to the fractional power of the Poisson kernel. We extend the fractional power of the Poisson kernel to a non-integrable kernel and investigate the boundary behavior of associated Poisson integrals. The existence of certain tangential limit (Fatou type theorem) as well as its sharpness (Littlewood type theorem) are given. The admissible tangency varies according to the integrability of the boundary function. Our Littlewood type theorem is new even for the fractional power of the Poisson kernel.