Riesz potential on the Heisenberg group and modified Morrey spaces

Abstract

In this paper we study the fractional maximal operator M-alpha, 0 <= alpha < Q and the Riesz potential operator F-alpha L 0 < alpha < Q on the Heisenberg group in the modified Morrey spaces L-p,L-lambda(H-n), where Q = 2n + 2 is the homogeneous dimension on H-n. We prove that the operators M-alpha and F-alpha are bounded from the modified Morrey space <(L)over tilde>(1,lambda)(H-n) to the weak modified Morrey space W (L) over tilde (q,lambda) (H-n) if and only if, alpha/Q <= 1 - 1/q <= alpha/(Q - lambda) and from (L) over tilde (p,lambda)(H-n) to (L) over tilde (q,lambda)(H-n) if and only if, alpha/Q <= 1/p - 1/q <= alpha/(Q - lambda). In the limiting case Q-lambda/alpha <= p <= Q/alpha we prove that the operator M-alpha is bounded from (L) over tilde (p,lambda)(H-n) to L-infinity (H-n) and the modified fractional integral operator (I) over tilde (alpha) is bounded from (L) over tilde (p,lambda)(H-n) to BMO(H-n). As applications of the properties of the fundamental solution of sub-Laplacian L on H-n, we prove two Sobolev-Stein embedding theorems on modified Morrey and Besov-modified Morrey spaces in the Heisenberg group setting. As an another application, we prove the boundedness of F alpha, from the Besov-modified Morrey spaces B (L) over tilde (s)(p theta),(lambda)(H-n) to B (L) over tilde (s)(q theta),lambda(H-n).