Necessary and sufficient condition for the boundedness of the Gegenbauer-Riesz potential on Morrey spaces

Abstract

In this paper, we study the Riesz potential (G-Riesz potential) generated by the Gegenbauer differential operator G(lambda) = (x(2) - 1()1/2-lambda) d/dx (x(2) - 1)(lambda+1/2) d/dx, x is an element of(1, infinity), lambda is an element of(0, 1/2). We prove that the G-Riesz potential I-G(alpha), 0 < alpha < 2 lambda + 1, is bounded from the G-Morrey space L-p,L-lambda,L-y to L-q,L-lambda,L-y if and only if 1/p - 1/q = alpha/2 lambda+1-gamma, 1 < p < 2 lambda + 1 - gamma/alpha. Also, we prove that the G-Riesz potential I-G(alpha) is bounded from the G-Morrey space L-1,L-lambda,L-gamma to the weak G-Morrey space WLq,lambda,gamma if and only if 1 - 1/q = alpha/2 lambda+1-gamma.