NECESSARY AND SUFFICIENT CONDITIONS FOR THE BOUNDEDNESS OF THE RIESZ POTENTIAL IN MODIFIED MORREY SPACES

We prove that the fractional maximal operator M-alpha and the Riesz potential operator I-alpha, 0 < alpha < n are bounded from the modified Morrey space (L) over tilde (1,lambda) (R-n) to the weak modified Morrey space W (L) over tilde (q,lambda) (R-n) if and only if, alpha/n <= 1 - 1/q <= alpha/(n - lambda) and from (L) over tilde (p,lambda) (R-n) to (L) over tilde (q,lambda) (R-n) if and only if, alpha/n <= 1/p - 1/q <= alpha/(n - lambda).