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). As applications, we establish the boundedness of some Schodinger type operators on modified Morrey spaces related to certain nonnegative potentials belonging to the reverse Holder class. As an another application, we prove the boundedness of various operators on modified Morrey spaces which are estimated by Riesz potentials.