Parabolic Fractional Maximal Operator in Modified Parabolic Morrey Spaces

We prove that the parabolic fractional maximal operator M-alpha(P), 0 <= alpha < gamma, is bounded from the modified parabolic Morrey space <(M)over tilde>(1,lambda,P)(R-n) to the weak modified parabolic Morrey space W (M) over tilde (q,lambda,P)(R-n) if and only if alpha/gamma <= 1 - 1/q <= alpha/(gamma-lambda) and from (M) over tilde (p,lambda,P)(R-n) to (M) over tilde (q,lambda,P)(R-n) if and only if alpha/gamma <= 1/p - 1/q <= alpha/(gamma-lambda). Here gamma = trP is the homogeneous dimension on R-n. In the limiting case (gamma-lambda)/alpha <= p <= gamma/alpha we prove that the operator M-alpha(P) is bounded from (M) over tilde (p,lambda,P)(R-n) to L infinity (R-n). As an application, we prove the boundedness of M-alpha(P) from the parabolic Besov- modified Morrey spaces (BM) over tilde (s)(p,theta,lambda)(R-n) to (BM) over tilde (s)(q,theta,lambda)(R-n). As other applications, we establish the boundedness of some Schr " odinger- ype operators on modified parabolic Morrey spaces related to certain nonnegative potentials belonging to the reverse H " older class.