Parabolic Fractional Maximal Operator in Modified Parabolic Morrey Spaces

Abstract

We prove that the parabolic fractional maximal operator MαP, 0≤α<γ, is bounded from the modified parabolic Morrey space M̃1,λ,P(ℝn) to the weak modified parabolic Morrey space WM̃q,λ,P(ℝn) if and only if α/γ≤1-1/q≤α/(γ-λ) and from M̃p,λ,P(ℝn) to M̃q,λ,P(ℝn) if and only if α/γ≤1/p-1/q≤α/(γ-λ). Here γ=trP is the homogeneous dimension on ℝn. In the limiting case (γ-λ)/α≤p≤γ/α we prove that the operator MαP is bounded from M̃p,λ,P(ℝn) to L∞(ℝn). As an application, we prove the boundedness of MαP from the parabolic Besov-modified Morrey spaces BM̃pθ,λs(ℝn) to BM̃qθ,λs(ℝn). As other applications, we establish the boundedness of some Schrödinger-ype operators on modified parabolic Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class.