Parabolic fractional maximal operator in modified parabolic Morrey spaces (Q1936452)

Summary: We prove that the parabolic fractional maximal operator \(M^P_\alpha\), \(0 \leq \alpha < \gamma\), is bounded from the modified parabolic Morrey space \(\widetilde{M}_{1, \lambda, P}(\mathbb R^n)\) to the weak modified parabolic Morrey space \(W \widetilde{M}_{q, \lambda, P}(\mathbb R^n)\) if and only if \(\alpha/\gamma \leq 1 - 1/q \leq \alpha/(\gamma - \lambda)\) and from \(\widetilde{M}_{p, \lambda, P}(\mathbb R^n)\) to \(\widetilde{M}_{q, \lambda, P}(\mathbb R^n)\) if and only if \(\alpha/\gamma \leq 1/p - 1/q \leq \alpha/(\gamma - \lambda)\). Here \(\gamma = \operatorname{tr}P\) is the homogeneous dimension on \(\mathbb R^n\). In the limiting case \((\gamma - \lambda)/\alpha \leq p \leq \gamma/\alpha\), we prove that the operator \(M^P_\alpha\) is bounded from \(\widetilde{M}_{p, \lambda, P}(\mathbb R^n)\) to \(L_\infty(\mathbb R^n)\). As an application, we prove the boundedness of \(M^P_\alpha\) from the parabolic Besov-modified Morrey spaces \(\widetilde{BM}^s_{p\theta, \lambda}(\mathbb R^n)\) to \(\widetilde{BM}^s_{q\theta, \lambda} (\mathbb R^n)\). As another application, we establish the boundedness of some Schrödinger-type operators on modified parabolic Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class.