Sharp maximal function estimates and $H^{p}$ continuities of pseudo-differential operators

TL;DR

This work advances sharp maximal function estimates and Hardy-space regularity for pseudo-differential operators with symbols in general Hörmander classes $S^{m}_{\varrho,\delta}$. By a Littlewood–Paley decomposition and detailed kernel bounds, the authors prove pointwise sharp-maximal estimates $M^{\sharp}(T_{a}f)\lesssim M_{p}f$ and analogous bounds for the dual $T^{*}_{a}$ across broad symbol orders, including endpoint cases. They then establish $(H^{p},H^{p})$ and $(H^{p},L^{p})$ continuities for $0<p\le1$ via atomic and molecular techniques, using vanishing moment conditions adapted to the Hardy-space setting. In addition, weighted $L^{p}$ and weak-type results for $T_{a}$ and $T^{*}_{a}$ are obtained through interpolation and Fefferman–Stein inequalities, enriching the theory of $L^{p}$ boundedness and duality for PDOs beyond the classical $L^{2}$ framework. The results generalize and extend prior work (e.g., Miyachi–Yabuta, Álvarez–Hounie) and provide a unified treatment that includes endpoint $p=1$ and dual operators, with potential extensions to Morrey-type spaces discussed as future directions.

Abstract

It is studied that pointwise estimates and continuities on Hardy spaces of pseudo-differential operators (PDOs for short) with the symbol in general Hörmander's classes. We get weighted weak-type $(1,1)$ estimate, weighted normal inequalities, $(H^{p},H^{p})$ continuities and $(H^{p},L^{p})$ continuities for PDOs, where $0<p\leq1$.

Theorems & Definitions (57)

• Theorem 1.1: Wang W
• Theorem 1.2
• Theorem 1.3: Miyachi and Yabuta MiyachiY
• Theorem 1.4: Michalowski, Rule and Staubach Michalowski
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8: Wang W
• Theorem 1.9
• Theorem 1.10