A maximal function characterization of $H^{p}$ on the space of homogeneous type

Abstract:

Let ${\psi _0}(x) \in {\mathcal {S(}}{R^n}{\text {)}}$ and let $\int _{{R^n}} {{\psi _0}(y) dy \ne 0}$.For $f \in {\mathcal {S}’}{R^n}$, $x \in {R^n}$ and $M \geqslant 0$, let \[ {f^ + }(x) = \sup \limits _{t > 0} \left | {f {\ast } {\psi _{0t}}(x)} \right |\] and let ${f^{{\ast }M}}(x) = \sup \{ \left | {f {\ast } {\psi _t}(x)} \right |: t > 0$, $\psi (y) \in {\mathcal {S(}}{R^n})$, $\operatorname {supp} \psi \subset \{ y \in {R^n}: \left | y \right | < 1\}$, ${\left \| {{D^\alpha }\psi } \right \|_{{L^\infty }}} \leqslant 1$ for any multi-index $\alpha = ({\alpha _1}, \ldots , {\alpha _n})$ such that $\Sigma _{i = 1}^n {\alpha _i} \leqslant M\}$ where ${\psi _t}(y) = {t^{ - n}}\psi (y/t)$. Fefferman-Stein [11] showed Theorem A. Let $p > 0$. Then there exists $M(p, n)$, depending only on p and n, such that if $M \geqslant M(p, n)$, then \[ c\left \| {{f^ + }} \right \|{L^p} \leqslant \left \| {{f^{{\ast }M}}} \right \|{L^p} \leqslant {\textbf {C}}\left \| {{f^ + }} \right \|{L^p}\] for any $f \in {\mathcal {S}’}({R^n})$, where c and C are positive constants depending only on ${\psi _0}$, p, M and n. We investigate this on the space of homogeneous type with certain assumptions.