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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

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


Author: Akihito Uchiyama
Journal: Trans. Amer. Math. Soc. 262 (1980), 579-592
MSC: Primary 46J15; Secondary 46E30
MathSciNet review: 586737
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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

$\displaystyle {f^ + }(x)\, = \,\mathop {\sup }\limits_{t\, > \,0} \,\left\vert {f\,{\ast}\,{\psi _{0t}}(x)} \right\vert$

and let $ {f^{{\ast}M}}(x)\, = \,\sup \{ \left\vert {f\,{\ast}\,{\psi _t}(x)} \right\vert:\,t\, > \,0$, $ \psi (y)\, \in \,{\mathcal{S(}}{R^n})$, $ \operatorname{supp} \,\psi \, \subset \,\{ y\, \in \,{R^n}:\,\left\vert y \right\vert\, < \,1\} $, $ {\left\Vert {{D^\alpha }\psi } \right\Vert _{{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

$\displaystyle c\left\Vert {{f^ + }} \right\Vert{L^p}\, \leqslant \,\left\Vert {... ...right\Vert{L^p}\, \leqslant \,{\textbf{C}}\left\Vert {{f^ + }} \right\Vert{L^p}$

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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1980-0586737-4
PII: S 0002-9947(1980)0586737-4
Keywords: $ {H^p}$, space of homogeneous type, BMO, Lipschitz space, maximal function, Poisson kernel, Poisson-Szegö kernel
Article copyright: © Copyright 1980 American Mathematical Society