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

Harmonic analysis and ultracontractivity


Authors: Michael Cowling and Stefano Meda
Journal: Trans. Amer. Math. Soc. 340 (1993), 733-752
MSC: Primary 47D03; Secondary 43A99
MathSciNet review: 1127154
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Abstract: Let $ {({T_t})_{t > 0}}$ be a symmetric contraction semigroup on the spaces $ {L^p}(M)\;(1 \leq p \leq \infty )$, and let the functions $ \phi $ and $ \psi $ be "regularly related". We show that $ {({T_t})_{t > 0}}$ is $ \phi $-ultracontractive, i.e., that $ {({T_t})_{t > 0}}$ satisfies the condition $ {\left\Vert {{T_t}f} \right\Vert _\infty } \leq C\phi {(t)^{ - 1}}{\left\Vert f \right\Vert _1}$ for all $ f$ in $ {L^1}(M)$ and all $ t$ in $ {{\mathbf{R}}^ + }$, if and only if the infinitesimal generator $ \mathcal{G}$ has Sobolev embedding properties, namely, $ {\left\Vert {\psi {{(\mathcal{G})}^{ - \alpha }}f} \right\Vert _q} \leq C{\left\Vert f \right\Vert _p}$ for all $ f$ in $ {L^p}(M)$, whenever $ 1 < p < q < \infty $ and $ \alpha = 1/p - 1/q$ . We establish some new spectral multiplier theorems and maximal function estimates. In particular, we give sufficient conditions on $ m$ for $ m(\mathcal{G})$ to map $ {L^p}(M)$ to $ {L^q}(M)$, and for the example where there exists $ \mu $ in $ {{\mathbf{R}}^ + }$ such that $ \phi (t) = {t^\mu }$ for all $ t$ in $ {{\mathbf{R}}^ + }$ , we give conditions which ensure that the maximal function $ {\sup _{t > 0}}\vert{t^\alpha }{T_t}f( \bullet )\vert$ is bounded.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1993-1127154-7
PII: S 0002-9947(1993)1127154-7
Article copyright: © Copyright 1993 American Mathematical Society