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MAXIMAL OPERATORS FOR THE HOLOMORPHIC ORNSTEIN–UHLENBECK SEMIGROUP

Published online by Cambridge University Press:  25 March 2003

J. GARCÍA-CUERVA
Affiliation:
Departamento de Matemáticas, C-XV Universidad Autónoma, 28049 Madrid, [email protected]
G. MAUCERI
Affiliation:
Dipartimento di Matematica, via Dodecaneso 35, 16146 Genova IT-16146, [email protected]
S. MEDA
Affiliation:
Dipartimento di Matematica, Università di Milano-Bicocca, via Bicocca degli Arcimboldi 8, IT-20126 Milano, [email protected]
P. SJÖGREN
Affiliation:
Department of Mathematics, Chalmers University of Technology and Göteborg University, S-412 96 Göteborg, [email protected]
J. L. TORREA
Affiliation:
Departamento de Matemáticas, C-XV Universidad Autónoma, 28049 Madrid, [email protected]
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Abstract

For each $p$ in $[1, \infty)$ let ${\bf E}_p$ denote the closure of the region of holomorphy of the Ornstein–Uhlenbeck semigroup $\{{\cal H}_t : t >0\}$ on $L^p$ with respect to the Gaussian measure. Sharp weak type and strong type estimates are proved for the maximal operator $f \mapsto {{\cal H}^*}_pf=\sup\{\vert {\cal H}_zf\vert :z\in {\bf E}_p\}$ and for a class of related operators. As a consequence, a new and simpler proof of the weak type 1 estimate is given for the maximal operator associated to the Mehler kernel.

Type
Notes and Papers
Copyright
The London Mathematical Society, 2003

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