The local part and the strong type for operators related to the Gaussian measure.
type: Item
instance of: scholarly article
title: The local part and the strong type for operators related to the Gaussian measure.
zbMATH Open document ID: 1058.42015
DOI: 10.1007/BF02922016
author: Sonsoles Pérez
published in: The Journal of Geometric Analysis
publication date: 2002-08-08T00:00:00Z
review text: The author gives results implying weak type \(1\) estimates for the local part of a linear operator associated with solutions of a differential equation given by the Ornstein-Uhlenbeck semigroup. Let \(L = {1 \over 2} \Delta_x - \langle x, \nabla_x\rangle\) be considered on the Euclidean space \(R^n\) where the solution of \({\partial u \over \partial t} = Lu, u(x, 0) = f(x)\) is given by the Ornstein-Uhlenbeck semigroup \[ u(x, t) = \pi^{-1/n} \int_{R^n} M_{e^{-t}}(x, y) f(y) \, dy, \] with \[ M_{e^{-t}}(x, y) = {\exp \left( -{ | y - e^{-t} x| ^2 \over 1 - e^{-2t} } \right) \over (1 - e^{-2t})^{n/2} }. \] The natural measure is the Gaussian measure \(\gamma(x) \, dx = e^{-| x| ^2} \, dx\), and in general, if \(T\) is a sublinear operator, and \(B_h(x) = \{ y: | y -x| \leq n(1 \wedge 1/| x| )\) is a hyperbolic ball about \(x\), the local part of \(T\) is \(T_0 f(x) = T(f \chi_{B_h(x)})(x)\), and its global part is \(T_1 f(x) = T(f \chi_{B_h^c(x)})(x)\). She shows under growth conditions on the kernel and its derivatives of order \(1\) on hyperbolic balls about the diagonal, that if the principal value integral maps \(L^p(\gamma)\) into itself, then its local part is of weak type \(1\) with respect to \(\gamma\). She verifies that the usual Riesz transform operators satisfy the growth conditions. She then verifies that for kernels of the form \[ K_F(x, y) = \int_0^1 r^{m - 1} \left( {-\log r \over 1 - r^2} \right)^{m-2 \over 2} F\left({y - rx \over \sqrt {1 - r^2}}\right) {e^{- | y - r x| ^2 \over 1 - r^2} \over (1 - r^2)^{n/2 + 1}} \, dr, \] where \(F\) is a \(C^1\) function of \(z\) orthogonal to \(\gamma\) that grows not faster than exponential of order \(2\), i.e., \[ \int_{R^n} F(z) e^{- | z| ^2} \, dz = 0, \] and for each \(\varepsilon > 0\), there is a \(C_{\varepsilon}, C_{1, \varepsilon} \) such that \( | F(z)| \leq C_{\varepsilon} e^{\varepsilon | z| ^2}, | \nabla F(z)| \leq C_{1,\varepsilon}e^{\varepsilon | z| ^2}, \) that the associated principal value op…
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 42B20
zbMATH DE Number: 1777178
zbMATH Keywords: Ornstein-Uhlenbeck semigroup
zbMATH Keywords: Gaussian measure
zbMATH Keywords: strong type estimates
zbMATH Keywords: weak type estimates
reviewed by: Raymond Johnson
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1777178