Spectral multipliers for the Ornstein-Uhlenbeck semigroup
The authors study the possibility to introduce functional calculus generated by the Laplacian corresponding to the multidimensional space with Gaussian measure. Though they prove, that the general result of \textit{S. Meda} [Proc. Am. Math. Soc. 110, No. 3, 639-647 (1990; Zbl 0760.42007)] cannot be applied to the case. They succeed in constructing the multipliers of Laplace transform type associated to the Laplacian. The boundednes on \(L^p\) for \(1<p<\infty\) and \((1,1)\)-boundedness of these operators are also proved. The authors also study the question of representability of the operators as principal value (or a weaker analog of the principal value) of a singular integral. Namely, if \(m\) is of Laplace transform type, then there exists a bounded function \(\alpha\) on \((0,\infty)\), such that \(m(L)f(x)=\lim_{\varepsilon\rightarrow 0+} (\alpha(\varepsilon)f(x)+\int_{|x-y|>\varepsilon} K(x,y)f(y)dy),\) for almost all \(x\), whenever \(f\) is a test function.
scientific article; zbMATH DE number 1369965
scientific article