Operators associated with the Hermite semigroup - a survey (Q1384980)

Let \(R^n\) be the Euclidean space equipped with the normalized Gaussian measure \(\pi^{-n/2} e^{-x^2} dx\). The symmetric Laplacian \(L\) on this space can be spectrally decomposed using the Hermite polynomials, and the heat operator \(e^{-tL}\) computed; a detailed derivation appears in this paper. The \(L^p\) and weak-type \((1,1)\) for the maximal operator \[ M^* f(x) = \sup_{t > 0} | e^{-tL} f(x)| \] and Riesz operators \(D^\alpha L^{-b} \Pi_0\) are summarized for both finite and infinite dimensions; here \(D\) is a differentiation operator \(b = 2| \alpha| \), and \(\Pi_0\) is the orthogonal projection to the complement of the \(0\) eigenspace. A new proof of the weak (1,1) type of second-order Riesz operators in finite dimensions is then proved, based on a precise computation of the kernel and its decomposition into local and global portions.