Maximal operators associated with generalized Hermite polynomial and function expansions (Q2855965)

The authors study the extension to \(\mathbb{R}^d\) of generalized Hermite polynomials (by a tensor-product definition) due to [\textit{G. Zsegő}, Orthogonal polynomials. 4th ed., American Mathematical Society Colloquium Publications. Vol. XXIII. Providence, R. I. (AMS) (1975; Zbl 0305.42011)].NEWLINENEWLINEThey associate these to the diffusion semigroup \(T^{\mu}_t=e^{\mathbf{L}_{\mu}}\) and look at the maximal operator NEWLINE\[NEWLINET^{\mu}_{\ast}f(x)=\sup_{t>0}\,|T^{\mu}_t f(x)|\;(f\in\text{L}^p(\mathbb{R}^d,\gamma_{\mu}),p\geq 1).NEWLINE\]NEWLINE (Here \(\gamma_{\mu}(dx)=\prod_{k=1}^d\,x_k^{2\mu_k}e^{-|x|^2}dx,\;x=(x_1,\ldots,x_d),\text{ and } \mathbf{L}_{\mu}= \sum_{k=1}^d\,L_{\mu_k}\), this differential-difference operator gives the connection with the heat-diffusion equation.)NEWLINENEWLINEThe main results are:NEWLINENEWLINE{Theorem 1.3.} For \(\mu\in (-1/2,\infty)^d\) the operator \(T^{\mu}_{\ast}\) is of weak-type \((1,1)\) and strong-type \((p,p)\) for \(p>1\) with respect to \(\gamma_{\mu}\).NEWLINENEWLINE{Corollary 1.4.} For \(\mu\in (-1/2,\infty)^d\) and every \(f\in\text{L}^p(\text{ R}^d,\gamma_{\mu})\), NEWLINE\[NEWLINE\lim_{t\downarrow 0}\,T^{\mu}_t f=f\;a.e.NEWLINE\]NEWLINENEWLINENEWLINE{Theorem 1.5.} For \(\mu\in (-1/2,\infty)^d\), the diffusion semigroup \(T_t^{\mu}\) is hypercontractive.NEWLINENEWLINE{Corollary 1.6.} For \(\mu\in (-1/2,\infty)^d\), the Bessel and Riesz potentials associated to \(\mathbf{L}_{\mu}\) are of strong-type \((p,p),\;1<p<\infty\), with respect to the measure \(\gamma_{\mu}\).NEWLINENEWLINEMoreover, they study \(d\)-dimensional Hermite functions defined through a tensor product of the one dimensional generalized case due to \textit{M. Rosenblum} [Oper. Theory, Adv. Appl. 73, 369--396 (1994; Zbl 0826.33005)].NEWLINENEWLINEIn view of the length of an ordinary review, the important new results from the paper under review ({Theorem 1.9}, {Theorem 1.11}) are not given here, but they are highly recommended to study in detail before embarking upon further generalizations.