On the \(L^p\) boundedness of the non-centered Gaussian Hardy-Littlewood maximal function (Q2750889)

Let \(d\gamma=e^{-|x|^{2}} dx\) be the Gaussian measure on \(\mathbb R^n\). The authors show that the non-centered maximal function \(M\) associated with \(d\gamma\) is \(L^p\)-bounded for all \(1<p\leq +\infty\). The proof relies on geometric observations, the most difficult case being balls with big radius far from the origin. NEWLINENEWLINENEWLINENotice that \textit{P. Sjögren} proved in [Am. J. Math. 105, 1231-1233 (1983; Zbl 0528.42007)] that \(M\) is not of weak type \((1,1)\) for \(n>1\), whereas, for \(n=1\), \(M\) is of weak type \((1,1)\). Recall also that the corresponding centered maximal function is of weak type \((1,1)\) [see \textit{M. de Guzman}, Lect. Notes Math. 541, 181-185 (1976; Zbl 0332.26009)]. NEWLINENEWLINENEWLINEFinally, Sjögren and Soria, in a forthcoming paper, prove results for the non-centered maximal operator associated with more general radial measures.