Weak type \((1,1)\) estimates for Caffarelli-Calderón generalized maximal operators for semigroups associated with Bessel and Laguerre operators (Q2862188)

The authors investigate generalized maximal operators in the sense of Caffarelli and Calderón associated to the heat semigroups of multidimensional Bessel and Laguerre operators. In the Bessel case it is proved that the relevant maximal operator is, in the framework of the measure space \(((0,\infty)^n, \mu_\lambda)\), \(n\in \mathbb N\), \(\lambda=(\lambda_1,\ldots, \lambda_n)\in(-1/2,\infty)^n\), \(d\mu_\lambda(x)=\prod_{j=1}^n x_j^{2\lambda_j}dx_j\), of weak type (1,1) and hence of strong type \((p,p)\), \(1< p\leq\infty\). The corresponding result in the Laguerre case follows since the integral kernel of the associated heat semigroup is subordinated to the integral kernel of the heat semigroup from the Bessel case.