Weak type \((1,1)\) estimates of the maximal function for the Laguerre semigroup in finite dimensions. (English) Zbl 0823.42011

Dinger, Ulla

Rev. Mat. Iberoam. 8, No. 1, 93-120 (1992).

Let \(\{T^ t\}_{0< t<\infty}\) be a symmetric diffusion semigroup. The associated maximal operator is defined by \(Mf(x)= \sup_{t> 0} | T^ t f(x)|\). The \(L^ p\), \(p> 1\), boundedness of this operator was proved by E. M. Stein more than twenty years ago. The weak type (1,1) estimate is more delicate and requires some additional properties of the semigroup. In this paper, the weak type (1,1) estimate is proved for the Laguerre semigroup in higher dimensions. Analogous theorems are known for ultraspherical harmonics (B. Muckenhoupt and E. M. Stein), the one- dimensional Laguerre semigroup (B. Muckenhoupt) and the Hermite semigroup in higher dimensions (P. Sjögren).

Reviewer: Do Ngoc Diep (Hanoi)

MSC:

42B25	Maximal functions, Littlewood-Paley theory
47D03	Groups and semigroups of linear operators
42C05	Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis

Keywords:

diffusion semigroup; maximal operator; weak type (1,1) estimate; Laguerre semigroup