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

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).

### 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
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