The multi-dimensional pencil phenomenon for Laguerre heat-diffusion maximal operators

We investigate in detail the mapping properties of the maximal operator associated with the heat-diffusion semigroup corresponding to expansions with respect to multi-dimensional standard Laguerre functions . Our interest is focused on the situation when at least one coordinate of the type multi-index α is smaller than 0. For such parameters α the Laguerre semigroup does not satisfy the general theory of semigroups, and the behavior of the associated maximal operator on L ^p spaces is found to depend strongly on both α and the dimension. A. Nowak was supported in part by MNiSW Grant N201 054 32/4285.