BMO spaces related to Laguerre semigroups

Abstract

For the system of Laguerre functions inline image we define a suitable BMO space from the atomic version of the Hardy space inline image considered by Dziubański in [7], where inline image is the maximal operator of the heat semigroup associated to that Laguerre system. We prove boundedness of inline image over a weighted version of that BMO, and we extend such result to other systems of Laguerre functions, namely inline image and inline image. To do that, we work with a more general family of weighted BMO-like spaces that includes those associated to all of the above mentioned Laguerre systems. In this setting, we prove that the local versions of the Hardy-Littlewood and the heat-diffusion maximal operators turn to be bounded over such family of spaces for inline image weights. This result plays a decisive role in proving the boundedness of Laguerre semigroup maximal operators.