Transference between Laguerre and Hermite settings

In this paper we present a new method in order to transfer boundedness results for operators associated with Hermite functions to boundedness results for operators associated with Laguerre functions. The technique relies on an exact point-wise identity relating the heat kernels of both systems. The method that we present here has the novelty that can be used backwards, that is, boundedness results for Laguerre systems can be also transfered to boundedness results for Hermite systems. We apply our method in order to get new properties of some operators in the Laguerre setting. Among others, we mention the description of Riesz transforms as principal value operators. As an application of the reversibility of the method we characterize the class of Banach spaces B for which the Riesz transforms (in the Laguerre setting) are bounded from into itself. It is shown that this class coincides with the UMD class.