Transference between Laguerre and Hermite settings

Abstract

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 L p B into itself. It is shown that this class coincides with the UMD class.

Lusin type and cotype for Laguerre g-functions

“…The subordinated Poisson semigroup {P H t } t>0 can be defined by using formula (2), just by replacing T u by W H u . Given a Banach space B and a Bvalued function f defined on R we define the g-function g H q (f ), 1 < q < ∞, by…”

“…The subordinated Poisson semigroup, {P α t } t>0 , and the g-function, g α q , are defined for functions defined in (0, ∞), in a parallel way to the Hermite case; see (2) and (4). We introduce the following notation.…”

“…This procedure goes back to Muckenhoupt (see [14] and [15]), and has been used in an essential way in [3], [8], [12], [17], [18] and [21]. (2) The other common strategy is to use different formulae relating the Hermite polynomials in dimension n with Laguerre polynomials in dimension 1 and α = n/2 − 1; see, for example, [23]. These formulae can be used in order to transfer results from the Hermite setting in dimension n to the one-dimensional Laguerre setting with α = n/2 − 1.…”

“…Then some considerations about the kernel allow us to prove the L p -estimates. However, in the Laguerre setting, our procedure to analyze L p -boundedness properties of g α q -functions is completely different from (1) and (2). It relies on a new pointwise relation (see (10)) between the heat kernels W α t and W H t .…”

Riesz Transform on the Dual of the Laguerre Hypergroup

“…To achieve our goal we use a procedure that will be described below and that was developed for the first time by Torrea and others in J. J. Betancor, J. C. Fariña, L. Rodriguez-Mesa, A, Sanabria-Garcia and J. L. Torrea (2008).…”