Riesz transforms and g-function for Laguerre expansions

Abstract

We analyze boundedness properties of some operators related to the heat-diffusion semigroup associated to Laguerre functions systems. In particular, for any α > −1, we introduce appropriate Laguerre Riesz Transforms and we obtain power-weighted L p inequalities, 1 < p < ∞. We achieve this result by taking advantage of the existing classical relationship between n-variable Hermite polynomials and Laguerre polynomials on the half line of type α = n/2 − 1. Such connection allows us to transfer known boundedness properties for Hermite operators to Laguerre operators corresponding to those specific values of α. To extend the results to any α > −1, we make use of transplantation and some weighted inequalities we obtain in the Hermite setting (which we beleive of independent interest).

What is a Sobolev space for the Laguerre function systems?

“…The function µ satisfies (10). Hence, by Theorem 6, T µ is bounded from L p ((0, ∞), y δ dy) into L p ((0, ∞), y δ dy) for δ satisfying (C α+1 ).…”

“…which satisfies (10). Then the proof follows the same lines as in the Hermite case in Theorem 3 using the multiplier Theorem 6 and Theorem 8 in order to control the operator R k α .…”

“…The multiplier µ(n) = (n + (α + 1)/2) −s satisfies (10). Now that we see that the spaces in Definition 2 are well defined, we proceed as in the Hermite context.…”

“…For some previous work containing the definition and power weighted L p -boundedness of the first order Riesz transforms, see [10] and [9] for the system L α k , and [12] for the system ϕ α k (see Section 5). Recently, power weighted L p -boundedness of the higher order Riesz transforms of the form (D α ) k L −k/2 for the system ϕ α k (see Section 5) has been proved in [2].…”

L2-Theory of Riesz Transforms for Orthogonal Expansions

“…We consider the resulting two multi-dimensional settings separately, and in this subsection the setting connected with {ϕ α k } is treated. The study of Riesz transforms associated with expansions based on {ϕ α k } was initiated by the authors in [28]; in a somewhat different (one-dimensional) settings Riesz transforms were also defined and investigated by Harboure, Torrea, and Viviani [18] and by Betancor, Fariña, and Sanabria [2].…”