Harmonic analysis operators related to symmetrized Jacobi expansions for all admissible parameters

Abstract

This is an ultimate completion of our earlier paper [Acta. Math. Hungar. 140 (2013), 248-292] where mapping properties of several fundamental harmonic analysis operators in the setting of symmetrized Jacobi trigonometric expansions were investigated under certain restrictions on the underlying parameters of type. In the present article we take advantage of very recent results due to Nowak, Sjögren and Szarek to fully release those restrictions, and also to provide shorter and more transparent proofs of the previous restricted results. Moreover, we also study mapping properties of analogous operators in the parallel context of symmetrized Jacobi function expansions. Furthermore, as a consequence of our main results we conclude some new results related to the classical non-symmetrized Jacobi polynomial and function expansions.

On Harmonic Analysis Operators in Laguerre-Dunkl and Laguerre-Symmetrized Settings

“…The proofs of the two latter results fall under the well-known scope of the general (vector-valued) Calderón-Zygmund theory for spaces of homogeneous type. Analogous approach based on the Calderón-Zygmund theory in similar situations can be found in [8,14,22,25,26,28,29,30,34,40,41,42,43,44], among many others. For the reader's convenience we recall the main ingredients of this theory in what follows.…”

“…As it was already mentioned, the general strategy of proving L p mapping properties of operators considered in both settings relies on reducing the analysis to a number of suitably defined auxiliary operators related to a smaller measure metric space. This strategy has its roots in the papers [29,30] and was then successfully applied in [14,22,25,26,43,44]. Accordingly, the proofs of the two main results, Theorems 3.1 and 3.7, are reduced to two auxiliary results, Theorems 3.2 and 3.8.…”

“…Our main result in the Laguerre-symmetrized framework reads as follows. It is worth mentioning that an analogue of Theorem 3.7 in the one-dimensional framework of Jacobi trigonometric polynomial expansions was proved recently by Langowski [22,25], though without including Lusin area integrals; see [22,Theorem 2.1] and [25,Theorem 3.1]. Apart from that, no other symmetrized settings seem to have been studied earlier from a similar perspective.…”

Lusin area integrals related to Jacobi expansions

“…δδ * δδ * δ N components , N ≥ 0 (by convention, δ 0 = D 0 = Id), respectively. We point out that the derivative δ N is used in [4,11,13] whereas D N appears in [4,9] and have its roots in the so-called symmetrization procedure proposed in [14]. Now we are ready to introduce the central objects of our study in this paper.…”

“…These objects have more complex structure than those mentioned above, therefore their treatment is considerably more involved and demands more effort and additional technical tools. We point out that analysis in various Jacobi settings received a considerable attention in recent years, see for instance [4,9,11,12,13] as well as numerous other references given there.…”