Maximal operators of exotic and non-exotic Laguerre and other semigroups associated with classical orthogonal expansions

Abstract

Abstract. Classical settings of discrete and continuous orthogonal expansions, like Laguerre, Bessel and Jacobi, are associated with second order differential operators playing the role of the Laplacian. These depend on certain parameters of type that are usually restricted to a half-line, or a product of half-lines if higher dimensions are considered. Following earlier research done by Hajmirzaahmad, we deal in this paper with Laplacians in the above-mentioned contexts with no restrictions on the type parameters and bring to attention naturally associated orthogonal systems that in fact involve the classical ones, but are different. This reveals new frameworks related to classical orthogonal expansions and thus new potentially rich research areas, at least from the harmonic analysis perspective. To support the last claim we focus on maximal operators of multi-dimensional Laguerre, Bessel and Jacobi semigroups, with unrestricted type parameters, and prove that they satisfy weak type (1, 1) estimates with respect to the appropriate measures. Generally, these measures are not (locally) finite, which makes a contrast with the classical situations and generates new difficulties. An important partial result of the paper is a new proof of the weak type (1, 1) estimate for the classical multi-dimensional Laguerre semigroup maximal operator.

Spectral properties of ordinary differential operators admitting special decompositions

“…It relies (in the Laguerre case), roughly, on reflecting the setting corresponding to α>−1 by a use of an appropriate intertwining operator to cover the case α < 1. This idea may be easily adapted in other settings of orthogonal expansions, see [21,1], and we do this in our discussion. Moreover, the idea may be also adapted in continuous orthogonal settings; we comment this in Subsection 6.1.…”

“…Here for the divergent form operators we apply the principles described after (2.2) and for operators in Liouville form we use (6.5). We also mention that we borrow the concept of notation from [21] and use the labels cls and exo to distinguish between classic and exotic self-adjoint extensions (and other objects). In addition we shall use the mathbb font to denote self-adjoint extensions resulting from Theorem 5.2.…”

“…with Dom(L α ) = C ∞ c (0, ∞), that acts on L 2 (dµ α ) := L 2 ((0, ∞), x α e −x dx). Spectral properties of this operator were discussed, among others, in [12], [22,Section 7.2] (under the restriction α > −1), [9,Section 27], and in the introductory section of [21]. Decomposition (7.1) holds for L = L α with a = 0 and…”

The atomic Hardy space for a general Bessel operator

“…When ν ∈ (−1, ∞) d there exists a classical self-adjoint extension of B ν (acting initially on C 2 c (R d + )), from now on denoted by B cls ν [16,19] W cls t,ν j (x j , y j ), for x, y ∈ R d + and t > 0, where I τ denotes the modified Bessel function of the first kind and order τ > −1, cf. [26].…”

“…Recently, harmonic analysis related to the classical Bessel operator has been extensively developed, see e.g. [1][2][3][4][5][6]8,12,14,19,20] and references therein. In particular, the Hardy space…”