Calderón–Zygmund operators in the Bessel setting

Abstract

We study several fundamental operators in harmonic analysis related to Bessel operators, including maximal operators related to heat and Poisson semigroups, Littlewood-Paley-Stein square functions, multipliers of Laplace transform type and Riesz transforms. We show that these are (vector-valued) Calderón-Zygmund operators in the sense of the associated space of homogeneous type, and hence their mapping properties follow from the general theory.2000 Mathematics Subject Classification. 42C05 (primary), 42C20 (secondary).

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

“…Observe that for E = ∅ we recover the classical multi-dimensional Bessel context considered in [4,5,6,7,15], among many other papers. Otherwise, that is when E = ∅, the exotic situation occurs.…”

“…that correspond to the (modified) Hankel transform on R + and the classical Jacobi polynomials on (−1, 1), respectively. Harmonic analysis of self-adjoint 'Laplacians' emerging from B ν and J α,β in the classical ranges of the parameters ν, α, β > −1 has been widely investigated, see for instance [3,4,5,6,7,8,9,10,11,13,14,15,16,19,38,41,44] and also references therein, which is only a small part of the related literature. In both contexts, we consider the classical and exotic self-adjoint operators and introduce the resulting general multi-dimensional Jacobi and Bessel 'Laplacians'.…”

Bellman functions and dimension free Lp-estimates for the Riesz transforms in Bessel settings

“…and P α t denotes the Poisson kernel associated to B α (see Section 2 for definitions). The Bessel-Riesz transform R α turns out to be a Calderón-Zygmund operator in the homogeneous type space ((0, ∞), x 2α dx, ρ) where ρ represents the Euclidean metric on (0, ∞) (see [5]).…”

Sharp Multiplier Theorem for Multidimensional Bessel Operators

“…Now we provide a short argument for (1.5). In [5,Sec. 4.3] it is proved that B i b is associated with the kernel…”