Hardy spaces for Fourier-Bessel expansions

Abstract

Abstract. We study Hardy spaces for Fourier-Bessel expansions associated with Bessel operators on ((0, 1), x 2ν+1 dx) and ((0, 1), dx). We define Hardy spaces H 1 as the sets of L 1 -functions for which their maximal functions for the corresponding Poisson semigroups belong to L 1 . Atomic characterizations are obtained.

Local atomic decompositions for multidimensional Hardy spaces

“…Then, many atomic characterizations were proved for various operators including operators with Gaussian (or Davies-Gaffney) estimates, operators on spaces of homogeneous type, operators related to orthogonal expansions, Schrödinger operators, and others. The reader is referred to [1,2,6,[9][10][11]17,21,22] and references therein.…”

Variation operators for semigroups associated with Fourier-Bessel expansions

“…However, the harmonic analysis related to ∆ ν and S ν has important differences. For instance, as it is shown in [15] (see also [9]) the Hardy spaces H p ((0, 1), ∆ ν ) and H p ((0, 1), S ν ) have different properties. Here we will prove that the variation operators defined by using semigroups associated with ∆ ν and S ν have different L p -boundedness properties.…”

“…Hardy spaces for Fourier-Bessel expansions, H 1 ((0, 1), ∆ ν ), have been considered in [9,15]. For a function f ∈ L 1 ((0, 1), x 2ν+1 dx), in [15] it is said that f ∈ H 1 ((0, 1), ∆ ν ) provided that P ν * (f ) ∈ L 1 ((0, 1), x 2ν+1 dx), and in [9,Theorem 4.26] it is proven that…”

“…{W ν t } t>0 is a symmetric diffusion semigroup associated with ∆ ν on ((0, ∞), x 2ν+1 dx). As in [15], we consider a function ρ ∈ C ∞ (0, ∞) such that ρ(x) we can write…”

“…We are going to proceed as in [15,Case 3 Observe that m ν (I) ∼ 2 −N and m ν (I * j ) ∼ 2 −j , j ∈ N 0 . We write…”