Square functions and spectral multipliers for Bessel operators in UMD spaces

Abstract

Abstract. In this paper we consider square functions (also called Littlewood-Paley g-functions) associated to Hankel convolutions acting on functions in the Bochner Lebesgue spacewhere B is a UMD Banach space. As special cases we study square functions defined by fractional derivatives of the Poisson semigroup for the Bessel operatorWe characterize the UMD property for a Banach space B by using L p ((0, ∞), B)-boundedness properties of g-functions defined by Bessel-Poisson semigroups. As a by product we prove that the fact that the imaginary power ∆ iω λ , ω ∈ R \ {0}, of the Bessel operator ∆ λ is bounded in L p ((0, ∞), B), 1 < p < ∞, characterizes the UMD property for the Banach space B. As applications of our results for square functions we establish the boundedness in L p ((0, ∞), B) of spectral multipliers m(∆ λ ) of Bessel operators defined by functions m which are holomorphic in sectors Σ ϑ .

Hörmander functional calculus on UMD lattice valued Lp spaces under generalized Gaussian estimates

“…Here, s(A) = inf σ(A) is the spectral bound of A in L 2 (R d ). Thus, according to Corollaries 4.9 and 4.13, A − s(A) + ǫ has a H on = (0, ∞) [BCRM,p. 343].…”

UMD-Valued Square Functions Associated with Bessel Operators in Hardy and BMO Spaces

“…Square functions defined by the Bessel Poisson semigroup have been studied in [6], [13], [14] and [39], amongst others. In [13] it was considered generalized Littlewood-Paley functions associated with {P λ t } t>0 in a Banach valued setting.…”

“…According to [6,Theorem 1.3], by using vector-valued Calderón-Zygmund theory ( [34]) we infer that the operator defined by (58) can be extended from S λ (0, ∞) ⊗ B to L 1 ((0, ∞), B) as a bounded operator from L 1 ((0, ∞), B) into L 1,∞ ((0, ∞), γ (H, B)) and from H 1 ((0, ∞) γ(H, B)). By proceeding as in the proof of Lemma 2.1 we can conclude that the operator G λ B is bounded from L 1 ((0, ∞) γ(H, B)) and from H 1 ((0, ∞), B) into L 1 ((0, ∞), γ(H, B)).…”

“…We prove in this section that (iii) =⇒ (i). In order to see this, we use that the Banach space B is UMD if, and only if, for every γ ∈ R \ {0}, the imaginary power ∆ iγ λ can be extended to L p ((0, ∞), B) as a bounded operator from L p ((0, ∞), B) into itself, for some (equivalently, for every) 1 < p < ∞ (see [6,Proposition 5.1]).…”

Dunkl spectral multipliers with values in UMD lattices

“…As a predecessor of all these results, we mention the UMD space valued singular integrals of convolution type [5,12]. More recently, in [7,8] the boundedness of vector valued, but holomorphic, spectral multipliers associated with Hermite and Bessel operators is established, in the case that Y = [H, X] θ is a complex interpolation space between a Hilbert space and a UMD space. Note that according to [60,Corollary p. 216], this case is an intermediate case between the bare UMD space case and the UMD lattice case.…”