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).

In this paper we prove L p -boundedness properties of spectral multipliers associated with multidimensional Bessel operators. In order to do this we estimate the L p -norm of the imaginary powers of Bessel operators. We also prove that the Hankel multipliers of Laplace transform type on (0, ∞) n are principal value integral operators of weak type (1, 1).

In this paper we adapt the technique developed in [17] to show that many harmonic analysis operators in the Bessel setting, including maximal operators, Littlewood-Paley-Stein type square functions, multipliers of Laplace or Laplace-Stieltjes transform type and Riesz transforms are, or can be viewed as, Calderón-Zygmund operators for all possible values of type parameter λ in this context. This extends the results obtained recently in [7], which are valid only for a restricted range of λ.2010 Mathematics Subject Classification. 42C05 (primary), 42B20 (secondary).

We consider parabolic operators of the formWe assume that A is a (n + 1) × (n + 1)-dimensional matrix which is bounded, measurable, uniformly elliptic and complex, and we assume, in addition, that the entries of A are independent of the spatial coordinate x n+1 as well as of the time coordinate t. We prove that the boundedness of associated single layer potentials, with data in L 2 , can be reduced to two crucial estimates (Theorem 1.1), one being a square function estimate involving the single layer potential. By establishing a local parabolic Tb-theorem for square functions we are then able to verify the two crucial estimates in the case of real, symmetric operators (Theorem 1.2). As part of this argument we establish a scale-invariant reverse Hölder inequality for the parabolic Poisson kernel (Theorem 1.3). Our results are important when addressing the solvability of the classical Dirichlet, Neumann and Regularity problems for the operator+ , and by way of layer potentials. Mathematics Subject Classification 35K20 · 31B10Communicated by Y. Giga.

In this paper we study discrete harmonic analysis associated with ultraspherical orthogonal functions. We establish weighted p -boundedness properties of maximal operators and Littlewood-Paley g-functions defined by Poisson and heat semigroups generated by the difference operator ∆ λ f (n) := a λ n f (n + 1) − 2f (n) + a λ n−1 f (n − 1), n ∈ N, λ > 0, where a λ n := {(2λ+n)(n+1)/[(n+λ)(n+1+λ)]} 1/2 , n ∈ N, and a λ −1 := 0. We also prove weighted p -boundedness properties of transplantation operators associated with the system {ϕ λ n } n∈N of ultraspherical functions, a family of eigenfunctions of ∆ λ . In order to show our results we previously establish a vector-valued local Calderón-Zygmund theorem in our discrete setting.