Alejandro J. Castro

Harmonic analysis operators associated with multidimensional Bessel operators

In this paper we establish that the maximal operator and the Littlewood-Paley g-function associated with the heat semigroup defined by multidimensional Bessel operators are of weak type (1,1). Also, we prove that Riesz transforms in the multidimensional Bessel setting are of strong type (p,p), for every

Square functions and spectral multipliers for Bessel operators in UMD spaces

1<p<\infty

Full length article: The fractional Bessel equation in Hölder spaces

, and of weak type (1,1).

Weak type (1,1) estimates for Caffarelli-Calderón generalized maximal operators for semigroups associated with Bessel and Laguerre operators

In this paper, we consider square functions (also called Littlewood-Paley g-functions) associated to Hankel convolutions acting on functions in the Bochner Lebesgue space L-p((0, infinity), B), whe ...

Variable exponent Hardy spaces associated with discrete Laplacians on graphs

Motivated by the Poisson equation for the fractional Laplacian on the whole space with radial right hand side, we study global Holder and Schauder estimates for a fractional Bessel equation. Our methods stand on the so-called semigroup language. Indeed, by using the solution to the Bessel heat equation we derive pointwise formulas for the fractional operators. Appropriate Holder spaces, which can be seen as Campanato-type spaces, are characterized through Bessel harmonic extensions and fractional Carleson measures. From here the regularity estimates for the fractional Bessel equations follow. In particular, we obtain regularity estimates for radial solutions to the fractional Laplacian.

Calderón–Zygmund operators in the Bessel setting

is bounded from L(Ω, μ) into itself, for every 1 < p < ∞. As far as we know there is not a result showing the behavior of T∗ on L (Ω, μ) for every diffusion semigroup {Tt}t>0. The behavior of T∗ on L(Ω, μ) must be established by taking into account the intrinsic properties of {Tt}t>0. The usual result says that T∗ is bounded from L(Ω, μ) into L1,∞(Ω, μ), but not bounded from L(Ω, μ) into L(Ω, μ). In order to analyze T∗ in L (Ω, μ), in many cases this maximal operator is controlled by a Hardy-Littlewood type maximal operator, and also, the vector valued CalderónZygmund theory ([14]) can be used. These procedures have been employed to study the maximal operators associated to the classical heat semigroup [17, p. 57], to Hermite operators ([10], [15] and [20]), to Laguerre operators ([8], [9], [10], [13] and [19]), to Bessel operators ([1], [2], [3], [11] and [18]) and to Jacobi operators ([11] and [12]), amongst others.

Spectral Multipliers for Multidimensional Bessel Operators

In this paper we develop the theory of variable exponent Hardy spaces associated with discrete Laplacians on infinite graphs. Our Hardy spaces are defined by square integrals, atomic and molecular decompositions. Also we study boundedness properties of Littlewood-Paley functions, Riesz transforms, and spectral multipliers for discrete Laplacians on variable exponent Hardy spaces.