Luz Roncal

Quantitative weighted estimates for rough homogeneous singular integrals

We consider homogeneous singular kernels, whose angular part is bounded, but need not have any continuity. For the norm of the corresponding singular integral operators on the weighted space L2(w), we obtain a bound that is quadratic in A2 constant

Full length article: The Bochner-Riesz means for Fourier-Bessel expansions: Norm inequalities for the maximal operator and almost everywhere convergence

{\left[ w \right]_{{A_2}}}

On sharp heat and subordinated kernel estimates in the Fourier- Bessel setting

[w]A2. We do not know if this is sharp, but it is the best known quantitative result for this class of operators. The proof relies on a classical decomposition of these operators into smooth pieces, for which we use a quantitative elaboration of Laceys dyadic decomposition of Dini-continuous operators: the dependence of constants on the Dini norm of the kernels is crucial to control the summability of the series expansion of the rough operator. We conclude with applications and conjectures related to weighted bounds for powers of the Beurling transform.

The Riesz Transform for the Harmonic Oscillator in Spherical Coordinates

It is well known that the fundamental solution of

An Extension Problem and Trace Hardy Inequality for the Sublaplacian on H-Type Groups

{u_t}\left( {n,t} \right) = u\left( {n + 1,t} \right) - 2u\left( {n,t} \right) + u\left( {n - 1,t} \right),n \in \mathbb{Z},

The multiplier of the interval [−1, 1] for the Dunkl transform of arbitrary order on the real line

ut(n,t)=u(n+1,t)−2u(n,t)+u(n−1,t),n∈ℤ, with u(n, 0) = δnm for every fixed m ∈ Z is given by u(n, t) = e−2tIn−m(2t), where Ik(t) is the Bessel function of imaginary argument. In other words, the heat semigroup of the discrete Laplacian is described by the formal series Wtf(n) = Σm∈Ze−2tIn−m(2t)f(m). This formula allows us to analyze some operators associated with the discrete Laplacian using semigroup theory. In particular, we obtain the maximum principle for the discrete fractional Laplacian, weighted ℓp(Z)-boundedness of conjugate harmonic functions, Riesz transforms and square functions of Littlewood-Paley. We also show that the Riesz transforms essentially coincide with the so-called discrete Hilbert transform defined by D. Hilbert at the beginning of the twentieth century. We also see that these Riesz transforms are limits of the conjugate harmonic functions. The results rely on a careful use of several properties of Bessel functions.