José L. Torrea

Extension Problem and Harnack's Inequality for Some Fractional Operators

The fractional Laplacian can be obtained as a Dirichlet-to-Neumann map via an extension problem to the upper half space. In this paper we prove the same type of characterization for the fractional powers of second order partial differential operators in some class. We also get a Poisson formula and a system of Cauchy–Riemann equations for the extension. The method is applied to the fractional harmonic oscillator H σ = (− Δ + |x|2)σ to deduce a Harnacks inequality. A pointwise formula for H σ f(x) and some maximum and comparison principles are derived.

Poisson integrals and Riesz transforms for Hermite function expansions with weights

We consider expansions with respect to the multi-dimensional Hermite functions which are eigenfunctions of the harmonic oscillator L=−Δ+|x|2. For the heat-diffusion and Poisson semigroups corresponding to a self-adjoint extension L of L we investigate their boundary behaviour and mapping properties. All this is done for functions from Lp(w), 1⩽p<∞, w∈Ap. Then Riesz transforms and conjugate Poisson integrals are considered. The Riesz transforms occur to be Calderon–Zygmund operators hence their mapping properties follow by using results from a general theory.

Weighted inequalities for commutators of fractional and singular integrals

We dedicate this paper to the memory of José Luis Rubio de Francia, who developed the theory of extrapolation and gave beautiful applications of vectorial methods in harmonic analysis . Through this paper we shall work on Rn, endowed with the Lebesgue measure . Given a Banach space E we shall denote by LE(Rn) or LE the BochnerLebesgue space of E-valued strongly measurable functions such that

Higher-order Riesz operators for the Ornstein-Uhlenbeck semigroup

We prove that the second-order Riesz transforms associated to the Ornstein–Uhlenbeck semigroup are weak type (1,1) with respect to the Gaussian measure in finite dimension. We also show that they are given by a principal value integral plus a constant multiple of the identity. For the Riesz transforms of order three or higher, we present a counterexample showing that the weak type (1,1) estimate fails.

Spectral multipliers for the Ornstein-Uhlenbeck semigroup

The setting of this paper is Euclidean space with the Gaussian measure. We letL be the associated Laplacian, by means of which the Ornstein-Uhlenbeck semigroup is defined. The main result is a multiplier theorem, saying that a function ofL which is of Laplace transform type defines an operator of weak type (1,1) for the Gaussian measure. The (distribution) kernel of this operator is determined, in terms of an integral involving the kernel of the Ornstein-Uhlenbeck semigroup. This applies in particular to the imaginary powers ofL. It is also verified that the weak type constant of these powers increases exponentially with the absolute value of the exponent.

Dimension free estimates for the oscillation of riesz transforms

In this paper we establish dimension freeLp(ℝn,|x|α) norm inequalities (1<p<∞) for the oscillation and variation of the Riesz transforms in ℝn. In doing so we findAp-weighted norm inequalities for the oscillation and the variation of the Hilbert transform in ℝ. Some weighted transference results are also proved.

Higher order commutators for vector-valued Calderón-Zygmund operators

Weighted norm estimates for higher order commutators are obtained. The proof, that remain valid in the vector-valued case, are obtained as an application of some extrapolation results. The vector-valued version of the commutator theorem is applied to the Carleson operator, U.M.D. Banach spaces, approximate identities and maximal operators

A vector-valued approach to tent spaces

In this paper we present a new point of view to study the tent spaces introduced by Coifman, Meyer and Stein ([CMS1] and [CMS2]) by immersing them into vector-valued Lebesgue, Hardy and BMO spaces. This approach allows us to derive many of the known properties for tent spaces in a very simple manner. In fact most of the results are obtained as a consequence of similar results for those vector-valued spaces where they are immersed. The main tool we use to obtain such immersions and the applications to boundedness of operators, given in § 4, is the vector-valued Calderón-Zygmund theory.

Maximal operators for the holomorphic Ornstein-Uhlenbeck semigroup

For each p in [1, ∞) let Ep denote the closure of the region of holomorphy of the Ornstein-Uhlenbeck semigroup {Ht : t> 0} on L p with respect to the Gaussian measure. We prove sharp weak type and strong type estimates for the maximal operator f �→ H ∗ f = sup{|Hzf | : z ∈ Ep} and for a class of related operators. As a consequence of our methods, we give a new and simpler proof of the weak type 1 estimate for the maximal operator associated to the Mehler kernel.

Riesz transforms related to Bessel operators

Riesz transforms