arXiv subjects

P. R. Stinga

Publications and source records attributed to P. R. Stinga.

43 records · Page 3Linked to original sources

arXiv2012

Regularity estimates in Hölder spaces for Schrödinger operators via a T1 theorem

We derive Hölder regularity estimates for operators associated with a time independent Schrödinger operator of the form $-Δ+V$. The results are obtained by checking a certain condition on the function $T1$. Our general method applies to get regularity estimates for maximal operators and square functions of the heat and Poisson semigroups, for Laplace transform type multipliers and also for Riesz transforms and negative powers $(-Δ+V)^{-γ/2}$, all of them in a unified way.

math.AP↗

arXiv2012

Harnack's inequality for fractional nonlocal equations

We prove interior Harnack's inequalities for solutions of fractional nonlocal equations. Our examples include fractional powers of divergence form elliptic operators with potentials, operators arising in classical orthogonal expansions and the radial Laplacian. To get the results we use an analytic method based on a generalization of the Caffarelli--Silvestre extension problem, the Harnack's inequality for degenerate Schrödinger operators proved by C. E. Gutiérrez, and a transference method. In this manner we apply local PDE techniques to nonlocal operators. On the way a maximum principle and a Liouville theorem for some fractional nonlocal equations are obtained.

math.AP↗

arXiv2011

Regularity properties of Schrödinger operators

Let L be a Schrödinger operator of the form L=-Δ+V, where the nonnegative potential V satisfies a reverse Hölder inequality. Using the method of L-harmonic extensions we study regularity estimates at the scale of adapted Hölder spaces. We give a pointwise description of L-Hölder spaces and provide some characterizations in terms of the growth of fractional derivatives of any order and Carleson measures. Applications to fractional powers of L and multipliers of Laplace transform type developed.

math.AP↗

arXiv2011

A T1 criterion for Hermite-Calderon-Zygmund operators on the BMO_H(R^n) space and applications

In this paper we establish a T1 criterion for the boundedness of Hermite-Calderon-Zygmund operators on the BMO_H(R^n) space naturally associated to the Hermite operator H. We apply this criterion in a systematic way to prove the boundedness on BMO_H(R^n) of certain harmonic analysis operators related to H (Riesz transforms, maximal operators, Littlewood-Paley g-functions and variation operators).

math.CA↗

arXiv2011

Regularity theory for the fractional harmonic oscillator

In this paper we develop the theory of Schauder estimates for the fractional harmonic oscillator $H^σ=(-Δ+|x|^2)^σ$, $0<σ<1$. More precisely, a new class of smooth functions $C^{k,α}_H$ is defined, in which we study the action of $H^σ$. It turns out that these spaces are the suited ones for this type of regularity estimates. In order to prove our results, an analysis of the interaction of the Hermite-Riesz transforms with the Hölder spaces $C^{k,α}_H$ is needed, that we believe of independent interest. The parallel results for the fractional powers of the Laplacian $(-Δ)^σ$ were applied by Caffarelli, Salsa and Silvestre to the study of the regularity of the obstacle problem for the fractional Laplacian.

math.AP↗

arXiv2011

Square functions associated to Schrodinger operators

We characterize geometric properties of Banach spaces in terms of boundedness of square functions associated to general Schrodinger operators of the form $L=-Δ+V$, where the nonnegative potential $V$ satisfies a reverse Holder inequality. The main idea is to sharpen the well known localization method introduced by Z. Shen. Our results can be regarded as alternative proofs of the boundedness in $H^1$, $L^p$ and $BMO$ of classical $L$-square functions.

math.CA↗

arXiv2010

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 Harnack's inequality. A pointwise formula for $H^σf(x)$ and some maximum and comparison principles are derived.

math.AP↗