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J. L. Torrea

Publications and source records attributed to J. L. Torrea.

At least 19 recordsLinked to original sources
arXiv2022

Fractional powers of first order differential operators and new families of polynomials associated to inverse measures

First, we establish the theory of fractional powers of first order differential operators with zero order terms, obtaining PDE properties and analyzing the corresponding fractional Sobolev spaces. In particular, our study shows that Lebesgue and Sobolev spaces with inverse measures (like the inverse Gaussian measure) play a fundamental role in the theory of fractional powers of the first order operators. Second, and motivated in part by such a theory, we lay out the foundations for the development of the harmonic analysis for \emph{inverse} measures. We discover new families of polynomials related to the inverse Gaussian, Laguerre, and Jacobi measures, and characterize them using generating and Rodrigues formulas, and three-term recurrence relations. Moreover, we prove boundedness of several fundamental singular integral operators in these inverse measure settings.

math.CA↗
arXiv2020

Vector Valued Transference

Our principal result is the following. Let $X$ and $Y$ be Banach spaces, let $G$ be a locally compact abelian group, and let $K$ be an operator valued kernel defined on $G$ with values in the space of bounded linear operators from $X$ to $Y$. Suppose that $R$ and $\tilde{R}$ are representations of $G$ on $X$ and $Y$ respectively that intertwine the values of $K$. Then, under suitable boundedness conditions on $R, \tilde{R}$ and $K$, the formula $$T_Kx = \int_GK(u)R_{-u}xdu $$ defines a bounded linear operator $T_K$ from $X$ to $Y$ with norm controlled by norm of convolution by $K$ as a mapping from $L^p_X(G)$ into $L^p_Y(G)$, (for all values of $p$ in the range $1\le p < \infty$.) A number of applications to the geometry of Banach spaces are given. Several results are proved in the setting of abstract commutative harmonic analysis. We outline the proof of the affirmative resolution of a conjecture of Rubio de Francia. This technique of transference is used to obtain dimension free estimates for certain operators in an $\Bbb{R}^n$ setting.

math.CA↗
arXiv2019

Hölder, Sobolev, weak-type and $BMO$ estimates in mixed-norm with weights for parabolic equations

We prove weighted mixed-norm $L^q_t(W^{2,p}_x)$ and $L^q_t(C^{2,α}_x)$ estimates for $1 0,~x\in\mathbb{R}^n \\ v(0,x)=g&\hbox{for}~x\in\mathbb{R}^n. \end{cases}$$ The coefficients $a(t)=(a^{ij}(t))$ are just bounded, measurable, symmetric and uniformly elliptic. Furthermore, we show strong, weak type and $BMO$-Sobolev estimates with parabolic Muckenhoupt weights. It is quite remarkable that most of our results are new even for the classical heat equation $$\partial_tu-Δu+u=f.$$

math.AP↗
arXiv2018

Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications

The analysis of nonlocal discrete equations driven by fractional powers of the discrete Laplacian on a mesh of size $h>0$ \[ (-Δ_h)^su=f, \] for $u,f:\mathbb{Z}_h\to\mathbb{R}$, $0<s<1$, is performed. The pointwise nonlocal formula for $(-Δ_h)^su$ and the nonlocal discrete mean value property for discrete $s$-harmonic functions are obtained. We observe that a characterization of $(-Δ_h)^s$ as the Dirichlet-to-Neumann operator for a semidiscrete degenerate elliptic local extension problem is valid. Regularity properties and Schauder estimates in discrete Hölder spaces as well as existence and uniqueness of solutions to the nonlocal Dirichlet problem are shown. For the latter, the fractional discrete Sobolev embedding and the fractional discrete Poincaré inequality are proved, which are of independent interest. We introduce the negative power (fundamental solution) \[ u=(-Δ_h)^{-s}f, \] which can be seen as the Neumann-to-Dirichlet map for the semidiscrete extension problem. We then prove the discrete Hardy--Littlewood--Sobolev inequality for $(-Δ_h)^{-s}$. As applications, the convergence of our fractional discrete Laplacian to the (continuous) fractional Laplacian as $h\to0$ in Hölder spaces is analyzed. Indeed, uniform estimates for the error of the approximation in terms of $h$ under minimal regularity assumptions are obtained. We finally prove that solutions to the Poisson problem for the fractional Laplacian \[ (-Δ)^sU=F, \] in $\mathbb{R}$, can be approximated by solutions to the Dirichlet problem for our fractional discrete Laplacian, with explicit uniform error estimates in terms of~$h$.

math.AP↗
arXiv2017

Regularity theory and extension problem for fractional nonlocal parabolic equations and the master equation

We develop the regularity theory for solutions to space-time nonlocal equations driven by fractional powers of the heat operator $$(\partial_t-Δ)^su(t,x)=f(t,x),\quad\hbox{for}~0<s<1.$$ This nonlocal equation of order $s$ in time and $2s$ in space arises in Nonlinear Elasticity, Semipermeable Membranes, Continuous Time Random Walks and Mathematical Biology. It plays for space-time nonlocal equations like the generalized master equation the same role as the fractional Laplacian for nonlocal in space equations. We obtain a pointwise integro-differential formula for $(\partial_t-Δ)^su(t,x)$ and parabolic maximum principles. A novel extension problem to characterize this nonlocal equation with a local degenerate parabolic equation is proved. We show parabolic interior and boundary Harnack inequalities, and an Almgrem-type monotonicity formula. Hölder and Schauder estimates for the space-time Poisson problem are deduced using a new characterization of parabolic Hölder spaces. Our methods involve the \textit{parabolic language of semigroups} and the Cauchy Integral Theorem, which are original to define the fractional powers of $\partial_t-Δ$. Though we mainly focus in the equation $(\partial_t-Δ)^su=f$, applications of our ideas to variable coefficients, discrete Laplacians and Riemannian manifolds are stressed out.

math.AP↗
arXiv2017

On weighted mixed-norm Sobolev estimates for some basic parabolic equations

Novel global weighted parabolic Sobolev estimates, weighted mixed-norm estimates and a.e. convergence results of singular integrals for evolution equations are obtained. Our results include the classical heat equation, the harmonic oscillator evolution equation $$\partial_tu=Δu-|x|^2u+f,$$ and their corresponding Cauchy problems. We also show weighted mixed-norm estimates for solutions to degenerate parabolic extension problems arising in connection with the fractional space-time nonlocal equations $(\partial_t-Δ)^su=f$ and $(\partial_t-Δ+|x|^2)^su=f$, for $0<s<1$.

math.AP↗
arXiv2015

Maximum principles, extension problem and inversion for nonlocal one-sided equations

We study one-sided nonlocal equations of the form $$\int_{x_0}^\infty\frac{u(x)-u(x_0)}{(x-x_0)^{1+α}} dx=f(x_0),$$ on the real line. Notice that to compute this nonlocal operator of order $0<α<1$ at a point $x_0$ we need to know the values of $u(x)$ to the right of $x_0$, that is, for $x\geq x_0$. We show that the operator above corresponds to a fractional power of a one-sided first order derivative. Maximum principles and a characterization with an extension problem in the spirit of Caffarelli--Silvestre and Stinga--Torrea are proved. It is also shown that these fractional equations can be solved in the general setting of weighted one-sided spaces. In this regard we present suitable inversion results. Along the way we are able to unify and clarify several notions of fractional derivatives found in the literature.

math.AP↗
arXiv2015

Fractional discrete Laplacian versus discretized fractional Laplacian

We define and study some properties of the fractional powers of the discrete Laplacian $$(-Δ_h)^s,\quad\hbox{on}~\mathbb{Z}_h = h\mathbb{Z},$$ for $h>0$ and $0<s<1$. A comparison between our fractional discrete Laplacian and the \textit{discretized} continuous fractional Laplacian as $h\to0$ is carried out. We get estimates in $\ell^\infty$ for the error of the approximation in terms of $h$ under minimal regularity assumptions. Moreover, we provide a pointwise formula with an explicit kernel and deduce Hölder estimates for $(-Δ_h)^s$. A study of the negative powers (or discrete fractional integral) $(-Δ_h)^{-s}$ is also sketched. Our analysis is mainly performed in dimension one. Nevertheless, we show certain asymptotic estimates for the kernel in dimension two that can be extended to higher dimensions. Some examples are plotted to illustrate the comparison in both one and two dimensions.

math.AP↗
arXiv2014

Harmonic Analysis associated with a discrete Laplacian

It is well-known that the fundamental solution of $$ u_t(n,t)= u(n+1,t)-2u(n,t)+u(n-1,t), \quad n\in\mathbb{Z}, $$ with $u(n,0) =δ_{nm}$ for every fixed $m \in\mathbb{Z}$, is given by $u(n,t) = e^{-2t}I_{n-m}(2t)$, where $I_k(t)$ is the Bessel function of imaginary argument. In other words, the heat semigroup of the discrete Laplacian is described by the formal series $$ W_tf(n) = \sum_{m\in\mathbb{Z}} e^{-2t} I_{n-m}(2t) f(m). $$ By using semigroup theory, this formula allows us to analyze some operators associated with the discrete Laplacian. In particular, we obtain the maximum principle for the discrete fractional Laplacian, weighted $\ell^p(\mathbb{Z})$-boundedness of conjugate harmonic functions, Riesz transforms and square functions of Littlewood-Paley. Interestingly, it is shown that the Riesz transforms coincide essentially with the so called discrete Hilbert transform defined by D. Hilbert at the beginning of the XX 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.

math.CA↗
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↗
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↗
arXiv2009

On the boundary convergence of solutions to the Hermite-Schrödinger equation

In the half-space $\mathbb{R}^d \times \mathbb{R}_+$, we consider the Hermite-Schrödinger equation $i\partial u/\partial t = - Δu + |x|^2 u$, with given boundary values on $\mathbb{R}^d$. We prove a formula that links the solution of this problem to that of the classical Schrödinger equation. It shows that mixed norm estimates for the Hermite-Schrödinger equation can be obtained immediately from those known in the classical case. In one space dimension, we deduce sharp pointwise convergence results at the boundary, by means of this link.

math.AP↗
arXiv2008

The Littlewood--Paley--Rubio de Francia property of a Banach space for the case of equal intervals

Let $X$ be a Banach space. It is proved that an analogue of the Rubio de Francia square function estimate for partial sums of the Fourier series of $X$-valued functions holds true for all disjoint collections of subintervals of the set of integers of equal length and for all exponents $p$ greater or equal than 2 if and only if the space $X$ is a UMD space of type 2. The same criterion is obtained for the case of subintervals of the real line and Fourier integrals instead of Fourier series.

math.FA↗