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We obtain sharp interior Schauder estimates for solutions to nonlocal Poisson problems driven by fractional powers of nondivergence form elliptic operators $(-a^{ij}(x) \partial_{ij})^s$, for $0

We prove sharp characterizations of higher order fractional powers $(-L)^s$, where $s>0$ is noninteger, ofgenerators $L$ of uniformly bounded $C_0$-semigroups on Banach spaces via extension problems, which in particular include results of Caffarelli-Silvestre, Stinga-Torrea and Gal\'e-Miana-Stinga when $00$, in terms of boundary derivatives of $U$ at $y=0$, under the sharp hypothesis that $u$ is in the domain of $(-L)^s$. Our results resolve the question of setting up the correct initial conditions that guarantee well-posedness of both extension problems. Furthermore, we discover new explicit subordination formulas for the solution $U$ in terms of the semigroup $\{e^{tL}\}_{t\geq0}$ generated by $L$., Comment: 20 pages. To appear in Journal of Functional Analysis

We develop an analytic theory of existence and regularity of surfaces (given by graphs) arising from the geometric minimization problem $$\min_{\mathcal{M}}\frac{1}{2}\int_{\mathcal{M}}|\nabla_{\mathcal{M}}H|^2\,dA$$ where $\mathcal{M}$ ranges over all $n$-dimensional manifolds in $\mathbb{R}^{n+1}$ with prescribed boundary, $\nabla_{\mathcal{M}}H$ is the tangential gradient along $\mathcal{M}$ of the mean curvature $H$ of $\mathcal{M}$ and $dA$ is the differential of surface area. The minimizers, called surfaces of minimum mean curvature variation, are central in applications of computer-aided design, computer-aided manufacturing and mechanics. Our main results show the existence of both smooth surfaces and of variational solutions to the minimization problem together with geometric regularity results. These are the first analytic results available on the literature for this problem., Comment: 13 pages. To appear in Archive for Rational Mechanics and Analysis

This paper, that will appear in the Notices of the AMS, begins with a brief historical account of the beginnings of fractional calculus and the crucial roles played by Leibniz and Fourier. Fourier's definition of fractional derivative is introduced and unpacked by using the modern technique known as the method of semigroups. Recent advances on the theory of fractional derivatives are presented. Furthermore, we address some questions that have been raised by some in the scientific community. Finally, we present three different applications: population growth with memory, viscoleastic materials and anomalous diffusions., Comment: 14 pages, 2 figures. To appear in Notices of the American Mathematical Society

We develop the regularity theory of viscosity solutions to transmission problems for fully nonlinear second order uniformly elliptic equations. Our results give a complete theory of existence, uniqueness, comparison principle, and regularity of solutions to flat interface transmission problems; and the $C^{0,\alpha}$, $C^{1,\alpha}$ and $C^{2,\alpha}$ regularity of viscosity solutions up to the transmission surface for the case of curved interfaces., Comment: 38 pages. To appear in Advances in Mathematics

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., Comment: 33 pages

We define the fractional powers $L^s=(-a^{ij}(x)\partial_{ij})^s$, $0 < s < 1$, of nondivergence form elliptic operators $L=-a^{ij}(x)\partial_{ij}$ in bounded domains $\Omega\subset\mathbb{R}^n$, under minimal regularity assumptions on the coefficients $a^{ij}(x)$ and on the boundary $\partial\Omega$. We show that these fractional operators appear in several applications such as fractional Monge--Amp\`ere equations, elasticity, and finance. The solution $u$ to the nonlocal Poisson problem $$\begin{cases} (-a^{ij}(x) \partial_{ij})^su = f&\hbox{in}~\Omega\\ u=0&\hbox{on}~\partial\Omega \end{cases}$$ is characterized by a local degenerate/singular extension problem. We develop the method of sliding paraboloids in the Monge--Amp\`ere geometry and prove the interior Harnack inequality and H\"older estimates for solutions to the extension problem when the coefficients $a^{ij}(x)$ are bounded, measurable functions. This in turn implies the interior Harnack inequality and H\"older estimates for solutions $u$ to the fractional problem., Comment: 55 pages. To appear in Journal de Math\'ematiques Pures et Appliqu\'ees

We study existence, uniqueness, and optimal regularity of solutions to transmission problems for harmonic functions with $C^{1,\alpha}$ interfaces. For this, we develop a novel geometric stability argument based on the mean value property., Comment: 25 pages, 2 figures

We obtain sharp parabolic interior and global Schauder estimates for solutions to nonlocal space-time master equations $(\partial_t +L)^su = f$ in $\mathbb{R} \times \Omega$, where $L$ is an elliptic operator in divergence form, subject to homogeneous Dirichlet and Neumann boundary conditions. In particular, we establish the precise behavior of solutions near the boundary. Along the way, we prove a characterization of the correct intermediate parabolic H\"older spaces in the spirit of Sergio Campanato., Comment: 49 pages. To appear in Journal of Evolution Equations

We characterize one-sided weighted Sobolev spaces $W^{1,p}(\mathbb{R},\omega)$, where $\omega$ is a one-sided Sawyer weight, in terms of a.e.~and weighted $L^p$ limits as $\alpha\to1^-$ of Marchaud fractional derivatives of order $\alpha$. Similar results for weighted Sobolev spaces $W^{2,p}(\mathbb{R}^n,\nu)$, where $\nu$ is an $A_p$-Muckenhoupt weight, are proved in terms of limits as $s\to1^-$ of fractional Laplacians $(-\Delta)^s$. These are Bourgain--Brezis--Mironescu-type characterizations for weighted Sobolev spaces. We also complement their work by studying a.e.~and weighted $L^p$ limits as $\alpha,s\to0^+$., Comment: 28 pages. We are grateful to Francisco J. Mart\'in-Reyes for pointing out to us that Lemma 1.3 was not correct, so we erased it and updated the related results accordingly. To appear in Nonlinear Analysis

The \textit{method of semigroups} is a unifying, widely applicable, general technique to formulate and analyze fundamental aspects of fractional powers of operators $L$ and their regularity properties in related functional spaces. The approach was introduced by the author and Jos\'e L.~Torrea in 2009 (arXiv:0910.2569v1). The aim of this chapter is to show how the method works in the particular case of the fractional Laplacian $L^s=(-\Delta)^s$, $0

We prove weighted mixed-norm $L^q_t(W^{2,p}_x)$ and $L^q_t(C^{2,\alpha}_x)$ estimates for $10,~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-\Delta u+u=f.$$, Comment: 27 pages. To appear in Science China Mathematics

We study master equations of the form $$(\partial_t+L)^su=f\quad\hbox{in}~\mathbb{R}\times\Omega$$ where $L$ is a divergence form elliptic operator and $\Omega\subseteq\mathbb{R}^n$. These are nonlocal equations of order $2s$ in space and $s$ in time that take into account the values of $u$ everywhere in $\Omega$ and for past times. We show parabolic interior and boundary Harnack inequalities and local parabolic H\"older continuity of solutions. To this end, we prove a characterization of fractional powers of parabolic operators $\partial_t+L$ with a degenerate parabolic extension problem., Comment: 28 pages. To appear in SIAM Journal of Mathematical Analysis

This article takes at heart Boolos' intentions when presenting `The Hardest Logic Puzzle Ever.' We stress the fact that Boolos instructs us to solve the puzzle by asking three yes-no questions. It is a requirement that all the gods, including Random -whose behavior is determined by the outcome of flipping a coin in its head-, are always obliged to answer. Moreover, Boolos' solution implies that the meaning of `Da' and `Ja' is irrelevant to solve the puzzle, and, at the same time, that all of this would be in virtue of the irreducible fundamentality of `the Law of the Excluded Middle.' The purpose of this paper is, then, twofold. First, we prove that Boolos' original puzzle cannot be solved, in an absolute deterministic way, in less than three yes-no \textit{admissible} questions. This does not mean that one could identify the gods' identities in less than three questions, which chances, second, we compute as well., Comment: 10 pages, reference added, new Section 4

We study the obstacle problem for a nonlocal, degenerate elliptic Monge--Amp\`ere equation. We show existence and regularity of a unique classical solution to the problem and regularity of the free boundary., Comment: 23 pages. To appear in Communications in Partial Differential Equations

In this paper we show novel underlying connections between fractional powers of the Laplacian on the unit sphere and functions from analytic number theory and differential geometry, like the Hurwitz zeta function and the Minakshisundaram zeta function. Inspired by Minakshisundaram's ideas, we find a precise pointwise description of $(-\Delta_{\mathbb{S}^{n-1}})^s u(x)$ in terms of fractional powers of the Dirichlet-to-Neumann map on the sphere. The Poisson kernel for the unit ball will be essential for this part of the analysis. On the other hand, by using the heat semigroup on the sphere, additional pointwise integro-differential formulas are obtained. Finally, we prove a characterization with a local extension problem and the interior Harnack inequality., Comment: 23 pages. To appear in Contemporary Mathematics

For $0

The analysis of nonlocal discrete equations driven by fractional powers of the discrete Laplacian on a mesh of size $h>0$ \[ (-\Delta_h)^su=f, \] for $u,f:\mathbb{Z}_h\to\mathbb{R}$, $0

The fractional nonlocal linearized Monge--Amp\`ere equation is introduced. A Harnack inequality for nonnegative solutions to the Poisson problem on Monge--Amp\`ere sections is proved., Comment: 46 pages, 2 figures. To appear in Calculus of Variations and Partial Differential 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 $$\partial_tu=\Delta u-|x|^2u+f,$$ 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 $(\partial_t-\Delta)^su=f$ and $(\partial_t-\Delta+|x|^2)^su=f$, for $0

We develop the regularity theory for solutions to space-time nonlocal equations driven by fractional powers of the heat operator $$(\partial_t-\Delta)^su(t,x)=f(t,x),\quad\hbox{for}~0

We define and study some properties of the fractional powers of the discrete Laplacian $$(-\Delta_h)^s,\quad\hbox{on}~\mathbb{Z}_h = h\mathbb{Z},$$ for $h>0$ and $0

We study one-sided nonlocal equations of the form $$\int_{x_0}^\infty\frac{u(x)-u(x_0)}{(x-x_0)^{1+\alpha}} dx=f(x_0),$$ on the real line. Notice that to compute this nonlocal operator of order $0<\alpha<1$ 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., Comment: 20 pages. To appear in Journal of Differential Equations

Let $L=-\operatorname{div}_x(A(x)\nabla_x)$ be a uniformly elliptic operator in divergence form in a bounded domain $\Omega$. We consider the fractional nonlocal equations $$\begin{cases} L^su=f,&\hbox{in}~\Omega,\\ u=0,&\hbox{on}~\partial\Omega, \end{cases}\quad \hbox{and}\quad \begin{cases} L^su=f,&\hbox{in}~\Omega,\\ \partial_Au=0,&\hbox{on}~\partial\Omega. \end{cases}$$ Here $L^s$, $0

We consider the following fractional semilinear Neumann problem on a smooth bounded domain $\Omega\subset\mathbb{R}^n$, $n\geq2$, $$\begin{cases} (-\varepsilon\Delta)^{1/2}u+u=u^{p},&\hbox{in}~\Omega,\\ \partial_\nu u=0,&\hbox{on}~\partial\Omega,\\ u>0,&\hbox{in}~\Omega, \end{cases}$$ where $\varepsilon>0$ and $1

In this note we show how to obtain regularity estimates for the fractional Laplacian on the multidimensional torus $\mathbb{T}^n$ from the fractional Laplacian on $\mathbb{R}^n$. Though at first glance this may seem quite natural, it must be carefully precised. A reason for that is the simple fact that $L^2$ functions on the torus can not be identified with $L^2$ functions on $\mathbb{R}^n$. The transference is achieved through a formula that holds in the distributional sense. Such an identity allows us to transfer Harnack inequalities, to relate the extension problems, and to obtain pointwise formulas and H\"older regularity estimates., Comment: 7 pages. To appear in Special Functions, Partial Differential Equations and Harmonic Analysis. Proceedings of the conference in honor of Calixto P. Calder\'on, Roosevelt University at Chicago, November 16-18, 2012. C. Georgakis, A. Stokolos and W. Urbina (eds)

In this paper we prove mixed norm estimates for Riesz transforms related to Laplace--Beltrami operators on compact Riemannian symmetric spaces of rank one. These operators are closely related to the Riesz transforms for Jacobi polynomials expansions. The key point is to obtain sharp estimates for the kernel of the Jacobi--Riesz transforms with uniform control on the parameters, together with an adaptation of Rubio de Francia's extrapolation theorem. The latter results are of independent interest., Comment: 19 pages. To appear in Milan Journal of Mathematics

Motivated by the Poisson equation for the fractional Laplacian on the whole space with radial right hand side, we study global H\"older 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 H\"older 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., Comment: 36 pages. To appear in Journal of Approximation Theory

We study Hardy spaces for Fourier--Bessel expansions associated with Bessel operators on $((0,1), x^{2\nu+1}\, dx)$ and $((0,1), dx)$. We define Hardy spaces $H^1$ as the sets of $L^1$-functions for which their maximal functions for the corresponding Poisson semigroups belong to $L^1$. Atomic characterizations are obtained., Comment: 21 pages, 2 figures. To appear in Journal d'Analyse Mathematique

We study the fractional Laplacian $(-\Delta)^{\sigma/2}$ on the $n$-dimensional torus $\mathbb{T}^n$, $n\geq1$. First, we present a general extension problem that describes \textit{any} fractional power $L^\gamma$, $\gamma>0$, where $L$ is a general nonnegative selfadjoint operator defined in an $L^2$-space. This generalizes to all $\gamma>0$ and to a large class of operators the previous known results by Caffarelli and Silvestre. In particular it applies to the fractional Laplacian on the torus. The extension problem is used to prove interior and boundary Harnack's inequalities for $(-\Delta)^{\sigma/2}$, when $0<\sigma<2$. We deduce regularity estimates on H\"older, Lipschitz and Zygmund spaces. Finally, we obtain the pointwise integro-differential formula for the operator. Our method is based on the semigroup language approach., Comment: 18 pages, 2 figures. To appear in Communications in Contemporary Mathematics

In this paper we show that the rate of convergence of Wong-Zakai approximations for stochastic partial differential equations driven by Wiener processes is essentially the same as the rate of convergence of the driving processes W_n approximating the Wiener process, provided the area processes of W_n also converge to those of W with that rate. We consider non-degenerate and also degenerate stochastic PDEs with time dependent coefficients., Comment: 27 pages. To appear in the Proceedings of the Seventh Seminar on Stochastic Analysis, Random Fields and Applications, Ascona 2011

We extend results of Caffarelli--Silvestre and Stinga--Torrea regarding a characterization of fractional powers of differential operators via an extension problem. Our results apply to generators of integrated families of operators, in particular to infinitesimal generators of bounded $C_0$ semigroups and operators with purely imaginary symbol. We give integral representations to the extension problem in terms of solutions to the heat equation and the wave equation., Comment: 19 pages, revised version, appendix added. To appear in Journal of Evolution Equations

The weighted Lebesgue spaces of initial data for which almost everywhere convergence of the heat equation holds was only very recently characterized. In this note we show that the same weighted space of initial data is optimal for the heat--diffusion parabolic equations involving the harmonic oscillator and the Ornstein--Uhlenbeck operator., Comment: 6 pages. To appear in Potential Analysis

In this paper we study mixed norm boundedness for fractional integrals related to Laplace--Beltrami operators on compact Riemannian symmetric spaces of rank one. The key point is the analysis of weighted inequalities for fractional integral operators associated to trigonometric Jacobi polynomials expansions. In particular, we find a novel sharp estimate for the Jacobi fractional integral kernel with explicit dependence on the type parameters., Comment: 16 pages. To appear in Advances in Mathematics

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 Schr\"odinger operators proved by C. E. Guti\'errez, 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., Comment: 15 pages, new title, references added. To appear in Discrete and Continuous Dynamical Systems - Series A

We derive H\"older regularity estimates for operators associated with a time independent Schr\"odinger operator of the form $-\Delta+V$. 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 $(-\Delta+V)^{-\gamma/2}$, all of them in a unified way., Comment: 22 pages, revised version. To appear in Annali di Matematica Pura ed Applicata

Let L be a Schr\"odinger operator of the form L=-\Delta+V, where the nonnegative potential V satisfies a reverse H\"older inequality. Using the method of L-harmonic extensions we study regularity estimates at the scale of adapted H\"older spaces. We give a pointwise description of L-H\"older 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., Comment: 20 pages. To appear in Journal of Mathematical Analysis and Applications

We characterize geometric properties of Banach spaces in terms of boundedness of square functions associated to general Schrodinger operators of the form $L=-\Delta+V$, 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., Comment: 17 pages, revised version following referee's comments, to appear in Studia Mathematica

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)., Comment: 20 pages. To appear in Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

In this paper we develop the theory of Schauder estimates for the fractional harmonic oscillator $H^\sigma=(-\Delta+|x|^2)^\sigma$, $0<\sigma<1$. More precisely, a new class of smooth functions $C^{k,\alpha}_H$ is defined, in which we study the action of $H^\sigma$. 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 H\"older spaces $C^{k,\alpha}_H$ is needed, that we believe of independent interest. The parallel results for the fractional powers of the Laplacian $(-\Delta)^\sigma$ were applied by Caffarelli, Salsa and Silvestre to the study of the regularity of the obstacle problem for the fractional Laplacian., Comment: 23 pages, references added, to appear in Journal of Functional Analysis

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 $H^\sigma=(-\Delta+|x|^2)^\sigma$ to deduce a Harnack's inequality. A pointwise formula for $H^\sigma f(x)$ and some maximum and comparison principles are derived., Comment: 24 pages. Main results improved to full generality thanks to referee comments. To appear in Communications in Partial Differential Equations.