User’s guide to the fractional Laplacian and the method of semigroups

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 Jose L.~Torrea in 2009 . The aim of this chapter is to show how the method works in the particular case of the fractional Laplacian $L^s=(-\Delta)^s$, $0<s<1$. The starting point is the semigroup formula for the fractional Laplacian. From here, the classical heat kernel permits us to obtain the pointwise formula for $(-\Delta)^su(x)$. One of the key advantages is that our technique relies on the use of heat kernels, which allows for applications in settings where the Fourier transform is not the most suitable tool. In addition, it provides explicit constants that are key to prove, under minimal conditions on $u$, the validity of the pointwise limits $$\lim_{s\to1^-}(-\Delta)^su(x)=-\Delta u(x)\quad\hbox{and}\quad\lim_{s\to0^+}(-\Delta)^su(x)=u(x).$$ The formula for the solution to the Poisson problem $(-\Delta)^su=f$ is found through the semigroup approach as the inverse of the fractional Laplacian $u(x)=(-\Delta)^{-s}f(x)$ (fundamental solution). We then present the Caffarelli--Silvestre extension problem, whose explicit solution is given by the semigroup formulas that were first discovered by the author and Torrea. With the extension technique, an interior Harnack inequality and derivative estimates for fractional harmonic functions can be obtained. The classical Holder and Schauder estimates $$(-\Delta)^{\pm s}:C^\alpha\to C^{\alpha\mp 2s}$$ are proved with the method of semigroups in a rather quick, elegant way. The crucial point for this will be the characterization of Holder and Zygmund spaces with heat semigroups.

Related papers:

title: Potential Theory on Locally Compact Abelian Groups
abstract: I. Harmonic Analysis.- 1. Notation and Preliminaries.- 2. Some Basic Results From Harmonic Analysis.- 3. Positive Definite  Positive Definite Functions on ?.- 6. Periodicity.- II. Negative Definite Functions and Semigroups.- 7. Negative Definite Functions.- 8. Convolution Semigroups.- 9. Completely Monotone Functions and Bernstein Functions.- 10. Examples of Negative Definite Functions and Convolution Semigroups.- 11. Contraction Semigroups.- 12. Translation Invariant Contraction Semigroups.- III. Potential Theory for Transient Convolution Semigroups.- 13. Transient Convolution Semigroups.- 14. Transient Convolution Semigroups on the Half-Axis and Integrals of Convolution Semigroups.- 15. Convergence Lemmas and Potential Theoretic Principles.- 16. Excessive Measures.- 17. Fundamental Families Associated With Potential Kernels.- 18. The Levy Measure for a Convolution Semigroup.- Symbols.- General Index.

title: Potential theory of subordinate killed Brownian motion in a domain
abstract: Subordination of a killed Brownian motion in a bounded domain D⊂ℝd via an α/2-stable subordinator gives a process Zt whose infinitesimal generator is −(−Δ|D)α/2, the fractional power of the negative Dirichlet Laplacian. In this paper we study the properties of the process Zt in a Lipschitz domain D by comparing the process with the rotationally invariant α-stable process killed upon exiting D. We show that these processes have comparable killing measures, prove the intrinsic ultracontractivity of the generator of Zt, prove the intrinsic ultracontractivity of the semigroup of Zt, and, in the case when D is a bounded C1,1 domain, obtain bounds on the Green function and the jumping kernel of Zt.

title: Harnack Inequalities for Jump Processes
abstract: We consider a class of pure jump Markov processes in Rd whose jump kernels are comparable to those of symmetric stable processes. We establish a Harnack inequality for nonnegative functions that are harmonic with respect to these processes. We also establish regularity for the solutions to certain integral equations.

title: Harmonic Analysis And The Theory Of Probability
abstract: Thank you for downloading harmonic analysis and the theory of probability. As you may know, people have look numerous times for their chosen books like this harmonic analysis and the theory of probability, but end up in malicious downloads. Rather than enjoying a good book with a cup of tea in the afternoon, instead they juggled with some infectious virus inside their computer. harmonic analysis and the theory of probability is available in our digital library an online access to it is set as public so you can download it instantly. Our digital library spans in multiple countries, allowing you to get the most less latency time to download any of our books like this one. Kindly say, the harmonic analysis and the theory of probability is universally compatible with any devices to read.

title: On fractional differentiation and integration on spaces of homogeneous type
abstract: In this paper we define derivatives of fractional order on spaces of homogeneous type by generalizing a classical formula for the fractional powers of the Laplacean [S1], [S2], [SZ] and introducing suitable quasidistances related to an approximation of the identity. We define integration of fractional order as in [GV] but using quasidistances related to the approximation of the identity mentioned before.
We show that these operators act on Lipschitz spaces as in the classical cases. We prove that the composition Ta of a fractional integral Ia and a fractional derivative Da of the same order and its transpose (a fractional derivative composed with a fractional integral of the same order) are Calderon-Zygmund operators. We also prove that for small order a, Ta is an invertible operator in L2. In order to prove that Ta is invertible we obtain Nahmod type representations for Ia and Da and then we follow the method of her thesis [N1], [N2].

title: Hypersingular Integrals and Their Applications
abstract: Part I: Hypersingular Integrals and Their Applications. Some Basics from the Theory of Special Functions and Operator Theory. The Reisz Potential Operator and the Lizorkin Type Invarient Spaces. Hypersingular Integrals with Constant Characteristics Potentials and Hypersingular Integrals with Non-Homogenous Characteristics. Hypersingular Integrals on the Unit Sphere. Part II: Applications of Hypersingular Integrals. Hypersingular Integrals in the Theory of Functional Spaces. Solution of Multidimensional Integral Equations of the First Kind with a Potential Type Kernel. Hypersingular Operators as Positive Fractional Powers of Some Operators of Mathematical Physics. Regularization of Integral Equations of the First Kind with a Potential Type Kernel Applications to Function Spaces on a Sphere. Some Modifications of Hypersingular Integrals and Their Applications.

title: Fractional Integrals and Derivatives: Theory and Applications
abstract: Fractional integrals and derivatives on an interval fractional integrals and derivatives on the real axis and half-axis further properties of fractional integrals and derivatives other forms of fractional integrals and derivatives fractional integrodifferentiation of functions of many variables applications to integral equations of the first kind with power and power-logarithmic kernels integral equations fo the first kind with special function kernels applications to differential equations.

title: SUBORDINATION IN THE SENSE OF BOCHNER AND A RELATED FUNCTIONAL CALCULUS
abstract: We prove a new representation of the generator of a subordinate semigroup as limit of bounded operators. Our construction yields, in particular, a characterization of the domain of the generator. The generator of a subordinate semigroup can be viewed as a function of the generator of the original semigroup. For a large class these functions we show that operations at the level of functions has its counterpart at the level of operators.

title: The solution of the Kato square root problem for second order elliptic operators on Rn
abstract: We prove the Kato conjecture for elliptic operators on Jfin. More precisely, we establish that the domain of the square root of a uniformly complex elliptic operator L =-div (AV) with bounded measurable coefficients in IEtn iS the

title: Topics in Harmonic Analysis Related to the Littlewood-Paley Theory.
abstract: This work deals with an extension of the classical Littlewood-Paley theory in the context of symmetric diffusion semigroups. In this general setting there are applications to a variety of problems, such as those arising in the study of the expansions coming from second order elliptic operators. A review of background material in Lie groups and martingale theory is included to make the monograph more accessible to the student.

title: Singular Integrals and Di?erentiability Properties of Functions
abstract: A plurality of disks include solid laserable material and are in parallel spaced apart relation within a transparent tubular enclosure. Spaces between the disks constitute portions of a collant fluid passage within the tubular enclosure.

title: Regularity of spectral fractional Dirichlet and Neumann problems
abstract: Consider the fractional powers and of the Dirichlet and Neumann realizations of a second-order strongly elliptic differential operator A on a smooth bounded subset Ω of . Recalling the results on complex powers and complex interpolation of domains of elliptic boundary value problems by Seeley in the 1970's, we demonstrate how they imply regularity properties in full scales of -Sobolev spaces and Holder spaces, for the solutions of the associated equations. Extensions to nonsmooth situations for low values of s are derived by use of recent results on -calculus. We also include an overview of the various Dirichlet- and Neumann-type boundary problems associated with the fractional Laplacian.

title: Fractional powers of operators.
abstract: A definition of fractional (or complex) powers A°>, a e C, is given for closed linear operators A in a Banach space X with the resolvent set containing the negative real ray (—oo, 0) and such that {λ(λ + A)"; 0 < λ < oo} is bounded; fundamental properties such as additivity (A*A = AP), coincidence with the iterations A* = A for integers a = n, and analytic dependence on a are discussed. Since the fractional powers A* are generally unbounded in both of the cases Re a > 0 and Re a < 0, attention is paid to the domains D{A), which are related to the spaces D and R of xeX defined by the regularity of (Λ + A)"^ at oo and 0. When —A generates a bounded continuous semi-group or a bounded analytic semi-group, more detailed results are obtained.

title: Analytic Methods In The Theory Of Differential And Pseudo-Differential Equations Of Parabolic Type
abstract: 1 Equations. Problems.    1.1 Differential equations.- 1.2 Pseudo-differential equations.- 1.3 Main lemmas.- 2 Parabolic Equations of a Quasi-Homogeneous Structure.- 2.1 Fundamental solution of the Cauchy problem for equations with bounded coefficients.- 2.2 Cauchy problem for equations with bounded coefficients.- 2.3 Equations with growing coefficients.- 2.4 Equations with degenerations on the initial hyperplane.- 2.5 Comments.- 3 Degenerate Equations of the Kolmogorov Type.- 3.1 Fundamental solution of the Cauchy problem.- 3.2 Cauchy problem.- 3.3 Properties of solutions of the Fokker-Planck-Kolmogorov equations.- 3.4 Comments.- 4 Pseudo-Differential Parabolic Equations with Quasi-Homogeneous Symbols.- 4.1 Fundamental solution of the Cauchy problem.- 4.2 Cauchy problem.- 4.3 On qualitative properties of solutions of some equations with constant symbols.- 4.4 Comments.- 5 Fractional Diffusion Equations.- 5.1 Fractional derivatives.- 5.2 Fundamental solution of the Cauchy problem.- 5.3 The  Existence and representation of solutions.- 5.4 Uniqueness theorems.- 5.5 Comments.- Appendix. Fox's H-Functions.- Notation.

title: Pseudo-Differential Operators and Markov Processes
abstract: Markov processes and pseudo-dofferential operators - why? Levy processes and continuous negative definite functions pseudo-differential operators and Dirichlet forms constructing Feller semi-groups from pseudo-differential operators the martingale problem for pseudo-differential operators subordination in the sense of Bochner transition functions (existence, estimates) path properties and symbol Balayage theory and the Dirichlet problem for pseudo-differential operators.

title: Pseudo-Differential Operators and Markov Processes
abstract: Markov processes and pseudo-dofferential operators - why? Levy processes and continuous negative definite functions pseudo-differential operators and Dirichlet forms constructing Feller semi-groups from pseudo-differential operators the martingale problem for pseudo-differential operators subordination in the sense of Bochner transition functions (existence, estimates) path properties and symbol Balayage theory and the Dirichlet problem for pseudo-differential operators.

title: Levy-Type Processes and Pseudodifferential Operators
abstract: Our aim in this survey is to show how pseudodifferential operators arise naturally in the theory of Markov processes and that this opens the way to use Fourier analytic techniques for general Markov processes. In order to keep things simple, we will only consider ℝ n as state space. We will see in Section 1 that one can identify the characteristic exponent ψ(ξ) of a Levy process with the symbol of its infinitesimal generator which is a pseudodifferential operator with constant coefficients. The key observation now is that we can—under reasonable assumptions (e.g., Feller)— associate with any Markov process {X t } t≥0 on ℝ n a function q(x, ξ) which turns out to be the symbol of the generator for {X t } t≥0 which is a pseudodifferential operator. This function q(x, ξ) is given by 
 
$$ q(x,\xi ): = - \mathop {\lim }\limits_{t \to 0} {{{E^x}({e^{i({X_t} - x) \cdot \xi }}) - 1} \over t}. $$ 
 
Needless to say that this formula will produce ψ(ξ) in the case of a Levy process. Our programme is simple to state: study the process through its symbol.

title: An Extension Problem Related to the Fractional Laplacian
abstract: The operator square root of the Laplacian (− ▵)1/2 can be obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to the Neumann condition. In this article, we obtain similar characterizations for general fractional powers of the Laplacian and other integro-differential operators. From those characterizations we derive some properties of these integro-differential equations from purely local arguments in the extension problems.

title: Regularity Results for Nonlocal Equations by Approximation
abstract: We obtain C1,α regularity estimates for nonlocal elliptic equations that are not necessarily translation-invariant using compactness and perturbative methods and our previous regularity results for the translation-invariant case.

title: Regularity theory for fully nonlinear integro‐differential equations
abstract: We consider nonlinear integro-differential equations, like the ones that arise from stochastic control problems with purely jump Levy processes. We obtain a nonlocal version of the ABP estimate, Harnack inequality, and interior C 1,� regularity for general fully nonlinear integro- differential equations. Our estimates remain uniform as the degree of the equation approaches two, so they can be seen as a natural extension of the regularity theory for elliptic partial differential equations.

title: Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation
abstract: Motivated by the critical dissipative quasi-geostrophic equation, we prove that drift-diffusion equations with L 2 initial data and minimal assumptions on the drift are locally Holder continuous. As an application we show that solutions of the quasi-geostrophic equation with initial L 2 data and critical diffusion (-Δ) 1/2 are locally smooth for any space dimension.

title: A fractional free boundary problem related to a plasma problem
abstract: We study a fractional analogue of a plasma problem arising from physics. Specifically, for a fixed bounded domain $\Omega$ we study solutions to the eigenfunction equation \[ 
(- \Delta)^s u = \lambda(u- \gamma)_+ 
\] with $u \equiv 0$ on $\partial \Omega$.

title: On the regularity of the non-dynamic parabolic fractional obstacle problem
abstract: In the class of the so called non-dynamic Fractional Obstacle Problems of parabolic type, it is shown how to obtain higher regularity as well as optimal regularity of the space derivatives of the solution. Furthermore, at free boundary points of positive parabolic density, it is proven that the time derivative of the solution is H\"{o}lder continuous. Finally, at regular free boundary points, space-time regularity of the corresponding free boundary is obtained for any fraction $s\in(0,1)$.

title: Fractional elliptic equations, Caccioppoli estimates and regularity
abstract: 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 $L^s$, $0<s<1$, is the fractional power of $L$ and $\partial_Au$ is the conormal derivative of $u$ with respect to the coefficients $A(x)$. We reproduce Caccioppoli type estimates that allow us to develop the regularity theory. Indeed, we prove interior and boundary Schauder regularity estimates depending on the smoothness of the coefficients $A(x)$, the right hand side $f$ and the boundary of the domain. Moreover, we establish estimates for fundamental solutions in the spirit of the classical result by Littman--Stampacchia--Weinberger and we obtain nonlocal integro-differential formulas for $L^su(x)$. Essential tools in the analysis are the semigroup language approach and the extension problem.

title: Regularity Theory and Extension Problem for Fractional Nonlocal Parabolic Equations and the Master Equation
abstract: We develop the regularity theory for solutions to space-time nonlocal equations driven by fractional powers of the heat operator $$(\partial_t-\Delta)^su(t,x)=f(t,x),\quad\hbox{for}~0<s<1.$$ 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 $(\partial_t-\Delta)^su(t,x)$ 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. H\"older and Schauder estimates for the space-time Poisson problem are deduced using a new characterization of parabolic H\"older spaces. Our methods involve the \textit{parabolic language of semigroups} and the Cauchy Integral Theorem, which are original to define the fractional powers of $\partial_t-\Delta$. Though we mainly focus in the equation $(\partial_t-\Delta)^su=f$, applications of our ideas to variable coefficients, discrete Laplacians and Riemannian manifolds are stressed out.

title: Critical SQG in Bounded Domains
abstract: We consider the critical dissipative SQG equation in bounded domains, with the square root of the Dirichlet Laplacian dissipation. We prove global a priori interior $$C^{\alpha }$$Cα and Lipschitz bounds for large data.

title: The random walk's guide to anomalous diffusion: a fractional dynamics approach
abstract: Fractional kinetic equations of the diffusion, diffusion–advection, and Fokker–Planck type are presented as a useful approach for the description of transport dynamics in complex systems which are governed by anomalous diffusion and non-exponential relaxation patterns. These fractional equations are derived asymptotically from basic random walk models, and from a generalised master equation. Several physical consequences are discussed which are relevant to dynamical processes in complex systems. Methods of solution are introduced and for some special cases exact solutions are calculated. This report demonstrates that fractional equations have come of age as a complementary tool in the description of anomalous transport processes.

title: Chaos, fractional kinetics, and anomalous transport
abstract: Chaotic dynamics can be considered as a physical phenomenon that bridges the regular evolution of systems with the random one. These two alternative states of physical processes are, typically, described by the corresponding alternative methods: quasiperiodic or other regular functions in the first case, and kinetic or other probabilistic equations in the second case. What kind of kinetics should be for chaotic dynamics that is intermediate between completely regular (integrable) and completely random (noisy) cases? What features of the dynamics and in what way should they be represented in the kinetics of chaos? These are the subjects of this paper, where the new concept of fractional kinetics is reviewed for systems with Hamiltonian chaos. Particularly, we show how the notions of dynamical quasi-traps, Poincare recurrences, Levy flights, exit time distributions, phase space topology prove to be important in the construction of kinetics. The concept of fractional kinetics enters a different area of applications, such as particle dynamics in different potentials, particle advection in fluids, plasma physics and fusion devices, quantum optics, and many others. New characteristics of the kinetics are involved to fractional kinetics and the most important are anomalous transport, superdiffusion, weak mixing, and others. The fractional kinetics does not look as the usual one since some moments of the distribution function are infinite and fluctuations from the equilibrium state do not have any finite time of relaxation. Different important physical phenomena: cooling of particles and signals, particle and wave traps, Maxwell's Demon, etc. represent some domains where fractional kinetics proves to be valuable.

title: Fractional quantum mechanics
abstract: A path integral approach to quantum physics has been developed. Fractional path integrals over the paths of the Levy flights are defined. It is shown that if the fractality of the Brownian trajectories leads to standard quantum and statistical mechanics, then the fractality of the Levy paths leads to fractional quantum mechanics and fractional statistical mechanics. The fractional quantum and statistical mechanics have been developed via our fractional path integral approach. A fractional generalization of the Schrodinger equation has been found. A relationship between the energy and the momentum of the nonrelativistic quantum-mechanical particle has been established. The equation for the fractional plane wave function has been obtained. We have derived a free particle quantum-mechanical kernel using Fox's H function. A fractional generalization of the Heisenberg uncertainty relation has been established. Fractional statistical mechanics has been developed via the path integral approach. A fractional generalization of the motion equation for the density matrix has been found. The density matrix of a free particle has been expressed in terms of the Fox's H function. We also discuss the relationships between fractional and the well-known Feynman path integral approaches to quantum and statistical mechanics.

title: Fractional diffusion models of cardiac electrical propagation: role of structural heterogeneity in dispersion of repolarization
abstract: Impulse propagation in biological tissues is known to be modulated by structural heterogeneity. In cardiac muscle, improved understanding on how this heterogeneity influences electrical spread is key to advancing our interpretation of dispersion of repolarization. We propose fractional diffusion models as a novel mathematical description of structurally heterogeneous excitable media, as a means of representing the modulation of the total electric field by the secondary electrical sources associated with tissue inhomogeneities. Our results, analysed against in vivo human recordings and experimental data of different animal species, indicate that structural heterogeneity underlies relevant characteristics of cardiac electrical propagation at tissue level. These include conduction effects on action potential (AP) morphology, the shortening of AP duration along the activation pathway and the progressive modulation by premature beats of spatial patterns of dispersion of repolarization. The proposed approach may also have important implications in other research fields involving excitable complex media.

title: The influence of a line with fast diffusion on Fisher-KPP propagation
abstract: We propose here a new model to describe biological invasions in the plane when a strong diffusion takes place on a line. We establish the main properties of the system, and also derive the asymptotic speed of spreading in the direction of the line. For low diffusion, the line has no effect, whereas, past a threshold, the line enhances global diffusion in the plane and the propagation is directed by diffusion on the line. It is shown here that the global asymptotic speed of spreading in the plane, in the direction of the line, grows as the square root of the diffusion on the line. The model is much relevant to account for the effects of fast diffusion lines such as roads on spreading of invasive species.

title: Some constructions for the fractional Laplacian on noncompact manifolds
abstract: We give a denition of the fractional Laplacian on some noncompact manifolds, through an extension problem introduced by Caarelli-Silvest re. While this denition in the compact case is straightforward, in the noncompact setting one needs to have a precise control of the behavior of the metric at innity and geometry plays a crucial role. First we give explicit calculations in the hyperbolic space, including a formula for the kernel and a trace Sobolev inequality. Then we consider more general noncompact manifolds, where the problem reduces to obtain suitable upper bounds for the heat kernel.

title: Fractional Laplacians on the sphere, the
 Minakshisundaram zeta function and
 semigroups
abstract: 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.

title: Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications
abstract: 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 $u,f:\mathbb{Z}_h\to\mathbb{R}$, $0<s<1$, is performed. The pointwise nonlocal formula for $(-\Delta_h)^su$ and the nonlocal discrete mean value property for discrete $s$-harmonic functions are obtained. We observe that a characterization of $(-\Delta_h)^s$ as the Dirichlet-to-Neumann operator for a semidiscrete degenerate elliptic local extension problem is valid. Regularity properties and Schauder estimates in discrete Holder 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 Poincare inequality are proved, which are of independent interest. We introduce the negative power (fundamental solution) \[ u=(-\Delta_h)^{-s}f, \] 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 $(-\Delta_h)^{-s}$. 
As applications, the convergence of our fractional discrete Laplacian to the (continuous) fractional Laplacian as $h\to0$ in Holder 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 \[ (-\Delta)^sU=F, \] 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~$h$.

title: Harmonic analysis associated with a discrete Laplacian
abstract: It is well known that the fundamental solution of $${u_t}\left( {n,t} \right) = u\left( {n + 1,t} \right) - 2u\left( {n,t} \right) + u\left( {n - 1,t} \right),n \in \mathbb{Z},$$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.

title: Fractional integrals on compact Riemannian symmetric spaces of rank one
abstract: 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.

title: Fractional powers of the parabolic Hermite operator. Regularity properties
abstract: Let $\mathcal{L}= \partial_t- \Delta_x+|x|^2$. Consider its Poisson semigroup $e^{-y\sqrt{\mathcal{L}}}$. For $\alpha >0$ define the Parabolic Hermite-Zygmund spaces 
$$ 
\Lambda^\alpha_{\mathcal{L}}=\left\{f: \:f\in L^\infty(\mathbb{R}^{n+1})\:\;  \:\; \left\|\partial_y^k e^{-y\sqrt{\mathcal{L}}} f \right\|_{L^\infty(\mathbb{R}^{n+1})}\leq C_k y^{-k+\alpha},\;\: {\rm with }\, k=[\alpha]+1, y>0. \right\}, 
$$ 
with the obvious norm. It is shown that these spaces have a pointwise description of H\"older type. 
The fractional powers $\mathcal{L}^{\pm \beta}$ are well defined in these spaces and the following regularity properties are proved: 
\begin{eqnarray*} 
\alpha, \beta >0, \quad \|\mathcal{L}^{-\beta} f\|_{ \Lambda^{\alpha+2\beta}_{\mathcal{L}}}\le C \|f\|_{ \Lambda^\alpha_{\mathcal{L}}}. 
\end{eqnarray*} 
\begin{eqnarray*} 
0< 2\beta < \alpha, \quad \|\mathcal{L}^\beta f\|_{\Lambda_{\mathcal{L}}^{\alpha-2\beta}}\le C \|f\|_{\Lambda^\alpha_{\mathcal{L}}}. 
\end{eqnarray*} 
Parallel results are obtained for the Hermite operator $- \Delta +|x|^2.$ The proofs use in a fundamental way the semigroup definition of the operators $\mathcal{L}^{\pm \beta}$ and $(-\Delta+|x|^2)^{\pm \beta}$. The non-convolution structure of the operators produce an extra difficulty of the arguments.

title: The fractional nonlocal Ornstein-Uhlenbeck equation, Gaussian symmetrization and regularity
abstract: For \begin{document}$0 , we consider the Dirichlet problem for the fractional nonlocal Ornstein-Uhlenbeck equation \begin{document}$\begin{cases} (-Δ+x·\nabla)^su = f,&\hbox{in}~Ω,\\ u = 0,&\hbox{on}~\partialΩ, \end{cases}$ \end{document} where \begin{document}$Ω$\end{document} is a possibly unbounded open subset of \begin{document}$\mathbb{R}^n$\end{document} , \begin{document}$n≥2$\end{document} . The appropriate functional settings for this nonlocal equation and its corresponding extension problem are developed. We apply Gaussian symmetrization techniques to derive a concentration comparison estimate for solutions. As consequences, novel \begin{document}$L^p$\end{document} and \begin{document}$L^p(\log L)^α$\end{document} regularity estimates in terms of the datum \begin{document}$f$\end{document} are obtained by comparing \begin{document}$u$\end{document} with half-space solutions.

title: Harnack inequality for the fractional nonlocal linearized Monge–Ampère equation
abstract: The fractional nonlocal linearized Monge–Ampère equation is introduced. A Harnack inequality for nonnegative solutions to the Poisson problem on Monge–Ampère sections is proved.

title: Extension Problem and Harnack's Inequality for Some Fractional Operators
abstract: 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 σ f(x) and some maximum and comparison principles are derived.

title: Regularity theory for the fractional harmonic oscillator
abstract: 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.

title: Fractional semilinear Neumann problems arising from a fractional Keller–Segel model
abstract: We consider the following fractional semilinear Neumann problem on a smooth bounded domain $$\Omega \subset \mathbb {R}^n$$Ω⊂Rn, $$n\ge 2$$n≥2, $$\begin{aligned} ^{1/2}u+u=u^{p},&{}\quad \hbox {in}~\Omega ,\\ \partial _\nu u=0,&{}\quad \hbox {on}~\partial \Omega ,\\ u>0,&{}\quad \hbox {in}~\Omega , \end{array}\right. } \end{aligned}$$(-εΔ)1/2u+u=up,inΩ,∂νu=0,on∂Ω,u>0,inΩ,where $$\varepsilon >0$$ε>0 and $$1<p<(n+1)/(n-1)$$1<p<(n+1)/(n-1). This is the fractional version of the semilinear Neumann problem studied by Lin–Ni–Takagi in the late 1980’s. The problem arises by considering steady states of the Keller–Segel model with nonlocal chemical concentration diffusion. Using the semigroup language for the extension method and variational techniques, we prove existence of nonconstant smooth solutions for small $$\varepsilon $$ε, which are obtained by minimizing a suitable energy functional. In the case of large $$\varepsilon $$ε we obtain nonexistence of nonconstant solutions. It is also shown that as $$\varepsilon \rightarrow 0$$ε→0 the solutions $$u_\varepsilon $$uε tend to zero in measure on $$\Omega $$Ω, while they form spikes in $$\overline{\Omega }$$Ω¯. The regularity estimates of the fractional Neumann Laplacian that we develop here are essential for the analysis. The latter results are of independent interest.

title: Non-local fractional derivatives. Discrete and continuous
abstract: We prove maximum and comparison principles for fractional discrete derivatives in the integers. Regularity results when the space is a mesh of length $h$, and approximation theorems to the continuous fractional derivatives are shown. When the functions are good enough, these approximation procedures give a measure of the order of approximation. These results also allows us to prove the coincidence, for good enough functions, of the Marchaud and Gr\"unwald-Letnikov derivatives in every point and the speed of convergence to the Gr\"unwald-Letnikov derivative. The fractional discrete derivative will be also described as a Neumann-Dirichlet operator defined by a semi-discrete extension problem. Some operators related to the Harmonic Analysis associated to the discrete derivative will be also considered, in particular their behavior in the Lebesgue spaces $\ell^p(\mathbb{Z}).$

title: Maximum principles, extension problem and inversion for nonlocal one-sided equations
abstract: 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.

title: Extension problem and fractional operators: semigroups and wave equations
abstract: 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 C0 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.

title: Harnack's inequality for fractional nonlocal equations
abstract: 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.

title: Unique Continuation for Fractional Orders of Elliptic Equations
abstract: We establish the strong unique continuation property of fractional orders of linear elliptic equations with Lipschitz coefficients by establishing almost monotonicity for an Almgren-type frequency functional via an extension procedure.

title: On Some Pointwise Inequalities Involving Nonlocal Operators
abstract: The purpose of this paper is threefold: first, we survey on several known pointwise identities involving fractional operators; second, we propose a unified way to deal with those identities; third, we prove some new pointwise identities in different frameworks in particular geometric and infinite-dimensional ones.

title: Fractional Laplacians, extension problems and Lie groups
abstract: We generalize some results concerning the fractional powers of the Laplace operator to the setting of nilpotent Lie Groups and we study its relationship with the solutions to a partial differential equation in the spirit of the articles of Caffarelli & Silvestre  and Stinga & Torrea [7] .

title: Harnack inequality for fractional sub-Laplacians in Carnot groups
abstract: In this paper we prove an invariant Harnack inequality on Carnot–Carathéodory balls for fractional powers of sub-Laplacians in Carnot groups. The proof relies on an “abstract” formulation of a technique recently introduced by Caffarelli and Silvestre. In addition, we write explicitly the Poisson kernel for a class of degenerate subelliptic equations in product-type Carnot groups.

title: A symmetry result for degenerate elliptic equations on the Wiener space with nonlinear boundary conditions and applications
abstract: The purpose of this paper is to study a boundary reaction problem on the space $X \times {\mathbb R}$, where $X$ is an abstract 
Wiener space. We prove that smooth bounded solutions enjoy a symmetry property, i.e., are one-dimensional in a suitable 
sense. As a corollary of our result, we obtain a symmetry property for some solutions of the following equation 
$$ 
(-\Delta_\gamma)^s u= f(u), 
$$ 
with $s\in (0,1)$, where $(-\Delta_\gamma)^s$ denotes a fractional power 
of the Ornstein-Uhlenbeck operator, and we prove that for any $s \in (0,1)$ 
monotone solutions are one-dimensional.

title: Comparison and regularity results for the fractional Laplacian via symmetrization methods
abstract: In this paper we establish a comparison result through symmetrization for solutions to some boundary value problems involving the fractional Laplacian. This allows to get sharp estimates for the solutions, obtained by comparing them with solutions of suitable radial problems. Furthermore, we use such result to prove a priori estimates for solutions in terms of the data, providing several regularity results which extend the well known ones for the classical Laplacian.

title: A PDE Approach to Fractional Diffusion in General Domains: A Priori Error Analysis
abstract: The purpose of this work is to study solution techniques for problems involving fractional powers of symmetric coercive elliptic operators in a bounded domain with Dirichlet boundary conditions. These operators can be realized as the Dirichlet-to-Neumann map for a degenerate/singular elliptic problem posed on a semi-infinite cylinder, which we analyze in the framework of weighted Sobolev spaces. Motivated by the rapid decay of the solution to this problem, we propose a truncation that is suitable for numerical approximation. We discretize this truncation using first degree tensor product finite elements. We derive a priori error estimates in weighted Sobolev spaces. The estimates exhibit optimal regularity but suboptimal order for quasi-uniform meshes. For anisotropic meshes instead, they are quasi-optimal in both order and regularity. We present numerical experiments to illustrate the method’s performance.

title: The Bayesian formulation and well-posedness of fractional elliptic inverse problems
abstract: We study the inverse problem of recovering the order and the diffusion coefficient of an elliptic fractional partial differential equation from a finite number of noisy observations of the solution. We work in a Bayesian framework and show conditions under which the posterior distribution is given by a change of measure from the prior. Moreover, we show well-posedness of the inverse problem, in the sense that small perturbations of the observed solution lead to small Hellinger perturbations of the associated posterior measures. We thus provide a mathematical foundation to the Bayesian learning of the order—and other inputs—of fractional models.

title: The Calderón problem for variable coefficients nonlocal elliptic operators
abstract: In this paper, we introduce an inverse problem of a Schrödinger type variable nonlocal elliptic operator (−∇⋅(A(x)∇))s+q), for 0<s<1. We determine the unknown bounded potential q from the exterior partial measurements associated with the nonlocal Dirichlet-to-Neumann map for any dimension n≥2. Our results generalize the recent initiative  of introducing and solving inverse problem for fractional Schrödinger operator ((−Δ)s+q) for 0<s<1. We also prove some regularity results of the direct problem corresponding to the variable coefficients fractional differential operator and the associated degenerate elliptic operator.

title: On higher order extensions for the fractional Laplacian
abstract: The technique of Caffarelli and Silvestre, characterizing the fractional Laplacian as the Dirichlet-to-Neumann map for a function U satisfying an elliptic equation in the upper half space with one extra spatial dimension, is shown to hold for general positive, non-integer orders of the fractional Laplace operator, by showing an equivalence between the H^s norm on the boundary and a suitable higher-order seminorm of U.

title: WAVE EXTENSION PROBLEM FOR THE FRACTIONAL LAPLACIAN
abstract: We show that the fractional Laplacian can be viewed as a Dirichlet-to-Neumann map for a degenerate hyperbolic problem, namely, the wave equation with an additional diffusion term that blows up at time zero. A solution to this wave extension problem is obtained from the Schrodinger group by means of an oscillatory subordination formula, which also allows us to find kernel representations for such solutions. Asymptotics of related oscillatory integrals are analysed in order to determine the correct domains for initial data in the general extension problem involving non-negative self-adjoint operators. An alternative approach using Bessel functions is also described.

title: Regularity properties of Schrödinger operators
abstract: Let L be a Schrodinger operator of the form L = − Δ + V , where the nonnegative potential V satisfies a reverse Holder inequality. Using the method of L -harmonic extensions we study regularity estimates at the scale of adapted Holder spaces. We give a pointwise description of L -Holder 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.

title: Fully Nonlinear Elliptic Equations
abstract: Preliminaries Viscosity solutions of elliptic equations Alexandroff estimate and maximum principle Harnack inequality Uniqueness of solutions Concave equations $W^{2,p}$ regularity Holder regularity The Dirichlet problem for concave equations Bibliography Index.

title: Nonlinear Potential Theory and Weighted Sobolev Spaces
abstract: The book systematically develops nonlinear potential theory and the Sobolev space theory covers results concerning approximation, extension, and interpolation, Sobolev and Poincare inequalities, Ma ...

title: A new formulation of Harnack?s inequality for nonlocal operators
abstract: We provide a new formulation of Harnackʼs inequality for nonlocal operators. In contrast to previous versions we do not assume harmonic functions to have a sign. The version of Harnackʼs inequality given here generalizes Harnackʼs classical result from 1887 to nonlocal situations. As a consequence we derive Holder regularity estimates by an extension of Moserʼs method. The inequality that we propose is equivalent to Harnackʼs original formulation but seems to be new even for the Laplace operator.

title: A characterization of the Gaussian Lipschitz space and sharp estimates for the Ornstein-Uhlenbeck Poisson kernel
abstract: The Gaussian Lipschitz space was defined by Gatto and Urbina, by means of the Ornstein-Uhlenbeck Poisson kernel. We give a characterization of this space in terms of a combination of ordinary Lipschitz continuity conditions. The main tools used in the proof are sharp estimates of the Ornstein-Uhlenbeck Poisson kernel and some of its derivatives.

title: On Gaussian Lipschitz spaces and the boundedness of fractional integrals and fractional derivatives on them
abstract: The main purpose of this paper is to study the boundedness of Gaussian fractional integrals and derivatives associated to Hermite polynomial expansions on Gaussian Lipschitz spaces Lipα (γ). To get these results we introduce formulas for these operators in terms of the Hermite-Poisson semigroup as well as the Gaussian Lipschitz spaces. This approach was originally developed for the classical Poisson integral. These proofs can also be extended to the case of Laguerre and Jacobi expansions. In subsequent papers we will study the same operators on Gaussian Besov-Lipschitz and Triebel-Lizorkin spaces.