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We construct the heat kernel of the 1/2-order Laplacian perturbed by a first-order gradient term in Hölder spaces and a zero-order potential term in a generalized Kato class, and obtain sharp two-sided estimates as well as a gradient estimate of the heat kernel, where the proof of the lower bound is based on a probabilistic approach.

Heat kernel estimates for the Dirichlet fractional Laplacian

Zhen-Qing Chen, Panki Kim, Renming Song (2010)

Journal of the European Mathematical Society

We consider the fractional Laplacian $- {(- Δ)}^{α / 2}$ on an open subset in $ℝ^{d}$ with zero exterior condition. We establish sharp two-sided estimates for the heat kernel of such a Dirichlet fractional Laplacian in $C^{1, 1}$ open sets. This heat kernel is also the transition density of a rotationally symmetric $α$ -stable process killed upon leaving a $C^{1, 1}$ open set. Our results are the first sharp twosided estimates for the Dirichlet heat kernel of a non-local operator on open sets.

Higher order Riesz transforms for the Dunkl Ornstein-Uhlenbeck operator

Walid Nefzi (2019)

Czechoslovak Mathematical Journal

The aim of this paper is to extend the study of Riesz transforms associated to Dunkl Ornstein-Uhlenbeck operator considered by A. Nowak, L. Roncal and K. Stempak to higher order.

Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations

Guy Barles, Emmanuel Chasseigne, Cyril Imbert (2011)

Journal of the European Mathematical Society

This paper is concerned with the Hölder regularity of viscosity solutions of second-order, fully non-linear elliptic integro-differential equations. Our results rely on two key ingredients: first we assume that, at each point of the domain, either the equation is strictly elliptic in the classical fully non-linear sense, or (and this is the most original part of our work) the equation is strictly elliptic in a non-local non-linear sense we make precise. Next we impose some regularity and growth...

Hölder weight estimates of Riesz-Bessel singular integrals generated by a generalized shift operator.

Ekincioglu, I., Keskin, C., Er, S. (2011)

Analele Ştiinţifice ale Universităţii “Ovidius" Constanţa. Seria: Matematică

Hypoellipticité et paramétrix pour des opérateurs pseudodifférentiels à caractéristiques doubles

Alain Grigis (1975)

Journées équations aux dérivées partielles

Improved Muckenhoupt-Wheeden inequality and weighted inequalities for potential operators.

Y. Rakotondratsimba (1995)

Publicacions Matemàtiques

By a variant of the standard good λ inequality, we prove the Muckenhoupt-Wheeden inequality for measures which are not necessarily in the Muckenhoupt class. Moreover we can deal with a general potential operator, and consequently we obtain a suitable approach to the two weight inequality for such an operator when one of the weight functions satisfies a reverse doubling condition.

Index and dynamics of quantized contact transformations

Steven Zelditch (1997)

Annales de l'institut Fourier

Quantized contact transformations are Toeplitz operators over a contact manifold $(X, α)$ of the form $U_{χ} = Π A χ Π$ , where $Π : H^{2} (X) \to L^{2} (X)$ is a Szegö projector, where $χ$ is a contact transformation and where $A$ is a pseudodifferential operator over $X$ . They provide a flexible alternative to the Kähler quantization of symplectic maps, and encompass many of the examples in the physics literature, e.g. quantized cat maps and kicked rotors. The index problem is to determine $ind (U_{χ})$ when the principal symbol is unitary, or equivalently to determine...

Index formulas for pseudodifferential operators with discontinuous symbols

Grigori V. Rozenblum (1994)

Journées équations aux dérivées partielles

Index theory for multivariable Wiener-Hopf operators.

Harald Upmeier (1988)

Journal für die reine und angewandte Mathematik

Inequalities for differentiable reproducing kernels and an application to positive integral operators.

Buescu, Jorge, Paixão, A.C. (2006)

Journal of Inequalities and Applications [electronic only]

Instabilité spectrale semiclassique pour des opérateurs non-autoadjoints I : un modèle

Mildred Hager (2006)

Annales de la faculté des sciences de Toulouse Mathématiques

Dans ce travail, nous considérons un opérateur différentiel simple ainsi que des perturbations. Alors que le spectre de l’opérateur non-perturbé est confiné à une droite à l’intérieur du pseudospectre, nous montrons pour les opérateurs perturbés que les valeurs propres se distribuent à l’intérieur du pseudospectre d’après une loi de Weyl.

Integral and derivative operators of functional order on generalized Besov and Triebel-Lizorkin spaces in the setting of spaces of homogeneous type

Silvia I. Hartzstein, Beatriz E. Viviani (2002)

Commentationes Mathematicae Universitatis Carolinae

In the setting of spaces of homogeneous-type, we define the Integral, $I_{φ}$ , and Derivative, $D_{φ}$ , operators of order $φ$ , where $φ$ is a function of positive lower type and upper type less than $1$ , and show that $I_{φ}$ and $D_{φ}$ are bounded from Lipschitz spaces $Λ^{ξ}$ to $Λ^{ξ φ}$ and $Λ^{ξ / φ}$ respectively, with suitable restrictions on the quasi-increasing function $ξ$ in each case. We also prove that $I_{φ}$ and $D_{φ}$ are bounded from the generalized Besov ${\dot{B}}_{p}^{ψ, q}$ , with $1 \leq p, q < \infty$ , and Triebel-Lizorkin spaces ${\dot{F}}_{p}^{ψ, q}$ , with $1 < p, q < \infty$ , of order $ψ$ to those of order $φ ψ$ and $ψ / φ$ respectively,...

Integral and pseudointegral operators.

Tibilov, K.T. (1999)

Vladikavkazskiĭ Matematicheskiĭ Zhurnal

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