We study stochastic differential equations with jumps with no diffusion part. We provide some basic stochastic characterizations of solutions of the corresponding non-local partial differential equations and prove the Harnack inequality for a class of these operators. We also establish key connections between the recurrence properties of these jump processes and the non-local partial differenti...

We define the Littlewood-Paley decomposition associated with the Dunkl operators; from this decomposition we give the characterization of the Sobolev, Hölder and Lebesgue spaces associated with the Dunkl operators. We construct the paraproduct operators associated with the Dunkl operators similar to those defined by J.M. Bony in [1]. Using the LittlewoodPaley decomposition we establish the Sobo...

In this note, we study a general version of a problem posed by Feng Qi in [10] in the context of a measured space endowed with a positive finite measure. For other studies and results, one can consult the papers [2], [3], [5], [8], [9], [12], [13] and [14]. Our basic tool is the classical Hölder inequality. By the convexity method (see [3]) we give an interpretation of the lower bound occuring ...

We give a different proof of a recent result of Klartag [12] concerning the concentration of the volume of a convex body within a thin Euclidean shell and proving a conjecture of Anttila, Ball and Perissinaki [1]. It is based on the study of the Lp-centroid bodies. We prove an almost isometric reverse Hölder inequality for their mean width and a refined form of a stability result.

In this paper, we focus on variational representations of some matrix symmetric norm functions that are related to the quantum Rényi relative entropy. Concretely, obtain function ( A , B ) ? | q / 2 K ? p s for norms by using Hölder inequality and Young inequality. These expressions enable us make proofs convexity/concavity trace Tr more clear give a new extension.

In this paper, we study the existence and uniqueness of solutions to the Cauchy problem for nonautonomous fractional differential equations involving Caputo derivative in Banach spaces. Definition for the solution in the Carathéodory sense and fundamental lemma are introduced. Some sufficient conditions for the existence and uniqueness of solutions are established by means of fractional calculu...

*Correspondence: [email protected] 2School of Information & Statistics, Guangxi University of Financial and Economics, Nanning, Guangxi 530003, China Full list of author information is available at the end of the article Abstract In this paper, we establish some generalizations and refinements of the Hölder inequality on time scales via the diamond-α dynamic integral, which is defined as a li...

CONTENTS Introduction. 0. Pseudodifferential operators and linear PDE. §0.1 The Fourier integral representation and symbol classes §0.2 Schwartz kernels of pseudodifferential operators §0.3 Adjoints and products §0.4 Elliptic operators and parametrices §0.5 L 2 estimates §0.6 Gårding's inequality §0.7 The sharp Gårding inequality §0.8 Hyperbolic evolution equations §0.9 Egorov's theorem §0.10 M...

Motivated by its practical success, we show that the 2D total variation denoiser satisfies a sharp oracle inequality that leads to near optimal rates of estimation for a large class of image models such as bi-isotonic, Hölder smooth and cartoons. Our analysis hinges on properties of the unnormalized Laplacian of the two-dimensional grid such as eigenvector delocalization and spectral decay. We ...