We prove the equivalence of Hardyand Sobolev-type inequalities, certain uniform bounds on the heat kernel and some spectral regularity properties of the Neumann Laplacian associated with an arbitrary region of finite measure in Euclidean space. We also prove that if one perturbs the boundary of the region within a uniform Hölder category then the eigenvalues of the Neumann Laplacian change by a...

L 2 boundary decay properties of the heat kernel and spectral density. We deduce bounds on the rate of convergence of the eigenvalues when the region is slightly reduced in size. It is remarkable that several of the bounds do not involve the space dimension. AMS subject classifications: 35P99, 35P20, 47A75, 47B25 keywords: boundary decay, Laplacian, Hardy inequality, eigenfunctions, heat kernel...

Here we will consider examples of conformally flat manifolds that are conformally equivalent to open subsets of the sphere S. For such manifolds we shall introduce a Cauchy kernel and Cauchy integral formula for sections taking values in a spinor bundle and annihilated by a Dirac operator, or generalized Cauchy-Riemann operator. Basic properties of this kernel are examined, in particular we exa...

Weighted norm estimates and representation formulas are proved for nonhomogeneous singular integrals with no regularity condition on the kernel and only an L logL integrability condition. The representation formulas involve averages over a starshaped set naturally associated with the kernel. The proof of the norm estimates is based on the representation formulas, some new variations of the Hard...

In this paper, we prove the following reversed Hardy–Littlewood–Sobolev inequality with extension kernel: $$\int _{\mathbb R_+^n}\int _{\partial\mathbb R^n_+}\frac{x_n^\beta}{|x-y|^{n-\alpha}}f(y)g(x)\,dy\,dx\geq C_{n,\alpha ,\beta ,p}\|f\|_{L^{p}(\partia

Abstract. Let TΩ,α (0 ≤ α < n) be the singular and fractional integrals with variable kernel Ω(x,z), and [b,TΩ,α ] be the commutator generated by TΩ,α and a Lipschitz function b. In this paper, the authors study the boundedness of [b,TΩ,α ] on the Hardy spaces, under some assumptions such as the Lr-Dini condition. Similar results and the weak type estimates at the end-point cases are also given...

We begin by studying certain semigroup estimates which are more singular than those implied by a Sobolev embedding theorem but which are equivalent to certain logarithmic Sobolev inequalities. We then give a method for proving that such log–Sobolev inequalities hold for Euclidean regions which satisfy a particular Hardy–type inequality. Our main application is to show that domains which have ex...

We consider second-order ergodic Mean-Field Games systems in the whole space $\mathbb{R}^N$ with coercive potential and aggregating nonlocal coupling, defined terms of a Riesz interaction kernel. These MFG describe Nash equilibria games large population indistinguishable rational players attracted toward regions where is highly distributed. Equilibria solve system PDEs an Hamilton-Jacobi-Bellma...