Harnack’s inequality for generalized subelliptic Schrödinger operators

We prove a uniform Harnack inequality for nonnegative solutions of Δ_G u − μu = 0, where Δ_G is a sublaplacian, μ is a non-negative Radon measure and satisfying scale-invariant Kato condition.      

Log-Sobolev inequalities for semi-direct product operators and applications

In this paper, we prove some type of logarithmic Sobolev inequalities (with parameters) for operators in semi-direct product forms (see Sect. 1 for a precise definition). This generalizes the tensorization procedure for this type of inequalities and allows to deal with some operators with varying coefficients. We provide many examples of applications and obtain ultracontractive bounds for some of these operators by using appropriate Hardy’s type inequalities necessary for our method. This theory is developed in the setting of carré du champ with diffusion property.     

Sobolev inequalities with remainder terms for sublaplacians and other subelliptic operators

We prove that on bounded domains Ω, the usual Sobolev inequality for sublaplacians on Carnot groups can be improved by adding a remainder term, in striking analogy with the euclidean case. We also show analogous results for subelliptic operators like $$ {\user1{\mathcal{L}}} = \Delta _{x} + |x|^{{2\alpha }} \Delta _{y} ,\,\alpha \gt 0. $$     

Sharp spectral multipliers for operators satisfying generalized Gaussian estimates

Let L   be a non-negative self-adjoint operator acting on L²(X)$L^2(X)$ where X is a space of homogeneous type. Assume that L   generates a holomorphic semigroup e^−tL$e^{-tL}$ which satisfies generalized m-th order Gaussian estimates. In this article, we study singular and dyadically supported spectral multipliers for abstract self-adjoint operators. We show that in this setting sharp spectral multiplier results follow from Plancherel or Stein–Tomas type estimates. These results are applicable to spectral multipliers for a large class of operators including m-th order elliptic differential operators with constant coefficients, biharmonic operators with rough potentials and Laplace type operators acting on fractals.     

Harnack inequality for two‐weight subelliptic p ‐Laplace operators

In this note, we prove a Harnack inequality for two‐weight subelliptic p ‐Laplace operators together with an upper bound of the Harnack constant associated with such inequality. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

广义Baskakov型算子的强逆不等式

利用二阶Ditzian-Totik模考虑一类算子的强逆不等式，这类不等式曾经被许多者用不同的方法研究过。本文将采用一种统一的方法来处理一大类算子的强逆不等式，得到了它们的Ditzian-Ivanov结果。本方法适用于更广的算子。     

Boundary behaviour of the mappings satisfying generalized inverse modular inequalities

We study the geometric properties of the mappings for which generalized inverse modular inequalities hold. We generalize in this way known theorems from the theory of analytic mappings and the theory of quasiregular mappings, like the theorems of Fatou, M. and F. Riesz, Beurling and Lindelöf and their extensions given for quasiregular mappings by Martio, Rickman and Vuorinen.     

Fundamental solutions for hermite and subelliptic operators

In this article, we introduce a geometric method based on multipliers to compute heat kernels for operators with potentials. Using the heat kernel, we compute the fundamental solution for the Hermite operator with singularity at an arbitrary point on Euclidean space and on Heisenberg groups. As a consequence, we obtain the fundamental solutions for the sub-laplacian □ _J in a family of quadratic submanifolds. The research is partially supported by a William Fulbright Reserch Grant and a Competitive Research Grant at Georgetown University.     

On the lightness of the mappings satisfying generalized inverse modular inequalities

We study the lightness of some mappings satisfying generalized inverse modular inequalities and we apply the results to obtain conditions of lightness, openness and discreetness of some Sobolev mappings.     

Hölder estimates for subelliptic operators

We will study optimal Hölder estimates for equations of type

(∗)     

Local Poincaré inequalities in non-negative curvature and finite dimension

In this paper, we derive a new set of Poincaré inequalities on the sphere, with respect to some Markov kernels parameterized by a point in the ball. When this point goes to the boundary, those Poincaré inequalities are shown to give the curvature-dimension inequality of the sphere, and when it is at the center they reduce to the usual Poincaré inequality. We then extend them to Riemannian manifolds, giving a sequence of inequalities which are equivalent to the curvature-dimension inequality, and interpolate between this inequality and the Poincaré inequality for the invariant measure. This inequality is optimal in the case of the spheres.     

Non-compact generalized variational inequalities for quasi-monotone and hemi-continuous operators with applications

Some results are obtained for non-compact cases in topological vector spaces for the existence problem of solutions for some set-valued variational inequalities with quasi-monotone and lower hemi-continuous operators, and with quasi-semi-monotone and upper hemi-continuous operators. Some applications are given in non-reflexive Banach spaces for these existence problems of solutions and for perturbation problems for these set-valued variational inequalities with quasi-monotone and quasi-semi-monotone operators. This revised version was published online in June 2006 with corrections to the Cover Date.     

Dirichlet norms,capacities and generalized isoperimetric inequalities for Markov operators

We define the notion of p-capacity for a reversible Markov operator on a general measure space and prove that uniform estimates for the ratio of capacity and measure are equivalent to certain imbedding theorems for the Orlicz and Dirichlet norms. As a corollary we get results on connections between embedding theorems and isoperimetric properties for general Markov operators and, particularly, a generalization of the Kesten theorem on the spectral radius of random walks on amenable groups for the case of arbitrary graphs with non-finitely supported transition probabilities.     

Projection methods for a generalized system of nonconvex variational inequalities with different nonlinear operators

In this paper, we introduce and consider a new generalized system of nonconvex variational inequalities with different nonlinear operators. We establish the equivalence between the generalized system of nonconvex variational inequalities and the fixed point problems using the projection technique. This equivalent alternative formulation is used to suggest and analyze a general explicit projection method for solving the generalized system of nonconvex variational inequalities. Our results can be viewed as a refinement and improvement of the previously known results for variational inequalities.     

On power-bounded operators and operators satisfying a resolvent condition

We obtain, by means of a classification of the eigenvalues, local estimates for holomorphic. functions of a class of linear operators on a finite dimensional linear vector space. We apply these methods to find new proofs of some theorems ofKreiss andMorton, and in addition we give a local estimate of the powers of the inverse of any nonsingular operator in this class.     

An existence theorem for generalized variational inequalities with discontinuous and pseudomonotone operators

This paper gives a solution existence theorem for a generalized variational inequality problem with an operator which is defined on an infinite dimensional space, which is C-pseudomonotone in the sense of Inoan and Kolumbán [D. Inoan, J. Kolumbán, On pseudomonotone set-valued mappings, Nonlinear Analysis 68 (2008) 47-53], but which may not be upper semicontinuous on finite dimensional subspaces. The proof of the theorem provides a new technique which reduces infinite variational inequality problems to finite ones. Two examples are given and analyzed to illustrate the theorem. Moreover, an example is presented to show that the C-pseudomonotonicity of the operator cannot be omitted in the theorem.     

A weak type inequality for generalized maximal operators on spaces of homogeneous type

A condition is given for a certain generalized maximal operator to be of weak type (ps, qs), where 1≤p≤q<∞, 1≤s<∞. This operator unifies various results about the Poisson integral operators cited in the literature.