Weighted integral inequalities for conjugate A-harmonic tensors

We first prove a local weighted integral inequality for conjugate A-harmonic tensors. Then, as an application of our local result, we prove a global weighted integral inequality for conjugate A-harmonic tensors in L^s(μ)-averaging domains, which can be considered as a generalization of the classical result. Finally, we give applications of the above results to quasiregular mappings.     

Best constants in Sobolev inequalities on manifolds with boundary in the presence of symmetries and applications

In this paper we establish the best constants for a Sobolev inequality and a Sobolev trace inequality on compact Riemannian manifolds with boundary, the functions being invariant under the action of a compact subgroup G of the isometry group I(M,g) and we give applications to some nonlinear PDEs with upper critical Sobolev exponent.     

On a parabolic logarithmic Sobolev inequality

In order to extend the blow-up criterion of solutions to the Euler equations, Kozono and Taniuchi [H. Kozono, Y. Taniuchi, Limiting case of the Sobolev inequality in BMO, with application to the Euler equations, Comm. Math. Phys. 214 (2000) 191-200] have proved a logarithmic Sobolev inequality by means of isotropic (elliptic) BMO norm. In this paper, we show a parabolic version of the Kozono-Taniuchi inequality by means of anisotropic (parabolic) BMO norm. More precisely we give an upper bound for the L^∞ norm of a function in terms of its parabolic BMO norm, up to a logarithmic correction involving its norm in some Sobolev space. As an application, we also explain how to apply this inequality in order to establish a long-time existence result for a class of nonlinear parabolic problems.     

Duality in ε-variational inequality problems

In this paper a general analysis of duality for an extended ε-variational inequality problem based on the notions of ε-convexity and ε-conjugacy is performed. Optimal solutions of both the primal and dual problems are also related to the saddle point of an associated Lagrangian. Gap functions for these problems are proposed. An existence theorem for the extended ε-variational inequality is also established by means of the KKM lemma.     

N-Capacity, N-harmonic radius and N-harmonic transplantation

For the N-Laplacian ΔN on a regular domain the N-Green's function exists. This allows us to define the N-Robin's function and the N-harmonic radius. We show their basic properties and extend the method of the harmonic transplantation to that of the N-harmonic transplantation. The method is used to estimate the first eigenvalue of the N-Laplacian ΔN. We also give another proof of the isoperimetric inequality for the N-capacity and give the isoperimetric inequality for the N-harmonic radius.     

Littlewood-Paley inequalities in uniformly convex and uniformly smooth Banach spaces

It is proved that the inequality δX(ε)?cεp, p?2, where δX is the modulus of convexity of X, is sufficient and necessary for the inequality

     

Riesz transforms related to Schrödinger operators acting on BMO type spaces

In this work we obtain boundedness on suitable weighted BMO type spaces of Riesz transforms, and their adjoints, associated to the Schrödinger operator −Δ+V, where V satisfies a reverse Hölder inequality. Our results are new even in the unweighted case.     

Φ-entropy inequality and application for SDEs with jumps

By using the Φ-entropy inequality derived in [16] and [2] for Poisson measures, the same type of inequality is established for a class of stochastic differential equations driven by purely jump Lévy processes. This inequality implies the exponential convergence in Φ-entropy of the associated Markov semigroup. The semigroup Φ-entropy inequality for SDEs driven by Poisson point processes is also considered.     

Estimates of the dilatation function of Beurling-Ahlfors extension

In this paper we estimate the dilatation function of the Beurling-Ahlfors extension in the most general case. By introducing ?h,m-function, we obtain an inequality which is sharp up to a constant.     

Infinitely Many Critical Points of Non-Differentiable Functions and Applications to a Neumann-Type Problem Involving the p-Laplacian

For a family of functionals in a Banach space, which are possibly non-smooth and depend also on a positive real parameter, the existence of a sequence of critical points (according to Motreanu and Panagiotopoulos (“Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities,” Nonconvex Optimization Applications, Vol. 29, Kluwer, Dordrecht, 1998, Chap. 3)) is established by mainly adapting a new technique due to Ricceri (2000, J. Comput. Appl. Math.113, 401-410). Two applications are then presented. Both of them treat the Neumann problem for an elliptic variational-hemivariational inequality with p-Laplacian.     

On h-convexity

We introduce a class of h-convex functions which generalize convex, s-convex, Godunova-Levin functions and P-functions. Namely, the h-convex function is defined as a non-negative function which satisfies f(αx+(1−α)y)?h(α)f(x)+h(1−α)f(y), where h is a non-negative function, α∈(0,1) and x,y∈J. Some properties of h-convex functions are discussed. Also, the Schur-type inequality is given.     

Generalized 3G theorem and application to relativistic stable process on non-smooth open sets

Let G(x,y) and GD(x,y) be the Green functions of rotationally invariant symmetric α-stable process in Rd and in an open set D, respectively, where 0<α<2. The inequality GD(x,y)GD(y,z)/GD(x,z)?c(G(x,y)+G(y,z)) is a very useful tool in studying (local) Schrödinger operators. When the above inequality is true with c=c(D)∈(0,∞), then we say that the 3G theorem holds in D. In this paper, we establish a generalized version of 3G theorem when D is a bounded κ-fat open set, which includes a bounded John domain. The 3G we consider is of the form GD(x,y)GD(z,w)/GD(x,w), where y may be different from z. When y=z, we recover the usual 3G. The 3G form GD(x,y)GD(z,w)/GD(x,w) appears in non-local Schrödinger operator theory. Using our generalized 3G theorem, we give a concrete class of functions belonging to the non-local Kato class, introduced by Chen and Song, on κ-fat open sets. As an application, we discuss relativistic α-stable processes (relativistic Hamiltonian when α=1) in κ-fat open sets. We identify the Martin boundary and the minimal Martin boundary with the Euclidean boundary for relativistic α-stable processes in κ-fat open sets. Furthermore, we show that relative Fatou type theorem is true for relativistic stable processes in κ-fat open sets. The main results of this paper hold for a large class of symmetric Markov processes, as are illustrated in the last section of this paper. We also discuss the generalized 3G theorem for a large class of symmetric stable Lévy processes.     

On some new generalized sequence spaces

In this paper we define the sequence sets ?_∞(u,Δ²,p), c(u,Δ²,p) and c₀(u,Δ²,p), and give α- and β-duals of these sets. Further we investigate matrix transformations in the spaces and give a characterization of the class (?_∞(u,Δ²,p),?_∞).     

Nonlinear diffusions, hypercontractivity and the optimal L-Euclidean logarithmic Sobolev inequality

The equation u_t=Δ_p(u^1/(p−1)) for p>1 is a nonlinear generalization of the heat equation which is also homogeneous, of degree 1. For large time asymptotics, its links with the optimal L^p-Euclidean logarithmic Sobolev inequality have recently been investigated. Here we focus on the existence and the uniqueness of the solutions to the Cauchy problem and on the regularization properties (hypercontractivity and ultracontractivity) of the equation using the L^p-Euclidean logarithmic Sobolev inequality. A large deviation result based on a Hamilton-Jacobi equation and also related to the L^p-Euclidean logarithmic Sobolev inequality is then stated.     

A novel use of t-packings to construct d-disjunct matrices

A t-packing is an ordered pair (V,P) where V is a v-set and P is a collection of k-subsets (blocks) of V such that each t-subset of V occurs in at most one block of P. If each t-subset of V occurs in exactly one block of P, then (V,P) is known as a Steiner (t,k,v)-design. In this paper, we explore a novel use of t-packings to construct d-disjunct matrices.     

A class of B-(p,r)-invex functions and mathematical programming

Invexity of a function is generalized. The new class of nonconvex functions, called B-(p,r)-invex functions with respect to η and b, being introduced, includes many well-known classes of generalized invex functions as its subclasses. Some properties of the introduced class of B-(p,r)-invex functions with respect to η and b are studied. Further, mathematical programming problems involving B-(p,r)-invex functions with respect to η and b are considered. The equivalence between saddle points and optima, and different type duality theorems are established for this type of optimization problems.     

On equilibrium problem of abstract economy, generalized quasi-variational inequality, and an optimization problem in locally H-convex spaces

In this paper, the equilibrium existence problem for abstract economies, the existence problem for solution of generalized quasi-variational inequality, and an optimization problem in locally H-convex spaces are researched.     

Some fixed-point theorems and minimax inequalities in FC-space

By applying the existence theorem of maximal elements, some new collectively fixed-point theorems for a family of set-valued mappings defined on the product space of noncompact FC-space are proved and some new theorems about minimax inequality involving two functions are given to show the relations of fixed-point theorem and minimax inequality in FC-spaces. These results improve and generalize many important results in the recent literature.     

Generalized difference sequence spaces and their dual spaces

The definition of the pα-, pβ- and pγ-duals of a sequence space was defined by Et [Internat. J. Math. Math. Sci. 24 (2000) 785-791]. In this paper we compute pα- and N-duals of the sequence spaces Δ^m_v(X) for X=?_∞, c and c₀, and compute β- and γ-duals of the sequence spaces Δ^m_v(X) for X=?_∞, c and c₀.