Generalization of Hardy–Littlewood maximal inequality with variable exponent

Some functional inequalities in variable exponent spaces with a more generalization of uniform continuity condition

‎Some functional inequalities‎ ‎in variable exponent Lebesgue spaces are presented‎. ‎The bi-weighted modular inequality with variable exponent $p(.)$ for the Hardy operator restricted to non‎- ‎increasing function which is‎‎$$‎‎int_0^infty (frac{1}{x}int_0^x f(t)dt)^{p(x)}v(x)dxleq‎‎Cint_0^infty f(x)^{p(x)}u(x)dx‎,‎$$‎ ‎is studied‎. ‎We show that the exponent $p(.)$ for which these modular ine...

Poincaré-type Inequality for Variable Exponent Spaces of Differential Forms

We prove both local and global Poincaré inequalities with the variable exponent for differential forms in the John domains and s L -averaging domains, which can be considered as generalizations of the existing versions of Poincaré inequalities.

Korn’s First Inequality with variable coefficients and its generalization

If Ω ⊂ Rn is a bounded domain with Lipschitz boundary ∂Ω and Γ is an open subset of ∂Ω, we prove that the following inequality Z Ω |A(x)∇u(x)| dx 1/p + Z Γ |u(x)| dH(x) 1/p ≥ c ‖u‖W1,p(Ω) holds for all u ∈ W 1,p(Ω;Rm) and 1 < p < ∞, where (A(x)∇u(x))k = m X i=1 n X j=1 a k (x) ∂ui ∂xj (x) (k = 1, 2, . . . , r; r ≥ m) defines an elliptic differential operator of first order with continuous coeff...

On The Sobolev-type Inequality for Lebesgue Spaces with a Variable Exponent

A Caffarelli - Kohn - Nirenberg type inequality with variable exponent and applications to PDE ’ s ∗

Given Ω ⊂ R (N ≥ 2) a bounded smooth domain we establish a Caffarelli-Kohn-Nirenberg type inequality on Ω in the case when a variable exponent p(x), of class C, is involved. Our main result is proved under the assumption that there exists a smooth vector function − →a : Ω → R , satisfying div−→a (x) > 0 and −→a (x) · ∇p(x) = 0 for any x ∈ Ω. Particularly, we supplement a result of X. Fan, Q. Zh...