Weighted Sobolev theorem in Lebesgue spaces with variable exponent

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

The Sampling Theorem in Variable Lebesgue Spaces

hold. The facts above are well-known as the classical Shannon sampling theorem initially proved by Ogura [10]. Ashino and Mandai [1] generalized the sampling theorem in Lebesgue spaces L0(R) for 1 < p0 < ∞. Their generalized sampling theorem is the following. Theorem 1.1 ([1]). Let r > 0 and 1 < p0 < ∞. Then for all f ∈ L 0(R) with supp f̂ ⊂ [−rπ, rπ], we have the norm inequality C p r ‖f‖Lp0(Rn...

Nonlinear eigenvalue problems in Sobolev spaces with variable exponent

Abstract. We study the boundary value problem −div((|∇u|1 + |∇u|2)∇u) = f(x, u) in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in R . We focus on the cases when f±(x, u) = ±(−λ|u| u+ |u|u), where m(x) := max{p1(x), p2(x)} < q(x) < N ·m(x) N−m(x) for any x ∈ Ω. In the first case we show the existence of infinitely many weak solutions for any λ > 0. In the second case we prove that if λ is...

Vector-valued Inequalities on Herz Spaces and Characterizations of Herz–sobolev Spaces with Variable Exponent

The origin of Herz spaces is the study of characterization of functions and multipliers on the classical Hardy spaces ([1, 8]). By virtue of many authors’ works Herz spaces have became one of the remarkable classes of function spaces in harmonic analysis now. One of the important problems on the spaces is boundedness of sublinear operators satisfying proper conditions. Hernández, Li, Lu and Yan...

A Cantor-lebesgue Theorem with Variable "coefficients"

If {qn} is a lacunary sequence of integers, and if for each n, cn(x) and c-n(x) are trigonometric polynomials of degree n, then {Cn(X)} must tend to zero for almost every x whenever {cn(x)ei?nX + c-n(-x)e-i?'nX} does. We conjecture that a similar result ought to hold even when the sequence {f On} has much slower growth. However, there is a sequence of integers {nj } and trigonometric polynomial...