Sobolev embeddings for Riesz potential spaces of variable exponents near 1 and Sobolev's exponent

Traces for Fractional Sobolev Spaces with Variable Exponents

In this note we prove a trace theorem in fractional spaces with variable exponents. To be more precise, we show that if p : Ω × Ω → (1,∞) and q : ∂Ω→ (1,∞) are continuous functions such that (n− 1)p(x, x) n− sp(x, x) > q(x) in ∂Ω ∩ {x ∈ Ω: n− sp(x, x) > 0}, then the inequality ‖f‖Lq(·)(∂Ω) ≤ C { ‖f‖Lp̄(·)(Ω) + [f ]s,p(·,·) } holds. Here p̄(x) = p(x, x) and [f ]s,p(·,·) denotes the fractional semi...

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...

Riesz Bases in Sobolev Spaces 1792 Hb

Riesz and Wolff potentials and elliptic equations in variable exponent weak Lebesgue spaces ∗

We prove optimal integrability results for solutions of the p(·)-Laplace equation in the scale of (weak) Lebesgue spaces. To obtain this, we show that variable exponent Riesz and Wolff potentials map L to variable exponent weak Lebesgue spaces.