The present paper deals with a Kirchhoff problem under homogeneous Dirichlet boundary conditions, set in a bounded smooth domain Ω of R^{N}. The problem studied is a stationary version of the orig inal Kirchhoff equation, involving the p(x)-Lap lacian operator, in the framework of the variable exponent Lebesgue and Sobolev spaces. The question of the existence of weak solutions is treated. Appl...

We find optimal conditions on m-linear Fourier multipliers to give rise to bounded operators from a product of Hardy spaces Hj , 0 < pj ≤ 1, to Lebesgue spaces Lp. The conditions we obtain are necessary and sufficient for boundedness and are expressed in terms of L2-based Sobolev spaces. Our results extend those obtained in the linear case (m = 1) by Calderón and Torchinsky [1] and in the bilin...

We establish conditions on ψ under which the small-scale affine system {ψ(ajx− k) : j ≥ J, k ∈ Zd} spans the Lebesgue space L(R) and the Sobolev space W(R), for 1 ≤ p < ∞ and J ∈ Z. The dilation matrices aj are expanding (meaning limj→∞ ‖a−1 j ‖ = 0) but they need not be diagonal. For spanning L our result assumes ∫ Rd ψ dx 6= 0 and, when p > 1, that the periodization of |ψ| or of 1{ψ 6=0} is b...

We study the Sobolev spaces of exponential type associated with the Dunkl-Bessel Laplace operator. Some properties including completeness and the imbedding theorem are proved. We next introduce a class of symbols of exponential type and the associated pseudodifferential-difference operators, which naturally act on the generalized Dunkl-Sobolev spaces of exponential type. Finally, using the theo...

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...

In 1994, Lions, Perthame and Tadmor conjectured the maximal smoothing effect for multidimensional scalar conservation laws in Sobolev spaces. For strictly smooth convex flux and the one-dimensional case we detail the proof of this conjecture in the framework of Sobolev fractional spaces W s,1, and in fractional BV spaces: BV s. The BV s smoothing effect is more precise and optimal. It implies t...

A comprehensive analysis of Sobolev-type inequalities for the Ornstein-Uhlenbeck operator in Gauss space is offered. unified approach proposed, providing one with criteria their validity class rearrangement-invariant function norms. Optimal target and domain norms relevant are characterized via a reduction principle to one-dimensional Calderón type integral patterned on Gaussian isoperimetric f...