Sobolev Embedding Theorems for Spaces Wk, p(x)(Ω)

By Xianling Fan, J. Shen, Dun Zhao. 2001

673 citations recorded by the source corpus.

Abstract This paper gives a Sobolev-type embedding theorem for the generalized Lebesgue–Sobolev space W k, p(x) (Ω), where Ω is an open domain in R N ( N ≥ 2) with cone property, and p ( x ) is a Lipschitz continuous function defined on Ω satisfying 1 p − ≤ p + N k . The main result can be stated as follows: for any measurable function q ( x )( x ∈ [formula]) with p(x) ≤ q(x) ≤ p*(x) := Np ( x ) N − kp ( x ) , there exists a continuous embedding from W k , p ( x ) (Ω) to L q ( x ) (Ω).

View Paper: https://www.semanticscholar.org/paper/adf6b0082d683a64dcac75a263bc9868061651d1

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Citations in the corpus

Lebesgue points in variable exponent spaces. (2004, unknown): “[7], [8], [9], [10], [13], [14], [15], [19], [26].”
Maximal function on Musielak–Orlicz spaces and generalized Lebesgue spaces (2005, result): “We refer to Hudzik [16], Ková̌cik, Rákosník [19], Samko [30], Edmunds, Lang, Nekvinda [1 Růžička [29], Edmunds, Rákosník [12], Fan, Shen, Zhao [14], Diening [5,6] for basic p erties of the spaces Lp(·) andWk,p(·) such as reflexivity, denseness of smooth functio and Sobolev type embeddings.”
Quasilinear elliptic problems with nonstandard growth. (2011, background): “Samko [16, 17, 18] working based on earlier Russian work (Sharapudinov [19] and Zhikov [20]), Fan and collaborators [5, 6, 7, 8] drawing inspiration from the study of differential equations(e.”
Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators (2005, background): “We refer to [5], [6], [4], [ 3 ] for the basics of the spaces Lp(¢) with variable exponent.”