Weighted Sobolev theorem in Lebesgue spaces with variable exponent

For the Riesz potential operator Iα there are proved weighted estimates

     

Boundary trace embedding theorems for variable exponent Sobolev spaces

We study boundary trace embedding theorems for variable exponent Sobolev space W1,p(⋅)(Ω). Let Ω be an open (bounded or unbounded) domain in RN satisfying strong local Lipschitz condition. Under the hypotheses that p∈L^∞(Ω), 1?infp(x)?supp(x)<N, |∇p|∈Lγ(⋅)(Ω), where γ∈L^∞(Ω) and infγ(x)>N, we prove that there is a continuous boundary trace embedding W1,p(⋅)(Ω)→Lq(⋅)(∂Ω) provided q(⋅), a measurable function on ∂Ω, satisfies condition for x∈∂Ω.     

A mass transportation approach for Sobolev inequalities in variable exponent spaces

In this paper we provide a proof of the Sobolev–Poincaré inequality for variable exponent spaces by means of mass transportation methods, in the spirit of Cordero-Erausquin et al. (Adv Math 182(2):307–332, 2004). The importance of this approach is that the method is flexible enough to deal with different inequalities. As an application, we also deduce the Sobolev-trace inequality improving the result of Fan (J Math Anal Appl 339(2):1395–1412, 2008) by obtaining an explicit dependence of the exponent in the constant.     

On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent

We consider the nonlinear eigenvalue problem

in , on , where is a bounded open set in with smooth boundary and , are continuous functions on such that , , and for all . The main result of this paper establishes that any sufficiently small is an eigenvalue of the above nonhomogeneous quasilinear problem. The proof relies on simple variational arguments based on Ekeland's variational principle.

     

On the boundary limits of monotone Sobolev functions in variable exponent Orlicz spaces

Our aim in this note is to deal with boundary limits of monotone Sobolev functions with ▽u∈ Lp(·)logLq(·)(B) for the unit ball BRn. Here p(·) and q(·) are variable exponents satisfyingthe log-Hlder and the log log-Hlder conditions, respectively.     

On the Sobolev trace Theorem for variable exponent spaces in the critical range

In this paper, we study the Sobolev trace Theorem for variable exponent spaces with critical exponents. We find conditions on the best constant in order to guaranty the existence of extremals. Then, we give local conditions on the exponents and on the domain (in the spirit of Adimurthy and Yadava) in order to satisfy such conditions and therefore to ensure the existence of extremals.     

On the Sobolev embedding theorem for variable exponent spaces in the critical range

In this paper we study the Sobolev embedding theorem for variable exponent spaces with critical exponents. We find conditions on the best constant in order to guaranty the existence of extremals. The proof is based on a suitable refinement of the estimates in the Concentration–Compactness Theorem for variable exponents and an adaptation of a convexity argument due to P.L. Lions, F. Pacella and M. Tricarico.     

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

Our aim in this paper is to deal with Sobolev embeddings for Riesz potentials of order α for functions f satisfying the Orlicz type condition

     

Embedding of Sobolev spaces with limit exponent revisited

An embedding of the Sobolev spaces W _p ^s (? ⁿ ) in Lizorkin-type spaces of locally integrable functions of smoothness zero is obtained; a similar assertion for Riesz and Bessel potentials is presented. The embedding theorem is extended to Sobolev spaces on irregular domains in n-dimensional Euclidean space. The statement of the theorem depends on geometric parameters of the domain of functions.     

Interpolation theorems for variable exponent Lebesgue spaces

When Hardy-Littlewood maximal operator is bounded on Lp(⋅)(Rn) space we prove θ[Lp(⋅)(Rn),BMO(Rn)]=Lq(⋅)(Rn) where q(⋅)=p(⋅)/(1−θ) and θ[Lp(⋅)(Rn),H¹(Rn)]=Lq(⋅)(Rn) where 1/q(⋅)=θ+(1−θ)/p(⋅).     

Iterated maximal functions in variable exponent Lebesgue spaces

In this paper we study the iterated Hardy?CLittlewood maximal operator in variable exponent Lebesgue spaces with exponent allowed to reach the value 1. We use modulars where the L ^p(·)-modular is perturbed by a logarithmic-type function, and the results hold also in the more general context of such Musielak?COrlicz spaces.     

Relative rearrangement and Lebesgue spaces with variable exponent

We apply the techniques of monotone and relative rearrangements to the nonrearrangement invariant spaces L^p()(Ω) with variable exponent. In particular, we show that the maps uL^p()(Ω)→k(t)u_*L^p_*()(0,measΩ) and uL^p()(Ω)→u_*L^p*()(0,measΩ) are locally -Hölderian (u_* (resp. p^*) is the decreasing (resp. increasing) rearrangement of u (resp. p)). The pointwise relations for the relative rearrangement are applied to derive the Sobolev embedding with eventually discontinuous exponents.     

Gagliardo-Nirenberg type inequality for variable exponent Lebesgue spaces

We prove analogies of the classical Gagliardo-Nirenberg inequalities

     

Sobolev embeddings in metric measure spaces with variable dimension

In this article we study metric measure spaces with variable dimension. We consider Lebesgue spaces on these sets, and embeddings of the Riesz potential in these spaces. We also investigate Hajłasz-type Sobolev spaces, and prove Sobolev and Trudinger inequalities with optimal exponents. All of these questions lead naturally to function spaces with variable exponents. Supported the Research Council of Norway, Project 160192/V30.     

Compactness for the weighted Hardy operator in variable exponent spaces

In this paper, we prove a necessary and sufficiency condition for the weighted Hardy operator

$H_{\upsilon ,\omega }f(x)=\upsilon (x)\underset{0}{\overset{x}{\int }}f(t)\omega (t)\mathrm{d}t$

to be compactly acting from $L^{p(?)}(0,\infty )$ to $L^{q(?)}(0,\infty )$.     

Boundedness of commutators on Herz spaces with variable exponent

Our aim in the present paper is to prove the boundedness of vector-valued commutators on Herz spaces with variable exponent. In order to obtain the result, we clarify a relation between variable exponent and BMO norms.