Variable Exponent Lebesgue Spaces Research Articles

Characterizations of variable exponent Hardy spaces via Riesz transforms

Let $$p(\cdot ): \mathbb R^n\rightarrow (0,\infty )$$ be a variable exponent function satisfying that there exists a constant $$p_0\in (0,p_-)$$ , where $$p_-:=\hbox {ess inf}_{x\in \mathbb R^n}p(x)$$ , such that the Hardy–Littlewood maximal operator is bounded on the variable exponent Lebesgue space $$L^{p(\cdot )/p_0}(\mathbb R^n)$$ . In this article, via investigating relations between boundary values of harmonic functions on the upper half space and elements of variable exponent Hardy spaces $$H^{p(\cdot )}(\mathbb R^n)$$ introduced by E. Nakai and Y. Sawano and, independently, by D. Cruz-Uribe and L.-A. D. Wang, the authors characterize $$H^{p(\cdot )}(\mathbb R^n)$$ via the first order Riesz transforms when $$p_-\in (\frac{n-1}{n},\infty )$$ , and via compositions of all the first order Riesz transforms when $$p_-\in (0,\frac{n-1}{n})$$ .

Approximation by trigonometric polynomials in the framework of variable exponent grand Lebesgue spaces

Abstract In this paper we establish direct and inverse theorems on approximation by trigonometric polynomials for the functions of the closure of the variable exponent Lebesgue space in the variable exponent grand Lebesgue space.

Estimates of Fractional Integral Operators on Variable Exponent Lebesgue Spaces

By some estimates for the variable fractional maximal operator, the authors prove that the fractional integral operator is bounded and satisfies the weak-type inequality on variable exponent Lebesgue spaces.

On the upper and lower estimates of norms in variable exponent spaces

In the present paper we investigate some geometrical properties of the norms in Banach function spaces. Particularly there is shown that if exponent 1/p(·) belongs to BLO1/ log then for the norm of corresponding variable exponent Lebesgue space we have the following lower estimate ∥ ∥∑χQ‖ f χQ‖p(·)/‖χQ‖p(·) ∥ ∥ p(·) C‖ f‖p(·) where {Q} defines disjoint partition of [0;1] . Also we have constructed variable exponent Lebesgue space with above property which does not possess following upper estimation ‖ f‖p(·) C ∥ ∥∑χQ‖ f χQ‖p(·)/‖χQ‖p(·) ∥ ∥ p(·) . Mathematics subject classification (2010): 42B35, 42B20, 46B45, 42B25.

Some norm inequalities in Musielak-Orlicz spaces

There has been a considerable amount of studies on the variable exponent Lebesgue spaces L; see [5, 7] etc. for exhaustive account of this direction of research. In those studies, various kinds of norm inequalites were discussed, including those which show the boundedness of important operators. Cruz-Uribe, Fiorenza, Martell and Perez [6] gave a method to obtain L-norm inequalities from L0(w)-norm inequalities with a constant exponent p0 and weights w. In fact, they proved [6, Theorem 1.3]:

Nonlinear Parabolic Equations with Variable Exponent Growth in Nonsmooth Domains

We study the Calderon--Zygmund theory for nonlinear parabolic problems in the setting of the variable exponent Lebesgue spaces. In particular, we prove the global $L^{s(\cdot)}$ integrability of the gradient of solutions to parabolic equations with $p(\cdot)$ growth in nonsmooth domains with $s(\cdot)>p(\cdot)$. In addition, we present precise regularity conditions on the variable exponent functions, the nonlinearity, and the boundary of a domain to enable us to obtain the desired result.

Interpolation on variable Morrey spaces defined on quasi-metric measure spaces

In this paper, we show the validity of a Riesz–Thorin type interpolation theorem for linear operators acting from variable exponent Lebesgue spaces into variable exponent Morrey space in the framework of quasi-metric measure spaces.

The maximal operator on generalized Orlicz spaces

In this note I present a sufficient condition for the boundedness of the maximal operator on generalized Orlicz spaces. The result includes as special cases the optimal condition for Orlicz spaces as well as the essentially optimal conditions for variable exponent Lebesgue spaces and the double-phase functional.

Poincaré and Sobolev inequalities for vector fields satisfying Hörmander’s condition in variable exponent Sobolev spaces

In this paper, we will establish Poincare inequalities in variable exponent non-isotropic Sobolev spaces. The crucial part is that we prove the boundedness of the fractional integral operator on variable exponent Lebesgue spaces on spaces of homogeneous type. We obtain the first order Poincare inequalities for vector fields satisfying Hormander’s condition in variable non-isotropic Sobolev spaces. We also set up the higher order Poincare inequalities with variable exponents on stratified Lie groups. Moreover, we get the Sobolev inequalities in variable exponent Sobolev spaces on whole stratified Lie groups. These inequalities are important and basic tools in studying nonlinear subelliptic PDEs with variable exponents such as the p(x)-subLaplacian. Our results are only stated and proved for vector fields satisfying Hormander’s condition, but they also hold for Grushin vector fields as well with obvious modifications.

Hodge decomposition of variable exponent spaces of Clifford-valued functions and applications to Dirac and Stokes equations

In this paper, we establish a Hodge-type decomposition of variable exponent Lebesgue spaces of Clifford-valued functions, where one of the subspaces is the space of all monogenic Lp(x)-functions. Using this decomposition, we obtain the existence and uniqueness of solutions to the homogeneous A-Dirac equations with variable growth under certain appropriate conditions and to the Stokes equations in the setting of variable exponent spaces of Clifford-valued functions.

Approximation in Smirnov classes with variable exponent

In this work, the inverse problem of approximation theory in the variable exponent Smirnov classes of analytic functions, defined on the Jordan domains with a Dini-smooth boundaries, is studied. First, for this purpose, an inverse theorem in the variable exponent Lebesgue spaces of periodic functions is obtained. Later, using the special linear operators, this inverse theorem to the variable exponent Smirnov classes of analytic functions is moved.

Estimates of s-numbers of a Sobolev embedding involving spaces of variable exponent

Let Ω be a bounded open subset of Rd, suppose that p(⋅):Ω→(1,∞) is a bounded, log-Hölder continuous function, and let Lp(⋅)(Ω), W∘(Ω)p(⋅)1 be the usual variable exponent Lebesgue space and the corresponding Sobolev space. The natural embedding id:W∘(Ω)p(⋅)1→Lp(⋅)(Ω) is compact; when Ω is a bounded domain it is shown that there are positive constants K1,K2 such that for all n∈N,K1≤n1/dsn(id)≤K2, where sn(id) is the nth approximation, Bernstein, Gelfand or Kolmogorov number of id. When p is constant this result is familiar; for variable p and d>1 it appears to be the first result available for s-numbers of Sobolev embeddings. The paper also contains a sharp estimate of the norm of embeddings between Lp(⋅)(Ω) spaces which is interesting in its own right.

Maximal multiplier operators in [formula omitted] spaces

In this paper we study some estimates of norms in variable exponent Lebesgue spaces for maximal multiplier operators. We will consider the case when multiplier is the Fourier transform of a compactly supported Borel measure.

The Daugavet property in the Musielak–Orlicz spaces

We show that among all Musielak–Orlicz function spaces on a σ-finite non-atomic complete measure space equipped with either the Luxemburg norm or the Orlicz norm the only spaces with the Daugavet property are L1, L∞, L1⊕1L∞ and L1⊕∞L∞. In particular, we obtain complete characterizations of the Daugavet property in the weighted interpolation spaces, the variable exponent Lebesgue spaces (Nakano spaces) and the Orlicz spaces.

Navier-Stokes equations with variable viscosity in variable exponent spaces of Clifford-valued functions

In this paper we study the stationary generalized Navier-Stokes equations when the viscosity is not only a constant but also a function which depends on the position and the shear-velocity. For this we establish an improved decomposition of variable exponent Lebesgue spaces of Clifford-valued functions. Using this decomposition together with Clifford operator calculus, we obtain the existence, uniqueness and representation of solutions for the generalized Stokes equations and the generalized Navier-Stokes equations with variable viscosity in the setting of variable exponent spaces of Clifford-valued functions. Furthermore, the equivalences of solutions and weak solutions for the aforementioned equations are justified.

Γ-Convergence of Inhomogeneous Functionals in Orlicz–Sobolev Spaces

AbstractThe asymptotic behaviour of inhomogeneous power-law type functionals is undertaken via De Giorgi’s Γ-convergence. Our results generalize recent work dealing with the asymptotic behaviour of power-law functionals acting on fields belonging to variable exponent Lebesgue and Sobolev spaces to the Orlicz–Sobolev setting.

Interplay of Wiener--Hopf and Hankel operators with almost periodic Fourier symbols on standard and variable exponent Lebesgue spaces

Wiener--Hopf plus Hankel and Wiener--Hopf minus Hankel operators in both frameworks of standard and variable exponent Lebesgue spaces are considered in this paper. The main aim is to describe certain dependencies between the Fredholm property of some Wiener--Hopf operators acting between variable exponent Lebesgue spaces and the invertibility of Wiener--Hopf plus and minus Hankel operators on all the standard Lebesgue spaces. Different types of Fourier symbols will be used but special focus will be considered on the Wiener subclass of almost periodic matrix functions. In the first part of the paper we will give a survey of investigations on related results. This will be useful at the end of the paper to derive the above mentioned dependencies between the operators under study.

Function Spaces and Applications

Function spaces and its various applications are subjects of investigation in many research centers and we are glad that this special issue has received the remarkable attention by researchers in these areas. We have received papers from wide range of topics of function spaces such as interpolation methods for stochastic processes spaces, vector optimization problems in Banach spaces, Marcinkiewicz integral operator in Morrey spaces, Hardy’s operator in the variable exponent Lebesgue spaces, and differential equations of weakly parabolic type in Banach spaces. A number of papers are devoted to investigating different kinds of problems in Hardy spaces or Hardy subspaces, for example, Calderon-Zygumnd singular integral operators in Hardy spaces, weighted Hardy spaces related to the Schrodinger operator, and composition operator in hyperbolic Bloch-type spaces. Geometric problems such as modulus of convexity in some Banach sequence spaces, p-uniform convexity, and quniform smoothness in C spaces are also covered by this issue.

Multilinear Fourier multipliers on variable Lebesgue spaces

In this paper, we study the properties of a bilinear multiplier space. We give a necessary condition for a continuous bounded function to be a bilinear multiplier on variable exponent Lebesgue spaces, and we prove the localization theorem of multipliers on variable exponent Lebesgue spaces. Moreover, we present a Mihlin-Hormander type theorem for multilinear Fourier multipliers on weighted variable Lebesgue spaces and give some applications.

Maximal and Calderón–Zygmund operators in grand variable exponent Lebesgue spaces

Abstract New function spaces L p ( · ) , θ $L^{ p(\,\cdot \,), \theta }$ , ℒ p ( · ) , θ ${\mathcal {L}}^{ p(\,\cdot \,), \theta }$ unifying grand Lebesgue spaces and variable exponent Lebesgue spaces are introduced. The boundedness of maximal and Calderón–Zygmund operators in these spaces defined on spaces of homogeneous type are derived. The Sobolev type theorem for fractional integrals is also established in the class of functions which is narrower than the space L p ( · ) , θ $L^{ p(\,\cdot \,),\theta }$ .