Variable Exponent Lebesgue Spaces Research Articles

English

We obtain the factorization theorem for Hardy space via the variable exponent Lebesgue spaces. As an application, it is proved that if the commutator of Coifman, Rochberg and Weiss [b, T] is bounded on the variable exponent Lebesgue spaces, then b is a bounded mean oscillation (BMO) function.

Note on the variable exponent Lebesgue function spaces close to L∞

In this paper we characterize those exponents p ( ⋅ ) for which corresponding variable exponent Lebesgue space L p ( ⋅ ) ( [ 0 ; 1 ] ) has in common with L ∞ the property that the space of continuous functions is a closed linear subspace in it. In particular, we obtain necessary and sufficient condition on decreasing rearrangement of exponent p ( ⋅ ) for which exists equimeasurable exponent of p ( ⋅ ) which corresponding variable exponent Lebesgue space have the above mentioned property.

Uniform boundedness of Kantorovich operators in variable exponent Lebesgue spaces

In this paper, the Kantorovich operators Kn, n ? N are shown to be uniformly bounded in variable exponent Lebesgue spaces on the closed interval [0; 1]. Also an upper estimate is obtained for the difference Kn(f)-f for functions f of regularity of order 1 and 2 measured in variable exponent Lebesgue spaces, which is of interest on its own and can be applied to other problems related to the Kantorovich operators.

Approximation by Zygmund means in variable exponent Lebesque spaces

In the present work we investigate the approximation of the functions by the Zygmund means in variable exponent Lebesgue spaces. Here the estimate which is obtained depends on sequence of the best approximation in Lebesgue spaces with variable exponent. Also, these results were applied to estimates of approximations of Zygmund sums in Smirnov classes with variable exponent defined on simply connected domains of the complex plane.

Quantitative Microwave Imaging Method in Lebesgue Spaces With Nonconstant Exponents

An innovative nonlinear inverse scattering approach in variable exponent Lebesgue spaces is proposed for microwave imaging purposes. The main objective of the approach is to overcome one of the main problems associated with reconstruction procedures developed in Lebesgue spaces with constant exponent, i.e., the need for the selection of the optimum Lebesgue-space norm parameter, which is critical for obtaining accurate reconstructions, and no exact rules exist for this choice. The approach proposed in this paper is oriented to tomographic imaging applications at microwave frequencies, for which the adopted inversion procedure is based on a Gauss–Newton method, which has been reformulated in the unconventional variable exponent Lebesgue spaces. The capabilities of the approach are demonstrated by a set of numerical simulations, in which cylindrical dielectric targets are inspected. Moreover, an experimental result is reported.

A Characterization of Approximation of Hardy Operators in VLS

Variable exponent spaces and Hardy operator space have played an important role in recent harmonic analysis because they have an interesting norm including both local and global properties. The variable exponent Lebesgue spaces are of interest for their applications to modeling problems in physics, and to the study of variational integrals and partial differential equations with non-standard growth conditions. This studies also has been stimulated by problems of elasticity, fluid dynamics, calculus of variations, and differential equations with non-standard growth conditions. In this study, we will discuss a characterization of approximation of Hardy operators in variable Lebesgue spaces.

Riesz-Kolmogorov theorem in variable exponent Lebesgue spaces and its applications to Riemann-Liouville fractional differential equations

In this paper, we obtain the necessary and sufficient condition of the pre-compact sets in the variable exponent Lebesgue spaces, which is also called the Riesz-Kolmogorov theorem. The main novelty appearing in this approach is the constructive approximation which does not rely on the boundedness of the Hardy-Littlewood maximal operator in the considered spaces such that we do not need the log-Holder continuous conditions on the variable exponent. As applications, we establish the boundedness of Riemann-Liouville integral operators and prove the compactness of truncated Riemann-Liouville integral operators in the variable exponent Lebesgue spaces. Moreover, applying the Riesz-Kolmogorov theorem established in this paper, we obtain the existence and the uniqueness of solutions to a Cauchy type problem for fractional differential equations in variable exponent Lebesgue spaces.

Some Inverse and Simultaneous Approximation Theorems in Weighted Variable Exponent Lebesgue Spaces

In this work we prove some inverse and simultaneous approximation theorems in the weighted variable exponent Lebesgue space and discuss the constructive description problems in the generalized Lipschitz subclasses.

Characterization of boundedness of some commutators of maximal functions in terms of Lipschitz spaces

Our aim is to characterize the Lipschitz functions by variable exponent Lebesgue spaces. We give some characterizations of the boundedness of the maximal or nonlinear commutators of the Hardy-Littlewood maximal function and sharp maximal function in variable exponent Lebesgue spaces when the symbols $b$ belong to the Lipschitz spaces, by which some new characterizations of Lipschitz spaces and nonnegative Lipschitz functions are obtained. Some equivalent relations between the Lipschitz norm and the variable exponent Lebesgue norm are also given.

Approximation by integral functions of finite degree in variable exponent Lebesgue spaces on the real axis

We obtain several inequalities of approximation by integral functions of finite degree in generalized Lebesgue spaces with variable exponent defined on the real axis. Among them are direct, inverse, and simultaneous estimates of approximation by integral functions of finite degree in $L^{p\left( \cdot \right)}.$ An equivalence of modulus of continuity with Peetre's $K$ -functional is established. A constructive characterization of Lipschitz class is also obtained.

One-Sided Operators in Grand Variable Exponent Lebesgue Spaces

One-Sided Operators in Grand Variable Exponent Lebesgue Spaces

Weighted variable exponent Sobolev estimates for elliptic equations with non-standard growth and measure data

Consider the following nonlinear elliptic equation of p(x)-Laplacian type with nonstandard growth $$\begin{aligned} {\left\{ \begin{array}{ll} \mathrm{div}a(Du, x)=\mu &{}\quad \text {in} \quad \Omega ,\\ u=0 &{} \quad \text {on} \quad \partial \Omega , \end{array}\right. } \end{aligned}$$where \(\Omega \) is a Reifenberg domain in \(\mathbb {R}^n\), \(\mu \) is a Radon measure defined on \(\Omega \) with finite total mass and the nonlinearity \(a: \mathbb {R}^n\times \mathbb {R}^n\rightarrow \mathbb {R}^n\) is modeled upon the \(p(\cdot )\)-Laplacian. We prove the estimates on weighted variable exponent Lebesgue spaces for gradients of solutions to this equation in terms of Muckenhoupt–Wheeden type estimates. As a consequence, we obtain some new results such as the weighted \(L^q-L^r\) regularity (with constants \(q < r\)) and estimates on Morrey spaces for gradients of the solutions to this non-linear equation.

Density, Duality and Preduality in Grand Variable Exponent Lebesgue and Morrey Spaces

In this note some structural properties of grand variable exponent Lebesgue/Morrey spaces over spaces of homogeneous type are obtained. In particular, it is proved that the closure of $$L^{\infty }$$ and the closure of $$L^{p(\cdot ), \lambda (\cdot )}$$ in grand variable exponent Morrey space $$L^{p(\cdot ), \lambda (\cdot ), \theta }$$ coincide if the measure of the underlying space is finite. Moreover, we get two different characterizations of this class. Further, duality and preduality of grand variable exponent Lebesgue spaces defined on quasi-metric measure spaces with $$\sigma $$ -finite measure are obtained, which is new even when $$\mu $$ is finite.

Optimal Control Problem for Bianchi Equation in Variable Exponent Sobolev Spaces

In this paper, a necessary and sufficient condition, such as the Pontryagin’s maximum principle for an optimal control problem with distributed parameters, is given by the third-order Bianchi equation with coefficients from variable exponent Lebesgue spaces. The statement of an optimal control problem is studied by using a new version of the increment method that essentially uses the concept of the adjoint equation of the integral form.

The bounded approximation property of variable Lebesgue spaces and nuclearity

In this paper we prove the bounded approximation property for variable exponent Lebesgue spaces, study the concept of nuclearity on such spaces and apply it to trace formulae such as the Grothendieck-Lidskii formula. We apply the obtained results to derive criteria for nuclearity and trace formulae for periodic operators on $\mathbb{R}^n$ in terms of global symbols.

On the Rellich inequality in L p(·)(a,b)

Abstract In this note we prove a one-dimensional Rellich inequality in variable exponent Lebesgue spaces, with exponents that satisfy a condition of log-Hölder type.

Boundedness of the Bergman Projection and Some Properties of Bergman Type Spaces

We give a simple proof of the boundedness of Bergman projection in various Banach spaces of functions on the unit disc in the complex plain. The approach of the paper is based on the idea of Zaharyuta and Yudovich (Uspekhi Mat Nauk 19(2):139–142, 1964) where the boundedness of the Bergman projection in Lebesgue spaces was proved using Calderon–Zygmund operators. We exploit this approach and treat the cases of variable exponent Lebesgue space, Orlicz space and variable exponent generalized Morrey spaces. In the case of variable exponent Lebesgue space the boundedness result is known, so in that case we provide a simpler proof, whereas the other cases are new. The major idea of this paper is to show that the approach can be applied to a wide range of function spaces. We also study the rate of growth of functions near the boundary in spaces under consideration and their approximation by mollified dilations.

Approximation by Matrix Transforms in Weighted Lebesgue Spaces with Variable Exponent

In this work the approximation properties of the matrix transforms of functions in the weighted variable exponent Lebesgue spaces are investigated.

On approximation by Nörlund and Riesz submethods in variable exponent lebesgue spaces

In this study the results on the degree of approximation by the Nörlund and the Riesz submethods of the partial sums of their Fourier series of functions where in the variable exponent Lebesgue spaces are given by weakening the monotonicity conditions of sequences in the submethods. Therefore the results given in G¸ven and Israfilov (2010) are generalized according to · both the monotonicity conditions and both the methods.

Nondivergence parabolic equations in weighted variable exponent spaces

We prove the global Calderón-Zygmund estimates for second order parabolic equations in nondivergence form in weighted variable exponent Lebesgue spaces. We assume that the associated variable exponent is log-Hölder continuous, the weight is of a certain Muckenhoupt class with respect to the variable exponent, the coefficients of the equation are the functions of small bonded mean oscillation, and the underlying domain is a C 1 , 1 C^{1,1} -domain.