Variable Exponent Lebesgue Spaces Research Articles

Sharp maximal and weighted estimates for multilinear iterated commutators of multilinear integrals with generalized kernels

In this paper, the authors establish the sharp maximal estimates for the multilinear iterated commutators generated by BMO functions and multilinear singular integral operators with generalized kernels. As applications, the boundedness of this kind of multilinear iterated commutators on the product of weighted Lebesgue spaces and the product of variable exponent Lebesgue spaces can be obtained, respectively.

Approximation problems in the Lebesgue spaces with variable exponent

In the variable exponent Lebesgue space, the r-th modulus of smoothness (r=1,2,...) is defined and in this term, the direct and inverse theorems of approximation theory are proved. Moreover, the constructive characterization problems for the some subclasses are discussed.

Multilinear singular integral operators with generalized kernels and their multilinear commutators

In this paper, the authors study a class of multilinear singular integral operators with generalized kernels and their multilinear commutators with BMO functions. By establishing the sharp maximal estimates, the boundedness on product of weighted Lebesgue spaces and product of variable exponent Lebesgue spaces is obtained, respectively. Moreover, the endpoint estimate of this class of mutilinear singular integral operators is also established. These results can improve the corresponding known results of classical multilinear Calderon–Zygmund operators and multilinear Calderon–Zygmund operators with Dini type kernels.

On Boundedness of Bergman Projection Operators in Banach Spaces of Holomorphic Functions in Half-Plane and Harmonic Functions in Half-Space

We present a simple proof of the boundedness of holomorphic and harmonic Bergman projection operators on a half-plane and a half-space respectively on the Orlicz space, the variable exponent Lebesgue space, and the variable exponent generalized Morrey space. The approach is based on an idea due to V. P. Zaharyuta and V. I. Yudovich (1962) to use Calderon–Zygmund operators for proving the boundedness of the Bergman projection in Lebesgue spaces on the unit disc. We also study the rate of growth of functions near the boundary in the spaces under consideration.

A unified point of view on boundedness of Riesz type potentials

We introduce a natural extension of the Riesz potentials to quasi-metric measure spaces with an upper doubling measure. In particular, these operators are defined when the underlying space has components of differing dimensions. We study the behavior of the potential on classical and variable exponent Lebesgue spaces, obtaining necessary and sufficient conditions for its boundedness. The technique we use relies on a geometric property of the measure of the balls which holds both in the doubling and non-doubling situations, and allows us to present our results in a unified way.

On Some Properties of Convolutions in Variable Exponent Lebesgue Spaces

A convolution in the variable exponent Lebesgue spaces $$L_{2\pi }^{p\left( \cdot \right) }$$ is defined and its basic properties are investigated. It is also proved that this convolution can be approximated in $$L_{2\pi }^{p\left( \cdot \right) }$$ by the finite linear combinations of Steklov means of the original function.

Hardy–Littlewood Maximal Operator in Weighted Grand Variable Exponent Lebesgue Space

The boundedness of the Hardy–Littlewood maximal operator is proved in weighted grand variable exponent Lebesgue space with power weights.

Invertibility characterization of Wiener–Hopf plus Hankel operators on variable exponent Lebesgue spaces via even asymmetric factorization

Abstract We obtain invertibility and Fredholm criteria for the Wiener–Hopf plus Hankel operators acting between variable exponent Lebesgue spaces on the real line. Such characterizations are obtained via the so-called even asymmetric factorization, which is applied to the Fourier symbols of the operators under study.

The Riemann boundary value problem in the class of Cauchy type integrals with densities of grand variable exponent Lebesgue spaces

Abstract The present paper deals with the Riemann boundary value problem for analytic functions in the framework of the new function spaces introduced by the first two authors, the so-called grand variable exponent Lebesgue spaces which unify two non-standard type function spaces: variable exponent Lebesgue spaces and grand Lebesgue spaces.

Complex interpolation on variable exponent Campanato spaces of order k

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 Campanato spaces of order k.

Global W 1,p(·) estimate for renormalized solutions of quasilinear equations with measure data on Reifenberg domains

Abstract In this paper, we prove the gradient estimate for renormalized solutions to quasilinear elliptic equations with measure data on variable exponent Lebesgue spaces with BMO coefficients in a Reifenberg flat domain.

The Dirichlet problem for elliptic systems with data in Köthe function spaces

We show that the boundedness of the Hardy–Littlewood maximal operator on a Köthe function space \mathbb X and on its Köthe dual \mathbb X ' is equivalent to the well-posedness of the \mathbb X -Dirichlet and \mathbb X '-Dirichlet problems in \mathbb R^n_+ in the class of all second-order, homogeneous, elliptic systems, with constant complex coefficients. As a consequence, we obtain that the Dirichlet problem for such systems is well-posed for boundary data in Lebesgue spaces, variable exponent Lebesgue spaces, Lorentz spaces, Zygmund spaces, as well as their weighted versions. We also discuss a version of the aforementioned result which contains, as a particular case, the Dirichlet problem for elliptic systems with data in the classical Hardy space H^1 , and the Beurling-Hardy space HA ^p for p \in (1,\infty) . Based on the well-posedness of the L^p -Dirichlet problem we then prove the uniqueness of the Poisson kernel associated with such systems, as well as the fact that they generate a strongly continuous semigroup in natural settings. Finally, we establish a general Fatou type theorem guaranteeing the existence of the pointwise nontangential boundary trace for null-solutions of such systems.

Characterization of Lipschitz Functions via the Commutators of Singular and Fractional Integral Operators in Variable Lebesgue Spaces

We obtain characterizations of a variable version of Lipschitz spaces in terms of the boundedness of commutators of Calderon-Zygmund and fractional type operators in the context of the variable exponent Lebesgue spaces Lp(⋅), where the symbols of the commutators belong to the Lipschitz spaces. A useful tool is a pointwise estimate involving the sharp maximal operator of the commutator and certain associated maximal operators, which is new even in the classical context. Some boundedness properties of the commutators between Lebesgue and Lipschitz spaces in the variable context are also proved.

Some functional inequalities in variable exponent spaces with a more generalization of uniform continuity condition

Some functional inequalities in variable exponent Lebesgue spaces are presented. The bi-weighted modular inequality with variable exponent p(:) for the Hardy operator restricted to non-increasing function which is Z 1 0 f(x) p(x) u(x)dx; is studied. We show that the exponent p(:) for which these modular inequalities hold must have constant oscillation. Also we study the boundedness of integral operator Tf(x) = R K(x;y)f(x)dy on L p(:) when the variable exponent p(:) satises some uniform continuity condition that is named -controller condition and so multiple interesting results which can be seen as a generalization of the same classical results in the constant exponent case, derived.

From Arzelà–Ascoli to Riesz–Kolmogorov

In this paper we study totally bounded sets in Banach function spaces (BFS), from which we characterize compact sets (via Hausdorff criterion) in some non-standard function spaces which fall under the umbrella of BFS. We obtain a Riesz–Kolmogorov compactness theorem for the grand variable exponent Lebesgue spaces.

On supercritical Sobolev type inequalities and related elliptic equations

Sobolev type embeddings for radial functions into variable exponent Lebesgue spaces are studied. In particular, the following inequality is proved: let $$B \subset \mathbb R^N, N \ge 3$$ , be the unit ball, and let $$H_{0,\mathrm rad}^1(B)$$ denote the first order Sobolev space of radial functions, and $$2^* = 2N/(N-2)$$ the corresponding critical Sobolev embeddding exponent. Let $$r = |x|$$ , and $$p(r)=2^*+r^{\alpha }$$ , with $$\alpha > 0$$ ; then 0.1 $$\begin{aligned} \sup \left\{ \int _{B} |u|^{p(r)} \; dx \big | u \in H^1_{0,\mathrm{rad}}(B), \Vert \nabla u \Vert _2=1 \right\} < +\infty . \end{aligned}$$ We point out that the growth of p(r) is strictly larger than $$2^*$$ , except in the origin. Furthermore, we show that for $$p(r) = 2^* + r^\alpha $$ , with $$0< \alpha < \min \{\frac{N}{2};N-2\}$$ , the supremum in (0.1) is attained. Finally, we prove that associated elliptic equations admit nontrivial radial solutions. This is somewhat surprising since the nonlinearities have strictly supercritical growth except in the origin.

Approximation by Faber–Laurent rational functions in Lebesgue spaces with variable exponent

Let Γ be a rectifiable Dini-smooth Jordan curve in the complex plane C. In this work the approximation properties of the Faber–Laurent series expansions in the variable exponent Lebesgue spaces defined on the curve Γ are investigated.

On maximal and potential operators with rough kernels in variable exponent spaces

In the framework of variable exponent Lebesgue and Morrey spaces we prove some boundedness results for operators with rough kernels, such as the maximal operator, fractional maximal operator, sharp maximal operators and fractional operators. The approach is based on some pointwise estimates.

Extension of the best polynomial approximation operator in variable exponent Lebesgue spaces

Extension of the best polynomial approximation operator in variable exponent Lebesgue spaces

Ergodic theorem in variable Lebesgue spaces

We prove an Ergodic Theorem in variable exponent Lebesgue spaces, whenever the exponent is invariant under the transformation. Moreover, a counterexample is provided which shows that the norm convergence fails to hold for an arbitrary exponent.