Generalized Lebesgue Spaces Research Articles

Nonlocal hyperbolic Stokes system with variable exponent of nonlinearity

In this paper, we study the problem for a nonlinear hyperbolic Stokes system of the second order with an integral term.Sufficient conditions for the uniqueness of the weak solution of this problem are found in a bounded domain. The nonlinear term of the system contains a variable exponent of nonlinearity, which is a function of spatial variables.The problem is studied in ordinary Sobolev spaces and generalized Lebesgue spaces, which is quite natural in this case.

Approximation properties of Bernstein singular integrals in variable exponent Lebesgue spaces on the real axis

In generalized Lebesgue spaces $L^{p(.)}$ with variable exponent $p(.)$ defined on the real axis, we obtain several inequalities of approximation by integral functions of finite degree. Approximation properties of Bernstein singular integrals in these spaces are obtained. Estimates of simultaneous approximation by integral functions of finite degree in $L^{p(.)}$ are proved.

Uniform estimates with data from generalized Lebesgue spaces in periodic structures

We study various types of uniform Calderón–Zygmund estimates for weak solutions to elliptic equations in periodic homogenization. A global regularity is obtained with respect to the nonhomogeneous term from weighted Lebesgue spaces, Orlicz spaces, and weighted Orlicz spaces, which are generalized Lebesgue spaces, provided that the coefficients have small BMO seminorms and the domains are δ-Reifenberg domains.

On Hausdorff-Young inequalities in generalized Lebesgue spaces

On Hausdorff-Young inequalities in generalized Lebesgue spaces

Quasilinear elliptic problem without Ambrosetti–Rabinowitz condition involving a potential in Musielak–Sobolev spaces setting

In this paper, we consider the quasilinear elliptic problem with potential where Ω is a smooth bounded domain in (), V is a given function in a generalized Lebesgue space , and is a Carathéodory function satisfying suitable growth conditions. Using variational arguments, we study the existence of weak solutions for in the framework of Musielak–Sobolev spaces. The main difficulty here is that the nonlinearity considered does not satisfy the well-known Ambrosetti–Rabinowitz condition.

A New Approach For Weighted Hardy’s Operator In VELS

Abstract A considerable number of research has been carried out on the generalized Lebesgue spaces L p(x) and boundedness of different integral operators therein. In this study, a new approach for weighted increasing near the origin and decreasing near infinity exponent function that provides a boundedness of the Hardy’s operator in variable exponent space is 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.

Derivatives of a polynomial of best approximation and modulus of smoothness in generalized Lebesgue spaces with variable exponent

Abstract The relation between derivatives of a polynomial of best approximation and the best approximation of the function is investigated in generalized Lebesgue spaces with variable exponent. In addition, the relationship between the fractional modulus of smoothness of the function and its de la Vallée-Poussin sums is studied.

W1,p(⋅) regularity for quasilinear problems with irregular obstacles on Reifenberg domains

In this paper, we prove the global gradient estimates on the generalized Lebesgue spaces for weak solutions to elliptic quasilinear obstacle problems. It is worth noticing that the coefficients related to the obstacle problems are merely measurable with small BMO norms and the underlying domain does not satisfy any smoothness conditions.

SPECTRAL THEOREMS ASSOCIATED TO THE DUNKL OPERATORS

In this paper, we characterize the support for the Dunkl transform on the generalized Lebesgue spaces via the Dunkl resolvent function. The behavior of the sequence of $L^{p}_{k}-$norms of iterated Dunkl potentials is studied depending on the support of their Dunkl transform. We systematically develop real Paley-Wiener theory for the Dunkl transform on $\mathbb{R}^{d}$ for distributions, in an elementary treatment based on the inversion theorem. Next, we improve the Roe's theorem associated to the Dunkl operators.

Approximations of periodic functions by analogue of Zygmund sums in the spaces Lp(•)

We found order estimates for the upper bounds of the deviations of analogue of Zygmund?s sums on the classes of (?;?)-differentiable functions in the metrics of generalized Lebesgue spaces with variable exponent.

EXPONENTIAL INTEGRABILITY FOR LOGARITHMIC POTENTIALS OF FUNCTIONS IN GENERALIZED LEBESGUE SPACES $L(\log L)^{q(\cdot)}$ OVER NON-DOUBLING MEASURE SPACES

In this paper, we are concerned with exponential integrability for logarithmic potentials of functions in generalized Lebesgue spaces $L(\log L)^{q(\cdot)}$ over non-doubling measure spaces. Here $q$ satisfies the loglog-Holder condition.

On basicity of the system of solutions of the Cauchy problem for the Sturm-Liouville equation in the generalized Lebesgue space

On basicity of the system of solutions of the Cauchy problem for the Sturm-Liouville equation in the generalized Lebesgue space

Maximal function on generalized martingale Lebesgue spaces with variable exponent

We prove the boundedness of the maximal operator for martingales on generalized Lebesgue spaces with variable exponent over probability spaces. We consider pointwise multipliers on martingale BMO as the variable exponent.

Best approximations of periodic functions in generalized lebesgue spaces

In generalized Lebesgue spaces with variable exponent, we determine the orders of the best approximations in the classes of (ψ; β)-differentiable 2π-periodic functions, deduce an analog of the well-known Bernstein inequality for the (ψ; β)-derivative, and apply this inequality to prove the inverse theorems of approximation theory in these classes.

On equivalent conditions for the general weighted Hardy type inequality in space $L^{p(\cdot)}$

We study the Hardy type two-weighted inequality for the multidimensional Hardy operator in the norms of generalized Lebesgue spaces L^{p(\cdot)}(\mathbb{R}^{n}) . In this way we prove equivalent conditions for L^{p(\cdot)}\rightarrow L^{q(\cdot)} boundedness of Hardy operator in the case of exponents q(0)\geq p(0) , q(\infty)\geq p\left(\infty \right) . We also prove that the condition for such inequality to hold coincides with condition for validity of two weighted Hardy inequalities with constant exponents, if we require the exponents to be regular near zero and at infinity.

Boundedness of functional calculi of Schrödinger operators on generalized Lebesgue spaces $${L^{p(\cdot)}(\mathbb{R}^n)}$$

Let L = −Δ + V be a Schrodinger operator on \({\mathbb{R}^n}\) , where \({V\in L^1_{\rm loc}({\mathbb R}^n)}\) is a nonnegative function on \({{\mathbb{R}^n}}\) . Let \({L^{p(\cdot)}(\mathbb{R}^n)}\) be the generalized Lebesgue spaces. In this article we use a technical atomic decomposition of Hardy space associated with L to show that the Lp(·)-norm of f can be controlled by the sum of the Lp(·)-norms of two variants of sharp maximal functions of f. As a result, we obtain boundedness of functional calculi of Schrodinger operators on generalized Lebesgue spaces \({L^{p(\cdot)}(\mathbb{R}^n)}\) .

Trigonometric approximation of functions in generalized Lebesgue spaces with variable exponent

We investigate the approximation properties of the trigonometric system in $ L_{2\pi }^{p\left( \cdot \right)} $ . We consider the moduli of smoothness of fractional order and obtain direct and inverse approximation theorems together with a constructive characterization of a Lipschitz-type class.

Some parabolic variational inequalities with variable exponent of nonlinearity: unique solvability and comparison theorems

We consider parabolic variational inequalities in Sobolev spaces and generalized Lebesgue spaces. Sufficient conditions of existence and uniqueness of solution of these inequalities have been established. We have also proved some comparison theorems.

Approximate identities and Young type inequalities in variable Lebesgue-Orlicz spaces Lp(.)(log L)q(.)

Our aim in this paper is to deal with approximate identities in generalized Lebesgue spaces L p(¢) (logL) q(¢) (R n ). As a related topic, we also study Young type inequalities for convolution with respect to norms in such spaces.