We show that the Lebesgue space with a variable exponent $L_{p(\cdot )}$ is a rearrangement--invariant space if and only if $p$ is constant. In addition, we give a necessary and …
We show that the Lebesgue space with a variable exponent $L_{p(\cdot )}$ is a rearrangement--invariant space if and only if $p$ is constant. In addition, we give a necessary and sufficient condition on a variable exponent for a martingale inequality to hold.
In this chapter, we present the variable exponent Lebesgue spaces defined by Orlicz1) in 1931. Although the variable exponent Lebesgue spaces are introduced in 1931, it began to be actively …
In this chapter, we present the variable exponent Lebesgue spaces defined by Orlicz1) in 1931. Although the variable exponent Lebesgue spaces are introduced in 1931, it began to be actively studied in 1990s. We especially used the resources by Diening, Harjulehto, Hasto, Ruzicka 2011; Fan, Zhao 2001; Kovacik, Rakosnik 1991; Cruz-Uribe, Fiorenza 2013; Radulescu and Repovs 2015 for this section. The notion variable exponent Lebesgue and Sobolev spaces is directly related to the classical Lebesgue and Sobolev spaces where the constant p is replaced with the function p(.) which may depend on a variable. Further properties of these spaces are introduced and analyzed in that chapter.
In this article, we begin with classical Lebesgue spaces Lp with p being constant and review the various properties such as completeness and duality of the space. To this end, …
In this article, we begin with classical Lebesgue spaces Lp with p being constant and review the various properties such as completeness and duality of the space. To this end, we also discuss the boundedness of Hardy-Littlewood maximal function and interpolation on such spaces. Finally, we focus our attention on variable exponent Lebesgue spaces and review various results on it. Moreover, we also see the differences in between these Lebesgue spaces.
Let M be a measure space and 1 ≤ p ≤ ∞. A real-valued function f on M is said to be pth-power integrable, or belong to L p , …
Let M be a measure space and 1 ≤ p ≤ ∞. A real-valued function f on M is said to be pth-power integrable, or belong to L p , if f is measurable, and |f| p is integrable if p < ∞, while if p = ∞, it is required that there exist a null set in M on whose complement f is bounded. The class of all such functions f is denoted by L p (M), or just L p , when the measure space is fixed. For any f ∈ L p (M), the norm of f, denoted by ‖f‖ p , is defined as follows: (a) if p < ∞, then ‖f‖ p = (∫|f|)1/p; (b) if p = ∞, <math display='block'> <mrow> <msub> <mrow> <mrow><mo>‖</mo> <mi>f</mi> <mo>‖</mo></mrow> </mrow> <mi>∞</mi> </msub> <mo>=</mo><munder> <mrow> <mi>inf</mi> </mrow> <mi>N</mi> </munder> <munder> <mrow> <mi>sup</mi> </mrow> <mrow> <mi>p</mi><mo>∉</mo><mi>N</mi> </mrow> </munder> <mrow><mo>|</mo> <mrow> <mi>f</mi><mrow><mo>(</mo> <mi>p</mi> <mo>)</mo></mrow> </mrow> <mo>|</mo></mrow> </mrow> </math> $${\left\| f \right\|_\infty } = \mathop {\inf }\limits_N \mathop {\sup }\limits_{p \notin N} \left| {f\left( p \right)} \right|$$ , where N ranges over all the null sets in M.
1 Introduction.- 2 A framework for function spaces.- 3 Variable exponent Lebesgue spaces.- 4 The maximal operator.- 5 The generalized Muckenhoupt condition*.- 6 Classical operators.- 7 Transfer techniques.- 8 Introduction …
1 Introduction.- 2 A framework for function spaces.- 3 Variable exponent Lebesgue spaces.- 4 The maximal operator.- 5 The generalized Muckenhoupt condition*.- 6 Classical operators.- 7 Transfer techniques.- 8 Introduction to Sobolev spaces.- 9. Density of regular functions.- 10. Capacities.- 11 Fine properties of Sobolev functions.- 12 Other spaces of differentiable functions.- 13 Dirichlet energy integral and Laplace equation.- 14 PDEs and fluid dynamics
We extend the theory of matrix weights to the variable Lebesgue spaces. The theory of matrix [Formula: see text] weights has attracted considerable attention beginning with the work of Nazarov, …
We extend the theory of matrix weights to the variable Lebesgue spaces. The theory of matrix [Formula: see text] weights has attracted considerable attention beginning with the work of Nazarov, Treil, and Volberg in the 1990s. We extend this theory by generalizing the matrix [Formula: see text] condition to the variable exponent setting. We prove boundedness of the convolution operator [Formula: see text] for [Formula: see text], and show that the approximate identity defined using [Formula: see text] converges in matrix weighted, variable Lebesgue spaces [Formula: see text] for [Formula: see text] in matrix [Formula: see text]. Our approach to this problem is through averaging operators; these results are of interest in their own right. As an application of our work, we prove a version of the classical [Formula: see text] theorem for matrix weighted, variable exponent Sobolev spaces.
Leibniz-type rules for Coifman–Meyer multiplier operators are studied in the settings of Triebel–Lizorkin and Besov spaces associated with weights in the Muckenhoupt classes. Even in the unweighted case, improvements on …
Leibniz-type rules for Coifman–Meyer multiplier operators are studied in the settings of Triebel–Lizorkin and Besov spaces associated with weights in the Muckenhoupt classes. Even in the unweighted case, improvements on the currently known estimates are obtained. The flexibility of the methods of proofs allows one to prove Leibniz-type rules in a variety of function spaces that include Triebel–Lizorkin and Besov spaces based on weighted Lebesgue, Lorentz, and Morrey spaces as well as variable-exponent Lebesgue spaces. Applications to scattering properties of solutions to certain systems of partial differential equations involving fractional powers of the Laplacian are presented.
We define the new central Morrey space with variable exponent and investigate its relation to the Morrey-Herz spaces with variable exponent. As applications, we obtain the boundedness of the homogeneous …
We define the new central Morrey space with variable exponent and investigate its relation to the Morrey-Herz spaces with variable exponent. As applications, we obtain the boundedness of the homogeneous fractional integral operator <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:msub><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Ω</mml:mi><mml:mo>,</mml:mo><mml:mi>σ</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math> and its commutator <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mo stretchy="false">[</mml:mo><mml:mi>b</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Ω</mml:mi><mml:mo>,</mml:mo><mml:mi>σ</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">]</mml:mo></mml:math> on Morrey-Herz space with variable exponent, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mi mathvariant="normal">Ω</mml:mi><mml:mo>∈</mml:mo><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math> for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mi>s</mml:mi><mml:mo>≥</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:math> is a homogeneous function of degree zero, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mn mathvariant="normal">0</mml:mn><mml:mo><</mml:mo><mml:mi>σ</mml:mi><mml:mo><</mml:mo><mml:mi>n</mml:mi></mml:math>, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:mrow><mml:mi>b</mml:mi></mml:mrow></mml:math> is a BMO function.
Abstract In this paper, applying the properties of variable exponent analysis and rough kernel, we study the mapping properties of rough singular integral operators. Then, we show the boundedness of …
Abstract In this paper, applying the properties of variable exponent analysis and rough kernel, we study the mapping properties of rough singular integral operators. Then, we show the boundedness of rough Calderón–Zygmund type singular integral operator, rough Hardy–Littlewood maximal operator, as well as the corresponding commutators in variable exponent vanishing generalized Morrey spaces on bounded sets. In fact, the results above are generalizations of some known results on an operator basis.
The perturbed system of exponents with a piecewise linear phase, consisting of eigenfunctions of a discontinuous differential operator, is considered in this work. Under certain conditions on the weight function …
The perturbed system of exponents with a piecewise linear phase, consisting of eigenfunctions of a discontinuous differential operator, is considered in this work. Under certain conditions on the weight function of the form of a power function, sufficient conditions for the basicity of this system are obtained in generalized weighted Lebesgue space.
The wavelet transform is an useful mathematical tool. It is a mapping of a time signal to the time-scale joint representation. The wavelet transform is generated from a wavelet function …
The wavelet transform is an useful mathematical tool. It is a mapping of a time signal to the time-scale joint representation. The wavelet transform is generated from a wavelet function by dilation and translation. This wavelet function satisfies an admissible condition so that the original signal can be reconstructed by the inverse wavelet transform. In this study, we firstly give some basic properties of the weighted variable exponent amalgam spaces. Then we investigate the convergence of the θ-means of f in these spaces under some conditions. Finally, using these results the convergence of the inverse continuous wavelet transform is considered in these spaces.
We consider weighted norm inequalities for multilinear multipliers whose symbols satisfy a product-type Hörmander condition. Our approach is to consider a more general family of multilinear singular integral operators associated …
We consider weighted norm inequalities for multilinear multipliers whose symbols satisfy a product-type Hörmander condition. Our approach is to consider a more general family of multilinear singular integral operators associated to a family of smooth kernels that satisfy an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript r"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>r</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">L^r</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Schwartz regularity condition. We give conditions for these operators to satisfy weighted Hardy space estimates and derive our results for multipliers as a special case. As an additional application, we use Rubio de Francia extrapolation to prove multilinear estimates on the variable exponent Hardy spaces.
In the present paper we find optimal conditions separating the regular case from the one with Lavrentiev gap for the borderline case of double phase potencial and related general classes …
In the present paper we find optimal conditions separating the regular case from the one with Lavrentiev gap for the borderline case of double phase potencial and related general classes of integrands. We present new results on density of smooth functions.
Abstract We define a new class of functions of variable smoothness that are analytic in the unit disc and continuous in the closed disc. We construct the theory of the …
Abstract We define a new class of functions of variable smoothness that are analytic in the unit disc and continuous in the closed disc. We construct the theory of the Nevanlinna outer-inner factorization, taking into account the influence of the inner factor on the outer function, for functions of the new class.
UDC 517.5 In the framework of variable exponent Morrey and Morrey–Herz spaces, we prove some boundedness results for the commutator of Marcinkiewicz integrals with rough kernels. The approach is based …
UDC 517.5 In the framework of variable exponent Morrey and Morrey–Herz spaces, we prove some boundedness results for the commutator of Marcinkiewicz integrals with rough kernels. The approach is based on the theory of variable exponent and on generalization of the BMO-norms.
The grand-Lebesgue space is defined. Based on the shift operator, a separable subspace is determined in which continuous functions are dense. The concepts of frame and atomic decomposition are defined. …
The grand-Lebesgue space is defined. Based on the shift operator, a separable subspace is determined in which continuous functions are dense. The concepts of frame and atomic decomposition are defined. An atomic decomposition of double and unary systems of functions in grand-Lebesgue spaces is considered. Relationship between atomic decomposition of these systems in grand-Lebesgue spaces is established.
In the present paper, we study $p(x)-$biharmonic problem involving $q(x)-$Hardy type potential with no-flux boundary condition. By using the mountain pass type theorem and Ekeland variatoinal principle, we obtain at …
In the present paper, we study $p(x)-$biharmonic problem involving $q(x)-$Hardy type potential with no-flux boundary condition. By using the mountain pass type theorem and Ekeland variatoinal principle, we obtain at least two nontrivial weak solutions.
Our aim in this paper is to deal with Sobolev inequalities for Riesz potentials of functions in Lebesgue spaces of variable exponents near Sobolev’s exponent over nondoubling metric measure spaces.
Our aim in this paper is to deal with Sobolev inequalities for Riesz potentials of functions in Lebesgue spaces of variable exponents near Sobolev’s exponent over nondoubling metric measure spaces.
This paper aims to show that the fractional Hardy operator and its adjoint operator are bounded on central Morrey space with variable exponent. Similar results for their commutators are obtained …
This paper aims to show that the fractional Hardy operator and its adjoint operator are bounded on central Morrey space with variable exponent. Similar results for their commutators are obtained when the symbol functions belong to λ ‐central bounded mean oscillation ( λ ‐central BMO) space with variable exponent.
In this paper, we analyze multi-dimensional weighted ergodic components in general metric.We provide numerous illustrative examples, theoretical results and applications to the abstract Volterra integro-differential equations.
In this paper, we analyze multi-dimensional weighted ergodic components in general metric.We provide numerous illustrative examples, theoretical results and applications to the abstract Volterra integro-differential equations.
This work deals with the Orlicz space and the Hardy-Orlicz classes generated by this space, which consist of analytic functions inside and outside the unit disk. The homogeneous Riemann boundary …
This work deals with the Orlicz space and the Hardy-Orlicz classes generated by this space, which consist of analytic functions inside and outside the unit disk. The homogeneous Riemann boundary value problems with piecewise continuous coefficients are considered in these classes. New characteristic of Orlicz space is defined which depends on whether the power function belongs to this space or not. Relationship between this characteristic and Boyd indices of Orlicz space is established. The concept of canonical solution of homogeneous problem is defined, which depends on the argument of the coefficient. In terms of the above characteristic, a condition on the jumps of the argument is found which is sufficient for solvability of these problems, and, in case of solvability, a general solution is constructed. It is established the basicity of the parts of exponential system in Hardy-Orlicz classes.
In this paper, a class of nonlinear differential boundary value problems with variable exponent is investigated. The existence of at least one non-zero solution is established, without assuming on the …
In this paper, a class of nonlinear differential boundary value problems with variable exponent is investigated. The existence of at least one non-zero solution is established, without assuming on the nonlinear term any condition either at zero or at infinity. The approach is developed within the framework of the Orlicz-Sobolev spaces with variable exponent and it is based on a local minimum theorem for differentiable functions.
In this paper, we prove a Hölder’s type inequality for fully measurable grand Lebesgue spaces, which involves the notion of fully measurable small Lebesgue spaces. It is proved that these …
In this paper, we prove a Hölder’s type inequality for fully measurable grand Lebesgue spaces, which involves the notion of fully measurable small Lebesgue spaces. It is proved that these spaces are non-reflexive rearrangement invariant Banach function spaces. Moreover, under certain continuity assumptions, along with several properties of fully measurable small Lebesgue spaces, we establish Levi’s theorem for monotone convergence and that grand and small spaces are associated to each other.
We study quasilinear elliptic double obstacle problems with a variable exponent growth when the right-hand side is a measure. A global Calder\'{o}n-Zygmund estimate for the gradient of an approximable solution …
We study quasilinear elliptic double obstacle problems with a variable exponent growth when the right-hand side is a measure. A global Calder\'{o}n-Zygmund estimate for the gradient of an approximable solution is obtained in terms of the associated double obstacles and a given measure, identifying minimal requirements for the regularity estimate.
We show that many classical operators in harmonic analysis|such as maximal operators, singular integrals, commutators and fractional integrals|are bounded on the variable Lebesgue space L p( ) whenever the Hardy{Littlewood …
We show that many classical operators in harmonic analysis|such as maximal operators, singular integrals, commutators and fractional integrals|are bounded on the variable Lebesgue space L p( ) whenever the Hardy{Littlewood maximal operator is bounded on L p( ) . Further, we show that such operators satisfy vector-valued inequalities. We do so by applying the theory of weighted norm inequalities and extrapolation. As applications we prove the Calder on{Zygmund inequality for solutions of 4u = f in variable Lebesgue spaces, and prove the Calder on extension theorem for variable Sobolev spaces.
In this article a new method for moving from local to global results in variable exponent function spaces is presented.Several applications of the method are also given: Sobolev and trace …
In this article a new method for moving from local to global results in variable exponent function spaces is presented.Several applications of the method are also given: Sobolev and trace embeddings; variable Riesz potential estimates; and maximal function inequalities in Morrey spaces are derived for unbounded domains.
We give continuity conditions on the exponent function p(x) which are su-- cient for the Hardy{Littlewood maximal operator to be bounded on the variable Lebesgue space L p(x) (›) , …
We give continuity conditions on the exponent function p(x) which are su-- cient for the Hardy{Littlewood maximal operator to be bounded on the variable Lebesgue space L p(x) (›) , where › is any open subset of R n . Further, our conditions are necessary on R. Our result extends the recent work of Pick and R••a (20), Diening (3) and Nekvinda (19). We also show that under much weaker assumptions on p(x) , the maximal operator satisfles a weak-type modular inequality.
We consider maximal operators M B with respect to a basis B. In the case when M B satisfies a reversed weak type inequality, we obtain a boundedness criterion for …
We consider maximal operators M B with respect to a basis B. In the case when M B satisfies a reversed weak type inequality, we obtain a boundedness criterion for M B on an arbitrary quasi-Banach function space X.Being applied to specific B and X this criterion yields new and short proofs of a number of well-known results.Our principal application is related to an open problem on the boundedness of the two-dimensional one-sided maximal function M + on L p w .
Abstract This paper represents a broadened version of the plenary lecture presented by the author at the conference Analytic Methods of Analysis and Differential Equations (AMADE-2003), September 4–9, 2003, Minsk, …
Abstract This paper represents a broadened version of the plenary lecture presented by the author at the conference Analytic Methods of Analysis and Differential Equations (AMADE-2003), September 4–9, 2003, Minsk, Belarus. We give a survey of investigations on 'the variable exponent business', concentrating mainly on recent advances in the operator theory and harmonic analysis in the generalized Lebesgue and Sobolev spaces L p(·) and W m, p(·). Keywords: Variable exponentMaximal operatorsSingular operatorsPotential operatorsHardy operatorsGeneralized Lebesgue and Sobolev spaces
We prove the boundedness of the Hardy-Littlewood maximal function on the generalized Lebesgue space L p(•) (R d ) under a continuity assumption on p that is weaker than uniform …
We prove the boundedness of the Hardy-Littlewood maximal function on the generalized Lebesgue space L p(•) (R d ) under a continuity assumption on p that is weaker than uniform Hölder continuity.We deduce continuity of mollifying sequences and density of C ∞ (Ω) in W 1,p(•) (Ω) .
We consider Hardy-Littlewood maximal operator on the general Lebesgue space L p(x) (R n ) with variable exponent.A sufficient condition on the function p is known for the boundedness of …
We consider Hardy-Littlewood maximal operator on the general Lebesgue space L p(x) (R n ) with variable exponent.A sufficient condition on the function p is known for the boundedness of the maximal operator on L p(x) (Ω) with an open bounded Ω .Our main aim is to find an additional condition to p to guarantee the boundedness of the maximal operator on L p(x) (R n ) .From this point of view we put an emphasis on the behavior of functions p near the infinity.We find a sufficient condition on p such that the maximal operator is bounded on L p(x) (R n ) .We also construct a function p for which the maximal operator is unbounded.