Generalized Lebesgue Spaces Research Articles

On a Hardy Type General Weighted Inequality in Spaces L p(·)

A Hardy type two-weighted inequality is investigated for the multidimensional Hardy operator in the norms of generalized Lebesgue spaces Lp(·). Equivalent necessary and sufficient conditions are found for the \({L^{p(\cdot)} \longrightarrow L^{q(\cdot)}}\) boundedness of the Hardy operator when exponents q(0) < p(0), q(∞) < p(∞). It is proved that the condition for such an inequality to hold coincides with the condition for the validity of two-weighted Hardy inequalities with constant exponents if we require of the exponents to be regular near zero and at infinity.

On geometric properties of the spaces L p(x)

We study the reflexivity, the uniform convexity, the Daugavet property and the Radon-Nikodym property of the generalized Lebesgue spaces Lp(x).

Trigonometric approximation in generalized Lebesgue spaces L^p(x)

The approximation properties of Norlund (Nn) and Riesz (Rn) means of trigono- metric Fourier series are investigated in generalized Lebesgue spaces L p(x) . The deviations � f −Nn(f)� p(x) andf −Rn(f)� p(x) are estimated by n −α for f ∈ Lip(α,p(x)) (0 < α 1).

On Boundedness of Weighted Hardy Operator in and Regularity Condition

We give a new proof for power-type weighted Hardy inequality in the norms of generalized Lebesgue spaces . Assuming the logarithmic conditions of regularity in a neighborhood of zero and at infinity for the exponents , necessary and sufficient conditions are proved for the boundedness of the Hardy operator from into . Also a separate statement on the exactness of logarithmic conditions at zero and at infinity is given. This shows that logarithmic regularity conditions for the functions at the origin and infinity are essentially one.

GEOMETRY OF SOBOLEV SPACES WITH VARIABLE EXPONENT AND A GENERALIZATION OF THE p-LAPLACIAN

Let Ω ⊂ ℝN, N ≥ 2, be a smooth bounded domain. It is shown that: (a) if [Formula: see text] and ess infx ∈ yp(x) &gt; 1, then the generalized Lebesgue space (Lp (·)(Ω), ‖·‖p(·)) is smooth; (b) if [Formula: see text] and p(x) &gt; 1, [Formula: see text], then the generalized Sobolev space [Formula: see text] is smooth. In both situations, the formulae for the Gâteaux gradient of the norm corresponding to each of the above spaces are given; (c) if [Formula: see text] and p(x) ≥ 2, [Formula: see text], then [Formula: see text] is uniformly convex and smooth.

Geometry of Sobolev spaces with variable exponent: smoothness and uniform convexity

Let Ω ⊂ R N , N ⩾ 2 , be a smooth bounded domain. It is shown that: (a) if p ∈ L ∞ ( Ω ) and ess inf x ∈ Ω p ( x ) > 1 , then the generalized Lebesgue space ( L p ( ⋅ ) ( Ω ) , ‖ ‖ p ( ⋅ ) ) is smooth; (b) if p ∈ C ( Ω ¯ ) and p ( x ) > 1 , for all x ∈ Ω ¯ , then the generalized Sobolev space ( W 0 1 , p ( ⋅ ) ( Ω ) , ‖ ‖ 1 , p ( ⋅ ) ) is smooth. In both situations, the formulae giving the Gâteaux derivative of the norm, corresponding to each of the above spaces, are given; (c) if p ∈ C ( Ω ¯ ) and p ( x ) ⩾ 2 , for all x ∈ Ω ¯ , then ( W 0 1 , p ( ⋅ ) ( Ω ) , ‖ ‖ 1 , p ( ⋅ ) ) is uniformly convex and smooth. To cite this article: G. Dinca, P. Matei, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Sobolev's inequalities and vanishing integrability for Riesz potentials of functions in the generalized Lebesgue space [formula omitted

Our aim in this paper is to deal with Sobolev's inequalities for Riesz potentials of functions belonging to L p ( ⋅ ) ( log L ) q ( ⋅ ) . To do so, we study the boundedness of Hardy–Littlewood maximal functions and apply the Hedberg's trick. As an application, we treat vanishing integrability for Riesz potentials.

Integrability of maximal functions for generalized Lebesgue spaces with variable exponent

AbstractOur aim in this paper is to deal with integrability of maximal functions for generalized Lebesgue spaces with variable exponent. Our exponent approaches 1 on some part of the domain, and hence the integrability depends on the shape of that part and the speed of the exponent approaching 1. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Semi-Fredholm singular integral operators with piecewise continuous coefficients on weighted variable Lebesgue spaces are Fredholm

Suppose $\Gamma$ is a Carleson Jordan curve with logarithmic whirl points, $\varrho$ is a Khvedelidze weight, $p:\Gamma\to(1,\infty)$ is a continuous function satisfying $|p(\tau)-p(t)|\le -\mathrm{const}/\log|\tau-t|$ for $|\tau-t|\le 1/2$, and $L^{p(\cdot)}(\Gamma,\varrho)$ is a weighted generalized Lebesgue space with variable exponent. We prove that all semi-Fredholm operators in the algebra of singular integral operators with $N\times N$ matrix piecewise continuous coefficients are Fredholm on $L_N^{p(\cdot)}(\Gamma,\varrho)$.

Maximal function on Musielak–Orlicz spaces and generalized Lebesgue spaces

We consider the Hardy–Littlewood maximal operator M on Musielak–Orlicz Spaces L φ ( R d ) . We give a necessary condition for the continuity of M on L φ ( R d ) which generalizes the concept of Muckenhoupt classes. In the special case of generalized Lebesgue spaces L p ( ⋅ ) ( R d ) we show that this condition is also sufficient. Moreover, we show that the condition is “left-open” in the sense that not only M but also M q is continuous for some q > 1 , where M q f = ( M ( | f | q ) ) 1 q .

Sobolev's inequality for Riesz potentials with variable exponent satisfying a log-Hölder condition at infinity

Our aim in this paper is to deal with the boundedness of maximal functions in generalized Lebesgue spaces L p ( ⋅ ) when p ( ⋅ ) satisfies a log-Hölder condition at infinity that is weaker than that of Cruz-Uribe, Fiorenza and Neugebauer [D. Cruz-Uribe, A. Fiorenza, C.J. Neugebauer, The maximal function on variable L p spaces, Ann. Acad. Sci. Fenn. Math. 28 (2003) 223–238; 29 (2004) 247–249]. Our result extends the recent work of Diening [L. Diening, Maximal functions on generalized L p ( ⋅ ) spaces, Math. Inequal. Appl. 7 (2004) 245–254] and the authors Futamura and Mizuta [T. Futamura, Y. Mizuta, Sobolev embeddings for Riesz potential space of variable exponent, preprint]. As an application of the boundedness of maximal functions, we show Sobolev's inequality for Riesz potentials with variable exponent.

On the Boundedness and Compactness of Weighted Hardy Operators in Spaces 𝐿𝑝(𝑥)

Abstract Necessary conditions and sufficient conditions for the boundedness/ compactness of weighted Hardy operators are established in generalized Lebesgue spaces 𝐿𝑝(𝑥).

Integral operators on the halfspace in generalized Lebesgue spaces [formula omitted], part I

In this paper we generalize a version of the classical Calderón–Zygmund theorem on principle value integrals in generalized Lebesgue spaces L p ( ⋅ ) proved in [J. Reine Angew. Math. 563 (2003) 197–220], to kernels, which do not satisfy standard estimates on R d + 1 . This result will be used in part II of this paper to prove the classical theorem on halfspace estimates of Agmon, Douglis, and Nirenberg [Comm. Pure Appl. Math. 12 (1959) 623–727] for generalized Lebesgue spaces L p ( ⋅ ) .

Integral operators on the halfspace in generalized Lebesgue spaces [formula omitted], part II

In this paper we generalize, under slightly more restrictive conditions on the kernel K, the classical theorem on halfspace estimates of Agmon, Douglis and Nirenberg [Comm. Pure Appl. Math. 12 (1959) 623–727] to generalized Lebesgue spaces L p ( ⋅ ) . In particular this yields W 2 , p ( ⋅ ) -estimates in R ⩾ d + 1 for the Laplace equation and the Stokes system.

Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces Lp(·) and Wk,p(·)

AbstractWe study the Riesz potentials Iαf on the generalized Lebesgue spaces Lp(·)(ℝd), where 0 &lt; α &lt; d and Iαf(x) ≔ ∫ |f(y)| |x – y|α – d dy. Under the assumptions that p locally satisfies |p(x) – p(x)| ≤ C/(– ln |x – y|) and is constant outside some large ball, we prove that Iα : Lp(·)(ℝd) → Lp♯(·)(ℝd), where $ {\textstyle {1 \over {p ^{\sharp} (x)}} = {1 \over {p(x)}} - {\alpha \over d}} $. If p is given only on a bounded domain Ω with Lipschitz boundary we show how to extend p to $ \tilde p $ on ℝd such that there exists a bounded linear extension operator ℰ : W1,p(·)(Ω) ↪ $ W^{1, {\tilde p}} $(ℝd), while the bounds and the continuity condition of p are preserved. As an application of Riesz potentials we prove the optimal Sobolev embeddings Wk,p(·)(ℝd) ↪Lp*(·)(Rd) with $ {\textstyle {1 \over {p ^{\ast} (x)}} = {1 \over {p(x)}} - {k \over d}} $ and W1,p(·)(Ω) ↪ Lp*(·)(Ω) for k = 1. We show compactness of the embeddings W1,p(·)(Ω) ↪ Lq(·)(Ω), whenever q(x) ≤ p*(x) – ε for some ε &gt; 0. (© 2004 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Maximal function on generalized Lebesgue spaces L^p(⋅)

We prove the boundedness of the Hardy–Littlewood maximal function on the generalized Lebesgue space Lp(·)(Rd) under a continuity assumption on p that is weaker than uniform Holder continuity. We deduce continuity of mollifying sequences and density of C∞(Ω) in W1,p(·)(Ω) . Mathematics subject classification (2000): 42B25, 46E30.

Calderón-Zygmund operators on generalized Lebesgue spaces Lp(⋅) and problems related to fluid dynamics

Calderón-Zygmund operators on generalized Lebesgue spaces Lp(⋅) and problems related to fluid dynamics

Mixed Problem for an Ultraparabolic Equation in Unbounded Domain

We investigate a mixed problem for a nonlinear ultraparabolic equation in a certain domain Q unbounded in the space variables. This equation degenerates on a part of the lateral surface on which boundary conditions are given. We establish conditions for the existence and uniqueness of a solution of the mixed problem for the ultraparabolic equation; these conditions do not depend on the behavior of the solution at infinity. The problem is investigated in generalized Lebesgue spaces.

Averaging operators on l^{p_n} and L^p(x)

We consider the generalized Lebesgue space Lp(x) and its discrete analogue l{pn} , each given the appropriate Luxemburg norm. Let Tk be the averaging operator given by (Tka)n = 1 k (an + an+1 + · · · + an+k−1), a = {an} ∈ l{pn}. We show that the Tk are uniformly bounded from l {pn} into l{pn} under certain assumptions on pn and find a counter–example to show that Tk need not be bounded if these assumptions are not satisfied. Moreover, we construct a bounded Lipschitz function p(x) on [0,∞) such that the operator Ts given, for each Tsf (x) = 1 s ∫ s 0 f (t)dt is unbounded on Lp(x) for all s > 0 . Mathematics subject classification (2000): 46E30,26D15.

On the Spaces Lp(x)(Ω) and Wm, p(x)(Ω)

In this paper we present some basic results on the generalized Lebesgue spaces Lp(x)(Ω) and generalized Lebesgue–Sobolev spaces Wm,p(x)(Ω). These results provide the necessary framework for the study of variational problems and elliptic equations with non-standard p(x)-growth conditions.