Integrability of maximal functions for generalized Lebesgue spaces with variable exponent

Our 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 & Co. KGaA, Weinheim)     

Relative rearrangement and Lebesgue spaces with variable exponent

We apply the techniques of monotone and relative rearrangements to the nonrearrangement invariant spaces L^p()(Ω) with variable exponent. In particular, we show that the maps uL^p()(Ω)→k(t)u_*L^p_*()(0,measΩ) and uL^p()(Ω)→u_*L^p*()(0,measΩ) are locally -Hölderian (u_* (resp. p^*) is the decreasing (resp. increasing) rearrangement of u (resp. p)). The pointwise relations for the relative rearrangement are applied to derive the Sobolev embedding with eventually discontinuous exponents.     

Multilinear singular integrals and commutators in variable exponent Lebesgue spaces

Boundedness of multilinear singular integrals and their commutators from products of variable exponent Lebesgue spaces to variable exponent Lebesgue spaces are obtained. The vector-valued case is also considered.     

The boundedness of multilinear commutators of singular integrals on Lebesgue spaces with variable exponent

The boundednees of multilinear commutators of Calderón-Zygmund singular integrals on Lebesgue spaces with variable exponent is obtained. The multilinear commutators of generalized Hardy-Littlewood maximal operator are also considered.     

Generalized potentials in variable exponent Lebesgue spaces on homogeneous spaces

We consider generalized potential operators with the kernel on bounded quasimetric measure space (X, μ, d) with doubling measure μ satisfying the upper growth condition μB(x, r) ? Kr^N, N ∈ (0, ∞). Under some natural assumptions on a(r) in terms of almost monotonicity we prove that such potential operators are bounded from the variable exponent Lebesgue space L^p(?)(X, μ) into a certain Musielak‐Orlicz space L^p(X, μ) with the N‐function Φ(x, r) defined by the exponent p(x) and the function a(r). A reformulation of the obtained result in terms of the Matuszewska‐Orlicz indices of the function a(r) is also given. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Boundedness of maximal operators and potential operators on Carleson curves in Lebesgue spaces with variable exponent

We prove the boundedness of the maximal operator Mr in the spaces L^p（·）（Г,p） with variable exponent p（t） and power weight p on an arbitrary Carleson curve under the assumption that p（t） satisfies the log-condition on Г. We prove also weighted Sobolev type L^p（·）（Г, p） → L^q（·）（Г, p）-theorem for potential operators on Carleson curves.     

Weighted Sobolev theorem in Lebesgue spaces with variable exponent

For the Riesz potential operator Iα there are proved weighted estimates

     

On the interpolation constants for variable Lebesgue spaces

For $\theta \in (0,1)$ and variable exponents $p_0(·),q_0(·)$ and $p_1(·),q_1(·)$ with values in [1, ∞], let the variable exponents $p_{\theta }(·),q_{\theta }(·)$ be defined by $$1/p_{\theta }(·):=(1-\theta )/p_0(·)+\theta /p_1(·),1/q_{\theta }(·):=(1-\theta )/q_0(·)+\theta /q_1(·).$$ The Riesz–Thorin–type interpolation theorem for variable Lebesgue spaces says that if a linear operator T acts boundedly from the variable Lebesgue space $L^{p_j(·)}$ to the variable Lebesgue space $L^{q_j(·)}$ for $j=0,1$, then $${\parallel T\parallel }_{L^{p_{\theta }(·)}\to L^{q_{\theta }(·)}}\le {C\parallel T\parallel }_{L^{p_0(·)}\to L^{q_0(·)}}^{1-\theta }{\parallel T\parallel }_{L^{p_1(·)}\to L^{q_1(·)}}^{\theta },$$ where C is an interpolation constant independent of T. We consider two different modulars ${\varrho }^{\max }(·)$ and ${\varrho }^{\mathrm{sum}}(·)$ generating variable Lebesgue spaces and give upper estimates for the corresponding interpolation constants C_max and C_sum, which imply that $C_{\max }\le 2$ and $C_{\mathrm{sum}}\le 4$, as well as, lead to sufficient conditions for $C_{\max }=1$ and $C_{\mathrm{sum}}=1$. We also construct an example showing that, in many cases, our upper estimates are sharp and the interpolation constant is greater than one, even if one requires that $p_j(·)=q_j(·)$, $j=0,1$ are Lipschitz continuous and bounded away from one and infinity (in this case, ${\varrho }^{\max }(·)={\varrho }^{\mathrm{sum}}(·)$).     

Sobolev‐type inequalities for potentials in grand variable exponent Lebesgue spaces

We introduce a new scale of grand variable exponent Lebesgue spaces denoted by . These spaces unify two non‐standard classes of function spaces, namely, grand Lebesgue and variable exponent Lebesgue spaces. The boundedness of integral operators of Harmonic Analysis such as maximal, potential, Calderón–Zygmund operators and their commutators are established in these spaces. Among others, we prove Sobolev‐type theorems for fractional integrals in . The spaces and operators are defined, generally speaking, on quasi‐metric measure spaces with doubling measure. The results are new even for Euclidean spaces.     

Burkholder-Gundy-Davis inequality in martingale Hardy spaces with variable exponent

In this article, by extending classical Dellacherie's theorem on stochastic sequences to variable exponent spaces, we prove that the famous Burkholder-Gundy-Davis inequality holds for martingales in variable exponent Hardy spaces. We also obtain the variable exponent analogues of several martingale inequalities in classical theory, including convexity lemma, Chevalier's inequality and the equivalence of two kinds of martingale spaces with predictable control. Moreover, under the regular condition on σ-algebra sequence we prove the equivalence between five kinds of variable exponent martingale Hardy spaces.     

多线性Hardy型算子在变指数Herz-Morrey乘积空间上的有界性

本文证明了多线性分数次Hardy算子Hβ,m和H *β,m (m∈Z+且m≥1)在变指数Herz-Morrey乘积空间上的有界性.对多线性Hardy算子也建立了相应的结果.     

Compactness in the spaces of variable integrability and summability

We study totally bounded sets in the spaces of variable integrability and summability. The full characterization of these sets is given. Furthermore, the Sudakov theorem in the setting of the mixed Lebesgue sequence spaces is proven.     

二元Lagrange三角插值多项式在Orlicz空间内的逼近

研究了二元函数用一种组合型的三角插值多项式算子逼近的问题.借助连续模这一工具,给出了这类三角插值多项式在Orlicz空间内的逼近定理.     

Note on Nikodym maximal in the variable Lebesgue spaces

In this paper we consider the k‐plane Nikodym maximal estimates in the variable Lebesgue spaces . We first formulate the problem about the boundedness of the k‐plane Nikodym maximal and show that the maximal estimate in is equivalent to that in for . So, the optimal Nikodym maximal estimate in follows from Cordoba's estimate.     

Characterisation of zero trace functions in variable exponent Sobolev spaces

It is well known that if u belongs to the Sobolev space , where Ω is an open subset of and , then if belongs to weak , where dist . Results of this type are given here for Sobolev spaces with a variable exponent p , under the conditions that Ω is bounded and satisfies a mild regularity condition, and p is a bounded, log‐Hölder continuous function that is bounded away from 1. The outcome includes theorems that are new even when p is constant. In particular it is shown that if and only if and .     

Inversion of the Bessel potential operator in weighted variable Lebesgue spaces

We study the inversion problem of the Bessel potential operator within the frameworks of the weighted Lebesgue spaces with variable exponent. The inverse operator is constructed by using approximative inverse operators. This generalizes some classical results to the variable exponent setting.