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 Lp(⋅) 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 Lp 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 Lp(⋅) 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.     

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)     

Weighted Morrey spaces of variable exponent and Riesz potentials

We prove weighted inequalities for the Hardy‐Littlewood maximal operator on weighted Morrey spaces of variable exponent. As an application of the boundedness of the maximal operator, we establish weighted Sobolev's inequality for Riesz potentials. We are also concerned with weighted Trudinger's inequality for Riesz potentials.     

Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators

We prove Sobolev-type p(⋅)→q(⋅)-theorems for the Riesz potential operator Iα in the weighted Lebesgue generalized spaces Lp(⋅)(Rn,ρ) with the variable exponent p(x) and a two-parametrical power weight fixed to an arbitrary finite point and to infinity, as well as similar theorems for a spherical analogue of the Riesz potential operator in the corresponding weighted spaces Lp(⋅)(Sn,ρ) on the unit sphere Sn in Rn+1.     

The Dirichlet problem for nonlinear elliptic equations with variable exponent

In this paper we study the Dirichlet problem for nonlinear elliptic equations with variable exponents in Sobolev spaces with variable exponent. We show that for every continuous function $g$ on the boundary there exists a unique continuous extension of $g$.     

Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators, II

In [S.G. Samko, B.G. Vakulov, Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators, J. Math. Anal. Appl. 310 (2005) 229-246], Sobolev-type p(⋅)→q(⋅)-theorems were proved for the Riesz potential operator Iα in the weighted Lebesgue generalized spaces Lp(⋅)(Rn,ρ) with the variable exponent p(x) and a two-parameter power weight fixed to an arbitrary finite point x₀ and to infinity, under an additional condition relating the weight exponents at x₀ and at infinity. We show in this note that those theorems are valid without this additional condition. Similar theorems for a spherical analogue of the Riesz potential operator in the corresponding weighted spaces Lp(⋅)(Sn,ρ) on the unit sphere Sn in Rn+1 are also improved in the same way.     

Addendum to “On the Riesz potential operator of variable order from variable exponent Morrey space to variable exponent Campanato space”, Math Meth Appl Sci. 2020; 1–8

In the paper mentioned in the title, it is proved the boundedness of the Riesz potential operator of variable order α(x) from variable exponent Morrey space to variable exponent Campanato space, under certain assumptions on the variable exponents p(x) and λ(x) of the Morrey space. Assumptions on the exponents were different depending on whether $\frac{\alpha (x)p(x)?n+\lambda (x)}{p(x)}$ takes or not the critical values 0 or 1. In this note, we improve those results by unifying all the cases and covering the whole range $0?\frac{\alpha (x)p(x)?n+\lambda (x)}{p(x)}?1$. We also provide a correction to some minor technicality in the proof of Theorem 2 in the aforementioned paper.     

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.     

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.     

Integrability of maximal functions and Riesz potentials in Orlicz spaces of variable exponent

Our aim in this paper is to deal with Sobolev's type inequality, Hardy's type inequality and Trudinger's inequality for Riesz potentials of functions in Orlicz spaces of variable exponent. These results are based on the boundedness of maximal operators and so-called Hedberg's trick. Our methods can also be applied to the case of constant exponents with slight modifications.     

Homogeneous variable exponent Besov and Triebel–Lizorkin spaces

We introduce homogeneous Besov and Triebel–Lizorkin spaces with variable indexes. We show that their study reduces to the study of inhomogeneous variable exponent spaces and homogeneous constant exponent spaces. Corollaries include trace space characterizations and Sobolev embeddings.     

Existence of solutions of the parabolic variational inequality with variable exponent of nonlinearity

In this article, new properties of variable exponent Lebesgue and Sobolev spaces are examined. Using these properties we prove the existence of the solution of some parabolic variational inequality.     

变指数空间上多线性分数次积分的Lipschitz光滑性

本文研究了多线性分数次积分算子在变指数空间的有界性.利用多线性分数次积分转化为相对应的分数次积分的方法,获得了它从变指数强和弱Lebesgue空间到变指数Lipschitz空间的有界性,推广了先前的研究结果.     

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.     

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.     

Herz type Besov and Triebel-Lizorkin spaces with variable exponent

The Herz type Besov and Triebel-Lizorkin spaces with variable exponent are introduced. Then characterizations of these new spaces by maximal functions are given.     

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.     

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.     

Variable exponent Hardy spaces associated with discrete Laplacians on graphs

In this paper we develop the theory of variable exponent Hardy spaces associated with discrete Laplacians on infinite graphs. Our Hardy spaces are defined by square integrals, atomic and molecular decompositions. Also we study boundedness properties of Littlewood-Paley functions, Riesz transforms, and spectral multipliers for discrete Laplacians on variable exponent Hardy spaces.