Interpolation theorems for variable exponent Lebesgue spaces

In this note we prove weighted Rellich–Sobolev and Hardy–Sobolev inequalities in variable exponent Lebesgue spaces L p ⁢ ( ⋅ ) ⁢ ( 𝔾 ) {L^{p(\,\cdot\,)}(\mathbb{G})} defined on stratified homogeneous groups 𝔾 {\mathbb{G}} . To derive the main results, we rely on weighted estimates for the Riesz potential operators in L p ⁢ ( ⋅ ) ⁢ ( 𝔾 ) {L^{p(\,\cdot\,)}(\mathbb{G})} , where 𝔾 {{\mathbb{G}}} is a general homogeneous group. The results are new even for the Abelian (Euclidean) case 𝔾 = ( ℝ d , + ) {\mathbb{G}=(\mathbb{R}^{d},+)} and the Heisenberg groups 𝔾 = ℍ n {\mathbb{G}={\mathbb{H}}^{n}} . The main statements are obtained for variable exponents satisfying the condition that the Hardy–Littlewood maximal operator is bounded in appropriate variable exponent Lebesgue spaces. We also give some quantitative estimates for the norms of integral operators involved in derived estimates.

Maximal operator on variable Lebesgue spaces with radial exponent

Our aim is to characterize the Lipschitz functions by variable exponent Lebesgue spaces. We give some characterizations of the boundedness of the maximal or nonlinear commutators of the Hardy-Littlewood maximal function and sharp maximal function in variable exponent Lebesgue spaces when the symbols $b$ belong to the Lipschitz spaces, by which some new characterizations of Lipschitz spaces and nonnegative Lipschitz functions are obtained. Some equivalent relations between the Lipschitz norm and the variable exponent Lebesgue norm are also given.

In this paper we introduce the variable exponent Hormander spaces and we study some of their properties. In particular, it is shown that \({{(\mathcal{B}_{p_{(\cdot)}}^{c}(\Omega))^{\prime}}}\) is isomorphic to \({{\mathcal{B}^{loc}_{\widetilde{p^\prime(\cdot)}(\Omega)}}}\) (Ω open set in \({{\mathbb{R}^n, p? > 1}}\) and the Hardy–Littlewood maximal operator M is bounded in \({L_p(\cdot))}\) extending a Hormander’s result to our context. As a consequence, a number of results on sequence space representations of variable exponent Hormander spaces are given.

Let \(L^{q ( x ) } (\mathbb{R} ) \) be variable exponent Lebesgue space and \(\ell^{ \{ q_{n} \} }\) be discrete analog of this space. In this work we define the amalgam spaces W(L p(x),L q(x)) and \(W ( L^{p ( x ) },\ell^{ \{ q_{n} \} } ) \), and discuss some basic properties of these spaces. Since the global components \(L^{q ( x ) } (\mathbb{R} ) \) and \(\ell^{ \{ q_{n} \} }\) are not translation invariant, these spaces are not a Wiener amalgam space. But we show that there are similar properties of these spaces to the Wiener amalgam spaces. We also show that there is a variable exponent q(x) such that the sequence space \(\ell^{ \{ q_{n} \}}\) is the discrete space of \(L^{q ( x ) } (\mathbb{R} )\). By using this result we prove that \(W ( L^{p ( x ) },\ell ^{ \{ p_{n} \} } ) =L^{p ( x ) } (\mathbb{R} ) \). We also study the frame expansion in \(L^{p ( x ) } ( \mathbb{R} )\). At the end of this work we prove that the Hardy–Littlewood maximal operator from \(W ( L^{s ( x ) },\ell^{ \{t_{n} \} } ) \) into \(W ( L^{u ( x ) },\ell^{ \{v_{n} \} } ) \) is bounded under some assumptions.

In this paper, we investigate the Hardy–Littlewood maximal operator (in a sence of Bekjan ) on non-commutative symmetric spaces. We obtain an upper distributional estimate (by means of the Cesàro operator) of a generalized singular number of the non-commutative Hardy–Littlewood maximal operator. We also show boundedness of the Hardy–Littlewood maximal operator from a general non-commutative symmetric space to another.

We will introduce the times modified centered and uncentered Hardy-Littlewood maximal operators on nonhomogeneous spaces for. We will prove that the times modified centered Hardy-Littlewood maximal operator is weak type bounded with constant when if the Radon measure of the space has "continuity" in some sense. In the proof, we will use the outer measure associated with the Radon measure. We will also prove other results of Hardy-Littlewood maximal operators on homogeneous spaces and on the real line by using outer measures.

We investigate the behaviour of the classical (non-smooth) Hardy-Littlewood maximal operator in the context of Banach lattices. We are mainly concerned with end-point results for p = ∞. Naturally, the main role is played by the space BMO. We analyze the range of the maximal operator in BMOx. This turns out to depend strongly on the convexity of the Banach lattice . We apply these results to study the behaviour of the commutators associated to the maximal operator. We also consider the parallel results for the maximal fractional integral operator.

The purpose of this chapter is to present a series of the local and global estimates for some operators, including the homotopy operator T, the Laplace–Beltrami operator Δ = d d * + d * d, Green’s operator G, the gradient operator ∇, the Hardy–Littlewood maximal operator, and the differential operator, which act on the space of harmonic forms defined in a domain in R n , and the compositions of some of these operators. We introduce the Hardy–Littlewood maximal operator M s and the sharp maximal operator \({\rm M}_s^\#\) applied to differential forms in Section 7.1. We develop some basic estimates for Green’s operator ∇ ◦ T and d◦ T in Section 7.2. We establish some L s -estimates and imbedding inequalities for the compositions of homotopy operator T and Green’s operator G in Section 7.3. In Section 7.4, we prove some Poincare-type inequalities for T◦ G and G◦ T. In Section 7.5, we obtain Poincare-type inequalities for the homotopy operator T. In Section 7.6, we study various estimates for the composition T◦ H. In Section 7.7, we provide the estimates for the compositions of three operators. Finally, in Section 7.8, we offer some norm comparison theorems for the maximal operators.

<p>In this paper, we established some sufficient conditions for the boundedness of the Hardy–Littlewood maximal operators and the Hausdorff operators on $ p $-adic Herz spaces and $ p $-adic local block spaces with variable exponents. In particular, the boundedness of the $ p $-adic maximal commutator operators, the $ p $-adic Hardy–Littlewood average operators, and the $ p $-adic Hardy-Hilbert operators on such spaces was also discussed.</p>

Consider mathbb {R}^d with the euclidean distance, and let 0<alpha <1. We study the behavior of the averaging operators given by the radial density dmu (x)=|x|^{-alpha d}dx. When 1le p<infty is such that (1-2alpha )p<1-alpha , we show that the weak (p, p) bounds grow exponentially with the dimension d. As a consequence, the corresponding results follow for the centered Hardy–Littlewood maximal operator. The lower bounds obtained here are new for averaging operators, and when 0<alpha le 1/2 and p>1, they are also new for the maximal operator.

We study the regularity of Orlicz–Sobolev functions on metric measure spaces equipped with a doubling measure. We show that each Orlicz–Sobolev function is quasicontinuous and has Lebesgue points outside a set of capacity zero and that the discrete maximal operator is bounded in the Orlicz–Sobolev space. We also show that if the Hardy–Littlewood maximal operator is bounded in the Orlicz space \(L^{\Psi}(X)\) , then each Orlicz–Sobolev function can be approximated by a Hölder continuous function both in the Lusin sense and in norm.

Maximal functions and multiplier theorem for Fourier orthogonal series

Boundedness of the Commutators of Fractional Maximal Operator on Morrey-Herz Space on Spaces of Homogeneous Type

The aim of this paper is to prove the boundedness of the Hardy-Littlewood maximal operator on weighted Morrey spaces and multilinear maximal operator on multiple weighted Morrey spaces. In particular, the result includes the Komori-Shirai theorem and the Iida-Sato-Sawano-Tanaka theorem for the Hardy-Littlewood maximal operator and multilinear maximal function.