Spherical maximal function, maximal Bochner–Riesz mean and geometrical maximal function on Herz spaces with variable exponents

This paper extends the celebrated Stein–Weiss inequalities to the Herz spaces with variable exponents. It is an extension of the classical Stein–Weiss inequalities and the Stein–Weiss inequalities on Lebesgue spaces with variable exponents. The Stein–Weiss inequalities to the Herz spaces with variable exponents are presented as the boundedness of the fractional integral operators on the power weighted Herz spaces with variable exponents.

<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>

Our aim in this paper is to give the boundedness of the Marcinkiewicz integralμΩon Herz spacesK˙p(·)α(·),q(Rn)andKp(·)α(·),q(Rn), where the two main indices are variable. Meanwhile, we consider the boundedness of the higher order commutatorμΩ,bmgenerated byμΩand a functionbin BMO(Rn)on these spaces.

The aim of this paper is to deal with the boundedness of theθ-type Calderón-Zygmund operators and their commutators on Herz spaces with two variable exponentsp(⋅),q(⋅). It is proved that theθ-type Calderón-Zygmund operators are bounded on the homogeneous Herz space with variable exponents$\begin{array}{} \displaystyle \dot{K}^{\alpha,q(\cdot)}_{p(\cdot)}(\mathbb{R}^{n}). \end{array}$Furthermore, the boundedness of the corresponding commutators generated by BMO function and Lipschitz function is also obtained respectively.

In this article, we introduce the concept of two-weighted grand variable Herz spaces as a natural generalization of variable Herz spaces. We establish the boundedness of sublinear operators within this framework, offering significant insights into their functional behavior. Our findings, incorporating additional weight factors, broaden the scope and applicability of our results.

This paper establishes the mapping properties of pseudo-differential operators and the Fourier integral operators on the weighted Morrey spaces with variable exponents and the weighted Triebel–Lizorkin–Morrey spaces with variable exponents. We obtain these results by extending the extrapolation theory to the weighted Morrey spaces with variable exponents. This extension also gives the mapping properties of Calderón–Zygmund operators on the weighted Hardy–Morrey spaces with variable exponents and the wavelet characterizations of the weighted Hardy–Morrey spaces with variable exponents.

Hardy spaces with variable exponents and generalized Campanato spaces

Our aim in this paper is to deal with the boundedness of the central Hardy-Littlewood maximal operator on Herz spaces with radial variable exponent. As a special case, we show the boundedness of the maximal operator in weighted Lebesgue spaces with radial variable exponent. We also discuss the integrability of the maximal functions in such spaces.

In this paper, we establish the boundedness of the Calderón operator on local Morrey spaces with variable exponents. We obtain our result by extending the extrapolation theory of Rubio de Francia to the local Morrey spaces with variable exponents. The exponent functions of the local Morrey spaces with the exponent functions are only required to satisfy the log-Hölder continuity assumption at the origin and infinity only. As special cases of the main result, we have Hardy’s inequalities, the Hilbert inequalities and the boundedness of the Riemann–Liouville and Weyl averaging operators on local Morrey spaces with variable exponents.

Via the boundedness of intrinsic g-functions from the Hardy spaces with variable exponent, ℋp(·)(ℝn), into Lebesgue spaces with variable exponent, Lp(·)(ℝn), and establishing some estimates on a discrete Littlewood-Paley g-function and a Peetre-type maximal function, we obtain several equivalent characterizations of ℋp(·)(ℝn) in terms of wavelets, which extend the wavelet characterizations of the classical Hardy spaces. The main ingredients are that, we overcome the difficulties of the quasi-norms of ℋp(·)(ℝn) by elaborately using an observation and the Fefferman-Stein vector-valued maximal inequality on Lp(·)(ℝn), and also overcome the difficulty of the failure of q = 2 in the atomic decomposition of ℋp(·)(ℝn) by a known idea.

We study a two-weight problem for the Hardy–Littlewood maximal operator in variable exponent Lebesgue spaces L p (·). The exponential function satisfies some logarithmic type continuity conditions. Bibliography: 25 titles.

In this paper, we introduce Hardy spaces with variable exponents defined on a probability space and develop the martingale theory of variable Hardy spaces. We prove the weak-type and strong-type inequalities on Doob’s maximal operator, and we get a (1,p(⋅),∞)-atomic decomposition for Hardy martingale spaces associated with conditional square functions. As applications, we obtain a dual theorem and the John–Nirenberg inequalities in the frame of variable exponents. The key ingredient is that we find a condition with a probabilistic characterization of p(⋅) to replace the so-called log-Hölder continuity condition in Rn.

In this paper, we obtain the boundedness of bilinear commutators generated by the bilinear Hardy operator and BMO functions on products of Herz spaces and Herz-Morrey spaces with variable exponents.

Let , with , be a homogeneous function of degree zero. In this paper, we establish the boundedness of the homogeneous fractional integral operator on weighted Herz spaces with variable exponent .

We obtain the boundedness of Calderón–Zygmund operators on the entire Hardy–Morrey spaces with variable exponents. We obtain this result by refining the extrapolation theory. This refined extrapolation theory also gives the boundedness of Fourier multipliers, Bochner–Riesz means, and parametric Marcinkiewicz integrals on Hardy–Morrey spaces with variable exponents.