Hausdorff operators on the Sobolev spaces Wk,1W^{k,1}

This paper is served as a first contribution regarding the boundedness of Hausdorff operators on function spaces with smoothness. The sharp conditions are established for boundedness of Hausdorff operators on Sobolev spaces $W^{k,1}$. As applications, some bounded and unbounded properties of Hardy operator and adjoint Hardy operator on $W^{k,1}$ are deduced

On relatively compact sets in quasi-Banach function spaces

This paper is devoted to the study of the relatively compact sets in Quasi-Banach function spaces, providing an important improvement of the known results. As an application, we take the final step in establishing a relative compactness criteria for function spaces with any weight without any assumption.Comment: To appear in Proc. Amer. Math. So

Unimodular multipliers on α\alpha-modulation spaces: A revisit with new method under weaker conditions

By a new method derived from Nicola--Primo--Tabacco[24], we study the boundedness on $\alpha$-modulation spaces of unimodular multipliers with symbol $e^{i\mu(\xi)}$. Comparing with the previous results, the boundedness result is established for a larger family of unimodular multipliers under weaker assumptions

The unified theory for the necessity of bounded commutators and applications

The general methods which are powerful for the necessity of bounded commutators are given. As applications, some necessary conditions for bounded commutators are first obtained in certain endpoint cases, and several new characterizations of $BMO$ space, Lipschitz spaces and their weighted versions via boundedness of commutators in various function spaces are deduced.Comment: Some minor issue have been revise

Hausdorff operators on modulation and Wiener amalgam spaces

We give the sharp conditions for boundedness of Hausdorff operators on certain modulation and Wiener amalgam spaces.Comment: To appear in "Annals of Functional Analysis

Characterization of Some Properties on Weighted Modulation Spaces

In this paper, some properties on weighted modulation and Wiener amalgam spaces are characterized by the corresponding properties on weighted Lebesgue spaces. As applications, sharp conditions for product inequalities, convolution inequalities and embedding on weighted modulation and Wiener amalgam spaces are obtained. These applications improve and extend many known results

Boundedness and compactness of commutators associated with Lipschitz functions

Let $\alpha\in (0, 1]$, $\beta\in [0, n)$ and $T_{\Omega,\beta}$ be a singular or fractional integral operator with homogeneous kernel $\Omega$. In this article, a CMO type space ${\rm CMO}_\alpha(\mathbb R^n)$ is introduced and studied. In particular, the relationship between ${\rm CMO}_\alpha(\mathbb R^n)$ and the Lipchitz space $Lip_\alpha(\mathbb R^n)$ is discussed. Moreover, a necessary condition of restricted boundedness of the iterated commutator $(T_{\Omega,\beta})^m_b$ on weighted Lebesgue spaces via functions in $Lip_\alpha(\mathbb R^n)$, and an equivalent characterization of the compactness for $(T_{\Omega,\beta})^m_b$ via functions in ${\rm CMO}_\alpha(\mathbb R^n)$ are obtained. Some results are new even in the unweighted setting for the first order commutators.Comment: arXiv admin note: text overlap with arXiv:1712.0829

Full Characterization of embedding relations between alpha modulation spaces

In this paper, we consider the embedding relations between any two $\alpha$% -modulation spaces. Based on an observation that the $\alpha$-modulation space with smaller $\alpha$ can be regarded as a corresponding $\alpha$% -modulation space with larger $\alpha$, we give a complete characterization of the Fourier multipliers between $\alpha$-modulation spaces with different $\alpha$. Then we establish a full version of optimal embedding relations between $\alpha$-modulation spaces. As an application, we determine that the bounded operators commuting with translations between $\alpha$-modulation spaces are of convolution type

On the compactness of oscillation and variation of commutators

In this paper, we first establish the weighted compactness result for oscillation and variation associated with the truncated commutator of singular integral operators. Moreover, we establish a new $CMO(\mathbb{R}^n)$ characterization via the compactness of oscillation and variation of commutators on weighted Lebesgue spaces

Limiting weak-type behaviors for factional maximal operators and fractional integrals with rough kernel

By a reduction method, the limiting weak-type behaviors of factional maximal operators and fractional integrals are established without any smoothness assumption on the kernel, which essentially improve and extend previous results. As a byproduct, we characterize the boundedness of several operators by the membership of their kernel in Lebesgue space on sphere.Comment: Some typos are correcte