Let $\vec{b} =(b_{1},b_{2},\dots,b_{m})$ be a collection of locally integrable functions and $T_{_{\Sigma \vec{b}}}$ the commutator of multilinear singular integral operator $T$. Denote by $\mathbb{L}(\delta)$ and $\mathbb{L}(\delta(\cdot)) $ the Lipschitz spaces and the variable Lipschitz spaces, respectively. The main purpose of this paper is to establish some new characterizations of the (variable) Lipschitz spaces in terms of the boundedness of multilinear commutator $T_{_{\Sigma \vec{b}}}$ in the context of the variable exponent Lebesgue spaces, that is, the authors give the necessary and sufficient conditions for $b_{j}$ $(j=1,2,\dots,m)$ to be $\mathbb{L}(\delta)$ or $\mathbb{L}(\delta(\cdot)) $ via the boundedness of multilinear commutator from products of variable exponent Lebesgue spaces to variable exponent Lebesgue spaces. The authors do so by applying the Fourier series technique and some pointwise estimate for the commutators. The key tool in obtaining such pointwise estimate is a certain generalization of the classical sharp maximal operator.

In this paper, we investigate the $L^{2}$ boundedness of the Fourier integral operator $T_{\phi,a}$ with smooth and rough symbols and phase functions which satisfy certain non-degeneracy conditions. In particular, if the symbol $a\in L^{\infty}S^{m}_{\rho}$, the phase function $\phi$ satisfies some measure conditions and $\|\nabla^{k}_{\xi}\phi(\cdot,\xi)\|_{L^{\infty}}\leq C|\xi|^{\epsilon-k}$ for all $k\geq2, \xi\neq 0$, and some $\epsilon>0$, we obtain that $T_{\phi,a}$ is bounded on $L^2$ if $m<\frac n2\min\{\rho-1,-\frac\epsilon2\}$. This result is a generalization of a result of Kenig and Staubach on pseudo-differential operators and it improves a result of Dos Santos Ferreira and Staubach on Fourier integral operators. Moreover, the Fourier integral operator with rough symbols and inhomogeneous phase functions we study in this paper can be used to obtain the almost everywhere convergence of the fractional Schrödinger operator.

In this paper, we give a complete real-variable theory of local variable Hardy spaces. First, we present various real-variable characterization in terms of several local maximal functions. Next, the new atomic and the finite atomic decomposition for the local variable Hardy spaces are established. As an application, we also introduce the local variable Campanato space which is showed to be the dual space of the local variable Hardy spaces. Analogous to the homogeneous case, some equivalent definitions of the dual of local variable Hardy spaces are also considered. Finally, we show the boundedness of inhomogeneous Calderón–Zygmund singular integrals and local fractional integrals on local variable Hardy spaces and their duals.

In order to study the boundedness of some operators in general function spaces which include Lorentz spaces and Orlicz spaces as special examples, Lorentz introduced a new space called rearrangement invariant Banach function spaces, denoted by RIBFS. It is shown in this paper that variation operators of singular integrals and their commutators are bounded on RIBFS whenever the kernels satisfy the L^r-Hörmander conditions. Moreover, we obtain some quantitative weighted bounds in the quasi-Banach spaces and modular inequalities for above variation operators and their commutators.

In this paper, we show that the Boussinesq operator $\mathcal{B}_tf$ converges pointwise to its initial data $f\in H^s(\mathbb{R})$ as $t\to 0$ provided $s\geq\frac{1}{4}$--more precisely--on one hand, by constructing a counterexample in $\mathbb{R}$ we discover that the optimal convergence index $s_{c,1}=\frac14$; on the other hand, we find that the Hausdorff dimension of the divergence set for $\mathcal{B}_tf$ is ${\alpha _1},\beta \left( s \right) = \left\{ {\begin{array}{*{20}{c}} {1 -2s,}\\ {1,} \end{array}\begin{array}{*{20}{c}} {as\frac{1}{4} \le s \le \frac{1}{2};}\\ {as0 < s < \frac{1}{4}.} \end{array}} \right.$ Moreover, a higher dimensional lift was also obtained for $f$ being radial.

Let $({\mathcal X},\rho,\mu)$ be a space of homogeneous type in the sense of Coifman and Weiss, and $Y({\mathcal X})$ a ball quasi-Banach function space on ${\mathcal X}$, which supports both a Fefferman--Stein vector-valued maximal inequality and the boundedness of the powered Hardy--Littlewood maximal operator on its associate space. The authors first introduce the Hardy space $H_{Y}({\mathcal X})$ associated with $Y({\mathcal X})$, via the Lusin-area function, and then establish its various equivalent characterizations, respectively, in terms of atoms, molecules, and Littlewood--Paley $g$-functions and $g_{\lambda}^*$-functions. As an application, the authors obtain the boundedness of Calder\'on--Zygmund operators from $H_{Y}({\mathcal X})$ to $Y({\mathcal X})$, or to $H_{Y}({\mathcal X})$ via first establishing a boundedness criterion of linear operators on $H_{Y}({\mathcal X})$. All these results have a wide range of generality and, particularly, even when they are applied to variable Hardy spaces, the obtained results are also new. The major novelties of this article exist in that, to escape the reverse doubling condition of $\mu$ and the triangle inequality of $\rho$, the authors subtly use the wavelet reproducing formula, originally establish an admissible molecular characterization of $H_{Y}({\mathcal X})$, and fully apply the geometrical properties of ${\mathcal X}$ expressed by dyadic reference points or dyadic cubes.

Let $\mathcal{L}=-\Delta+\mathit{V}$ be a Schrödinger operator, where $\Delta$ is the Laplacian operator on $\mathbb{R}^{d}$ $(d\geq 3)$, while the nonnegative potential $\mathit{V}$ belongs to the reverse Hölder class $B_{q}, q>d/2$. In this paper, we study weighted compactness of commutators of some Schrödinger operators, which include Riesz transforms, standard Calderón--Zygmund operators and Littlewood--Paley functions. These results substantially generalize some well-known results.

In this article, the authors first establish the pointwise characterizations of Besov and Triebel-Lizorkin spaces with generalized smoothness on $\mathbb{R}^n$ via the Hajłlasz gradient sequences, which serve as a way to extend these spaces to more general metric measure spaces. Moreover, on metric spaces with doubling measures, the authors further prove that the Besov and the Triebel-Lizorkin spaces with generalized smoothness defined via Hajłlasz gradient sequences coincide with those defined via hyperbolic fillings. As an application, some trace theorems of these spaces on Ahlfors regular spaces are established.

In this paper, carrying on with our study of the Hardy-amalgam spaces $\mathcal H^{(q,p)}$ and $\mathcal{H}_{\mathrm{loc}}^{(q,p)}$ ($0< q,p <\infty$), we give a characterization of their dual spaces whenever $0< q\leq 1$ and $q\leq p<\infty$. Moreover, when $0< q\leq p\leq 1$, these characterizations coincide with those obtained in our earlier papers.

Let T be a strongly singular Calderón-Zygmund operator and b ∈ L_loc(Rⁿ). This article finds out a class of non-trivial subspaces BMOω,p,u(Rⁿ) of BMO(Rⁿ) for certain ω ∈ A₁, 0 < p ≤ 1 and 1 < u ≤ ∞, such that the commutator[b, T] is bounded from weighted Hardy space H_ω^p(Rⁿ) to weighted Lebesgue space L_ω^p(Rⁿ) if b ∈ BMOω,p,∞(Rⁿ), and is bounded from weighted Hardy space H_ω^p(Rⁿ) to itself if T^∗1=0 and b ∈ BMOω,p,u(Rⁿ) for 1 < u < 2.