Related papers: Weighted complete continuity for the commutator of…

Let $\Omega$ be homogeneous of degree zero, have mean value zero and integrable on the unit sphere, and $\mathcal{M}_{\Omega}$ be the higher-dimensional Marcinkiewicz integral associated with $\Omega$. In this paper, the authors proved that…

Let $\mu_{\Omega,\vec{b}}$ be the multilinear commutator generalized by $\mu_{\Omega}$, the $n$-dimensional Marcinkiewicz integral with the bounded kernel, and $b_{j}\in \Osc_{\exp L^{r_{j}}}(1\le j\le m)$. In this paper, the following…

Suppose that $\Omega \in L^{\infty}(\mathbb{S} ^{n-1})$ is homogeneous of degree zero with mean value zero. Then we consider a fractional type Marcinkiewicz integral operator $$\mu_{\Omega ,\beta }f(x) = \left ( \int_{0}^{\infty } \left |…

Let $\Omega\in L^q(S^{n-1})$ with $1<q\le\infty$ be homogeneous of degree zero and has mean value zero on $S^{n-1}$. In this paper, we will study the boundedness of homogeneous singular integrals and Marcinkiewicz integrals with rough…

Let $\mu_{\Omega,\vec{b}}$ be the multilinear commutator generalized by $\mu_{\Omega}$, the $n$-dimensional Marcinkiewicz integral, with $\Osc_{\exp L^{^{\tau}}}(\R^{n})$ functions for $\tau\ge 1$, where $\Osc_{\exp L^{^{\tau}}}(\R^{n})$ is…

Let $\varphi:\mathbb{R}^n\times[0,\,\infty) \rightarrow [0,\,\infty)$ satisfy that $\varphi(x,\,\cdot)$, for any given $x\in\mathbb{R}^n$, is an Orlicz function and $\varphi(\cdot\,,t)$ is a Muckenhoupt $A_\infty$ weight uniformly in…

Let $0<\rho<n$ and $\mu^{\rho}_{\Omega}$ be the parametric Marcinkiewicz integral. In this paper, by using the atomic decomposition theory of weighted Hardy and weak Hardy spaces, we will obtain the boundedness properties of…

Let $M_{\Omega,\alpha}$ and $T_{\Omega,\alpha}$ be the fractional maximal and integral operators with rough kernels, where $0<\alpha<n$. In this paper, we shall study the continuity properties of $M_{\Omega,\alpha}$ and $T_{\Omega,\alpha}$…

Let $T$ be a Calder\'on-Zygmund singular integral operator. In this paper, we will show some weighted boundedness properties of commutator $[b,T]$ on the weighted Morrey spaces $L^{p,\kappa}(w)$ under appropriate conditions on the weight…

We investigate the convergence of entire Lagrange interpolations and of Hermite interpolations of exponential type in weighted $L^p$-spaces on the real line. The weights are reciprocals of entire functions and depend on the type and may be…

The present paper deals with singular integrals with variable Caldero'n-Zygmund type kernels satisfying mixed homogeneity condition. The continuity of these operators in The Lebesgue spaces is well studied by Fabes and Rivie're. Our goal is…

We consider weighted $L^p(w)$ boundedness ($1<p<\infty $ and $w$ a Muckenhoupt $A_p$ weight) of the Calder\'{o}n commutator $\mathcal C_\Omega$ associated with rough homogeneous kernel, under the condition $\Omega\in L^q(\mathbb S^{n-1})$…

Let $\Omega$ be homogeneous of degree zero, have mean value zero and integrable on the unit sphere, and $\mu_{\Omega}$ be the higher-dimensional Marcinkiewicz integral defined by $$\mu_\Omega(f)(x)= \Big(\int_0^\infty\Big|\int_{|x-y|\leq…

Let $0<\alpha<n$ and $I_\alpha$ be the fractional integral operator. In this paper, we shall use a unified approach to show some boundedness properties of commutators $[b,I_\alpha]$ on the weighted Morrey spaces $L^{p,\kappa}(w)$ under…

Quantitative $A_1-A_\infty$ estimates for rough homogeneous singular integrals $T_{\Omega}$ and commutators of $BMO$ symbols and $T_{\Omega}$ are obtained. In particular the following estimates are proved: % \[ \|T_\Omega \|_{L^p(w)}\le…

In this paper, we prove that the weighted BMO space as follows $${\rm BMO}^{p}(\omega)=\Big\{f\in L^{1}_{\rm loc}:\sup_{Q}\|\chi_{Q}\|^{-1}_{L^{p}(\omega)}\big\|(f-f_{Q})\omega^{-1}\chi_{Q}\big\|_{L^{p}(\omega)}<\infty\Big\}$$ is…

We investigate the global continuity on $L^p$ spaces with $p\in [1,\infty]$ of Fourier integral operators with smooth and rough amplitudes and/or phase functions subject to certain non-degeneracy conditions. We initiate the investigation of…

We give analytic description for the completion of $C_0^\infty ( \mathbf{R}_+)$ in Dirichlet space $D^{1,p}(\mathbf{R}_+, \omega):= \{ u:\mathbf{R}_+\rightarrow \mathbf{R}: u\ \hbox{ is locally absolutely continuous on} \ \mathbf{R}_+ \…

Let $\mathcal{T}^*$ be the semi-group maximal function associated to the Schr\"odinger operator $-\Delta+V(x)$ with $V$ satisfying an appropriate reverse H\"{o}lder inequality. In this paper, we show that the commutator of $\mathcal{T}^*$…

In this paper we find the condition on function $\omega$ and weight $v$ which ensures the equivalency of norms of the Riesz potential and the fractional maximal function in generalized weighted Morrey spaces ${\mathcal…