In this paper, we introduce weighted Carleson measure spaces associated with different homogeneities and prove that these spaces are the dual spaces of weighted Hardy spaces studied in a forthcoming paper. As an application, we establish the boundedness of composition of two Calderón-Zygmund operators with different homogeneities on the weighted Carleson measure spaces; this, in particular, provides the weighted endpoint estimates for the operators studied by Phong-Stein.

Our goal in this work is to present some function spaces on the complex plane $\mathbb C$, $X(\mathbb C)$, for which the quasiregular solutions of the Beltrami equation, $\overline\partial f (z) = \mu(z) \partial f (z)$, have first derivatives locally in $X(\mathbb C)$, provided that the Beltrami coefficient $\mu$ belongs to $X(\mathbb C)$.

In this paper, we prove certain $L^2$-estimate for multilinear Fourier multiplier operators with multipliers of limited smoothness. As a result, we extend the result of Calderón and Torchinsky in the linear theory to the multilinear case. The sharpness of our results and some related estimates in Hardy spaces are also discussed.

Let $({\mathcal X}, d, \mu)$ be a separable metric measure space satisfying the known upper doubling condition, the geometrical doubling condition, and the non-atomic condition that $\mu(\{x\})=0$ for all $x\in{\mathcal X}$. In this paper, we show that the boundedness of a Calderón-Zygmund operator $T$ on $L^2(\mu)$ is equivalent to that of $T$ on $L^p(\mu)$ for some $p\in (1, \infty)$, and that of $T$ from $L^1(\mu)$ to $L^{1,\,\infty}(\mu).$ As an application, we prove that if $T$ is a Calderón-Zygmund operator bounded on $L^2(\mu)$, then its maximal operator is bounded on $L^p(\mu)$ for all $p\in (1, \infty)$ and from the space of all complex-valued Borel measures on ${\mathcal X}$ to $L^{1,\,\infty}(\mu)$. All these results generalize the corresponding results of Nazarov et al. on metric spaces with measures satisfying the so-called polynomial growth condition.

In this paper we characterize the compactness of the commutator $[b,T]$ for the singular integral operator on the Morrey spaces $L^{p,\lambda}(\mathbb R^n)$. More precisely, we prove that if $b\in \operatorname{VMO}(\mathbb R^n)$, the $\operatorname {BMO} (\mathbb R^n)$-closure of $C_c^\infty(\mathbb R^n)$, then $[b,T]$ is a compact operator on the Morrey spaces $L^{p,\lambda}(\mathbb R^n)$ for $1\lt p\lt \infty$ and $0\lt \lambda\lt n$. Conversely, if $b\in \operatorname{BMO}(\mathbb R^n)$ and $[b,T]$ is a compact operator on the $L^{p,\,\lambda}(\mathbb R^n)$ for some $p\ (1\lt p\lt \infty)$, then $b\in \operatorname {VMO}(\mathbb R^n)$. Moreover, the boundedness of a rough singular integral operator $T$ and its commutator $[b,T]$ on $L^{p,\,\lambda}(\mathbb R^n)$ are also given. We obtain a sufficient condition for a subset in Morrey space to be a strongly pre-compact set, which has interest in its own right.

We prove the $L^p(\mathbb{R}^d)$ ($1
Categories:42B20, 42B25

We prove $L^p(\mathbb T^2)$ boundedness, $1
Categories:42B20, 42B08

We consider the Neumann problem for the Schr\"odinger equations $-\Delta u+Vu=0$, with singular nonnegative potentials $V$ belonging to the reverse H\"older class $\B_n$, in a connected Lipschitz domain $\Omega\subset\mathbf{R}^n$. Given boundary data $g$ in $H^p$ or $L^p$ for $1-\epsilon
Keywords:Neumann problem, Schrödinger equation, Lipschitz, domain, reverse Hölder class, $H^p$ space
Categories:42B20, 35J10

We study the singular integral operator $$ T_{\Omega,\alpha}f(x) = \pv \int_{R^n} b(|y|) \Omega(y') |y|^{-n-\alpha} f(x-y)\,dy, $$ defined on all test functions $f$,where $b$ is a bounded function, $\alpha\geq 0$, $\Omega(y')$ is an integrable function on the unit sphere $S^{n-1}$ satisfying certain cancellation conditions. We prove that, for $1
Categories:42B20, 42B25, 42B15

We prove that the operators $\int_{\mathbb{R}_+^2} e^{ix^a \cdot y^b} \varphi (x,y) f(y)\, dy$ map $L^p(\mathbb{R}^2)$ into itself for $p \in J =\bigl[\frac{a_l+b_l}{a_l+(\frac{b_l r}{2})},\frac{a_l+b_l} {a_l(1-\frac{r}{2})}\bigr]$ if $a_l,b_l\ge 1$ and $\varphi(x,y)=|x-y|^{-r}$, $0\le r <2$, the result is sharp. Generalizations to dimensions $d>2$ are indicated.

$H^p$ estimate for the multilinear operators which are finite sums of pointwise products of singular integrals and fractional integrals is given. An application to Sobolev space and some examples are also given.

Given function $\Omega$ on ${\Bbb R^n}$, we define the fractional maximal operator and the fractional integral operator by $$ M_{\Omega,\alpha}\,f(x)=\sup_{r>0}\frac 1{r^{n-\alpha}} \int_{|\,y|1)$, homogeneous of degree zero.