73 research outputs found

### A non-homogeneous local $Tb$ theorem for Littlewood-Paley $g_{\lambda}^{*}$-function with $L^p$-testing condition

In this paper, we present a local $Tb$ theorem for the non-homogeneous
Littlewood-Paley $g_{\lambda}^{*}$-function with non-convolution type kernels
and upper power bound measure $\mu$. We show that, under the assumptions \supp
b_Q \subset Q, $|\int_Q b_Q d\mu| \gtrsim \mu(Q)$ and $||b_Q||^p_{L^p(\mu)}
\lesssim \mu(Q)$, the norm inequality $\big\| g_{\lambda}^{*}(f)
\big\|_{L^p(\mu)} \lesssim \big\| f \big\|_{L^p(\mu)}$ holds if and only if the
following testing condition holds : \sup_{Q : cubes \ in \ \Rn}
\frac{1}{\mu(Q)}\int_Q \bigg(\int_{0}^{\ell(Q)} \int_{\Rn}
\Big(\frac{t}{t+|x-y|}\Big)^{m\lambda}|\theta_t(b_Q)(y,t)|^2 \frac{d\mu(y)
dt}{t^{m+1}}\bigg)^{p/2} d\mu(x) < \infty. This is the first time to
investigate $g_\lambda^*$-function in the simultaneous presence of three
attributes : local, non-homogeneous and $L^p$-testing condition. It is
important to note that the testing condition here is $L^p$ type with $p \in
(1,2]$.Comment: 26 page

### The existence and boundedness of multilinear Marcinkiewicz integrals on Companato spaces

In this paper, we established the boundedness of m-linear Marcinkiewicz
integral on Campanato type spaces. We showed that if the $m$-linear
Marcinkiewicz integral is finite for one point, then it is finite almost
everywhere. Moreover, the following norm inequality holds,
$\|\mu(\vec{f})\|_{\mathcal{E}^{\alpha,p}} \leq
C\prod_{j=1}^m\|f_j\|_{\mathcal{E}^{\alpha_j,p_j}},$ where
$\mathcal{E}^{\alpha,p}$ is the classical Campanato spaces.Comment: 27 page

### $L^p$ boundedness of non-homogeneous Littlewood-Paley $g^*_{\lambda,\mu}$-function with non-doubling measures

It is well-known that the $L^p$ boundedness and weak $(1,1)$ estiamte
$(\lambda>2)$ of the classical Littlewood-Paley $g_{\lambda}^{*}$-function was
first studied by Stein, and the weak $(p,p)$ $(p>1)$ estimate was later given
by Fefferman for $\lambda=2/p$. In this paper, we investigated the $L^p(\mu)$
boundedness of the non-homogeneous Littlewood-Paley
$g_{\lambda,\mu}^{*}$-function with non-convolution type kernels and a power
bounded measure $\mu$: $g_{\lambda,\mu}^*(f)(x) = \bigg(\iint_{{\mathbb
R}^{n+1}_{+}} \Big(\frac{t}{t + |x - y|}\Big)^{m \lambda} |\theta_t^\mu f(y)|^2
\frac{d\mu(y) dt}{t^{m+1}}\bigg)^{1/2},\ x \in {\mathbb R}^n,\ \lambda > 1,$
where $\theta_t^\mu f(y) = \int_{{\mathbb R}^n} s_t(y,z) f(z) d\mu(z)$, and
$s_t$ is a non-convolution type kernel. Based on a big piece prior boundedness,
we first gave a sufficient condition for the $L^p(\mu)$ boundedness of
$g_{\lambda,\mu}^*$. This was done by means of the non-homogeneous good lambda
method. Then, using the methods of dyadic analysis, we demonstrated a big piece
global $Tb$ theorem. Finally, we obtaind a sufficient and necessary condition
for $L^p(\mu)$ boundedness of $g_{\lambda,\mu}^*$-function. It is worth noting
that our testing conditions are weak $(1,1)$ type with respect to measures.Comment: 32 page

### A Revisit on Commutators of linear and bilinear Fractional Integral Operator

Let $I_{\alpha}$ be the linear and $\mathcal{I}_{\alpha}$ be the bilinear
fractional integral operators. In the linear setting, it is known that the
two-weight inequality holds for the first order commutators of $I_{\alpha}$.
But the method can't be used to obtain the two weighted norm inequality for the
higher order commutators of $I_{\alpha}$. In this paper, we first give an
alternative proof for the first order commutators of $I_{\alpha}$. This new
approach allows us to consider the higher order commutators. This was done by
showing that the commutator $[b,I_{\alpha}]$ can be represented as a finite
linear combination of some paraproducts. Then, by using the Cauchy integral
theorem, we show that the two-weight inequality holds for the higher order
commutators of $I_{\alpha}$. In the bilinear setting, we present a dyadic proof
for the characterization between $BMO$ and the boundedness of
$[b,\mathcal{I}_{\alpha}]$. Moreover, some bilinear paraproducts are also
treated in order to obtain the boundedness of $[b,\mathcal{I}_{\alpha}]$.Comment: 17 page

### Boundedness of Bi-parameter Littlewood-Paley operators on product Hardy space

Let $n_1,n_2\ge 1, \lambda_1>1$ and $\lambda_2>1$. For any $x=(x_1,x_2) \in
\mathbb {R}^n\times\mathbb{R}^m$, let $g$ and $g_{\vec{\lambda}}^*$ be the
bi-parameter Littlewood-Paley square functions defined by \begin{align*}
g(f)(x)= \Big(\int_0^{\infty}\int_0^{\infty}|\theta_{t_1,t_2} f(x_1,x_2)|^2
\frac{dt_1}{t_1} \frac{dt_2}{t_2} \Big)^{1/2}, \hbox{and} \end{align*} $g_{\vec{\lambda}}^*(f)(x) = \Big(\iint_{\mathbb{R}^{m+1}_{+}}
\iint_{\mathbb{R}^{n+1}_{+}} \prod_{i=1}^2\Big(\frac{t_1}{t_i + |x_i -
y_i|}\Big)^{n_i \lambda_i} |\theta_{t_1,t_2} f(y_1,y_2)|^2 \frac{dy_1
dt_1}{t_1^{n+1}} \frac{dy_2 dt_2}{t_2^{m+1}} \Big)^{1/2},$ \noindent where
$\theta_{t_1,t_2} f(x_1, x_2) = \iint_{\mathbb{R}^n\times\mathbb{R}^m}
s_{t_1,t_2}(x_1,x_2,y_1,y_2)f(y_1,y_2) dy_1dy_2$. It is known that the $L^2$
boundedness of bi-parameter $g$ and $g_{\vec{\lambda}}^*$ have been established
recently by Martikainen, and Cao, Xue, respectively. In this paper, under
certain structure conditions assumed on the kernel $s_{t_1,t_2},$ we show that
both $g$ and $g_{\vec{\lambda}}^*$ are bounded from product Hardy space
$H^1(\mathbb{R}^n\times\mathbb{R}^m)$ to $L^1(\mathbb{R}^n\times\mathbb{R}^m)$.
As consequences, the $L^p$ boundedness of $g$ and $g_{\vec{\lambda}}^*$ will be
obtained for $1<p<2$.Comment: 26 page

### Non-homogeneous $Tb$ Theorem for Bi-parameter $g$-Function

The main result of this paper is a bi-parameter $Tb$ theorem for
Littlewood-Paley $g$-function, where $b$ is a tensor product of two
pseudo-accretive functions. Instead of the doubling measure, we work with a
product measure $\mu = \mu_n \times \mu_m$, where the measures $\mu_n$ and
$\mu_m$ are only assumed to be upper doubling. The main techniques of the proof
include a bi-parameter $b$-adapted Haar function decomposition and an averaging
identity over good double Whitney regions. Moreover, the non-homogeneous
analysis and probabilistic methods are used again.Comment: 16 page

### Weighted Estimates for the iterated Commutators of Multilinear Maximal and Fractional Type Operators

In this paper, the following iterated commutators $T_{*,\Pi b}$ of maximal
operator for multilinear singular integral operators and $I_{\alpha, \Pi b}$ of
multilinear fractional integral operator are introduced and studied \aligned
T_{*,\Pi
b}(\vec{f})(x)&=\sup_{\delta>0}\bigg|[b_1,[b_2,...[b_{m-1},[b_m,T_\delta]_m]_{m-1}...]_2]_1
(\vec{f})(x)\bigg|, \aligned I_{\alpha, \Pi
b}(\vec{f})(x)&=[b_1,[b_2,...[b_{m-1},[b_m,I_\alpha]_m]_{m-1}...]_2]_1
(\vec{f})(x), where $T_\delta$ are the smooth truncations of the multilinear
singular integral operators and $I_{\alpha}$ is the multilinear fractional
integral operator, $b_i\in BMO$ for $i=1,...,m$ and $\vec {f}=(f_1,...,f_m)$.
Weighted strong and $L(\log L)$ type end-point estimates for the above iterated
commutators associated with two class of multiple weights $A_{\vec{p}}$ and
$A_{(\vec{p}, q)}$ are obtained, respectively.Comment: 23 pages, Corrected typo

### On the Boundedness of Multilinear Fractional Strong Maximal Operator with multiple weights

In this paper, we investigated the boundedness of multilinear fractional
strong maximal operator $\mathcal{M}_{\mathcal{R},\alpha}$ associated with
rectangles or related to more general basis with multiple weights
$A_{(\vec{p},q),\mathcal{R}}$. In the rectangles setting, we first gave an
end-point estimate of $\mathcal{M}_{\mathcal{R},\alpha}$, which not only
extended the famous linear result of Jessen, Marcinkiewicz and Zygmund, but
also extended the multilinear result of Grafakos, Liu, P\'{e}rez and Torres
($\alpha=0$) to the case $0<\alpha<mn.$ Then, in one weight case, we gave
several equivalent characterizations between $\mathcal{M}_{\mathcal{R},\alpha}$
and $A_{(\vec{p},q),\mathcal{R}}$, by applying a different approach from what
we have used before. Moreover, a sufficient condition for the two weighted norm
inequality of $\mathcal{M}_{\mathcal{R},\alpha}$ was presented and a version of
vector-valued two weighted inequality for the strong maximal operator was
established when $m=1$. In the general basis setting, we further studied the
properties of the multiple weights $A_{(\vec{p},q),\mathcal{R}}$ conditions,
including the equivalent characterizations and monotonic properties, which
essentially extended one's previous understanding. Finally, a survey on
multiple strong Muckenhoupt weights was given, which demonstrates the
properties of multiple weights related to rectangles systematically.Comment: 19 page

### Certain Multi(sub)linear square functions

Let $d\ge 1, \ell\in\Z^d$, $m\in \mathbb Z^+$ and $\theta_i$, $i=1,\dots,m$
are fixed, distinct and nonzero real numbers. We show that the $m$-(sub)linear
version below of the Ratnakumar and Shrivastava \cite{RS1} Littlewood-Paley
square function $T(f_1,\dots ,
f_m)(x)=\Big(\sum\limits_{\ell\in\Z^d}|\int_{\mathbb{R}^d}f_1(x-\theta_1
y)\cdots f_m(x-\theta_m y)e^{2\pi i \ell \cdot y}K (y)dy|^2\Big)^{1/2}$ is
bounded from $L^{p_1}(\mathbb{R}^d) \times\cdots\times L^{p_m}(\mathbb{R}^d)$
to $L^p(\mathbb{R}^d)$ when $2\le p_i<\infty$ satisfy $1/p=1/p_1+\cdots+1/p_m$
and $1\le p<\infty$. Our proof is based on a modification of an inequality of
Guliyev and Nazirova \cite{GN} concerning multilinear convolutions.Comment: 10 page

### Compactness of commutators of bilinear maximal Calder\'{o}n-Zygmund singular integral operators

Let $T$ be a bilinear Calder\'{o}n-Zygmund singular integral operator and
$T_*$ be its corresponding truncated maximal operator. The commutators in the
$i$-$th$ entry and the iterated commutators of $T_*$ are defined by $T_{\ast,b,1}(f,g)(x)=\sup_{\delta>0}\bigg|\iint_{|x-y|+|x-z|>\delta}K(x,y,z)(b(y)-b(x))f(y)g(z)dydz\bigg|,$
$T_{\ast,b,2}(f,g)(x)=\sup_{\delta>0}\bigg|\iint_{|x-y|+|x-z|>\delta}K(x,y,z)(b(z)-b(x))f(y)g(z)dydz\bigg|,$
\begin{align*}
T_{\ast,(b_1,b_2)}(f,g)(x)=\sup\limits_{\delta>0}\bigg|\iint_{|x-y|+|x-z|>\delta}
K(x,y,z)(b_1(y)-b_1(x))(b_2(z)-b_2(x))f(y)g(z)dydz\bigg|. \end{align*} In this
paper, the compactness of the commutators $T_{\ast,b,1}$, $T_{\ast,b,2}$ and
$T_{\ast,(b_1,b_2)}$ on $L^r(\mathbb{R}^n))$ is established.Comment: New version, corrected the fomer mistake

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