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\title{\large\bf
            Endpoint Boundedness for Some Multilinear Integral Operators
\footnotetext{Supported by the NNSF}}
\author{\small LIU Lanzhe   \\
      \small College of Mathematics  \\
      \small Changsha University of Science and Technology \\
      \small  Changsha 410077, P.R. of China \\
      \small E-mail:[e-mail]}
\begin{document}
\maketitle
\par
 {\bf Abstract:} In this paper, we prove the endpoint boundedness for some multilinear operators related to certain
 non-convolution operators. The operators include Littlewood-Paley operator and Marcinkiewiecz operator.
\par
 {\bf Keywords:} Multilinear operator; Littlewood-Paley operator; Marcinkiewiecz operator; BMO space; Hardy space.
\par
 {\bf MR(1991) Subject Classification} \ \ 42B20, 42B25.
\vskip2mm
\par\noindent
    {\large\bf 1.Introduction}
\vskip2mm
\par
 Let $T$ be a Calder\'on-Zygmund operator, a classical result of Coifman, Rochberg and Weiss (see [8]) states that the
 commutator $[b,T](f)=T(bf)-bT(f)$(where $b\in BMO(R^n)$) is bounded on $L^p(R^n)$ for $1<p<\infty$; Chanillo (see [2])
 proves a similar result when $T$ is replaced by the fractional integral operator; In [11], the endpoint boundedness of
 the commutators are obtained. The main purpose of this paper is to study the endpoint boundedness of some multilinear
 operators related to certain non-convolution operators. In fact, we shall establish the boundedness in the extreme cases
 of $p$ for the multilinear operators related to certain non-convolution operator only under certain conditions on the size
 of the operators. As the applications, the endpoint boundedness of the multilinear operators related to the
 Littlewood-Paley operator and Marcinkiewicz operator are obtained.
\vskip2mm
\par\noindent
    {\large\bf 2. Preliminaries}
\vskip2mm
\par
 Throughout this paper, $Q$ will denote a cube of $R^n$ with side parallel to the axes. For a cube $Q$ and a locally integrable
 function $f$, let $f_Q=|Q|^{-1}\int_Qf(x)dx$ and $f^{\#}(x)=\sup\limits_{x\in Q}|Q|^{-1}\int_Q|f(y)-f_Q|dy$. Moreover, $f$
 is said to belong to $BMO(R^n)$ if $f^{\#}\in L^\infty(R^n)$ and define $||f||_{BMO}=||f^{\#}||_{L^\infty}$. Also, we give
 the concepts of the atom and $H^1$ space. A function $a$ is called a $H^1$ atom if there exists a cube $Q$ such that $a$ is
 supported on $Q$, $||a||_{L^\infty}\le |Q|^{-1}$ and $\int_{R^n}a(x)dx=0$. It is well known that the Hardy space $H^1(R^n)$
 has the atomic decomposition characterization(see[10]).
\par
 In this paper, we will consider a class of multilinear operators related to some sublinear operators, whose definition
 are following.
\par
 Fixed $\delta>0$. Let $m$ be a positive integer and $A$ be a function on $R^n$. Set
$$
  R_{m+1}(A;x,y)=A(x)-\sum_{|\alpha|\le m}\frac{1}{\alpha!}D^\alpha A(y)(x-y)^\alpha
$$
and
$$
  Q_{m+1}(A;x,y)=R_m(A;x,y)-\sum_{|\alpha|=m}\frac{1}{\alpha!}D^\alpha A(x)(x-y)^\alpha.
$$
\par
 {\bf Definition.}\ Let $\varepsilon>0$ and $\psi$ be a fixed function which satisfies the following properties:
\par
    (1) \ \ $\int_{R^n}\psi(x)dx=0$,
\par
    (2) \ \ $|\psi(x)|\le C(1+|x|)^{-(n+1-\delta)}$,
\par
    (3) \ \ $|\psi(x+y)-\psi(x)|\le C|y|^\varepsilon(1+|x|)^{-(n+1+\varepsilon-\delta)}$ when $2|y|<|x|$;
\par
 The multilinear Littlewood-Paley operator is defined by
$$
 g_\delta^A(f)(x)=\left(\int_0^\infty |F_t^A(f)(x)|^2\frac{dt}{t}\right)^{1/2},
$$
  where
$$
 F_t^A(f)(x)=\int_{R^n}\frac{R_{m+1}(A;x,y)}{|x-y|^m}\psi_t(x-y)f(y)dy
$$
 and $\psi_t(x)=t^{-n+\delta}\psi(x/t)$ for $t>0$. Set $F_t(f)=\psi_t\ast f$. We also define that
$$
 g_\delta(f)(x)=\left(\int_0^\infty|F_t(f)(x)|^2\frac{dt}{t}\right)^{1/2},
$$
 which is the Littlewood-Paley operator with $\delta=0$(see [19]).
\par
 We shall prove the following theorems in Section 3.
\par
  {\bf Theorem.}\ Let $0\le\delta<n$ and $D^\alpha A\in BMO(R^n)$ for $|\alpha|=m$. Then
\par
   (i)\   $g_\delta^A$ maps $L^{n/\delta}(R^n)$ continuously into $BMO(R^n)$;
\par
   (ii)\  $\tilde g_\delta^A$ maps $H^1(R^n)$ continuously into $L^{n/(n-\delta)}(R^n)$;
\par
  (iii)\ $g_\delta^A$ maps $H^1(R^n)$ continuously into weak $L^{n/(n-\delta)}(R^n)$;
\par
 (iv)\ If for any $H^1$-atom $a$ supported on certain cube $Q$ and $u\in 3Q\setminus 2Q$, there is
$$
 \int_{(4Q)^c}\left|\left|\sum_{|\alpha|=m}\frac{1}{\alpha!}\frac{(x-u)^\alpha}{|x-u|^m}\psi_t(x-u)
 \int_QD^\alpha A(y)a(y)dy\right|\right|^{n/(n-\delta)}dx\le C,
$$
 then $g_\delta^A$ is bounded from $H^1(R^n)$ to $L^{n/(n-\delta)}(R^n)$;
\par
 (v)\ If for any cube $Q$ and $u\in 3Q\setminus 2Q$, there is
$$
 \frac{1}{|Q|}\int_Q\left|\left|\sum_{|\alpha|=m}\frac{1}{\alpha!}(D^\alpha A(x)-(D^\alpha A)_Q)
 \int_{(4Q)^c}\frac{(u-y)^\alpha}{|u-y|^m}\psi_t(u-y)f(y)dy\right|\right|dx\le C||f||_{L^{n/\delta}},
$$
  then $\tilde g_\delta^A$ is bounded from $L^{n/\delta}(R^n)$ to $BMO(R^n)$.
\vskip2mm
\par\noindent
 {\large\bf 3. Main theorem and Proof}
\vskip2mm
\par
\par
To prove Theorem, we need the following lemma.
\par
 {\bf Lemma.}\ Let $0\le\delta<n$, $1<p<n/\delta$, $1/q=1/p-\delta/n$ and $D^\alpha A\in BMO(R^n)$ for $|\alpha|=m$. Then
 $g_\delta^A$ and $\mu_\delta^A$ are all bounded from $L^p(R^n)$ to $L^q(R^n)$.
\par
 {\bf Proof.}\ By the Minkowski inequality, we get
\begin{eqnarray*}
 g_\delta^A (f)(x)&\le& \int_{R^n}\frac{|f(y)||R_{m+1}(A;x,y)|}{|x-y|^m}\left(\int_0^\infty|\psi_t(x-y)|^2\frac{dt}{t}\right)^{1/2}dy  \\
 &\le& C\int_{R^n}\frac{|f(y)||R_{m+1}(A;x,y)|}{|x-y|^m}\left(\int_0^\infty\frac{t^{-2n+2\delta}}{(1+|x-y|/t)^{2(n+1-\delta)}}
 \frac{dt}{t}\right)^{1/2}dy    \\
 &\le&C\int_{R^n}\frac{|R_{m+1}(A;x,y)|}{|x-y|^{m+n-\delta}}|f(y)|dy, \\
 \mu_\delta^A(f)(x)&\le& \int_{R^n}\frac{|\Omega(x-y)||R_{m+1}(A;x,y)|}{|x-y|^{m+n-1-\delta}}|f(y)|
 \left(\int_{|x-y|}^\infty\frac{dt}{t^3}\right)^{1/2}dy  \\
 &\le& C\int_{R^n}\frac{|R_{m+1}(A;x,y)|}{|x-y|^{m+n-\delta}}|f(y)|dy,
\end{eqnarray*}
thus, the lemma follows from [9].
\par
 {\bf Proof of Theorem.}(i).\ First, by the proof of Lemma 2, we have
$$
g_\delta(f)(x)\le C\int_{R^n}\frac{|f(y)|}{|x-y|^{n-\delta}}dy,
$$
 thus, $g_\delta$ is $(L^p, L^q)$-bounded for $p>1$ and $1/q=1/p-\delta/n$ by [2]. Now, it suffices to verify that
 $g_\delta^A$ satisfies the size condition in Main Theorem. Let supp$f\subset(2Q)^c$ and $\tilde A(x)=A(x)
 -\sum\limits_{|\alpha|=m}\frac{1}{\alpha!}(D^\alpha A)_Qx^\alpha$. We write
\begin{eqnarray*}
 F_t^{\tilde A}(f)(x)-F_t^{\tilde A}(f)(x_0)&=&\int_{R^n}\left[\frac{\psi_t(x-y)}{|x-y|^m}-\frac{\psi_t(x_0-y)}{|x_0-y|^m}\right]R_m(\tilde A;x,y)f(y)dy  \\
 &\;& +\int_{R^n}\frac{\psi_t(x_0-y)f(y)}{|x_0-y|^m}[R_m(\tilde A; x, y)-R_m(\tilde A; x_0, y)]dy  \\
 &\;& -\sum_{|\alpha|=m}\frac{1}{\alpha!}\int_{R^n}\left(\frac{\psi_t(x-y)(x-y)^\alpha}{|x-y|^m}-\frac{\psi_t(x_0-y)(x_0-y)^\alpha}{|x_0-y|^m}\right)D^\alpha\tilde A(y)f(y)dy \\
 &:=& I_1+I_2+I_3.
\end{eqnarray*}
 Note that $|x-y|\approx |x_0-y|$ for $x\in Q$ and $y\in R^n\setminus \tilde Q$. By Lemma 1 and the following
 inequality(see[18]):
$$
 |b_{Q_1}-b_{Q_2}|\le C\log(|Q_2|/|Q_1|)||b||_{BMO}\  \mbox{for} \  Q_1 \subset Q_2,
$$
 we know that, for $x\in Q$ and $y\in 2^{k+1}\tilde Q\setminus 2^k\tilde Q$,
\begin{eqnarray*}
 |R_m(\tilde A;x,y)|&\le& C|x-y|^m\sum_{|\alpha|=m}(||D^\alpha A||_{BMO}+|(D^\alpha A)_{\tilde Q(x,y)}-(D^\alpha A)_{\tilde Q}|) \\
 &\le& C k|x-y|^m\sum_{|\alpha|=m}||D^\alpha A||_{BMO}.
\end{eqnarray*}
 We choose $r>1$ such that $1/r+\delta/n=1$, similar to the proof of Lemma 2, we have
\begin{eqnarray*}
 ||I_1||&\le& C\int_{R^n\setminus \tilde Q}\left(\frac{|x-x_0|}{|x_0-y|^{m+n+1-\delta}}+\frac{|x-x_0|^\varepsilon}
 {|x_0-y|^{m+n+\varepsilon-\delta}}\right)|R_m(\tilde A; x,y)||f(y)|dy   \\
 &\le& C\sum_{|\alpha|=m}||D^\alpha A||_{BMO}\sum_{k=0}^\infty\int_{2^{k+1}\tilde Q\setminus2^k\tilde Q}k\left(\frac{|x-x_0|}
 {|x_0-y|^{n+1-\delta}}+\frac{|x-x_0|^\varepsilon}{|x_0-y|^{n+\varepsilon-\delta}}\right)|f(y)|dy    \\
 &\le& C\sum_{|\alpha|=m}||D^\alpha A||_{BMO}||f||_{L^{n/\delta}}\sum_{k=1}^\infty k(2^{-k}+2^{-\varepsilon k})
 \le C\sum_{|\alpha|=m}||D^\alpha A||_{BMO}||f||_{L^{n/\delta}};
\end{eqnarray*}
 For $I_2$, by the formula(see [6]):
$$
 R_m(\tilde A; x, y)-R_m(\tilde A; x_0, y)=\sum_{|\beta|<m}\frac{1}{\beta!}R_{m-|\beta|}(D^\beta\tilde A; x, x_0)(x-y)^\beta
$$
 and Lemma 1, we have
$$
 |R_m(\tilde A; x, y)-R_m(\tilde A; x_0, y)|\le C\sum_{|\beta|<m}\sum_{|\alpha|=m}|x-x_0|^{m-|\beta|}|x-y|^{|\beta|}||D^\alpha A||_{BMO},
$$
 similar to the estimates of $I_1$, we get
$$
 ||I_2|| \le C\sum_{|\alpha|=m}||D^\alpha A||_{BMO}\sum_{k=0}^\infty\int_{2^{k+1}\tilde Q\setminus2^k\tilde Q}
 \frac{|x-x_0|}{|x_0-y|^{n+1-\delta}}|f(y)|dy\le C\sum_{|\alpha|=m}||D^\alpha A||_{BMO}||f||_{L^{n/\delta}};
$$
 For $I_3$, by taking $r>1$ such that $1/r+\delta/n=1$,similar to the estimates of $I_1$, we get
\begin{eqnarray*}
 ||I_3||&\le&C\sum_{|\alpha|=m}\sum_{k=0}^\infty\int_{2^{k+1}\tilde Q\setminus2^k\tilde Q}\left(\frac{|x-x_0|}{|x_0-y|^{n+1-\delta}}
 +\frac{|x-x_0|^\varepsilon}{|x_0-y|^{n+\varepsilon-\delta}}\right)|D^\alpha \tilde A(y)||f(y)|dy    \\
 &\le& C\sum_{|\alpha|=m}\sum_{k=1}^\infty (2^{-k}+2^{-\varepsilon k})\left(|2^k\tilde Q|^{-1}\int_{2^k\tilde Q}|D^\alpha A(y)
 -(D^\alpha A)_{\tilde Q}|^rdy\right)^{1/r}||f||_{L^{n/\delta}}     \\
 &\le& C\sum_{|\alpha|=m}||D^\alpha A||_{BMO}||f||_{L^{n/\delta}}.
\end{eqnarray*}
 Thus
$$
 ||F_t^{\tilde A}(f_2)(x)-F_t^{\tilde A}(f_2)(x_0)||\le C\sum_{|\alpha|=m}||D^\alpha A||_{BMO}||f||_{L^{n/\delta}}.
$$
\par
 (ii).\ It suffices to show that there exists a constant $C>0$ such that for every $H^1$-atom $a$, we have
$$
||\tilde g_\delta^A(a)||_{L^{n/(n-\delta)}}\le C.
$$
 We write
$$
 \int_{R^n}\left[\tilde g_\delta^A(a)(x)\right]^{n/{n-\delta}}dx=\left[\int_{|x-x_0|\le 2r}
 +\int_{|x-x_0|>2r}\right]\left[\tilde g_\delta^A(a)(x)\right]^{n/(n-\delta)}dx:=J+JJ.
$$
  For $J$, by the following equality
$$
 Q_{m+1}(A;x,y)=R_{m+1}(A;x,y)+\sum_{|\alpha|=m}\frac{1}{\alpha!}(x-y)^\alpha(D^\alpha A(x)-D^\alpha A(y)),
$$
we have, similar to the proof of Lemma 2,
$$
 \tilde g_\delta^A(a)(x)\le g_\delta^A(a)(x)+C\sum_{|\alpha|=m}\int_{R^n}\frac{|D^\alpha A(x)-D^\alpha A(y)|}{|x-y|^{n-\delta}}|a(y)|dy,
$$
 thus, $\tilde g_\delta^A$ is $(L^p,L^q)$-bounded by Lemma 2 and [2], where $1/q=1/p-\delta/n$. We see that
$$
 J\le C||\tilde g_\delta^A(a)||_{L^q}^{n/((n-\delta)q)}|2Q|^{1-n/((n-\delta)q)}\le C||a||_{L^p}^{n/(n-\delta)}|Q|^{1-n/((n-\delta)q)}\le C.
$$
 To obtain the estimate of $JJ$, we denote that $\tilde A(x)=A(x)-\sum_{|\alpha|=m}\frac{1}{\alpha!}(D^\alpha A)_{2Q}x^\alpha$.