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\title
        Weighted Weak Type $(H^1, L^1)$ Estimates for Commutators of
        Littlewood-Paley Operators
\endtitle
\affil
        Liu Lanzhe   \\
        College of Mathematics  \\
       Changsha University of Science and Technology \\
       Changsha 410077, P.R. of China \\
        E-mail:lanzheliu$\@$263.net
\endaffil
\rightheadtext{}
\endtopmatter
\document
      {\bf Abstract} \ \ We show the weighted weak type
      $(H^1, L^1)$ estimates for the commutator of
      Littlewood-Paley operators.
\vskip2mm
\par\noindent
     {\bf Key words:} \ \ Littlewood-Paley operator, Commutator, BMO($R^n$), $A_1$ weight.
\vskip2mm
\par\noindent
     {\bf 2000 MR Subject Classification} \ \ 42B25, 42B20.
\vskip5mm
\par\noindent
    {\bf 1. Introduction}
\par
    Let $\varepsilon>0$, fixed a given function
    $\psi$ satisfy the following properties:
\par
    (1) \ \ $\int \psi (x)dx=0$,
\par
    (2) \ \ $|\psi (x)|\le C(1+|x|)^{-(n+\varepsilon)}$,
\par
    (3) \ \ $|\psi(x+y)-\psi(x)|\le C|y|^\varepsilon(1+|x|)
    ^{-(n+1+\varepsilon)}$ when $2|y|<|x|$.
\par
    Let $b$ be a locally integrable function and $\Gamma(x)=
    \{(y,t)\in R_+^{n+1}: |x-y|<t\}$. The commutators of
    Littlewood-Paley operator are defined by
$$
\align
\;&   g_{\psi,b}(f)(x)=\left(\int_0^\infty |F_{b,t}(x)|^2
      \frac{dt}{t}\right)^{1/2},     \\
\;&   S_{\psi, b}(f)(x)=\left(\int_{\Gamma(x)} |F_{b,t}(x,y)|^2 \frac{dydt}
      {t^{1+n}} \right)^{1/2},   \\
\;&   g_{\mu,b}^\ast(f)(x)=\left[\int\int_{R_+^{n+1}}
      \left(\frac{t}{t+|x-y|}\right)^{n\mu}
      |F_{b,t}(x,y)|^2 \frac{dydt}{t^{1+n}} \right]^{1/2}, \ \ \ \ \mu>1,
\endalign
$$
     where
$$
     F_{b,t}(x)=\int_{R^n}\psi_t(x-y)f(y)(b(x)-b(y))dy,
$$
$$
     F_{b,t}(x,y)=\int_{R^n}\psi_t(y-z)f(z)(b(x)-b(z))dz,
$$
     and $\psi_t(x)=t^{-n}\psi(x/t)$ for $t>0$. We denote $F_t(f)(x)=f\ast\psi_t(x)$.
     We also define
$$
\align \;&   g_\psi(f)(x)=\left(\int_0^\infty |f\ast\psi_t(x)|^2
      \frac{dt}{t}\right)^{1/2},  \\
\;&   S_\psi(f)(x)=\left(\int_{\Gamma(x)} |f\ast \psi_t(y)|^2
      \frac{dydt}{t^{1+n}} \right)^{1/2},   \\
\;&   g_\mu^\ast(f)(x)=\left[\int\int_{R_+^{n+1}}\left(
      \frac{t}{t+|x-y|}\right)^{n\mu} |f\ast \psi_t(y)|^2
      \frac{dydt}{t^{1+n}}\right]^{1/2},
\endalign
$$
      which are Littlewood-Paley operator (see [7]). It
      is well known that these operators play important role
      in harmonic analysis (see [6]). In 1976, a classical result of
      Coifman, Rochberg and Weiss [3]
      proved that  the commutator $[b, T]$ generated by BMO($R^n$)
     functions and the Calderon-
     Zygmund operator is bounded on $L^p (R^n)$ ($1<p<\infty$).
     However, it was observed that $[b, T]$ is not bounded, in general,
     from $H^p(R^n)$ to $L^p(R^n)$ and from $L^1(R^n)$ to $L^{1,\infty}$
     $(R^n)$ for $p\le 1$.
     The main purpose of this paper is to establish the
     weighted boundedness of the commutators related to Littlewood-Paley
     operator and $BMO(R^n)$ functions from $H^1$ space
     to weak $L^1$ space. Our result can be stated as follows.
\par
     {\bf Theorem.} \ \ Let $b\in BMO(R^n)$ and $w\in A_1$. Then the
     commutators $g_{\psi,b}$, $S_{\psi, b}$ and $g_{\mu,b}^\ast$ are
     all bounded from $H_w^1(R^n)$ to $L_w^{1,\infty}(R^n)$, i.e.,
     there exist constants $C$ such that for any
     $f\in H_w^1(R^n)$ and $\lambda>0$,
$$
     w(\{x\in R^n: g_{\psi,b}(f)(x)>\lambda\})\le C\lambda^{-1}
     ||b||_{BMO}||f||_{H_w^1(R^n)},
$$
$$
     w(\{x\in R^n: S_{\psi,b}(f)(x)>\lambda\})\le C\lambda^{-1}
     ||b||_{BMO}||f||_{H_w^1(R^n)},
$$
$$
     w(\{x\in R^n: g_\mu^\ast(f)(x)>\lambda\})\le C\lambda^{-1}
     ||b||_{BMO}||f||_{H_w^1(R^n)}.  \ \
$$

\vskip5mm
\par\noindent
    {\bf 2. Proof of Theorem}
\par
     Given $f\in H_w^1(R^n)$, by the atomic decomposition of
     $H_w^1(R^n)$ and a limiting argument, it suffices to prove the theorem for
     a finite sum for the atomic decomposition of $f=\sum\limits_Q
     \lambda_Q a_Q$ with supp$a_Q\subset Q$, $Q=Q(x_0,r)$ is the cube
     with center $x_0$ and side-length $r$, and
     $||a_Q||_\infty\le w(Q)^{-1}$, $\int a_Q(x)dx=0$,
     $\sum\limits_Q |\lambda_Q|\le C||f||_{H_w^1(R^n)}$. We
     may assume that each $Q$ is dyadic. For $\lambda>0$, by
     Lemma 4.1 of [3], there exists a collection of pairwise disjoint
     dyadic cubes $\{S\}$ such that
$$
     \sum_{Q\subset S}|\lambda_Q|\le C\lambda |S|, \ \ \text{for all} \ S,
$$
$$
     \sum_S|S|\le C\lambda^{-1}\sum_Q|\lambda Q|,
$$
$$
     \left|\left|\sum_{Q\not\subset any S}\lambda_Q |Q|^{-1}
     \chi_Q\right|\right|_\infty\le C\lambda.
$$
     Let $E=\bigcup\limits_S\overline S$, where, and in what follows,
     for a fixed cube $B$, $\overline B$ denotes the cube with the same
     center as $B$ but with the side-length $4\sqrt{n}$ times that
     of $B$. Then
$$
     |E|\le C\lambda^{-1}||f||_{H_w^1(R^n)}.
$$
     Set $M(x)=\sum\limits_S\sum\limits_{Q\subset S}\lambda_Q a_Q$,
     $N(x)=f(x)-M(x)$. By the $L^2$-boundedness of $g_{\psi, b}$,
     $S_{\psi, b}$ and $g_{\mu, b}^\ast$ (see [2]) and well-known
     arguments, it suffices to show that
$$
     \dot w(\{x\in E^c: T_b(M)(x)>\lambda\})\le C\lambda^{-1}
     ||f||_{H_w^1(R^n)},
$$
     where $T_b=g_{\psi, b}$ or $S_{\psi, b}$ or $g_{\mu,b}^\ast$.
\par
     For $g_{\psi, b}$ notice that $g_{\psi, b}(M)(x)\le \sum\limits_S
     \sum\limits_{Q\subset S}|\lambda_Q|g_{\psi, b}(a_Q)(x)$, by the
     vanishing condition of $a_Q$, and notice that
$$
     \int_0^\infty \frac{tdt}{(t+|x-x_0|)^{2(n+1+\varepsilon)}}=C
     |x-x_0|^{-2(n+\varepsilon)}.
$$
     We have, for $x\in (2Q)^C$,
$$
\align
       g_{\psi, b}(a_Q)(x)
\;&    \left[\int_0^\infty\left(\int_Q|\psi_t(x-y)-\psi_t(x-x_0)|
       |a_Q(y)| \ |b(x)-b(y)|dy\right)^2\frac{dt}{t}\right]^{1/2}   \\
\le&   C\left[\int_0^\infty\left(\int_Q t^{-n}|a_Q(y)| |b(x)-b(y)|
       \frac{(|y-x_0|/t)^\varepsilon}{(1+|x-x_0|/t)^{n+1+\varepsilon}}
       dy\right)^2\frac{dt}{t}\right]^{1/2}   \\
=&     C\left(\int_0^\infty\frac{tdt}{(t+|x-x_0|^{2(n+1+\varepsilon)})}
       \right)^{1/2}\left(\int_Q |y-x_0|^\varepsilon |a_Q(y)|
       |b(x)-b(y)|dy\right),    \\
\le&  C|x-x_0|^{-(n+\varepsilon)}|Q|^{\varepsilon/n}w(Q)^{-1}
       \int_Q|b(x)-b(y)|dy.  \\
\endalign
$$
     For $S_{\psi, b}$, we deduce that
$$
\align
\;&   S_{\psi,b}(a_Q)(x)   \\
\le&  \left[\int_{\Gamma(x)} \left(\int_Q |\psi_t(y-z)-\psi_t(y-x_0)| \
      |a_Q(z)| \ |b(x)-b(z)|dz \right)^2
      \frac{dydt}{t^{1+n}}\right]^{1/2}    \\
\le&  C\left[\int_{\Gamma(x)} \left(\int_Q t^{-n}|a_Q(z)| |b(x)-b(z)|
      \frac{(|x_0-z|/t)^\varepsilon}{(1+|x_0-y|/t)^{n+1+\varepsilon}}
      dy\right)^2\frac{dydt}{t^{1+n}}\right]^{1/2}    \\
=&    C\left[\int_{\Gamma(x)}\left(\int_Q\frac{|B|^{\varepsilon/n}
      w(Q)^{-1}t}{(t+|x_0-y|)^{n+1+\varepsilon}}|b(x)-b(z)|dz\right)^2
      \frac{dydt}{t^{1+n}}\right]^{1/2}   \\
\le&  C|Q|^{\varepsilon/n}w(Q)^{-1}\left[\int_{\Gamma(x)}\frac{t^{1-n}
      2^{2(n+1+\varepsilon)}}{(2t+2|x_0-y|)^{2(n+1+\varepsilon)}}
      \left(\int_Q |b(x)-b(z)|dz\right)^2 dydt\right]^{1/2},  \\
\le&  C|Q|^{\varepsilon/n} w(Q)^{-1}\left(\int_{\Gamma(x)}
      \frac{t^{1-n}dydt}{(t+|x-x_0|^{2(n+1+\varepsilon)})}\right)
      ^{1/2}\left(\int_Q |b(x)-b(z)|dz\right)      \\
\le&   C|Q|^{\varepsilon/n}w(Q)^{-1}\left(\int_0^\infty
       \frac{tdt}{(t+|x-x_0|)^{2(n+1+\varepsilon)}}\right)^{1/2}
       \left(\int_Q |b(x)-b(z)|dz\right)      \\
\le&   C|x-x_0|^{-(n+\varepsilon)}|Q|^{\varepsilon/n}w(Q)^{-1}
       \int_Q|b(x)-b(y)|dy.
\endalign
$$
      For $g_{\mu, b}^\ast$, notice that,
$$
\align
\;&   t^{-n}\int_{R^n}\left(\frac{t}{t+|x-y|}\right)^{n\mu}
      \frac{dy}{(t+|x_0-y|)^{2(n+1+\varepsilon)}}\le CM
      \left(\frac{1}{(t+|x_0-x|)^{2(n+1+\varepsilon)}}\right)   \\
\le&  C\left(\frac{1}{(t+|x_0-x|)^{2(n+1+\varepsilon)}}\right).
\endalign
$$
      We deduce that
$$
\align
\;&   g_{\mu,b}^\ast(a_Q)(x)\le\biggl[\int\int_{R_+^{n+1}}   \\
\;&   \left(\frac{t}{t+|x-y|}\right)^{n\mu}\biggl(\int_Q|\psi_t(y-z)
      -\psi_t(y-x_0)||a_Q(z)||b(x)-b(z)|dz\biggr)^2
      \frac{dydt}{t^{1+n}}\biggr]^{1/2}  \\
\le&  C\biggl[\int\int_{R_+^{n+1}}\left(\frac{t}{t+|x-y|}\right)^{n\mu} \\
\;&   \left(\int_Q t^{-n}|a_Q(z)||b(x)-b(z)|
      \frac{(|x_0-z|/t)^\varepsilon}
      {(1+|x_0-y|/t)^{n+1+\varepsilon}}dz\right)^2\frac{dydt}{t^{1+n}}
      \biggr]^{1/2}    \\
\endalign
$$
$$
\align
\le&  C|Q|^{\varepsilon/n}w(Q)^{-1}\biggl[\int\int_{R_+^{n+1}}\left(
      \frac{t}{t+|x-y|}\right)^{n\mu}\frac{t^2}{(t+|x_0-y|)^{2(n+1+
      \varepsilon)}}  \\
\;&   \left(\int_Q |b(x)-b(z)|dz\right)^2
      \frac{dydt}{t^{1+n}}\biggr]^{1/2}   \\
\le&   C|x-x_0|^{-(n+\varepsilon)}|Q|^{\varepsilon/n}w(Q)^{-1}
       \int_Q |b(x)-b(y)|dy.
\endalign
$$
     Thus, with $b_0=|Q|^{-1}\int_Q b(x)dx$,
$$
\align
\;&    w(\{x\in E^C: T_b(M)(x)>\lambda\})\le
       C\lambda^{-1}\int_{E^C} T_b(M)(x)w(x)dx  \\
\le&   C\lambda^{-1}\sum_{S}\sum_{Q\subset S}|\lambda_Q|
       \sum_{k=1}^\infty \int_{2^{k+1}\overline Q\setminus 2^k
       \overline Q}T_b(a_Q)(x)w(x)dx   \\
\le&   C\lambda^{-1}\sum_S\sum_{Q\subset S}|\lambda_Q||Q|^{\varepsilon/n}
       w(Q)^{-1}\sum_{k=1}^\infty\int_{2^{k+1}\overline Q\setminus 2^k
       \overline Q}|x-x_0|^{-(n+\varepsilon)}  \\
\;&    \left(\int_Q |b(x)-b(y)|dy\right)w(x)dx    \\
\le&   C\lambda^{-1}\sum_S\sum_{Q\subset S}|\lambda_Q||Q|^{\varepsilon/n}
       w(Q)^{-1}\sum_{k=1}^\infty\int_{2^{k+1}\overline Q\setminus 2^k
       \overline Q}|x-x_0|^{-(n+\varepsilon)}  \\
\;&    |Q|(|b(x)-b_0|+||b||_{BMO})w(x)dx    \\
\le&   C\lambda^{-1}\sum_S\sum_{Q\subset S}|\lambda_Q||Q|^{\varepsilon/n+1}
       w(Q)^{-1}\sum_{k=1}^\infty |2^k Q|^{-(1+\varepsilon/n)}
       \int_{2^{k+1}\overline Q} |b(x)-b_0|w(x)dx      \\
\;&    +C\lambda^{-1}\sum_S\sum_{Q\subset S}|\lambda_Q||Q|^{\varepsilon/n+1}
       w(Q)^{-1}\sum_{k=1}^\infty |2^k Q|^{-(1+\varepsilon/n)} ||b||_{BMO}
       w(2^k Q)  \\
=&     I_1+I_2.
\endalign
$$
     For $I_1$, taking $p>1$ and $1/p+1/p'=1$, using the properties of
     $BMO(R^n)$ function (see [6]), and noting $w\in A_1$, we get
     $\frac{w(B_2)}{|B_2|}\frac{|B_1|}{w(B_1)}\le C$ for all cubes
     $B_1, B_2$ with $B_1\subset B_2$. Thus, by Holder and reverse
     Holder inequality, we obtain
$$
\align
      I_1
\le&  C\lambda^{-1}\sum_S\sum_{Q\subset S}|\lambda_Q|
      \sum_{k=1}^\infty 2^{-k\varepsilon} |Q|w(Q)^{-1}
      \left(\frac{1}{|2^{k+1}\overline Q|}\int_{2^{k+1}\overline Q}
      |b(x)-b_0|^p dx\right)^{1/p}  \\
\;&   \left(\frac{1}{|2^{k+1}\overline Q|}
      \int_{2^{k+1}Q}w(x)^{p'}dx\right)^{1/p'}     \\
\le&  C\lambda^{-1}\sum_S\sum_{Q\subset S} |\lambda_Q|\sum_{k=1}^\infty k
      2^{-k\varepsilon} ||b||_{BMO}\left(\frac{w(2^k Q)}{|2^k Q|}
      \frac{|Q|}{w(Q)}\right)     \\
\endalign
$$
$$
\align
\le&  C\lambda^{-1}||b||_{BMO}\sum_S\sum_{Q\subset S}|\lambda_Q|  \\
\le&  C\lambda^{-1}||b||_{BMO}||f||_{H_w^1(R^n)}.
\endalign
$$
     For $I_2$, similar to the estimate of $I_1$, we get
     $I_2\le C\lambda^{-1}||b||_{BMO}$. This completes the proof of
     Theorem.

\vskip5mm \head{\bf References}
\endhead
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