\documentclass[12pt]{amsart} %\usepackage{amsmath,amssymb,amscd,amsthm} \usepackage{latexsym} \usepackage[dvips]{graphics,epsfig} \headheight=8pt \topmargin=0pt \textheight=624pt \textwidth=432pt \oddsidemargin=18pt \evensidemargin=18pt %\setlength{\textwidth}{6.5in} %\setlength{\textheight}{8.2in} %\setlength{\abovedisplayshortskip}{0.1937in} %\setlength{\belowdisplayshortskip}{0.1937in} %\setlength{\oddsidemargin}{0in} %\setlength{\evensidemargin}{0in} %\setlength{\topmargin}{-0.6in} %\setlength{\headheight}{7pt} \newtheorem{theorem}{Theorem} \newtheorem{lemma}{Lemma} \newtheorem{proposition}{Proposition} \newtheorem{remark}{Remark} \newtheorem{example}{Example} \newtheorem{exercise}{Exercise} \newtheorem{definition}{Definition} \newtheorem{corollary}{Corollary} \newtheorem{notation}{Notation} %SPACING \newcommand{\q}{\quad} \newcommand{\qq}{\quad\quad} \newcommand{\qqq}{\quad\quad\quad} \newcommand{\qqqq}{\quad\quad\quad\quad} \newcommand{\qqqqq}{\quad\quad\quad\quad\quad} \newcommand{\qqqqqq}{\quad\quad\quad\quad\quad\quad} %DELIMETERS \newcommand{\abs}[1]{\lvert#1\rvert} \newcommand{\norm}[2]{{\left\| #1 \right\|}_{#2}} %NUMBERS \newcommand{\ppr}{p'} \newcommand{\pnf}{+\infty} \newcommand{\mnf}{-\infty} \newcommand{\nf}{\infty} \newcommand{\oop}{\frac{1}{p}} \newcommand{\ooppr}{\frac{1}{p'}} \newcommand{\ooq}{\frac{1}{q}} \newcommand{\ooqpr}{\frac{1}{q'}} \newcommand{\oor}{\frac{1}{r}} \newcommand{\oorpr}{\frac{1}{r'}} %GREEK LETTERS \newcommand{\al}{\alpha} \newcommand{\be}{\beta} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\de}{\delta} \newcommand{\ben}{\beta_n} \newcommand{\De}{\Delta} \newcommand{\ve}{\varepsilon} \newcommand{\ze}{\zeta} \newcommand{\Th}{\Theta} \newcommand{\ka}{\kappa} \newcommand{\la}{\lambda} \newcommand{\laj}{\lambda_j} \newcommand{\lak}{\lambda_k} \newcommand{\La}{\Lambda} \newcommand{\si}{\sigma} \newcommand{\Si}{\Sigma} \newcommand{\vp}{\varphi} \newcommand{\om}{\omega} \newcommand{\Om}{\Omega} %EUCLIDEAN SPACES \newcommand{\rn}{{\mathbf R}^n} \newcommand{\rone}{\mathbf R^1} \newcommand{\rtwo}{\mathbf R^2} \newcommand{\rthree}{\mathbf R^3} \newcommand{\rfour}{\mathbf R^4} \newcommand{\tn}{\mathbf T^n} \newcommand{\tone}{\mathbf T^1} \newcommand{\ttwo}{\mathbf T^2} \newcommand{\tthree}{\mathbf T^3} \newcommand{\tfour}{\mathbf T^4} \newcommand{\zn}{\mathbf Z^n} \newcommand{\zp}{\mathbf Z^+} \newcommand{\zone}{\mathbf Z^1} \newcommand{\zz}{\mathbf Z} \newcommand{\ztwo}{\mathbf Z^2} \newcommand{\zthree}{\mathbf Z^3} \newcommand{\zfour}{\mathbf Z^4} \newcommand{\hn}{\mathbf H^n} \newcommand{\hone}{\mathbf H^1} \newcommand{\htwo}{\mathbf H^2} \newcommand{\hthree}{\mathbf H^3} \newcommand{\hfour}{\mathbf H^4} \newcommand{\cn}{\mathbf C^n} \newcommand{\cone}{\mathbf C^1} \newcommand{\ctwo}{\mathbf C^2} \newcommand{\cthree}{\mathbf C^3} \newcommand{\cfour}{\mathbf C^4} \newcommand{\sn}{\mathbf S^{n-1}} \newcommand{\sone}{\mathbf S^1} \newcommand{\stwo}{\mathbf S^2} \newcommand{\sthree}{\mathbf S^3} \newcommand{\sfour}{\mathbf S^4} %SPACES \newcommand{\lp}{L^{p}} \newcommand{\lppr}{L^{p'}} \newcommand{\lqq}{L^{q}} \newcommand{\lr}{L^{r}} \newcommand{\wlp}{L^{p,\infty}} \newcommand{\wlq}{L^{q,\infty}} \newcommand{\wlr}{L^{r,\infty}} \newcommand{\lprn}{L^{p}(\rn)} \newcommand{\lptn}{L^{p}(\tn)} \newcommand{\lpzn}{L^{p}(\zn)} \newcommand{\lpcn}{L^{p}(\cn)} \newcommand{\lphn}{L^{p}(\cn)} \newcommand{\lprone}{L^{p}(\rone)} \newcommand{\lptone}{L^{p}(\tone)} \newcommand{\lpzone}{L^{p}(\zone)} \newcommand{\lpcone}{L^{p}(\cone)} \newcommand{\lphone}{L^{p}(\hone)} \newcommand{\lqrn}{L^{q}(\rn)} \newcommand{\lqtn}{L^{q}(\tn)} \newcommand{\lqzn}{L^{q}(\zn)} \newcommand{\lqcn}{L^{q}(\cn)} \newcommand{\lqhn}{L^{q}(\hn)} \newcommand{\lo}{L^{1}} \newcommand{\lt}{L^{2}} \newcommand{\li}{L^{\infty}} \newcommand{\co}{C^{1}} \newcommand{\ci}{C^{\infty}} \newcommand{\coi}{C_0^{\infty}} %CALLIGRAPHIC \newcommand{\cs}{\mathcal S} \newcommand{\cm}{\mathcal M} \newcommand{\cb}{\mathcal B} \newcommand{\ce}{\mathcal E} \newcommand{\cd}{\mathcal D} \newcommand{\csp}{\mathcal S'} %GENERAL \newcommand{\pv}{\textup{p.v.}\,} \newcommand{\loc}{\textup{loc}} \newcommand{\intl}{\int\limits} \newcommand{\iintl}{\iint\limits} \newcommand{\dint}{\displaystyle\int} \newcommand{\diint}{\displaystyle\iint} \newcommand{\dintl}{\displaystyle\intl} \newcommand{\diintl}{\displaystyle\iintl} \newcommand{\liml}{\lim\limits} \newcommand{\suml}{\sum\limits} \newcommand{\supl}{\sup\limits} \newcommand{\f}{\frac} \newcommand{\df}{\displaystyle\frac} \newcommand{\p}{\partial} \newcommand{\Ar}{\textup{Arg}} \newcommand{\abssig}{\widehat{|\si_0|}} \newcommand{\abssigk}{\widehat{|\si_k|}} \newcommand{\tT}{\tilde{T}} \newcommand{\tV}{\tilde{V}} \newcommand{\wt}{\widetilde} %BEGIN END \newcommand{\lbl}{\label} \newcommand{\beq}{\begin{equation}} \newcommand{\eeq}{\end{equation}} \newcommand{\beqna}{\begin{eqnarray*}} \newcommand{\eeqna}{\end{eqnarray*}} \newcommand{\beqn}{\begin{equation*}} \newcommand{\eeqn}{\end{equation*}} \newcommand{\bp}{\begin{proof}} \newcommand{\ep}{\end{proof}} \newcommand{\bprop}{\begin{proposition}} \newcommand{\eprop}{\end{proposition}} \newcommand{\bt}{\begin{theorem}} \newcommand{\et}{\end{theorem}} \newcommand{\bex}{\begin{Example}} \newcommand{\eex}{\end{Example}} \newcommand{\bc}{\begin{corollary}} \newcommand{\ec}{\end{corollary}} \newcommand{\bl}{\begin{lemma}} \newcommand{\el}{\end{lemma}} \begin{document} \title [$L^p$ Bounds for singular integrals and maximal singular integrals] {$L^p$ bounds for singular integrals \\ and maximal singular integrals with rough kernels} \author{Loukas Grafakos} \address{ Loukas Grafakos\\ Department of Mathematics\\ University of Missouri\\ Columbia, MO 65211, USA} \curraddr{MSRI, Berkeley, CA 94720-5070} \email{loukas@math.missouri.edu} \author{Atanas Stefanov} \address{ Atanas Stefanov\\ Department of Mathematics\\ University of Missouri\\ Columbia, MO 65211, USA} \curraddr{MSRI, Berkeley, CA 94720-5070} \email{astefanov@pascal.math.missouri.edu} \thanks{Grafakos's research at MSRI partially supported by the NSF under grant DMS 9623120} %\date{\today} \subjclass{Primary 42B20. Secondary 42E30} \keywords{Calder\'on-Zygmund singular integrals, method of rotations} \begin{abstract} Convolution type Calder\'on-Zygmund singular integral operators with rough kernels $\pv \Om(x)/|x|^n$ are studied. A condition on $\Om$ implying that the corresponding singular integrals and maximal singular integrals map $L^p \to L^p$ for $1\ve} \f{\Om (y/|y|)}{|y|^n}f(x-y)\, dy = \pv \int_{\rn} \f{\Om (y/|y|)}{|y|^n}f(x-y)\, dy, \end{equation} for $f$ in the Schwartz class $\cs (\rn)$. The limit in (\ref{1}) is easily shown to exist for any $f$ a $C^1$ function on $\rn$ with some decay at infinity. For $\ve>0$, denote by $$ (T^{\ve}_\Om f)(x)= \int_{|y|>\ve} \f{\Om (y/|y|)}{|y|^n}f(x-y)\, dy $$ the truncated singular integral associated with $T_\Om$ and by $$ (T_\Om^* f)(x)=\sup_{\ve>0}|T^{\ve}_\Om f(x)| $$ the maximal singular integral operator corresponding to this $\Om$. Establishing the a priori bound $\|T_\Om^{\ve} f\|_{\lp}\le C \|f\|_{\lp}$ independently of $f\in\cs(\rn)$ and of $\ve >0$, leads to a (unique) extension of $T^{\ve}_{\Om}$ on $ \lprn$. Now, for $f\in \lprn$, $T^{\ve}_{\Om}f$ converges in $\lp$ as $\ve \to 0$ to some $T_\Om f$ (which extends $T_\Om f$ defined in (\ref{1}) for $f\in \cs(\rn)$), and by Fatou's lemma, $T_\Om$ is a bounded operator on $\lp$. A similar a priori bound for $T_\Om^*$ implies that for $f\in \lprn$, $T_{\ve}f$ converges (to $T_\Om f$) almost everywhere as $\ve \to 0$. We now discuss $\lp$ boundedness properties of these operators. It is well known that if $\Om$ has some smoothness, then both $T_\Om$ and $T_\Om^*$ extend to bounded operators on $\lprn$ for all $10$. We can show that if $\Om$ satisfies condition (\ref{1+alpha}) for some $\alpha >0$, then $T_\Om$ maps $\lprn$ into itself for some $p\neq 2$. More precisely, we have the following theorem: \bt Let $\al>0$. Let $\Om$ be a function in $L^1(\sn )$ with mean value zero which satisfies condition (\ref{1+alpha}) for some $\al>0$. Then $T_\Om$ extends to a bounded operator from $\lprn$ into itself for $(2+\al)/(1+\al)

0$, then it maps $\lprn$ into itself for all $11$. Let $\Om$ be a function in $L^1(\sn )$ with mean value zero which satisfies condition (\ref{1+alpha}) for some $\al>0$. Then $T^{*}_\Om$ extends to a bounded operator from $\lprn$ into itself for $1+3/(1+2\al)

0$, then $T^{*}_\Om$ maps $\lp$ to $\lp$ for all $1p_0$. In section 5, we show that condition (\ref{1+alpha}) for all $\al>0$ is indeed disjoint from the $H^1$ condition (\ref{h1}). \section{Boundedness of singular integrals} The theme of the proof of Theorem 1 is based on ideas developed by J. Duoandikoet- xea and J.-L. Rubio de Francia \cite{javirubio} to treat several other operators of this sort. Define $$ \si_k(x)= \f{\Om (x)}{|x|^n} \chi_{2^k\le |x|\le 2^{k+1}}, \qq k \in \zz. $$ Observe that $\widehat{\si_k}(\xi)= \widehat{\si_0}(2^k\xi)$. We calculate $\widehat{\si_0}(\xi)$. Set $\xi' =\xi/|\xi |$. Expressing $\widehat{\sigma_0}$ in polar coordinates, we obtain \beq\lbl{5657} \widehat{\si_0}(\xi)= \int_{\sn}\Om (\theta) \left[ \int_1^2 e^{2\pi ir|\xi | (\xi'\cdot \theta )} \f{dr}{r} \right]\, d\theta. \end{equation} Using that $\Om$ has mean value zero, we deduce that \beq\lbl{est1} |\widehat{\si_0}(\xi)|\le 2\pi (\ln 2) \|\Om \|_{L^1} |\xi |=C|\xi |, \end{equation} which is a good estimate for $|\xi |\le 2$. For $|\xi |\ge 2$ observe the following: The integral inside brackets in (\ref{5657}) is bounded by $\min \big(2, 3|\xi'\cdot \theta |^{-1} |\xi |^{-1} \big)$. (Pick a $\theta$ so that $\xi'\cdot \theta \neq 0$.) Therefore it must satisfy the estimate \begin{equation}\lbl{hnfiep} \left| \int_1^2 e^{2\pi ir|\xi | (\xi'\cdot \theta )} \f{dr}{r} \right| \le \f{2 \left(\ln (\f32 |\xi'\cdot \theta |^{-1}) \right)^{1+\al} } {( \ln |\xi | )^{1+\al} }. \end{equation} It follows from (\ref{hnfiep}) and (\ref{1+alpha}) that \beq\lbl{est2} |\widehat{\si_0}(\xi)|\le C (\ln |\xi |)^{-1-\al}\qq \text{for $|\xi |\ge 2$.} \end{equation} Since $\si_k$ is obtained from $\si_0$ by a suitable dilation, it follows that there exists a constant $C>0$, such that for all $k\in \zz$ the estimates below are valid: \begin{align}\begin{split}\lbl{estk} |\widehat{\si_k}(\xi)| &\le C (\ln |2^k \xi |)^{-1-\al},\q \text{for $2^k |\xi |\ge 2$, } \\ |\widehat{\si_k}(\xi)| &\le C 2^k |\xi |, \q \qqqq\,\text{for $2^k |\xi |\le 2$.} \end{split}\end{align} Now let $\psi$ be a $C^\nf$ function supported in $\{x\in \rn :\,3/4\le |x|\le 9/4\}$ such that $\sum_{j\in \zz} (\psi (2^{j} \xi ))^2 =1$. Let $S_j$ be the operator given on the Fourier transform by multiplication by $\psi_j(\xi )= \psi (2^{j} \xi )$. Define $$ T_jf= \sum_{k\in \zz} S_{j+k}(\si_k * S_{j+k}f). $$ It is easy to see that the identity $$ T_\Om f= \sum_{j \in \zz} T_jf $$ is valid at least for $f$ in the Schwartz class. Using a Fourier transform calculation, (\ref{estk}), and the fact that $\psi_{j+k}$ is supported near the annulus $|\xi | \sim 2^{-j-k}$, we obtain that $T_j$ are bounded on $\lt (\rn)$ with bound $C2^{-j}$ for $j\ge 0 $ and $C(|j|)^{-1-\al}$ for $j\le -1 $. In short \beq\lbl{ltest} \|T_jf\|_{\lt} \le C(1+ |j|)^{-1-\al}\|f\|_{\lt} \qq\text{for all $j\in \zz$.} \end{equation} We will also need estimates for the following maximal operator $$ f\to \si^*(f)= \sup_{k\in \zz} (|\si_k |*|f|). $$ Without loss of generality we can assume that $\norm{\Om}{\lo(\sn)}=1$. It follows that $\abssig(0)=1$. Introduce a radial function in the Schwartz class $\Phi$, such that $\widehat{\Phi}(\xi)=1$ for $ |\xi|\le 2$ and $\widehat{\Phi}(\xi )=0$ for $|\xi | >3$. Let us also introduce $\Phi_k$ defined by $\widehat{\Phi_k}(\xi)=\widehat{\Phi}(2^k \xi)$. Clearly we have \begin{equation}\label{maxfunc} \si^*(f) \leq \sup_{k\in \zz}|(|\si_k |-\Phi_k)*|f||+ \sup_{k\in \zz}|\Phi_k*|f||. \end{equation} Denote $\mu_k=|\si_k |-\Phi_k$. Since $\widehat{\mu_k}(0)=0$, the same proof giving (\ref{estk}) implies that \begin{align}\begin{split} \lbl{estkp} |\widehat{\mu_k}(\xi)| &\le C 2^k |\xi |, \q \qqqq\,\text{for $2^k |\xi |\le 2$,}\\ |\widehat{\mu_k}(\xi)| &\le C (\log |2^k \xi |)^{-1-\al},\q \text{for $2^k |\xi |\ge 2$. } \end{split}\end{align} Therefore we obtain from (\ref{maxfunc}) that \begin{equation}\label{mjd} \si^*(f) \leq \sup_{k\in \zz}(\mu_k*|f|)+{\mathcal M}f \leq \left(\sum_k |\mu_k*|f||^2\right)^{1/2}+ {\mathcal M}f, \end{equation} where ${\mathcal M}$ is the Hardy-Littlewood maximal function. Since for all $10$. At this point we recall the following lemma: \bl\lbl{lemma} (See \cite{javirubio} p. 544) If $\|\si^*(f)\|_{L^s}\le C \|f\|_{L^s}$ and $\df{1}{2s}= \left| \df12- \df{1}{q}\right|$, then for arbitrary functions $g_k$ we have \begin{equation*} \big\|( \sum_{k \in \zz} |\si_k*g_k|^2)^{1/2} \big\|_{L^{q}} \le C \big\|( \sum_{k \in \zz} |g_k|^2 )^{1/2} \big\|_{L^{q}}. \end{equation*} \end{lemma} Applying Lemma \ref{lemma} with $s=2$ and $ q=q_0=4$, we obtain that \beq\lbl{lpest} \|T_jf\|_{L^{q_0}} \le C \big\| (\sum_{k\in \zz} |\si_k*S_{j+k}f|^2)^{1/2} \big\|_{L^{q_0}} \le C \big\| (\sum_{k\in \zz} |S_{j+k}f|^2 )^{1/2} \big\|_{L^{q_0}} \le C \|f\|_{L^{q_0}}, \end{equation} where the middle inequality is a consequence of Lemma \ref{lemma} and the first and last inequalities follow from the Littlewood-Paley theorem. Interpolating between estimates (\ref{ltest}) and (\ref{lpest}) we obtain that $$ \|T_jf\|_{L^p}\le C (1+|j|)^{-(1+\al)\theta_p}\|f\|_{\lp}, $$ where $1/p= \theta_p/2+(1-\theta_p)/q_0$. Now observe that $T_\Om=\sum_{j\in \zz} T_j$ maps $\lp\to \lp$ for all $p$'s for which $p_1' < p< p_1$, where $p_1= (4+4\al)/(2+\al)$ is the unique solution of the equation $(1+\al)\theta_{p}=1$. The same argument also gives that $T_{\ve}f=\sum_k \ve_k (\mu_k*f)$ maps $\lp \to \lp$ for $p_1' < p< p_1$ uniformly on the choice of the signs $(\ve_j)$, $\ve_j=\pm 1.$ It follows that the square function in (\ref{13}) is also bounded on $\lp$ for this range of $p$'s and hence so is $\si^*(f)$ by the estimate in (\ref{mjd}). Thus we are in a position to apply Lemma \ref{lemma} again with $s$ in the interval $(p_1', p_1)$. Now continue this way. Fix $s_1\in (2, p_1)$ and let $q_{1}$ be the unique number bigger than $q_0=4$ which satisfies the equation $1/2s_1'=|1/2-1/q_1|$. Apply Lemma \ref{lemma} with $s=s_1'$ and $q=q_1$. As before we obtain that $T_\Om$ maps $\lp \to \lp$ for $p_2'< p< p_2$, where $p_2$ is the unique solution of the equation $(1+\al)\theta_{p}=1$, where $\theta_p$ is given by $1/p = \theta_p/2+(1-\theta_p)/q_1$ now. This bootstrapping argument leads to an inductive definition of three sequences $2=p_0 < p_1< \dots \, $, $2 < s_12^{k}} \f{\Om (y)}{|y|^n}f(x-y)\, dy =\sum_{j=k}^{\infty}(\si_j *f)(x), \\ & & (T^*f)(x)=\sup_k|(T_kf)(x)|. \eeqna If $2^{k-1}\leq\ve<2^k$, then $$ |(T_{\Om}^{\ve}f)(x)|\le|(T_k f)(x)|+\big|\int_{\ve<|y|<2^{k}} \f{\Om (y)}{|y|^n}f(x-y)\, dy\big| \leq |(T_k f)(x)|+(|\si_k|*|f|)(x). $$ From the proof of Theorem 1 we know that $\si^*$ maps $\lp\to\lp$ for $(2+\al)/(1+\al)0$ small, and the conclusion of Theorem 2 follows by summing on $j$. Now observe that $$ |Q_j f| \le \sup_k |\si_{j+k}*f|+\sup_k|\Phi_k*\si_{j+k}*f|\le C(\si^*(f)+ \cm(\si^*(f))). $$ Therefore $Q_j$ is bounded on $\lp$ whenever $\si^*$ is, that is $\norm{Q_jf}{\lp}\le C\norm{f}{\lp}$ when $2\le p<2+\al$ and (\ref{130}) is proved. To prove (\ref{131}) we need to exploit some orthogonality. We have $$ \|Q_j f\|_{\lt}^2\le \sum_k \|(\de-\Phi_k)*\si_{j+k}*f \|_{\lt}^2= \textup{Average} \| \sum_k \ve_k \big((\delta-\Phi_k)*\si_{j+k}*f \big) \|^2_{\lt}, $$ where $\ve =(\ve_k)_{k}$ is a sequence of $\pm 1$'s. For a fixed sequence $\ve_k= \pm 1$, let us denote by $$M_{j,k}f = \ve_k (\delta-\Phi_k)*\si_{j+k}*f.$$ We will need the following \bl Let $m\ge 1$, $j\ge 0$ and $ k_1\leq\ldots\leq k_{2m}$ be integers. Then $$ \norm{M_{j,k_1}\ldots M_{j,k_{2m}}}{2\to 2}\leq C^{2m} \prod_{i=1}^{2m} \left(\f{1}{1+j+k_i-k_1}\right)^{1+\al} . $$ \el \bp Since $\widehat{\Phi_{k_1}}(\xi)$ vanishes for $2^{k_1}|\xi |\le 2$ we have, \beqna \norm{M_{j,k_1}\ldots M_{j,k_{2m}}f}{L^2}^2&=&\int \prod_{i=1}^{2m}\left|1-\widehat{\Phi_{k_i}}(\xi)\right|^2 |\widehat{\si_{j+k_i}}(\xi)|^2|\widehat{f}(\xi)|^2 d\xi \\ &\leq& (C)^{2m}\int_{|\xi|\ge 2^{1-k_1}} \prod_{i=1}^{2m}\left[\f{1} {\log\left(2^{j+k_i} |\xi |\right)}\right]^{2+2\al} |\widehat{f}(\xi)|^2 d\xi \\ &\leq& C^{2m} \prod_{i=1}^{2m}\left[\f{1}{1+j+k_i-k_1}\right]^{2+2\al} \norm{f}{L^2}^2, \eeqna where we used the first estimate in (\ref{estk}) in the last inequality above. \ep Now we return to the proof of Theorem 2. We must show that $\norm{M_j^{\ve,N}}{2\to 2}\leq C(1+j)^{-\al}$ uniformly on $N$ and $\ve=(\ve_k)$, where $$ M_j^{\ve,N}f=\sum_{k=-N}^{N}\ve_k M_{j,k}. $$ Since $M_j^{\ve,N}$ are self adjoint operators, we have \begin{gather*} \norm{M_j^{\ve,N}}{2\to 2}^{2m}=\norm{(M_j^{\ve,N})^{2m}}{2\to 2}\leq \sum_{-N\le k_1\le \ldots\leq k_{2m}\le N} \norm{M_{j,k_1}\ldots M_{j,k_{2m}}}{2\to 2}\\ \leq\sum_{-N\le k_1\le \ldots\leq k_{2m}\le N}C^{2m}\prod_{i=1}^{2m} \left(\f{1}{1+j+k_i-k_1}\right)^{1+\al}\leq \f{N C^{2m}}{(1+j)^{1+\al}} \left(\f{1}{(1+j)^{\al}}\right)^{2m-1}\\ \le N \f{C^{2m}}{1+j} (1+j)^{-2m\al}. \end{gather*} Taking $(2m)^{\textup{th}}$ roots and letting $m\to \nf$ we obtain $$ \norm{M_j^{\ve,N}}{2\to 2}\leq C (1+j)^{-\al}. $$ This concludes the proof of (\ref{131}) and hence of Theorem 2. \section{Examples} It is easy to see that condition (\ref{1+alpha}) for all $\al>0$ contains the case $\Om \in L^q(\sn)$, $q>1$, considered by several authors, including \cite{javirubio}. However, it does not include the condition $\Om \in L\textup{log}L(\sn)$ of Calder\'on and Zygmund. It is therefore natural to ask whether there exist examples of $\Om\notin L\textup{log}L(\sn)$ which satisfy (\ref{1+alpha}) for all $\al>0$. In this section we prove something more. We construct an example to show that there exist integrable functions on $\sn$ with mean value zero which are not in $H^1(\sn)$ but which satisfy (\ref{1+alpha}) for all $\al>0$. Then we show that there exist functions that satisfy the converse. We begin with the converse which is easier. The function $$ \Om (\theta )= \sum_{k=2}^\infty \f{e^{ik\theta}}{(\log k)^2} $$ belongs to $H^1(\sone)$ but it fails to satisfy condition (\ref{1+alpha}) for any $\al >0$. Both assertions follow from the fact that $\Om (\theta )$ behaves like $\theta^{-1} \log^{-2}(\theta^{-1})$ as $\theta\to 0+$ (See \cite{Zyg} p. 189). We now construct an $\Om\in L^1(\sone) \setminus H^1(\sone)$ with mean value zero which satisfies condition (\ref{1+alpha}) for all $\al>0$. The example presented below is unavoidably complicated. The problem is that such a function must have an infinite number of spikes which are sufficiently far away from each other and which are (barely) integrable and have mean value zero. At this point we think of $\sone$ as the interval $[0,1]$ via the identification \begin{equation}\lbl{ident} \wt{\Om}(x) = \Om(\cos (2\pi x), \sin (2\pi x)), \end{equation} where $\Om$ is defined on $\sone$ and $\wt{\Om}$ on $[0,1]$. It is not hard to see that under the identification given in (\ref{ident}), the condition $\Om\notin H^1(\sn)$ is equivalent to the fact that the Hilbert transform of $\wt{\Om}\chi_{[0,1]}$ is not in $L^1(\rone)$, and condition (\ref{1+alpha}) is equivalent to \begin{equation}\lbl{1000} \supl_{0\leq z\leq 1} \int_0^1|\wt{\Om}(x)| \ln^{1+\al}\f{1}{|x-z|} dx \leq C_\al<\infty. \end{equation} For a detailed justification of these facts see \cite{Stefanov}. Now let \begin{align*} a_n &= (\ln n)^{-1} & b_n & = e^{-\ga_n}\\ \ga_n &= e^{(\ln n)^{1/2}} & \de_n &= e^{-\ga_n^{1/4}}\\ d_n &= a_n+\de_n & c_n &= a_n-\de_n \\ \be_n &= 1-(\ln n + \tfrac32 \ln \ga_n)\ga_n^{-1}&\q &\q \end{align*} Heuristically speaking, $a_n$ is a sequence that decays slowly to zero, $c_n$ and $d_n$ are symmetric points about $a_n$ at distance $\de_n$, $(c_n-b_n,c_n)$ and $(d_n-b_n, d_n)$ are small intervals near $c_n$ and $d_n$ with length $b_n=e^{-\ga_n}$, where $(\ln n )^\ve << \ga_n<0$, and the $\be_n$'s are powers that converge to one at a rate $\sim \ga_n^{-1}$. It is easy to see that \begin{equation}\lbl{1008} \f{b_n^{1-\be_n}}{1-\be_n}=\f{1}{n \ga_n^{1/2}(\ln n +\f32 (\ln n )^{1/2})} \sim \f{1}{n \ga_n^{1/2}\ln n }, \end{equation} for $n$ large. Now let $$ \wt{\Om}(x)=\suml_{n=10^9}^{\infty}\left(\f{1}{|x-c_n|^{\ben}} \chi_{(c_n-b_n,c_n)}(x) - \f{1}{|x-d_n|^{\ben}}\chi_{(d_n-b_n,d_n)}(x)\right). $$ We first verify that condition (\ref{1000}) holds for all $\al>0$. The worst possible $z$'s in (\ref{1000}) are the singularities of $\wt{\Om}$, i.e. the points $z=c_n$, $d_n$, and $z=0$. By symmetry we consider only $z=c_n$ and $z=0$. Fix $N\ge 10^9$ and consider $z=c_N$. We have $$ \int_0^1 |{\wt{\Om}}(x)| \ln^{1+\al}\f{1}{|x-c_N|} dx \le I_1(N)+I_2(N)+I_3(N)+I_4(N), $$ where \begin{eqnarray*} I_1 (N)& =& \suml_{n\neq N} \intl_{c_n-b_n}^{c_n} \f{1}{|x-c_n|^{\be_n}} \ln^{1+\al}\f{1}{|x-c_N|} dx, \\ I_2(N) &=&\suml_{n\neq N}\intl_{d_n-b_n}^{d_n} \f{1}{|x-d_n|^{\be_n}} \ln^{1+\al}\f{1}{|x-c_N|} dx, \\ I_3 (N)&=& \intl_{c_N-b_N}^{c_N} \f{1}{|x-c_N|^{\be_N}} \ln^{1+\al}\f{1}{|x-c_N|} dx ,\\ I_4 (N)&=& \intl_{d_N-b_N}^{d_N} \f{1}{|x-d_N|^{\be_N}} \ln^{1+\al}\f{1}{|x-c_N|} dx . \end{eqnarray*} Observe that $I_2(N)\le CI_1(N)$ and that $I_4(N)\le I_3(N)$. Also, it is easy to see that $$ \supl_{N\ge 10^9}I_3(N) \leq C\supl_{N\ge 10^9} \f{b_N^{1-\be_N}}{1-\be_N}\ln^{1+\al}\f{1}{b_N}\leq C \supl_{N\ge 10^9}\f{\ga_N^{1+\al}}{N \ga_N^{1/2}\ln N}\leq C_\al. $$ To control $\supl_{N\ge 10^9}I_1(N)$ we need to show that \begin{equation}\lbl{777777777} \supl_{N\ge 10^9} \left[\suml_{n\neq N}\intl_{c_n-b_n}^{c_n} \f{1}{|x-c_n|^{\be_n}} \ln^{1+\al}\f{1}{|x-c_N|} dx\right]\leq C_\al. \end{equation} Using that $|x-c_N|\sim|c_n-c_N|\sim |a_n-a_N|$ in the integrand above and (\ref{1008}), we conclude that (\ref{777777777}) will be a consequence of \begin{equation}\lbl{1001} \supl_{N\ge 10^9} \left[\suml_{n\neq N}\f{b_n^{1-\be_n}}{1-\be_n} \ln^{1+\al}\f{1}{|a_n-a_N|}\right]\leq C_\al. \end{equation} We have two cases. For $n > N, \q |a_n-a_N|\geq|a_{N+1}-a_N|\geq (N \ln^2 N)^{-1}$ and therefore $$ \supl_{N\ge 10^9} \left[\suml_{n > N}\f{b_n^{1-\be_n}}{1-\be_n} \ln^{1+\al}\f{1}{|a_n-a_N|}\right]\leq C \supl_{N\ge 10^9} \suml_{n > N}\f{\ln^{1+\al} (N \ln^2 N)} {n \ga_n^{1/2}\ln n} \leq C_\al, $$ the latter being an easy consequence of the integral test. For $10^9 \le n\le N-1$ we have \begin{align*} &\suml_{n=10^9}^{ N-1}\f{b_n^{1-\be_n}}{1-\be_n} \ln^{1+\al}\!\f{1}{|a_n-a_N|} \\ \le C &\suml_{n=10^9}^{ N-1} \f{1}{n \ga_n^{1/2}\ln n}\ln^{1+\al}\! \f{1}{|(\ln n)^{-1} - (\ln N)^{-1}|}=A(N)+B(N), \end{align*} where $A(N)$ is the sum above of over the indices $10^9\le n< \ga_N $ and $B(N)$ is the sum over the the indices $\ga_N \le n\le N-1$. On $A(N)$ we have $|(\ln n )^{-1}- (\ln N)^{-1}|^{-1} \le C \ln n$, and thus $A(N)$ is clearly bounded independently of $N$. On $B(N)$ we have $|(\ln n)^{-1} - (\ln N)^{-1}|^{-1} \le C N (\ln N)^2$. Now estimate $\supl_{N\ge 10^9} B(N)$ by $$ C\supl_{N\ge 10^9}\ln^{1+\al}( N^2)\,\sum_{n\ge \ga_N} \f{1}{n\ga_n^{1/2}\ln n } \le C \supl_{N\ge 10^9}\f{\ln^{1+\al} (N^2)}{\ga_{\ga_N}^{1/3}} \le C, $$ where we used the integral test to deduce the first inequality above. This concludes the proof of (\ref{1000}) when $z=c_N$. Condition (\ref{1000}) for $z=0$ is is equivalent to the following inequality $$ \suml_{n=10^9}^{\nf}\f{\ln^{1+\al}(\ln n)}{n \ga_n^{1/2}\ln n }\leq C_\al, $$ which is certainly correct by the choice of our parameters. This proves that $\wt{\Om}$ satisfies condition (\ref{1000}) for all $\al>0$. We now prove that $\wt{\Om}$ is not in the Hardy space $H^1$. Extend $\wt{\Om}$ to be equal to zero outside the interval $[0,1]$. Let $H$ be the usual Hilbert transform. Fix $N\ge 10^9$ and $y \in [d_N,d_N+b_N]$. Obviously \begin{equation}\lbl{6000} \pi |(H\wt{\Om})(y)| \ge K_N(y)-L_N(y), \end{equation} where \begin{eqnarray*} K_N(y) &=& \left|\,\intl_{c_N-b_N}^{c_N} \f{1}{|x-c_N|^{\be_N}}\f{1}{x-y}dx- \intl_{d_N-b_N}^{d_N}\f{1}{|x-d_N|^{\be_N}}\f{1}{x-y}dx\right|, \lbl{10022} \\ L_N(y)&=& \suml_{n\neq N}\left|\, \intl_{c_n-b_n}^{c_n}\f{1}{|x-c_n|^{\be_n}}\f{1}{x-y}dx- \intl_{d_n-b_n}^{d_n}\f{1}{|x-d_n|^{\be_n}}\f{1}{x-y}dx\right|.\lbl{1002} \end{eqnarray*} We first prove that \begin{equation}\lbl{6001} \supl_{N\ge 10^9}\supl_{y \in [d_N,d_N+b_N]}L_N(y) \le C \end{equation} Indeed, \begin{eqnarray*} & &\intl_{c_n-b_n}^{c_n}\f{1}{|x-c_n|^{\be_n}}\f{1}{x-y}dx= \f{b_n^{1-\be_n}}{1-\be_n} \f{1}{(-b_n+c_n-y)} + \textup{smaller term} \\ & & \intl_{d_n-b_n}^{d_n}\f{1}{|x-d_n|^{\be_n}}\f{1}{x-y}dx= \f{b_n^{1-\be_n}}{1-\be_n} \f{1}{(-b_n+d_n-y)} + \textup{smaller term} \end{eqnarray*} where the smaller terms are bounded by $ C b_n^{1-\be_n}/{(1-\be_n)}$ and $\sum\limits_{n\ge 10^9} b_n^{1-\be_n}/{(1-\be_n)}\leq C$. Therefore \begin{eqnarray*} L_N (y)\le C \suml_{n\neq N} \f{b_n^{1-\be_n}}{1-\be_n}\f{|d_n-c_n|}{|a_n-a_N|^2}\leq C \suml_{n\neq N}\f{b_n^{1-\be_n}}{1-\be_n}\f{\de_n}{|a_n-a_N|^2} \end{eqnarray*} and thus it remains to prove that \begin{equation}\lbl{4000} \supl_{N\ge 10^9} \suml_{n\neq N}\f{\de_n}{n \ga_n^{1/2}\ln n } \f{1}{((\ln n)^{-1}-(\ln N)^{-1})^2} \leq C. \end{equation} The sum in (\ref{4000}) for $n>N$ is bounded by $$ \suml_{n> N}\f{\de_n}{n \ga_n^{1/2}\ln n} \f{1}{((\ln n)^{-1}-(\ln N)^{-1})^2}\leq N^2 \ln^4 N \suml_{n> N}\f{\de_n}{n \ga_n^{1/2}\ln n } \leq C, $$ uniformly in $N\ge 10^9$. Split the sum in (\ref{4000}) for $n