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\TopicFR{Analyse harmonique}
\TopicEN{Harmonic analysis}

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\title{On Sharpness of $L$ log $L$ Criterion for Weak Type $(1,1)$ boundedness of rough operators}


\alttitle{Sur la netteté du critère $L$ {\upshape log} $L$ pour les faibles de type $(1, 1)$ continuité des opérateurs rugueux}

\author{\firstname{Ankit} \lastname{Bhojak}}

\address{Department of Mathematics, Indian Institute of Science Education and Research Bhopal, Bhopal-462066, India.}

\email{ankitb@iiserb.ac.in}

\thanks{The author is supported by Science and Engineering Research Board, Department of Science and Technology, Govt. of India, through the scheme Core Research Grant, file no. CRG/2021/000230.} 

\CDRGrant{CRG/2021/000230}

\keywords{\kwd{Singular Integrals} \kwd{Orlicz spaces}}
\altkeywords{\kwd{Intégrales singulières} \kwd{espaces d'Orlicz}}

\subjclass{42B20}

\begin{abstract} 
  In this note, we show that the $\Omega\in L\log L$ hypothesis is the strongest size condition on a function $\Omega$ on the unit sphere with mean value zero, which ensures that the corresponding singular integral $T_\Omega$ defined~by
  \[ T_{\Omega}f(x)=p.v.\int\frac{1}{|x-y|^d}\Omega\Bigl(\frac{x-y}{|x-y|}\Big)f(y)\, \mathrm{d} y,\]
  maps $L^1(\mathbb{R}^d)$ to weak $L^1(\mathbb{R}^d)$, provided $T_\Omega$ is bounded in $L^2(\mathbb{R}^d)$.
\end{abstract}


\begin{altabstract} 
   Dans cette note, nous montrons que l'hypothèse $\Omega\in L \log L$ est la condition de taille la plus forte sur une fonction $\Omega$ sur la sphère unitaire de valeur moyenne zéro, qui assure que l'intégrale singulière correspondante $T_\Omega$ définie par \[ T_{\Omega}f(x)=p.v.\int\frac{1}{|x-y|^d}\Omega\Bigl(\frac{x-y}{|x-y|}\Big)f(y)\, \mathrm{d} y,\] est borné de $L^1(\mathbb R^d)$ dans $L^1(\mathbb R^d)$ faibles, à condition que $T_\Omega$ soit bornée dans $L^2(\mathbb R^d)$.
\end{altabstract}

\dateposted{2024-11-05}
\begin{document}


% Use the \maketitle command after the abstract
\maketitle

\section{Introduction}
Let   $\Omega\in L^1(\mathbb S^{d-1})$ with $\int_{\mathbb S^{d-1}}\Omega(\theta) \dd \theta=0$, where $\dd \theta$ is the surface measure on $\mathbb S^{d-1}$. Calder\'on and Zygmund \cite{CZ} considered the rough singular integrals defined as,
\[ 
T_{\Omega}f(x)=p.v.\int\frac{1}{|x-y|^d}\Omega\Bigl(\frac{x-y}{|x-y|}\Big)f(y)\, \dd y,
\]

They showed that  $\Omega\in L \log L(\mathbb S^{d-1})$  i.e. $\int_{\Sp}|\Omega(\theta)|\log(e+|\Omega(\theta)|)<\infty$ implies that $T_\Omega$ is bounded on $L^p(\R^d)$ for $1<p<\infty$. The singular integral $T_\Omega$ was shown to be of weak type $(1,1)$ using $TT^*$ arguments by Christ and Rubio de Francia \cite{CR} in dimension $d=2$ (and independently by Hofmann \cite{Hof}). The case of general dimensions was resolved by Seeger \cite{S1} by showing that $T_\Omega$ is of weak type $(1,1)$ for $\Omega\in L\log L(\mathbb S^{d-1})$.

\par
%The case $p=1$  was more elusive. % It is well known that $T_\Omega$ does not map $L^1(\R^d)$ to $L^1(\R^d)$. 
%For $\Omega\in {\rm Lip}(\Sp^{d-1})$,  $T_{\Omega}$ is a standard Calder\'{o}n-Zygmund operator hence bounded from $L^1(\R^d)$ to $L^{1,\infty}(\R^d)$.   In \cite{C} and \cite{CR}, Christ and Rubio de Francia showed that $M_\Omega$ is weak (1,1) for $\Omega\in L\log L(\Sp^{d-1})$. It was shown that $T_\Omega$ is weak $(1,1)$ in dimension two independently in \cite{CR} and \cite{Hof}. Finally, the case  for all dimensions  for $T_\Omega$ was resolved by Seeger \cite{S1} using microlocal analysis. However, the weak $(1,1)$ boundedness for $T_{\Omega}^*$ is an open problem even for $\Omega\in L^\infty(\Sp^{d-1})$, see \cite{S2}.\\

It is of interest to know other sufficient conditions on $\Omega$ that ensures the weak type boundedness of the operator $T_\Omega$. In fact, during the inception of this problem, Calder\'on and Zygmund \cite{CZ} showed that $\Omega\in L\log L$ is ``almost'' a necessary size condition for $T_\Omega$ to be $L^2$ bounded. If we drop the condition that $\Omega\in L\log L$, then Calder\'on and Zygmund \cite{CZ} pointed out that $T_\Omega$ may even fail to be $L^2$ bounded. Infact, the examples of $\Omega$ constructed in \cite{WZ} lies outside the space $L\log L$ and the corresponding operator $T_\Omega$ is unbounded on $L^2(\R^d)$. Later on, it was shown in \cite{Co,RW} that $\Omega\in H^1(\Sp)$ in the sense of Coifman and Weiss \cite{CW} implies $T_\Omega:L^p(\R^d)\to L^p(\R^d),\;1<p<\infty$. For a detailed proof, we refer to \cite{GS1, GS2, RR}. It is still an open problem if $T_\Omega$ is of weak type $(1,1)$ for $\Omega\in H^1(\Sp)$. A partial result assuming additional conditions on $H^1$-atoms in dimension two was obtained by Stefanov \cite{Sv}.

In \cite{GHR, H2}, it was shown that $T_\Omega$ distinguishes $L^p$ spaces by considering a suitable quantity based on the Fourier transform of $\Omega$. However, we would like to know if there exists an Orlicz space $X\supsetneq L\log L$ which would ensure that the $L^2$ boundedness of $T_\Omega$ implies the weak $(1,1)$ boundeness of $T_\Omega$ when $\Omega\in X$. We will show that no such $X$ exists. To state our main result, we introduce the Orlicz spaces and discuss some of its basic properties.
\par
\begin{definition}[\cite{BS}]
	Let $\Phi:[0,\infty)\to [0,\infty)$ be a Young's function i.e. there exists an increasing and left continuous function $\phi:[0,\infty)\to [0,\infty)$ with $\phi(0)=0$ such that $\Phi(t)=\int_0^t\phi(u)\, \dd u$. We say $\Omega\in \Phi(L)(\Sp)$, if the quantity
	\begin{equation}\label{O1}
		\|\Omega\|_{\Phi(L)}=\int_{\mathbb{S}^1}\Phi(|\Omega(\theta)|)\, \dd\theta
	\end{equation}
	is finite.	
\end{definition}
We note that the function $\frac{\Phi(t)}{t}$ is non-decreasing.

The quantity in \eqref{O1} fails to be a norm and $\Phi(L)(\Sp)$ is not even a linear space. To remedy that, we define the set
\[L^\Phi(\Sp)=\{\Omega:\Sp\to\R: \exists k>0 \text{ such that }\norm{k^{-1}\Omega}_{\Phi(L)}<\infty\}.\]
We define the Luxemburg norm as
\[\vertiii{\Omega}_{\Phi(L)}=\inf\{k>0: \|k^{-1}\Omega\|_{\Phi(L)}\leq 1\}.\]
It is well-known that the Orlicz space $L^\Phi(\Sp)$ forms a Banach space with this norm. For details, we refer to \cite{BS}.

\section{Main result}
We state our main result for dimension two but the same also holds for higher dimensions using the methods in \cite{WZ,GHR}. Our main result is the following,
\begin{theorem}\label{logoptimal}
	Let $\Phi$ be a Young's function such that
	\begin{align}\label{hypothesis}
		\Psi(t)=\frac{t\log(e+t)}{\Phi(t)}&\to\infty,\;\quad \text{as}\;t\to\infty,
	\end{align}
	Then there exists an $\Omega\in \Phi(L)(\mathbb S^1)$ such that $T_\Omega$ is $L^p$ bounded iff $p=2$. In particular, $T_\Omega$ does not map $L^1(\R^2)$ to $L^{1,\infty}(\R^2)$.
\end{theorem}
We note that using the geometric construction in \cite{H2}, one can obtain the above theorem for the space $L(\log L)^{1-\epsilon}(\Sp),\;0<\epsilon\leq 1$. To obtain the general case, we will employ the construction in \cite{GHR} with a suitable modification to ensure that the resulting $\Omega$ lies in the required Orlicz space.

The proof of \Cref{logoptimal} is contained in \Cref{sec:proof}. We will require the following notations throughout the paper. We say $X\lesssim Y$ if there exists an absolute constant $C>0$ (not depending on $X$ and $Y$) such that $X\leq CY$. Similarly, we say $X\gtrsim Y$ if there exists an absolute constant $C>0$ (not depending on $X$ and $Y$) such that $X\geq CY$. We say $X\sim Y$ if $X\lesssim Y$ and $X\gtrsim Y$.
\section{Proof of \Cref{logoptimal}}\label{sec:proof}
To prove \Cref{logoptimal}, we will construct a sequence of even functions $\{\Omega_n\}\in \Phi(L)$ with mean value zero such that the $L^p$ norm of $T_{\Omega_n}$ is large for $p\neq 2$ while having bounded $\Phi(L)-$Orlicz norm uniformly in $n$. Moreover, the quantity $\|m(\Omega_n)\|_{L^\infty}$ grows slowly in terms of $n$. More precisely, we will show that $\Omega_n$ satisfies,
\begin{align*}
	\norm{T_{\Omega_n}}_{L^p(\R^2)\to L^p(\R^2)}&\gtrsim n^{\abs{\frac{1}{2}-\frac{1}{p}}},
	\intertext{and} %\quad%\\
	\vertiii{\Omega_n}_{\Phi(L)(\mathbb S^1)}+\|m(\Omega_n)\|_{L^\infty(\mathbb S^1)}&\lesssim \log n.
\end{align*}
This will lead to a contradiction by an application of uniform boundedness principle. The proof is divided into four crucial steps described below.

%\begin{steps}
%	\item 
\begin{proof}[\hypertarget{Step 1}{Step 1}. The geometric construction of functions $w_k$ and $\Omega_n$.] We will construct even functions $w_k$ and a sequence of even functions $\Omega_n$ on the unit circle $\mathbb S^1$ with mean value zero in this step.
	
	We fix a large $N\in\N$. Let $n\in\N$ be a number depending on $N$ to be chosen later (see \eqref{choiceofn}). 
	
	Let $s_n\in\N$ and $t_1,t_2,\dots,t_{2n}\in\Z$ be such that,
	\begin{itemize}
		\item The numbers $t_k$ are in arithmetic progression, i.e. $t_{k+1}-t_{k}=t_{k}-t_{k-1}$.
		\item Let $x_k=(t_k,s_n)\in\R^2$. Then $x_k,\; k=1,\dots,2n,$ lies in the second quadrant between the lines $y$-axis and $y=-x$.
		\item $\abs{\frac{x_{k+1}}{|x_{k+1}|}-\frac{x_k}{|x_k|}}\sim \frac{1}{n}$.
	\end{itemize}
	(We note that the points $x_k=(-kn,10n^2),\; k=1,\dots,2n,$ satisfies the above properties.)
	
	We denote $\tilde{x}_k$ to be the point on $\mathbb S^1$ obtained by rotating the point $\frac{x_k}{|x_k|}$ by $\frac{\pi}{2}$ radians clockwise. We consider $I_k,\;k=1,\dots,2n$, to be the arc on $\mathbb S^1$ with centre $\tilde x_k$ and arc length $N^{-1}$ and denote $\mathfrak R_{\alpha}(I_k)$ to be the arc obtained by rotating $I_k$ by $\alpha$ radians counterclockwise. We note that the arcs $I_k,\;k=1,\dots,2n,$ are disjoint for our choice of $n$; we will justify this in \hyperlink{Step 3}{Step 3}.
	
	We define $w_k$ as
	\[w_k(\theta)=c_{I_k}(-\chi_{_{I_k}}(\theta)+\chi_{_{\mathfrak R_{\frac{\pi}{2}}(I_k)}}(\theta)-\chi_{_{\mathfrak R_{\pi}(I_k)}}(\theta)+\chi_{_{\mathfrak R_{\frac{3\pi}{2}}(I_k)}}(\theta)),\]
	where the constants $c_{I_k}$ are determined in \hyperlink{Step 2}{Step 2}.
	
	We now set 
	\begin{equation*}
		\Omega_n=\sum_{k=1}^{2n}(-1)^k\epsilon_{\left[\frac{k+1}{2}\right]}w_k,
	\end{equation*}
	where $[\;]$ denotes the integer part and the coefficients $\epsilon_{[.]}$ are as in \Cref{Riesz} in \hyperlink{Step 3}{Step 3}. It is easy to see that $w_k$ and $\Omega_n$ are even functions with mean value zero for all $k=1,\dots,2n$.
\let\qed\relax
\end{proof}
	
\begin{proof}[\hypertarget{Step 2}{Step 2} Auxiliary properties of $m(w_k)$.] In this step, we will obtain some basic estimates for the quantity $m(w_k)$ and the Fourier transform of $w_k$. We recall that the Fourier transform of the kernel in $T_\Omega$ for any even $\Omega$ with mean value zero is given by
	\[\widehat K_\Omega(\xi)=\int_{\Sp}\Omega(\theta)\log\frac{1}{|\langle\xi,\theta\rangle|}\, \dd\theta.\]
	We define the larger quantity $m(\Omega)$ which will be useful for our purpose.
	\[
\postdisplaypenalty1000000	
	m(\Omega)(\xi):=\int_{\Sp}|\Omega(\theta)|\log\frac{1}{|\langle\xi,\theta\rangle|}\, \dd\theta.
	\]
	Clearly, $|\widehat{K}_\Omega(\xi)|\leq m(\Omega)(\xi)$.
		
	We choose $c_{I_k}$ such that $m(w_k)(\frac{x_k}{|x_k|})=1$.
	
	It is not difficult to see that $c_{I_k}$ and $\widehat K_{w_k}(\frac{x_k}{|x_k|})$ are independent of $k$. Moreover, we have the following estimates,
	\begin{Proposition}\label{auxiliary}
		For $k=1,\dots,2n$, the following holds true,
		\begin{enumerate}
			\item There exists an absolute constant $c>0$ such that $ \frac{N}{c\log N}\leq c_{I_k}\leq \frac{cN}{\log N}$.\label{constantest}
			\item $1\lesssim\sup_x|\widehat K_{w_k}(x)|=\Bigl|\widehat K_{w_k}\Bigl(\frac{x_k}{|x_k|}\Bigr)\Bigr|\leq\sup_x m(w_k)(x)=1$\label{fourierest}.
			\item Let $J_k$ be the arc centered at the point $\frac{x_k}{|x_k|}$ and of length $\frac{1}{100n}$. Then for $x\in\mathbb S^1$ lying in second quadrant between the lines $y$-axis and $y=-x$ with $x\notin\bigcup_{i=0}^3 \mathfrak{R}_{\frac{i\pi}{2}}(J_k)$, we have
			\begin{equation}
				m(w_k)(x)\lesssim \frac{\log n}{\log N}.\label{formula7}
			\end{equation}
			\item For $1\leq k\leq n$ and $x\in\mathbb S^1$ lying in second quadrant between the lines $y$-axis and $y=-x$ with $x\notin \bigl(\bigcup_{i=0}^3 \mathfrak{R}_{\frac{i\pi}{2}}(J_{2k})\bigr)\cup\bigl(\bigcup_{i=0}^3 \mathfrak{R}_{\frac{i\pi}{2}}(J_{2k-1})\bigr)$, we have
			\begin{equation}\label{formula8}
				|\widehat K_{w_{2k}}(x)-\widehat K_{w_{2k-1}}(x)|\lesssim \left(n\log N\left|\frac{x}{|x|}-\frac{x_{2k}}{|x_{2k}|}\right|\right)^{-1}.
			\end{equation}
		\end{enumerate}
	\end{Proposition}
	\begin{proof}
		First, we observe that it is enough to prove \eqref{fourierest} for $x\in\mathbb S^1$ as $\int_0^{2\pi}w_k(\expe^{\irm\theta})\, \dd\theta=0$. Since, $w_k$ is even, we have that for any $0\leq\gamma<2\pi$,
		\begin{align}
			\begin{split}\label{fourierformula}
				\widehat{K}_{w_k}(\expe^{\irm\gamma})&=\int_0^{2\pi}w_k(\expe^{\irm\theta})\log\frac{1}{|\expe^{\irm\theta}\cdot \expe^{\irm\gamma}|}\, \dd\theta\\
				&=\int_0^{2\pi}w_k(\expe^{\irm\theta})\log\frac{1}{\lvert\cos(\theta-\gamma)|}\, \dd\theta\\
				&=c_{I_k}\left[-\int_{A_k}+\int_{A_k+\frac{\pi}{2}}-\int_{A_k+\pi}+\int_{A_k+\frac{3\pi}{2}}\right]\log \frac{1}{\lvert\cos(\theta-\gamma)|}\, \dd\theta\\
				&=-2c_{I_k}\int_{-\frac{|A_k|}{2}}^{\frac{|A_k|}{2}}\log |\tan(\theta+\theta_k-\gamma)|\, \dd\theta,
			\end{split}
		\end{align}
		where $\expe^{\irm\theta_k}=\frac{x_k}{|x_k|}$ and $A_{k}$ be the interval in $(0,\frac{\pi}{4})$ such that $I_{k}-\frac{x_{k}}{|x_{k}|}=\{\expe^{\irm\theta}:\; \theta\in A_{k}\}$. Similarly, we obtain that,
		\begin{align*}
			c_{I_k}^{-1}&\sim-\int_{-\frac{|A_k|}{2}}^{\frac{|A_k|}{2}}\log \lvert\sin\theta|\, \dd\theta\\
			&=-2\int_{0}^{\frac{|A_k|}{2}}\log \sin\theta\, \dd\theta\\
			&\sim-\int_0^{\frac{|A_k|}{2}}\log t\, \dd t\\
			&\sim|A_k|\lvert\log |A_k||\sim\frac{\log N}{N},
		\end{align*}
		where we used the fact that $\sin \theta\sim\theta$ for $\theta\in(0,\frac{\pi}{4})$. Thus, we obtain \eqref{constantest} and the estimate \eqref{fourierest} follows similarly from \eqref{fourierformula}.
		
	The estimate \eqref{formula7} follows from the fact that for $\gamma\in (2I_k)^c\cap(0,\frac{\pi}{4})$, we have
	\begin{equation*}
		|m(w_{I_k})(e^{i\gamma})|\lesssim\frac{\lvert\log|\gamma-\tilde x_k||}{\lvert\log |I_k||}.
	\end{equation*}
	Indeed, for $\theta\in I_k$, we have $|\gamma-\tilde x_{k}|<|\theta-\tilde x_{k}|+|\theta-\gamma|<\frac{|I_k|}{2}+|\theta-\gamma|<|\gamma-\tilde x_{k}|/2+|\theta-\gamma|$. Thus $\frac{|\tilde x_{k}-\gamma|}{2}<|\theta-\gamma|$ and it follows that
	\begin{align*}
		|m(w_{I_k})(\expe^{\irm\gamma})|&\lesssim-c_{I_k}\int_{I_k}\log\lvert\sin(\theta-\gamma)|\, \dd\theta\\
		&\leq c_{I_k}|I_k|\lvert\log\left\lvert\sin\left(\frac{|\gamma-\tilde x_k|}{2}\right)\right|\\
		&\lesssim \frac{\lvert\log|\gamma-\tilde x_k||}{\lvert\log |I_k||}.
	\end{align*}
	We now prove the estimate \eqref{formula8}. Let $\expe^{\irm\theta_{2k}}=\frac{x_{2k}}{|x_{2k}|},\; \expe^{\irm\gamma}=\frac{x}{|x|}$ and $A_{2k}$ be the interval in $(-\frac{\pi}{4},\frac{\pi}{4})$ such that $I_{2k}-\frac{x_{2k}}{|x_{2k}|}=\{\expe^{\irm\theta}:\; \theta\in A_{2k}\}$. By using mean value theorem twice and the fact that $|\theta_{2k}-\theta_{2k-1}|$ is small, we have
	\begin{align*}
		{\color{white}eq}|\widehat K_{w_{2k}}(x)-\widehat K_{w_{2k-1}}(x)|&\lesssim c_{I_{2k}}\int_{A_{2k}}\left(\log \frac{1}{\lvert\tan (\theta+\theta_{2k}-\gamma)|}-\log\frac{1}{\lvert\tan (\theta+\theta_{2k-1}-\gamma)|}\right)\, \dd\theta\\
		&\lesssim c_{I_{2k}}\int_{A_{2k}}\frac{\lvert\tan(\theta+\theta_{2k}-\gamma)-\tan(\theta+\theta_{2k-1}-\gamma)|}{\lvert\tan(\theta+\theta_{2k}-\gamma)|}\, \dd\theta\\
		&\lesssim c_{I_{2k}}\int_{A_{2k}} \frac{|\theta_{2k}-\theta_{2k-1}|}{|\theta+\theta_{2k}-\gamma|}\, \dd\theta\\
		&\lesssim \frac{c_{I_{2k}}}{n}\int_{A_{2k}}\frac{1}{|\gamma-\theta_{2k}|}\, \dd\theta\\
		&\lesssim\left(n\log N\left|\frac{x}{|x|}-\frac{x_{2k}}{|x_{2k}|}\right|\right)^{-1},
	\end{align*}
	where we have used $|\gamma-\theta_{2k}|\leq 2|\theta+\theta_{2k}-\gamma|$ and $\tan \theta\sim\theta$ away from odd multiples of $\frac{\pi}{2}$.
\end{proof}
\let\qed\relax
\end{proof}
	
\begin{proof}[\hypertarget{Step 3}{Step 3}. The calculation of the $\vertiii{\Omega_n}_{\Phi(L)(\mathbb S^1)}$ and the $L^p$-norms of $T_{\Omega_n}$.] In this step, we compute the $\Phi(L)-$Orlicz norm of $\Omega_n$ and the $L^p-$ norm of the corresponding operator $T_{\Omega_n}$. We begin by choosing $n$ as follows,
		\begin{equation}\label{choiceofn}
			n=\left[\frac{N}{16\Phi\left(\frac{cN}{\log N}\right)}\right]+1,
		\end{equation}
	where $c>0$ is as in \Cref{auxiliary}\eqref{constantest}.
	By hypothesis \eqref{hypothesis}, we have $n\to\infty$ as $N\to\infty$. Moreover, we have $N^{-1}\lesssim n^{-1}$ as $\Phi$ is an increasing function. This implies that the corresponding arcs $I_k,\;k=1,\dots,2n,$ are disjoint. Hence, we have 
	\begin{align*}
		\norm{\Omega_n}_{\Phi(L)(\mathbb S^1)}&= \sum_{k=1}^{2n}\sum_{l=0}^3\int_{A_k+\frac{l\pi}{2}}\Phi\left(|\epsilon_{\left[\frac{k+1}{2}\right]}c_{I_k}|\right)\, \dd\theta\\
		&\leq \frac{8n}{N}\Phi\left(\frac{cN}{\log N}\right)\\
		&\leq 1.
	\end{align*}
	Thus, by the definition of the Luxemburg norm $\vertiii{\,\cdot\,}_{\Phi(L)}$, we have 
	\begin{equation}\label{Orlicznorm} \vertiii{\Omega_n}_{\Phi(L)(\mathbb S^1)}\leq 1.\end{equation}
	
	 We estimate the quantity $\|m(\Omega_n)\|_{L^\infty(\mathbb S^1)}$ by employing \Cref{auxiliary}\eqref{formula7}. Indeed, we have
	\begin{equation}\label{formula9}
		\norm{m(\Omega_n)}_{L^\infty(\mathbb S^1)}\lesssim 1+\frac{n\log n}{\log N}\leq\frac{\log n}{8c} \frac{\frac{cN}{\log N}}{\Phi\left(\frac{cN}{\log N}\right)}\lesssim \log n,
	\end{equation}
	where we used that $\frac{\Phi(t)}{t}$ is a non-decreasing function in the last step.
	
	We now compute the $L^p$-norms of the corresponding operator $T_{\Omega_n}$. The space of $L^p$ multipliers $M^p(\T)$ and $M^p(\R^2)$ are defined as
	\begin{align*}
		M^p(\T)&=\left\{\textbf{a}=\{a_n\}\in l^\infty(\Z):\; T_{\textbf{a}}f(x)=\sum_{n\in\Z}a_n\widehat f(n)\expe^{2\pi \irm nx} \text{ is bounded on } L^p(\T)\right\},\\
		M^p(\R^2)&=\left\{\gamma\in L^\infty(\R^2):\; T_{\gamma}f(x)=\int_{\R^2}\gamma(\xi)\widehat f(\xi)\expe^{2\pi \irm x\cdot \xi}\, \dd\xi \text{ is bounded on } L^p(\R^2)\right\}.
	\end{align*}
	We define $\norm{\textbf{a}}_{M^p(\T)}=\norm{T_{\textbf{a}}}_{L^p(\T)\to L^p(\T)}$ and $\norm{\gamma}_{M^p(\R^2)}=\norm{T_{\gamma}}_{L^p(\R^2)\to L^p(\R^2)}$.
	
	We state two lemmas from \cite{GHR} that will be useful in estimating the $L^p$-norms of $T_{\Omega_n}$. The first lemma states that there exist a sequence of multipliers $\{\{\dots,0,\epsilon_1,\epsilon_2,\dots,\epsilon_n,0,\dots\}:\;n\in\N\}$ on $\mathbb T$ whose $L^p$-norm blows up as $n$ tends to infinity for $p\neq2$. This was achieved in \cite{GHR} by employing the fact that $\{\expe^{2\pi \irm kx},\;k\in\Z\}$ is not an unconditional basis for $L^p(\mathbb T),\;p\neq 2$. Moreover, the quantity $\norm{\{\dots,0,\epsilon_1,\epsilon_2,\dots,\epsilon_n,0,\dots\}}_{M^p(\mathbb T)}$ grows atleast of the order $n^{\abs{\frac{1}{2}-\frac{1}{p}}}$. To justify this growth, we invoke Theorem 1 from \cite{R},
	
	\emph{For $n\in\N$, there exists $\{\epsilon_k\}_{k=1}^n$ with $\epsilon_k=\pm1$ such that $\norm{\sum_{k=1}^n\epsilon_k \expe^{2\pi \irm kx}}_{L^\infty(\mathbb T)}\leq 5n^{\frac{1}{2}}$},
	
	and the well-known fact (Exercise 3.1.6 from \cite{G}) that the $L^p$-norm of the Dirichlet kernel satisfies the following estimate:
	\[\left \lVert\sum_{k=1}^n\ \expe^{2\pi \irm kx}\right\rVert_{L^p(\mathbb T)}\sim n^{1-\frac{1}{p}}\;\text{for}\;1<p<\infty.\]
	Thus, we have,
	\begin{align*}
		\norm{\{\dots,0,\epsilon_1,\epsilon_2,\dots,\epsilon_n,0,\dots\}}_{M^p(\T)}&\geq\frac{\norm[\Big]{\sum_{k=1}^n\epsilon_k^2 \expe^{2\pi \irm kx}}_{L^p(\mathbb T)}}{\norm[\Big]{\sum_{k=1}^n \epsilon_k \expe^{2\pi \irm kx}}_{L^p(\mathbb T)}}\\
		&\geq \frac{\norm[\Big]{\sum_{k=1}^n\epsilon_k^2 \expe^{2\pi \irm kx}}_{L^p(\mathbb T)}}{\norm[\Big]{\sum_{k=1}^n \epsilon_k \expe^{2\pi \irm kx}}_{L^\infty(\mathbb T)}}\gtrsim n^{\frac{1}{2}-\frac{1}{p}}.
	\end{align*}
The inequality $\norm{\{\dots,0,\epsilon_1,\epsilon_2,\dots,\epsilon_n,0,\dots\}}_{M^p(\T)}\gtrsim n^{\frac{1}{p}-\frac{1}{2}}$ follows from 
\[
\norm{\{\dots,0,\epsilon_1,\epsilon_2,\dots,\epsilon_n,0,\dots\}}_{M^p(\T)}=\norm{\{\dots,0,\epsilon_1,\epsilon_2,\dots,\epsilon_n,0,\dots\}}_{M^{\frac{p}{p-1}}(\T)}
\] 
for $1<p<\infty$.
	\begin{lemma}[\cite{GHR}]\label{Riesz}
		For $p\neq 2$ and fixed $n\in\N$, there exists finite sequences $\{a_k\}_{k=1}^{n}$ and $\{\epsilon_k\}_{k=1}^{n}$ (depending on $n$) with $\epsilon_k\in\{-1,1\}$ such that
		\[
\left\Vert		
		\sum_{k=1}^n\epsilon_k a_k \expe^{2\pi \irm kx}
\right\Vert		
		_{L^p(\mathbb T)}
		\geq c_p n^{\abs{\frac{1}{2}-\frac{1}{p}}}
\left\Vert\sum_{k=1}^n a_k \expe^{2\pi \irm kx}\right\Vert_{L^p(\mathbb T)},\]
		where $c_p>0$ depends only on $p$. Consequently, $\norm{\{\dots,0,\epsilon_1,\epsilon_2,\dots,\epsilon_n,0,\dots\}}_{M^p(\T)}\gtrsim n^{\abs{\frac{1}{2}-\frac{1}{p}}}$. Moreover, we can choose $\epsilon_k$ such that
		\[\norm{\{\dots,0,\epsilon_1,\epsilon_2,\dots,\epsilon_n,0,\dots\}}_{M^p(\T)}=\sup\left\{\norm{\{\dots,0,\delta_1,\delta_2,\dots,\delta_n,0,\dots\}}_{M^p(\T)}:\;|\delta_k|\leq 1\right\}.\]
	\end{lemma}
	The second lemma (stated below) along with an application of \Cref{Riesz} provides us with a sequence of multipliers on the plane such that their $L^p$-norm blows up as $n$ tends to infinity for $p\neq2$. This lemma is based on a classical transference result of de Leeuw \cite{deLeeuw}. For a proof of the lemma, we refer to \cite{GHR}.
	\begin{lemma}[\cite{GHR}]\label{Transference}
		Let $1<p<\infty$ and $\gamma\in M^p(\R^2)$ be continuous on an arithmetic progression $\{x_k\}_{k=1}^n$ in\;$\R^2$ (i.e. there exists vector $v\in\R^2$ such that $x_k-x_{k-1}=v$). Then there exists a constant $C_p>0$ such that
		\[\norm{\gamma}_{M^p(\R^2)}\geq C_p \norm{\{\dots,0,\gamma(x_1),\gamma(x_2),\dots,\gamma(x_n),0,\dots\}}_{M^p(\T)}.\]
	\end{lemma}
	Now we turn to the estimate of $L^p-$bounds of $T_{\Omega_n}$. We claim that
	\begin{equation}\label{formula10}
		\norm{T_{\Omega_n}}_{L^p(\R^2)\to L^p(\R^2)}\gtrsim n^{\abs{\frac{1}{2}-\frac{1}{p}}}.
	\end{equation}
	For $1\leq k\leq n$, we have
	\[\widehat K_{\Omega_n}(x_{2k})=(-1)^{2k}\widehat K_{w_{2k}}(x_{2k})\epsilon_k+\sum_{1\leq i\neq 2k\leq 2n}(-1)^i\epsilon_{\left[\frac{i+1}{2}\right]}\widehat K_{w_i}(x_{2k})=D\epsilon_k+\delta_k,\]
	where $D=\widehat K_{w_{2k}}(x_{2k})$ and $\delta_k=\sum_{1\leq i\neq 2k\leq 2n}(-1)^i\epsilon_{\left[\frac{i+1}{2}\right]}\widehat K_{w_i}(x_{2k})$.
	
	Using \Cref{auxiliary}\eqref{formula7} for the term $i=2k-1$ and \Cref{auxiliary}\eqref{formula8} for the remaining terms (in pair), we get 
	\[|\delta_k|\leq C\left(\frac{\log n}{\log N}+\frac{1}{\log N}\sum_{i=1}^{2n}\frac{1}{i}\right)\leq \frac{C'\log n}{\log N}\leq \frac{|D|}{4} \text{ (for large }n).\] 
	Hence, by the choice of \Cref{Riesz}, we have
	\[
	\frac{1}{2}\norm{\{\dots,0,\epsilon_1,\epsilon_2,\dots,\epsilon_{n},0,\dots\}}_{M^p(\T)}\geq
	\left\Vert
	\left\{\dots,0,\frac{\delta_1}{D},\frac{\delta_2}{D},\dots,\frac{\delta_{n}}{D},0,\dots\right\}\right\Vert_{M^p(\T)}.
	\]
	Since $\widehat K_{\Omega_n}(\theta)$ is a circular convolution of a $L^1(\mathbb S^1)$ and $L^\infty(\mathbb S^1)$, it is continuous at the points $x_{2k},k=1,\dots,n,$ and applying \Cref{Transference}, we have
	\begin{align*}
		\norm{T_{\Omega_n}}_{L^p(\R^2)\to L^p(\R^2)}
		&=\norm{\widehat K_{\Omega_n}}_{M^p(\R^2)}\\
		&\gtrsim \norm{\{\dots,0,\widehat K_{\Omega_n}(x_2),\widehat K_{\Omega_n}(x_4),\dots,\widehat K_{\Omega_n}(x_{2n}),0,\dots\}}_{M^p(\T)}\\
		&\gtrsim|D|\left(\norm{\{\dots,0,\epsilon_1,\epsilon_2,\dots,\epsilon_{n},0,\dots\}}_{M^p(\T)}-
\left\Vert
\left\{\dots,0,\frac{\delta_1}{D},\frac{\delta_2}{D},\dots,\frac{\delta_{n}}{D},0,\dots\right\}\right\Vert_{M^p(\T)}\right)\\
		&\geq \frac{|D|}{2} \norm{\{\dots,0,\epsilon_1,\epsilon_2,\dots,\epsilon_{n},0,\dots\}}_{M^p(\T)}\\
		&\gtrsim n^{\abs{\frac{1}{2}-\frac{1}{p}}},
	\end{align*}
	where we used \Cref{Riesz} in the last step.
\let\qed\relax
\end{proof}
	
\begin{proof}[\hypertarget{Step 4}{Step 4}. The uniform boundedness principle and the conclusion.] We conclude the proof by an application of uniform boundedness principle. Indeed, We define the space,
	\[\mathfrak{B}:=\left\{\Omega:\mathbb S^1\to\R \text{ is even}:\int\Omega=0 \text{ and } \|\Omega\|_{\mathfrak{B'}}=\vertiii{\Omega}_{\Phi(L)(\mathbb S^1)}+\|m(\Omega)\|_{L^\infty(\mathbb S^1)}<\infty\right\}.\]
	The space $\mathfrak{B}$ forms a Banach space.
	
	Fix $p\neq 2$. For $\mathfrak{F}=\{f\in L^p(\R^2): \|f\|_p=1\}$, we define a collection of operators $\varTheta_f:\mathfrak{B}\to L^p$ as $\varTheta_f(\Omega)=T_\Omega(f)$. Suppose we have
	\[\|T_\Omega\|_{L^p(\R^2)\to L^p(\R^2)}=\sup_{f\in\mathfrak{F'}}\|T_\Omega f\|_{L^p(\R^2)}<\infty,\;\forall\Omega\in\mathfrak{B}.\]
	Then by uniform boundedness principle, there exists $M>0$ such that
	\[\|T_\Omega\|_{L^p(\R^2)\to L^p(\R^2)}=\sup_{f\in\mathfrak{F}}\|\varTheta_f(\Omega)\|_{L^p(\R^2)}<M\|\Omega\|_\mathfrak{B},\]
	which along with \eqref{Orlicznorm}, \eqref{formula9} and \eqref{formula10} implies that
	\begin{align*}
		n^{\abs{\frac{1}{2}-\frac{1}{p}}}&\lesssim\norm{T_{\Omega_n}}_{L^p(\R^2)\to L^p(\R^2)}\\
		&\lesssim\vertiii{\Omega_n}_{\Phi(L)(\mathbb S^1)}+\|m(\Omega_n)\|_{L^\infty(\mathbb S^1)}\\
		&\lesssim \log n.
	\end{align*}
	This is a contradiction for large $n$ and $p\neq 2$ and that concludes the proof of \Cref{logoptimal}.
\let\qed\relax
\end{proof}
	
\section*{Acknowledgement}
The author would like to thank Prof. Parasar Mohanty and Prof. Adimurthi for various useful discussions regarding the problem. The author is grateful to the anonymous referee for various useful comments and improving the exposition of the paper.


\section*{Declaration of interests}
The authors do not work for, advise, own shares in, or receive funds from
any organization that could benefit from this article, and have declared no affiliations other than their research organizations.


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