\documentclass[CRMATH,Unicode,biblatex,XML,published]{cedram} \TopicFR{Analyse harmonique} \TopicEN{Harmonic analysis} \providecommand\noopsort[1]{} \addbibresource{CRMATH_Bhojak_20230663.bib} %% Authors may use other style files (e.g. to create figures), as long as %% they do not alter the layout of the article. % \usepackage[all]{xypic} \usepackage{amsmath} \usepackage{amssymb} \usepackage{bbm} \usepackage{pdfsync} \usepackage{hyperref} \usepackage{mathtools} \usepackage{xcolor} \usepackage{xparse} \usepackage{enumitem} \newlist{steps}{enumerate}{1} \setlist[steps, 1]{label = Step \arabic*:} \usepackage[capitalize]{cleveref} \crefname{coro}{Corollary}{Corollaries} \crefname{defi}{Definition}{Definitions} \crefname{lemm}{Lemma}{Lemmas} \crefname{prop}{Proposition}{Propositions} \crefname{rema}{Remark}{Remarks} \crefname{exam}{Example}{Examples} \crefname{nota}{Notation}{Notations} \crefname{theo}{Theorem}{Theorems} \crefname{conj}{Conjecture}{Conjectures} \crefname{ques}{Question}{Questions} \crefname{table}{Table}{Tables} \crefname{figure}{Figure}{Figure} \crefname{cdrenonce}{Observation}{Observations} \newcommand{\irm}{\mathrm{i\,}} \newcommand{\dd}{\mathrm{d}} \DeclareMathOperator{\expe}{e} %% Insert here your own symbols, as the following ones: \newcommand{\R}{{\mathbb R}} % Field of real numbers \newcommand{\N}{{\mathbb N}} \newcommand{\T}{{\mathbb T}} \newcommand{\Sp}{{\mathbb S}^1} \newcommand{\Z}{{\mathbb Z}} % Ring of integer numbers \newcommand{\Toh}{{ T_{\Omega, h}}} \DeclarePairedDelimiter\abs{\lvert}{\rvert} \DeclarePairedDelimiter\norm{\lVert}{\rVert} \newcommand{\vertiii}[1]{{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert #1 \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}} \DeclarePairedDelimiterX{\set}[1]{\{}{\}}{\setargs{#1}} \NewDocumentCommand{\setargs}{>{\SplitArgument{1}{;}}m} {\setargsaux#1} \NewDocumentCommand{\setargsaux}{mm} {\IfNoValueTF{#2}{#1} {#1\,\delimsize|\,\mathopen{}#2}}%{#1\:;\:#2} \makeatletter \let\save@mathaccent\mathaccent \newcommand*\if@single[3]{% \setbox0\hbox{${\mathaccent"0362{#1}}^H$}% \setbox2\hbox{${\mathaccent"0362{\kern0pt#1}}^H$}% \ifdim\ht0=\ht2 #3\else #2\fi } %The bar will be moved to the right by a half of \macc@kerna, which is computed by amsmath: \newcommand*\rel@kern[1]{\kern#1\dimexpr\macc@kerna} %If there's a superscript following the bar, then no negative kern may follow the bar; %an additional {} makes sure that the superscript is high enough in this case: \newcommand*\widebar[1]{\@ifnextchar^{{\wide@bar{#1}{0}}}{\wide@bar{#1}{1}}} %Use a separate algorithm for single symbols: \newcommand*\wide@bar[2]{\if@single{#1}{\wide@bar@{#1}{#2}{1}}{\wide@bar@{#1}{#2}{2}}} \newcommand*\wide@bar@[3]{% \begingroup \def\mathaccent##1##2{% %Enable nesting of accents: \let\mathaccent\save@mathaccent %If there's more than a single symbol, use the first character instead (see below): \if#32 \let\macc@nucleus\first@char \fi %Determine the italic correction: \setbox\z@\hbox{$\macc@style{\macc@nucleus}_{}$}% \setbox\tw@\hbox{$\macc@style{\macc@nucleus}{}_{}$}% \dimen@\wd\tw@ \advance\dimen@-\wd\z@ %Now \dimen@ is the italic correction of the symbol. \divide\dimen@ 3 \@tempdima\wd\tw@ \advance\@tempdima-\scriptspace %Now \@tempdima is the width of the symbol. \divide\@tempdima 10 \advance\dimen@-\@tempdima %Now \dimen@ = (italic correction / 3) - (Breite / 10) \ifdim\dimen@>\z@ \dimen@0pt\fi %The bar will be shortened in the case \dimen@<0 ! 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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 $10 \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}\;10$ 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 $10$ 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)}