Let $chi$ a non negative continuous function whose support is included in $)1,1($ with $intchi=1$. Let $varphi$ a function in $L^{p’}$ non negative with compact support in $mathbb{R}setminus{0}$. Let $q,q’in)1,+infty($ such that $frac1q+frac1{q’}=1$.
 Show that $$limlimits_{ntoinfty}nint_{mathbb{R}^2}chi(nt’)varphi(t)frac{1}{vert tt’vert^{frac{1}{q}}}dt’dt=int_{mathbb{R}}varphi(t)frac{1}{vert tvert^{frac{1}{q}}}dt$$

Deduce that there does not exist a positive constant $C$ such that
$$forall(f,g)in L^1times L_omega^q,~Vert fstar gVert_{L^q}leqslant CVert fVert_{L^1}Vert gVert_{L_omega^q}$$
where we denote $L_omega^p$ the weak $L^q$ space endowed with the norm $Vert gVert_{L_omega^q}=suplimits_{lambda>0}lambda^qmu((vert gvert >lambda))$.

Deduce that there does not exist a positive constant $C$ such that
$$forall(f,g)in L^{q’}times L_omega^q,~Vert fstar gVert_{L^q}leqslant CVert fVert_{L^{q’}}Vert gVert_{L_omega^q}$$
My work
 Suppose $Supp(varphi)subset(K,varepsilon)cup(varepsilon,K)$ for $varepsilon,K>0$. Let $chi_n$ defined as $chi_n(x)=nchi(nx)$. Then $Supp(chi_n)subset)frac1n,frac1n($. $chi_n$ is an approximation of unity derived from $chi$. Let $n>frac2varepsilon$ so that for any $(t,t’)in Supp(phi)times Supp(chi_n)$, $vert tt’vert>fracvarepsilon2$. Note that if $mgeqslant n$, this is still true. Define $g:tmapstofrac{1}{vert tvert^{frac1q}}$ for $tin(Kfracvarepsilon2,fracvarepsilon2)cup(fracvarepsilon2,K+fracvarepsilon2)$ and $0$ everywhere else. Then $g$ is in every $L^p(mathbb{R})$, $1leqslant pleqslant +infty$.
In this way, the integral has a sense and we have:
$$nint_{mathbb{R}^2}chi(nt’)varphi(t)frac{1}{vert tt’vert^{frac{1}{q}}}dt’dt=int_{mathbb{R}}varphi(t)mathbf{1}_{(K,varepsilon)cup(varepsilon,K)}(t)int_{mathbb{R}}chi_n(t)mathbf{1}_{(fracvarepsilon2,fracvarepsilon2)}(t’)frac1{vert tt’vert^{frac1q}}dt’dt=int_{mathbb{R}}varphi(t)int_{mathbb{R}}chi_n(t)g(tt’)dt’dt=int_{mathbb{R}}varphi(t)(chi_nstar g)(t)dt$$
Because $gin L^1(mathbb{R})$ and $chi_n$ is an approximation of unity, $chi_nstar gxrightarrow(nto+infty){L^1}g$. This means that, up to an extraction, $chi_nstar gxrightarrow(nto+infty){} g$ almost everywhere.
In this way, we can apply the dominated convergence theorem: $varphi(t)(chi_nstar g)(t)$ converges almost everywhere to $varphi(t) g(t)$. And for almost all $t$, we have thanks to the Young inequality:
$$vertvarphi(t)(chi_nstar g)(t)vertleqslantvertvarphi(t)vertVertchi_nstar gVert_{infty}leqslantvertvarphi(t)vertunderbrace{Vertchi_nVert_{L^1}}_{=1}Vert gVert_{L^infty}$$
and $vertvarphivertVert gVert_{L^infty}$ is integrable thanks to the Hölder inequality:
$$int_{R}vertvarphi(t)Vert gVert_{L^infty}leqslantVert gVert_{L^infty}VertvarphiVert_{L^{q’}}Bigl(mu(Supp(varphi)Bigr)^{frac1q}$$
So we get the wanted convergence.
 First we show that $vertcdotvert^{frac{1}q}in L_omega^q(mathbb{R})$. Indeed, let $lambda>0$.
$$lambda^qmu((vertcdotvert>lambda))=lambda^qmu((vertcdotvert<frac{1}{lambda^q}))=2$$
Then, suppose this constant $C$ exists. Then from the first question we have $chi_nstar gin L^q$ as $chi_nin L^1$ and
$$int_{mathbb{R}}varphi(t)(gstarchi_n)(t)dtleqslant CVertvarphi Vert_{L^{q’}}Vert gVert_{L_omega^q}$$ which means that, when $n$ goes to infinity,
$$int_mathbb{R}varphi(t)g(t)dtleqslant CVertvarphi Vert_{L^{q’}}Vert gVert_{L_omega^q}$$
I tried using some well thought $varphi$ to find a contradiction but could not find one. For example, I tried characteristic functions, but it does not lead to a contradiction. Any hints ?