Muckenhoupt-Wheeden conjectures in higher dimensions (Q2809353)

Let \(f\) be a locally integrable function on \(\mathbb{R}^n\). Then the Hardy-Littlewood maximal function \(Mf\) is defined by setting, for any \(x\in\mathbb{R}^n\), NEWLINE\[NEWLINEMf(x):=\sup_{x\in Q}\frac{1}{|Q|}\int_Q|f(y)|\,dy, NEWLINE\]NEWLINE where the supremum is taken over all cubes \(Q\subset\mathbb{R}^n\) with sides parallel to the coordinate axes in \(\mathbb{R}^n\). Moreover, for \(f\in C^{\infty}_c(\mathbb{R}^n)\), the classical Calderón-Zygmund integral \(Tf\) is defined, for any \(x\in\mathbb{R}^n\), by NEWLINE\[NEWLINETf(x):=\mathrm{p.v.}\int_{\mathbb{R}^n}K(x,y)f(y)\,dy, NEWLINE\]NEWLINE where the kernel \(K\) has the form NEWLINE\[NEWLINEK(x,y)=\frac{\Omega(x-y)}{|x-y|^n}NEWLINE\]NEWLINE with \(\Omega\) a homogeneous function of degree 0 such that \(\Omega\in C^1(\mathbb{S}^{n-1})\) and \(\int_{\mathbb{S}^{n-1}}\Omega(x)\,d\sigma_{n-1}(x)=0\).NEWLINENEWLINEIn this article, the authors showed that, for any \(N>0\), there exists a weight function \(w\), a function \(f\in L^1(Mw)\) and a constant \(\lambda\in(0,\infty)\) such that NEWLINE\[NEWLINEw(\{x\in\mathbb{R}^n:\;|Tf(x)|>\lambda\})\geq\frac{N}{\lambda}\int_{\mathbb{R}^n}|f(x)|Mw(x)\,dx. NEWLINE\]NEWLINE Let \(p\in(1,\infty)\) and \(p'\) be the conjugate exponent of \(p\). The authors constructed weights \(u\) and \(v\) such that \(M\,: L^p(u)\rightarrow L^p(v)\) and \(M\,: L^{p'}(v^{1-p'})\rightarrow L^{p'}(u^{1-p'})\) but there exists a function \(f\in L^p(u)\) such that \(\|Tf\|_{L^p(v)}=\infty\). Furthermore, the authors also constructed a weight \(u\) and \(f\in L^p(u)\) such that \(M\) is bounded on \(L^p(u)\) but \(\|Tf\|_{L^p(u)}=\infty\).