On the Carleson duality (Q363495)

Let \(c_0>1\) and \(c_1>0\). A region \(W(t,x)\) in \(\mathbb R_+^{1+n}\) defined by NEWLINE\[NEWLINEW(t,x) =\{(s,y)\in \mathbb R_+^{1+n};\, |y-x|<c_1 t, 1/c_0<s/t<c_0 \}NEWLINE\]NEWLINE is said to be a Whitney region around \((t,x)\), and NEWLINE\[NEWLINE(W_qf)(t,x):=|W(t,x)|^{-1}\| f\|_{L^q(W(t,x))}NEWLINE\]NEWLINE is called \(L^q\) Whitney average around \((t,x)\). One defines the Calreson functionals NEWLINE\[NEWLINEC^rg(x):=\sup_{Q:\text{cube }\ni x}\bigl(|Q|^{-1}\int_{(0,\ell(Q))\times Q} |g(t,x)|^rdtdx\bigr)^{1/r},NEWLINE\]NEWLINE for \(1\leq r<\infty\). One of the authors' main theorems is the following: Let \(1/p+1/\tilde p=1/q+1/\tilde q=1/r\) with \(1\leq r\leq p<\infty\), \(r\leq q\leq\infty\). Then there is a constant \(0<C<\infty\) such that NEWLINE\[NEWLINE\|fg\|_{L^r(\mathbb R_+^{1+n})}\leq C\|N_*(W_qf)\|_{L^p(\mathbb R^n)} \|C^r(W_{\tilde q}g)\|_{L^{\tilde p}(\mathbb R^n)},NEWLINE\]NEWLINE NEWLINE\[NEWLINE\|N_*(W_qf)\|_{L^p(\mathbb R^n)}\leq C\sup_{\|C^r(W_{\tilde q}g)\|_{L^{\tilde p}(\mathbb R^n)}=1} \|fg\|_{L^r(\mathbb R_+^{1+n})},NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\|C^r(W_{\tilde q}g)\|_{L^{\tilde p}(\mathbb R^n)} \leq C\sup_{\|N_*(W_qf)\|_{L^p(\mathbb R^n)}=1}\|fg\|_{L^r(\mathbb R_+^{1+n})},NEWLINE\]NEWLINE where \(N_*f\) is the non-tangential maximal function of \(f\).NEWLINENEWLINEIn the case \(r=1\), this gives a concrete example of the Carleson duality in the following sense. \`\` Let \(X\) and \(Y\) be two Banach spaces. By a \textit{duality} \(\langle X,Y \rangle\), we mean a bilinear map \(X\times Y\ni (f,g) \to \langle f,g\rangle\in\mathbb R\) and a constant \(0<C<\infty\) such that \(\displaystyle|\langle f,g\rangle|\leq C\|f\|_X\|g\|_Y\), \(\|f\|_X\leq C\sup_{\|g\|_Y=1}|\langle f,g\rangle|\), and \(\|g\|_Y\leq C\sup_{\|f\|_X=1}|\langle f,g\rangle|\).'' This comes from the famous Carleson inequality in terms of Carleson measures on \(\mathbb R_+^{1+n}\). \(\|N_*(W_2f)\|_{L^2(\mathbb R^n)}\) was introduced by Kenig and Pipher as a tool for solving the Neumann problem for divergence-form equations. The authors answer questions which arose from recent studies of boundary value problems by Auscher and Rosén.NEWLINENEWLINETo show the above theorem, the authors first establish a duality theorem and its consequences in the dyadic case. They also discuss the relation between the above theorem and Coifman-Meyer-Stein tent spaces.