Note about square function estimates and uniformly rectifiable measures (Q2809200)

Let \(n\) and \(d\) be integers, with \(0<n<d\). Suppose \(S\) is a kernel satisfying the condition \(K_{n+\beta}(\mathbb R^d)\) for some \(\beta>0\). Let \(\mu\) be an \(n\)-UR (uniformly rectifiable) measure in \(\mathbb R^d\) and \(E=\text{spt} \mu\), where \(\text{diam}(E)=\infty\). Let \(T_{S,\mu}\) be the operator defined on a dense set as NEWLINE\[NEWLINE T_{S,\mu}f(x)=\int_ES(x,y)f(y)\,d\mu(y),\qquad x\in\mathbb R^d\setminus E. NEWLINE\]NEWLINE If a certain Carleson condition is satisfied, involving \(E\) and \(T1(x)=\int_LS(x,y)\,d\mathcal{H}^n(y)\) on \(n\)-planes \(L\subset\mathbb R^d\), then the following square function estimate holds for \(f\in L^2(\mu)\): NEWLINE\[NEWLINE \int_{\mathbb R^d\setminus E}|T_{S,\mu}f(x)|^2\text{dist}(x,E)^{2\beta-(d-n)}\,dx\lesssim\int_E|f(y)|^2\,d\mu(y). NEWLINE\]NEWLINENEWLINENEWLINEThis result for particular kernels and the measure \(\mu=\mathcal{H}^n|_E\), \(E\subset\mathbb R^{n+1}\) a given \(n\)-UR set, was already proved in [\textit{S. Hofmann} et al., Mem. Amer. Math. Soc. 245, No. 1159 (Published electronically in 2016)].