Local \(T_b\) theorem with \(L^2\) testing conditions and general measures: Calderón-Zygmund operators (Q2801754)

A local \(T_b\) theorem is proved for Calderón-Zygmund operators (CZOs) with respect to a measure \(\mu \) on \(\mathbb{R}^n\) that is \(m\)-dimensional, that is, \(\mu(B(x,r))\leq C r^m\) for all \(x\in\mathbb{R}^n\) and all \(r>0\), but need not satisfy a doubling condition. Let \(T\) be a Calderón-Zygmund operator, that is, \(T\) has a kernel \(K(x,y)\) defined for \(x\neq y\) satisfying \(|K(x,y)\leq C|x-y|^{-m}\), \(|K(x,y)-K(x',y)|\leq C|x-x'||x-y|^{-m-\alpha}\) when \(|x-x'|\geq 2|x-y|\) with a corresponding inequality for \(K'(x,y)=K(y,x)\), and with \(T\) defined by \(Tf(x)=\int_{\mathbb{R}^n} K(x,y)\, f(y)\, dy\) when \(x\notin\mathrm{supp } f\). The main result, Theorem 1.1, supposes that \(T:L^2(\mu)\to L^2(\mu)\) is a bounded CZO with adjoint \(T^\ast\). It is assumed that to each cube \(Q\), functions \(b_Q^T\) and \(b_Q^{T^\ast}\) (each supported in \(Q\)) are associated, whose \(L^1\) and \(L^2\) averages are bounded below, that is, with \(\langle f\rangle_Q=\mu(Q)^{-1}\int_Q f d\mu\), one has \(|\langle b_Q^T\rangle |\geq C\), \(\langle (b_Q^T)^2\rangle \geq C\), \(|\langle b_Q^{T^\ast}\rangle |\geq C\) and \(\langle (b_Q^{T^\ast})^2\rangle \geq C\). These conditions are referred to as \(L^2\)-type local testing conditions. The theorem concludes that the operator norm of \(T\) then depends only on the constants in the testing conditions.NEWLINENEWLINE NEWLINEA major component of the proof is a bound for non-homogeneous versions of so-called twisted martingale difference operators defined by \textit{P. Auscher} and \textit{E. Routin} [J. Geom. Anal. 23, No. 1, 303--374 (2013; Zbl 1266.42028); erratum ibid. 23, No. 1, 375--376 (2013)]. One begins with a standard dyadic grid \(\mathcal{D}\) of dyadic cubes and defines \(\mathcal{D}^T\) beginning with random squares \(Q^\ast\) and \(R^\ast\) at a fixed scale, defines \(\mathcal{F}^1_{Q^\ast}\) to consist of al cubes in \(\mathcal{D}^T\) contained in \(Q^\ast\) that are maximal subject to satisfying at least one of the conditions \(|\langle b_{Q^\ast}^T\rangle |<1/2\), \(\langle (M_\mu b_{Q^\ast}^T)^2 \rangle_Q>16 A^2\|M_\mu\|^2\) (where \(M_\mu\) is the centered maximal function) or \(\langle | T b_{Q^\ast}^T|^2 \rangle_Q>16 AB\) for certain fixed \(A,B\). One defines a next generation \(\mathcal{F}^2_{Q^\ast}\) by imposing this maximality criterion on cubes in \(\mathcal{D}^T\) that are contained in a fixed cube in \(\mathcal{F}^1_{Q^\ast}\) and defines subsequent generations accordingly. Finally, one defines \(\mathcal{F}_{Q^\ast}\) to be the union of \(\mathcal{F}^j_{Q^\ast}\). To a cube \(Q\in\mathcal{D}^T\) (\(Q\subset Q^\ast\) with children denoted \(\mathrm{ch}(Q)\)) one associates the twisted martingale difference operator NEWLINE\[NEWLINE\Delta_Q h=\sum_{Q'\in {\mathrm{ch}(Q)}} \Bigl(\frac{\langle h\rangle_{Q'}}{\langle b^T_{(Q')^a}\rangle_{Q'}} b^T_{(Q')^a}- \frac{\langle h\rangle_{Q}}{\langle b^T_{(Q)^a}\rangle_{Q}} b^T_{(Q)^a} \Bigr),NEWLINE\]NEWLINE where \(Q^a\) is the minimal cube in \(\mathcal{F}_{Q^\ast}\) such that \(Q\subset Q^a\). The main technical tool in proving Theorem 1.1 is Proposition 2.4, which states that if \(F\in \mathcal{F}_{Q^\ast}\), \(h\in L^2(\mu)\) and if \(\epsilon_Q\) (\(Q\in \mathcal{D}^T\)) are constants such that \(|\epsilon_Q|<1\), then one has the inequality NEWLINE\[NEWLINE\|\sum_{Q\in\mathcal{D}^T,Q^a=F } \epsilon_Q \Delta_Q h\|_{L^2(\mu)}^2\leq C\|h\|_{L^2(\mu)}^2\, . NEWLINE\]NEWLINE The authors bound \(\|T\|\) by means of uniform bounds for pairings \(|\langle Tf, g\rangle |\). To do so, they express \(f=\sum_{Q\in\mathcal{D}^T} \Delta_Q f+\langle f\rangle_{Q^\ast} b_{Q^\ast}^T\) with a corresponding expression for \(g\), and rely on splitting the sums of the corresponding terms in the inner products into good, bad and diagonal parts, and on a variety of techniques to estimate each of these pieces.