Invariance and stability of almost-orthogonal systems (Q2787979)

Set \(z_Q=(x_Q,\ell(Q))\), where \(Q\) is a dyadic cube in \(\mathbb{R}^d\) (i.e., a product of dyadic intervals \(I=[k/2^\ell, (k+1)/2^\ell)\), \(k,\ell\in\mathbb{Z}\)) with center \(x_Q\) and sidelength \(\ell(Q)\). For a sequence of functions \(\varphi^{(Q)}\) indexed by the dyadic cubes \(\mathcal{D}\), let \(\varphi_{z_Q}^{(Q)}=\varphi^{(Q)}((x-x_Q)/\ell(Q))\). If \(\varphi^{(Q)}\) is supported in \(B(0,1)\) then \(\varphi_{z_Q}^{(Q)}\) is supported in \(B(x_Q,\ell_Q)\). The rationale for this notation is to provide a perturbation result for families \(\Phi=\{\varphi^{(Q)}\}_{Q\in\mathcal{D}}\) of functions in \(\mathcal{C}^\alpha\) defined by the following conditions: (i) \(\varphi^{(Q)}\) is supported in \(B(0,1)\), (ii) \(|\varphi^{(Q)}(x)-\varphi^{(Q)}(y)|\leq |x-y|^\alpha\), and (iii) \(\int \varphi^{(Q)}(x)\, dx=0\). One says that the family \(\{\varphi_{z_Q}^{(Q)}\}\) is almost orthogonal with respect to \(\nu\) if there is a constant \(C>0\) such that for any finite subfamily \(\mathcal{F}\subset\mathcal{D}\) one has NEWLINE\[NEWLINE\int \Bigl|\sum_{Q\in\mathcal{F}} \lambda_Q \frac{\varphi_{z_Q}^{(Q)}}{\nu(Q)^{1/2}}\, \Bigr|^2 d\nu\leq C\sum_{Q\in\mathcal{F}} |\lambda_Q|^2\, . NEWLINE\]NEWLINE The first main theorem (Theorem 1) states that if \(\mu\) is a Muckenhoupt \(A_\infty\)-weight, then \(\{\varphi_{z_Q}^{(Q)}\}\) is almost orthogonal with respect to \(\mu\) if and only if \(\{\varphi_{z_Q}^{(Q)}\}\) is almost orthogonal with respect to the Lebesgue measure. Theorem 2 states that if \(\mu\) is doubling, then every family \(\{\varphi_{z_Q}^{(Q)}\}\subset\mathcal{C}^\alpha\) (\(\alpha\in (0,1]\)) is almost orthogonal with respect to \(\mu\) if and only if \(\mu\in A_\infty\). In fact, Theorems 9 and 10 extend this mutual almost orthogonality on \(C^\alpha\) families to an equivalence with the pair of doubling measures being mutually \(A_\infty\). One can perturb families \(\{\varphi_{z_Q}^{(Q)}\}\) by defining \(\widetilde\varphi^{(Q)}=\varphi^{(Q)}_{\zeta_Q}\), where \(\zeta_Q=(y_Q,r_Q)\) is such that the Euclidean distance \(|(y_Q,r_Q-1)|\leq\eta<1/2\) for \(\eta\) independent of \(Q\). For a pair of \(\mathcal{C}^\alpha\) families \(\Phi,\Psi\) and perturbations \(\{\zeta_Q\}\), define \(T_{\widetilde\Phi,\widetilde\Psi}^\mu(f)(x)=\sum_{Q\in\mathbb{D}}(\langle f,\widetilde\varphi_{z_Q}^{(Q)}\rangle/\mu(Q))\widetilde\psi_{z_Q}^{(Q)}\). Theorem 3 states that if \(\mu\in A_\infty\), then for every \(\tau\in (0,\alpha)\) and \(1<p<\infty\) there is a constant \(C>0\) such that NEWLINE\[NEWLINE\| T_{\Phi,\Psi}(f)-T_{\widetilde\Phi,\widetilde\Psi}(f)\|_{L^p_\mu}\leq C\eta^\alpha \|f\|_{L^p_\mu}\, . NEWLINE\]NEWLINE Technical tools include an equivalence between \(\mu\)-almost orthogonality and a Carleson inequality NEWLINE\[NEWLINE\sum_{Q'\subset Q}\Bigl|\frac{1}{\mu(Q')}\int_{Q'} \varphi_{z_Q}^{(Q')} d\mu\Bigr|^2\mu(Q')\leq C\mu(Q) NEWLINE\]NEWLINE and a statement that, if for some sequence \(\{c_Q\}\) one has the uniform inequality \(\frac{1}{\nu(Q)}\sum_{Q'\subset Q} c_{Q'}\nu(Q')\leq C_\nu\) for {\textit{some}} \(A_\infty\) measure \(\nu\) then one has a corresponding inequality for the same \(\{c_Q\}\) with any \(A_\infty\) measure.