Variation for singular integrals on Lipschitz graphs: \(L^{p}\) and endpoint estimates (Q2849029)

The author studies the \(\rho\)-variation and oscillation for Calderón-Zygmund singular integral operators with odd kernel defined on Lipschitz graphs and with respect to the Hausdorff measure.NEWLINENEWLINEGiven integers \(n\) and \(d\) with \(1\leq {n}<d\), \(\varepsilon>0\), and a Radon measure \(\mu\) in \({\mathbb{R}}^d\), consider NEWLINE\[NEWLINET_{\varepsilon}\mu(x):=\int_{|x-y|>\varepsilon} K(x-y)\,d\mu(y) \quad (x\in {\mathbb{R}}^d), NEWLINE\]NEWLINE where the kernel \(K: {\mathbb{R}}^d \setminus \{0\} \to {\mathbb{C}}\) satisfies for some constant \(C > 0\) NEWLINE\[NEWLINEK(x)\leq \frac{C}{|x|^n}, \;\;|\partial_{x_i}K(x)|\leq \frac{C}{|x|^{n+1}} \;\text{ and } \;|\partial_{x_i}\partial_{x_j}K(x)|\leq \frac{C}{|x|^{n+2}}, NEWLINE\]NEWLINE for all \(1\leq{i,j}\leq {d}\) and \(x=(x_1,\dots,x_d)\in {\mathbb{R}}^d \setminus \{0\}\), \(C>0\) is some constant, and moreover \(K(-x)=-K(x)\) for all \(x\neq 0\) (i.e., \(K\) is odd). Given \(f\in{L^1(\mu)}\), denote by \(T_{\varepsilon}^{\mu}f:=T_{\varepsilon}(f\mu)\). Set \(\mathcal{T}\mu:=\{T_{\varepsilon}\mu\}_{\varepsilon>0}\) and \(\mathcal{T}^\mu{f}:=\{T_{\varepsilon}^\mu{f}\}_{\varepsilon>0}\).NEWLINENEWLINELet \(\mathcal{F}:=\{F_{\varepsilon}\}_{\varepsilon>0}\) be a family of functions defined on \({\mathbb{R}}^d\). Given \(\rho>0\), the \(\rho\)-variation of \(\mathcal{F}\) at \(x\in {\mathbb{R}}^d\) is defined by NEWLINE\[NEWLINE\mathcal{V}_{\rho}(\mathcal{F})(x):=\sup_{\{\varepsilon_m\}} \bigg(\sum_{m\in{\mathbb{Z}}} |F_{\varepsilon_{m+1}}(x) - F_{\varepsilon_{m}}(x)|^{\rho} \bigg)^{1/\rho}, NEWLINE\]NEWLINE where the supremum is taken over all decreasing sequences \(\{\varepsilon_m\}_{m\in\mathbb{Z}}\subset (0,\infty)\).NEWLINENEWLINESet NEWLINE\[NEWLINE({\mathcal{V}}_{\rho}\circ{\mathcal{T}}){\mu}(x):= {\mathcal{V}}_{\rho}({\mathcal{T}}{\mu})(x) \quad {\text{and}} \quad ({\mathcal{V}}_{\rho}\circ{\mathcal{T}}^{\mu})f(x):= {\mathcal{V}}_{\rho}({\mathcal{T}}^{\mu}f)(x). NEWLINE\]NEWLINE In this paper, the author investigates the \(L^p\) and the endpoint estimates for \({\mathcal{V}}_{\rho}\circ\mathcal{T}\) and \({\mathcal{V}}_{\rho}\circ{\mathcal{T}}^{\mu}\). The two main results are as follows.NEWLINENEWLINELet \(\rho>2\) and \(\Gamma\subset {\mathbb{R}}^d\) be an \(n\)-dimensional Lipschitz graph with slope strictly less than \(1\) and set \(\mu:={\mathcal{H}}_{\Gamma}^n\), the \(n\)-dimensional Hausdorff measure restricted to \(\Gamma\). One of the main results states that \({\mathcal{V}}_{\rho}\circ{\mathcal{T}}^{\mu}\) is a bounded operator in \(L^p(\mu)\) for \(1<p<\infty\), from \(L^1(\mu)\) to \(L^{1,\infty}(\mu)\), and from \(L^{\infty}(\mu)\) to BMO\((\mu)\). The other one says \({\mathcal{V}}_{\rho}\circ{\mathcal{T}}\) is bounded from \(M({\mathbb{R}}^d)\) to \(L^{1,\infty}(\mu)\), where \(M({\mathbb{R}}^d)\) is the space of finite complex Radon measures on \({\mathbb{R}}^d\) equipped with the norm given by the variation of measures.