Cones, rectifiability, and singular integral operators. (English) Zbl 1502.42009
Let \(\mu\) be a Radon measure on \(\mathbb{R}^d\), \(p\in[1,\infty)\), and \(n\in(0,d)\). In this paper, the author studies the conical energy \(\mathcal{E}_{\mu,p}(x,V,\alpha)\), which quantifies the portion of \(\mu\) lying in the cone with the vertex \(x\in\mathbb{R}^d\), the direction \(G(d,n-d)\), and the aperture \(\alpha\in(0,1)\). Using these energies, the author characterizes rectifiability and the big pieces of Lipschitz graphs property. Moreover, if further assuming that \(\mu\) has polynomial growth, the author gives a sufficient condition for \(L^2(\mu)\)-boundedness of singular integral operators with smooth odd kernels of convolution type.
Reviewer: Sibei Yang (Lanzhou)
MSC:
| 42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |
| 28A75 | Length, area, volume, other geometric measure theory |
| 28A78 | Hausdorff and packing measures |
Keywords:
rectifiability; cone; singular integral operators; conical density; big pieces of Lipschitz graphsReferences:
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