Some remarks about curves in metric spaces
英文摘要:
Here we consider a few topics related to Lipschitz classes for functions and curves in metric spaces.
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Stephen Semmes
Here we consider a few topics related to Lipschitz classes for functions and curves in metric spaces.
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
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Recently, two of the authors obtained estimates for the adjoint restriction operator to finite type curves with respect to general measures. Strikingly, it turns out that some of such estimates are sharp, especially when the measures are given by surface measures under certain condition. A typical example is the surface measure on the sphere. We demonstrate sharpness of such estimates by constructing an example and, also, discuss related estimates over different type of surfaces.
We study an analogue of Marstrand's circle packing problem for curves in higher dimensions. We consider collections of curves which are generated by translation and dilation of a curve $γ$ in $\mathbb R^d$, i.e., $ x + t γ$, $(x,t) \in \mathbb R^d \times (0,\infty)$. For a Borel set $F \subset \mathbb R^d\times (0,\infty)$, we show the unions of curves $\bigcup_{(x,t) \in F} ( x+tγ)$ has Hausdorff dimension at least $α+1$ whenever $F$ has Hausdorff dimension bigger than $α$, $α\in (0, d-1)$. We also obtain results for unions of curves generated by multi-parameter dilation of $γ$. One of the ma…
This short survey has been prepared in connection with the workshop on discrete metric spaces and their applications at Princeton, August, 2003, and tries to convey some of the ways that one might look at functions on metric spaces in harmonic analysis, in terms of behavior at different scales and locations in the space.
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There seems to be quite a bit of room for interesting things related to surfaces M in C^m with real dimension m which are totally real and aspects of several complex variables on C^m around M. A basic case occurs when m = 1, with Cauchy integral operators along curves, holomorphic functions on the complements of curves, and so on. In the present article we briefly indicate a few points concerning these matters.
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