论文标题
正交投影和分析插值的混合符号对措施的维度
Mixed-norm of orthogonal projections and analytic interpolation on dimensions of measures
论文作者
论文摘要
假设在$ \ mathbb {r}^d $和g（d，n）$的$ \ mathbb {r}^d $和$ v \上，$μ，ν$是一个$ n $二维子空间。在本文中，我们会系统地研究混合 - - norm $ \ | | wyμ\ | _ {l^p（g（d，n））}^q \，dν（y），\ \ forall \ forall \，p，q \ in [1，\ infty），p，q in [1，\ infty），$π_v：\ dembb = $ de $ de $ de $ n r}投影和$$π^yμ（v）= \ int_ {y+v^\ perp}μ\，d \ nathcal {h}^{d-n} =π_vμ（π_vy），\ \ text {如果$ $ $ $具有持续密度}。 在证明中，我们首先考虑整数指数，然后在分析上插值，不仅在$ p，q $上，而且在度量方面。我们还引入了一个称为$ s $振幅的新数量，以呈现我们的结果并说明我们的想法。这种机制为操作员提供了措施的新观点，因此具有自身的兴趣。我们还提供了dąbrowski，orponen，villa的最新结果的替代证明。 以下后果也很有趣。 $ \ bullet $我们发现关键线段的跳跃不连续性在$ p $的范围内 $$ \ {（s_μ，s_ν）\在（0，d）^2：s_μ+s_ν= 2n，\，0 <s_ν<n \}，$$ $ \ \ $其中$s_μ，s_ν$是$μ，ν$的霜冻指数。这是出乎意料且令人惊讶的。 $ \ bullet $给定$ 1 \ leq m \ leq d-1 $和$ e，f \ subset \ mathbb {r}^d $，我们在f $中是否存在$ y \ y获得维度阈值 $γ_{d，m} \ {v \ in G（d，m）：v = \ operatorName {span} \ {x_1-y，\ dots，x_m-y \}：x_1，\ dots，\ dots，x_m \ $ \ \ $这概括了可见性问题（$ m = 1 $）。特别是，当$ m> \ frac {d} {2} $和$ \ dim _ {\ mathcal {h}} e $足够大时，特殊的集合具有Hausdorff dimension $ 0 $。
Suppose $μ, ν$ are compactly supported Radon measures on $\mathbb{R}^d$ and $V\in G(d,n)$ is an $n$-dimensional subspace. In this paper we systematically study the mixed-norm $$\int\|π^yμ\|_{L^p(G(d,n))}^q\,dν(y),\ \forall\,p,q\in[1,\infty),$$ where $π_V:\mathbb{R}^d\rightarrow V$ denotes the orthogonal projection and $$π^yμ(V)=\int_{y+V^\perp}μ\,d\mathcal{H}^{d-n}=π_Vμ(π_Vy),\ \text{if $μ$ has continuous density}.$$ When $n=d-1$ and $p=q$, our result significantly improves a previous result of Orponen. In the proof we consider integer exponents first, then interpolate analytically, not only on $p,q$, but also on dimensions of measures. We also introduce a new quantity called $s$-amplitude, to present our results and illustrate our ideas. This mechanism provides new perspectives on operators with measures, thus has its own interest. We also give an alternative proof of a recent result of Dąbrowski, Orponen, Villa on $\|π_Vμ\|_{L^p(\mathcal{H}^n\times G(d,n))}$. The following consequences are also interesting. $\bullet$ We discover jump discontinuities in the range of $p$ at the critical line segment $$\{(s_μ, s_ν)\in(0,d)^2: s_μ+s_ν=2n,\, 0<s_ν<n\},$$ $\ \ $ where $s_μ, s_ν$ are Frostman exponents of $μ, ν$ respectiely. This is unexpected and surprising. $\bullet$ Given $1\leq m\leq d-1$ and $E, F\subset\mathbb{R}^d$, we obtain dimensional threshold on whether there exists $y\in F$ such that $$γ_{d,m}\{V\in G(d,m): V=\operatorname{Span}\{x_1-y,\dots,x_m-y\}: x_1,\dots,x_m\in E\}>0.$$ $\ \ $ This generalizes the visibility problem ($m=1$). In particular, when $m>\frac{d}{2}$ and $\dim_{\mathcal{H}} E$ is large enough, the exceptional set has Hausdorff dimension $0$.
