The Riesz transform and quantitative rectifiability for general Radon measures
The Riesz transform and quantitative rectifiability for general Radon measures
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一般氡测量的 Riesz 变换和定量可校正性
DOI：
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
X. Tolsa
中科院分区：
文献类型：
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作者：
Daniel Girela;X. Tolsa
In this paper we show that if $$mu $$μ is a Borel measure in $${{mathbb {R}}}^{n+1}$$Rn+1 with growth of order n, such that the n-dimensional Riesz transform $${{mathcal {R}}}_mu $$Rμ is bounded in $$L^2(mu )$$L2(μ), and $$Bsubset {{mathbb {R}}}^{n+1}$$B⊂Rn+1 is a ball with $$mu (B)approx r(B)^n$$μ(B)≈r(B)n such that:(a)there is some n-plane L passing through the center of B such that for some $$delta >0$$δ>0 small enough, it holds $$egin{aligned}int _B frac{mathrm{dist}(x,L)}{r(B)},dmu (x)le delta ,mu (B),end{aligned}$$∫Bdist(x,L)r(B)dμ(x)≤δμ(B),(b)for some constant $${varepsilon }>0$$ε>0 small enough, $$egin{aligned}int _{B} |{{mathcal {R}}}_mu 1(x) - m_{mu ,B}({{mathcal {R}}}_mu 1)|^2,dmu (x) le {varepsilon },mu (B),end{aligned}$$∫B|Rμ1(x)-mμ,B(Rμ1)|2dμ(x)≤εμ(B), where $$m_{mu ,B}({{mathcal {R}}}_mu 1)$$mμ,B(Rμ1) stands for the mean of $${{mathcal {R}}}_mu 1$$Rμ1 on B with respect to $$mu $$μ, then there exists a uniformly n-rectifiable set $$Gamma $$Γ, with $$mu (Gamma cap B)gtrsim mu (B)$$μ(Γ∩B)≳μ(B), and such that $$mu |_Gamma $$μ|Γ is absolutely continuous with respect to $${{mathcal {H}}}^n|_Gamma $$Hn|Γ. This result is an essential tool to solve an old question on a two phase problem for harmonic measure in subsequent papers by Azzam, Mourgoglou, Tolsa, and Volberg.
In this paper we show that if $$mu $$μ is a Borel measure in $${{mathbb {R}}}^{n+1}$$Rn+1 with growth of order n, such that the n-dimensional Riesz transform $${{mathcal {R}}}_mu $$Rμ is bounded in $$L^2(mu )$$L2(μ), and $$Bsubset {{mathbb {R}}}^{n+1}$$B⊂Rn+1 is a ball with $$mu (B)approx r(B)^n$$μ(B)≈r(B)n such that:(a)there is some n-plane L passing through the center of B such that for some $$delta >0$$δ>0 small enough, it holds $$egin{aligned}int _B frac{mathrm{dist}(x,L)}{r(B)},dmu (x)le delta ,mu (B),end{aligned}$$∫Bdist(x,L)r(B)dμ(x)≤δμ(B),(b)for some constant $${varepsilon }>0$$ε>0 small enough, $$egin{aligned}int _{B} |{{mathcal {R}}}_mu 1(x) - m_{mu ,B}({{mathcal {R}}}_mu 1)|^2,dmu (x) le {varepsilon },mu (B),end{aligned}$$∫B|Rμ1(x)-mμ,B(Rμ1)|2dμ(x)≤εμ(B), where $$m_{mu ,B}({{mathcal {R}}}_mu 1)$$mμ,B(Rμ1) stands for the mean of $${{mathcal {R}}}_mu 1$$Rμ1 on B with respect to $$mu $$μ, then there exists a uniformly n-rectifiable set $$Gamma $$Γ, with $$mu (Gamma cap B)gtrsim mu (B)$$μ(Γ∩B)≳μ(B), and such that $$mu |_Gamma $$μ|Γ is absolutely continuous with respect to $${{mathcal {H}}}^n|_Gamma $$Hn|Γ. This result is an essential tool to solve an old question on a two phase problem for harmonic measure in subsequent papers by Azzam, Mourgoglou, Tolsa, and Volberg.