New criteria for the rectifiability of Radon measures in terms of Riesz transforms
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908821939552256 |
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| author | Tolsa, Xavier |
| author_facet | Tolsa, Xavier |
| contents | In this paper we explore the connection between quantitative rectifiability of measures and the $L^2$ boundedness of the codimension one Riesz transform. Among other things, we prove the following. Let $μ$ be a Radon measure in $\mathbb R^{n+1}$ with growth of degree $n$ such that the $n$-dimensional Riesz transform $R_μ$ is bounded in $L^2(μ)$, and let $B_0\subset\mathbb R^{n+1}$ be a suitably doubling ball such that:
(i) There exists some (small) ball $B_1$ centered in $B_0$ with $r(B_1)\leq δ_1 r(B_0)$ such that, for some constant $α>0$, $$\frac{μ(B_1)}{r(B_1)^n}\geq α\,\frac{μ(B_0)}{r(B_0)^n}.$$
(ii) For some $ε>0$, $$\int_{2B_0} |Rμ- m_{μ,2B_0}(Rμ)|^2\,dμ\leq ε\,\bigg(\frac{μ(B_0)}{r(B_0)^n}\bigg)^2\,μ(B_0).$$
If $δ_1$ is small enough, depending on $n$ and $α$, and $ε$ is small enough, then there exists a uniformly $n$-rectifiable set $Γ$ and some $τ>0$ such that $μ(Γ\cap B_0) \geqτ\,μ(B_0).$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_14534 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | New criteria for the rectifiability of Radon measures in terms of Riesz transforms Tolsa, Xavier Classical Analysis and ODEs 42B20, 28A75, 28A78, 49Q20 In this paper we explore the connection between quantitative rectifiability of measures and the $L^2$ boundedness of the codimension one Riesz transform. Among other things, we prove the following. Let $μ$ be a Radon measure in $\mathbb R^{n+1}$ with growth of degree $n$ such that the $n$-dimensional Riesz transform $R_μ$ is bounded in $L^2(μ)$, and let $B_0\subset\mathbb R^{n+1}$ be a suitably doubling ball such that: (i) There exists some (small) ball $B_1$ centered in $B_0$ with $r(B_1)\leq δ_1 r(B_0)$ such that, for some constant $α>0$, $$\frac{μ(B_1)}{r(B_1)^n}\geq α\,\frac{μ(B_0)}{r(B_0)^n}.$$ (ii) For some $ε>0$, $$\int_{2B_0} |Rμ- m_{μ,2B_0}(Rμ)|^2\,dμ\leq ε\,\bigg(\frac{μ(B_0)}{r(B_0)^n}\bigg)^2\,μ(B_0).$$ If $δ_1$ is small enough, depending on $n$ and $α$, and $ε$ is small enough, then there exists a uniformly $n$-rectifiable set $Γ$ and some $τ>0$ such that $μ(Γ\cap B_0) \geqτ\,μ(B_0).$ |
| title | New criteria for the rectifiability of Radon measures in terms of Riesz transforms |
| topic | Classical Analysis and ODEs 42B20, 28A75, 28A78, 49Q20 |
| url | https://arxiv.org/abs/2512.14534 |