The measures with $L^2$-bounded Riesz transform and the Painlevé problem
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908578420359168 |
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| author | Dąbrowski, Damian Tolsa, Xavier |
| author_facet | Dąbrowski, Damian Tolsa, Xavier |
| contents | In this work we provide a geometric characterization of the measures $μ$ in $\mathbb R^{n+1}$ with polynomial upper growth of degree $n$ such that the $n$-dimensional Riesz transform $Rμ(x) = \int \frac{x-y}{|x-y|^{n+1}}\,dμ(y)$ belongs to $L^2(μ)$. More precisely, it is shown that $$\|Rμ\|_{L^2(μ)}^2 + \|μ\|\approx \int\!\!\int_0^\infty β_{2,μ}(x,r)^2\,\frac{μ(B(x,r))}{r^n}\,\frac{dr}r\,dμ(x) + \|μ\|,$$ where $β_{μ,2}(x,r)^2 = \inf_L \frac1{r^n}\int_{B(x,r)} \left(\frac{\mathrm{dist}(y,L)}r\right)^2\,dμ(y),$ with the infimum taken over all affine $n$-planes $L\subset\mathbb R^{n+1}$. As a corollary, we obtain a characterization of the removable sets for Lipschitz harmonic functions in terms of a metric-geometric potential and we deduce that the class of removable sets for Lipschitz harmonic functions is invariant by bilipschitz mappings. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_08615 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The measures with $L^2$-bounded Riesz transform and the Painlevé problem Dąbrowski, Damian Tolsa, Xavier Classical Analysis and ODEs Analysis of PDEs 42B20, 28A75, 49Q15 In this work we provide a geometric characterization of the measures $μ$ in $\mathbb R^{n+1}$ with polynomial upper growth of degree $n$ such that the $n$-dimensional Riesz transform $Rμ(x) = \int \frac{x-y}{|x-y|^{n+1}}\,dμ(y)$ belongs to $L^2(μ)$. More precisely, it is shown that $$\|Rμ\|_{L^2(μ)}^2 + \|μ\|\approx \int\!\!\int_0^\infty β_{2,μ}(x,r)^2\,\frac{μ(B(x,r))}{r^n}\,\frac{dr}r\,dμ(x) + \|μ\|,$$ where $β_{μ,2}(x,r)^2 = \inf_L \frac1{r^n}\int_{B(x,r)} \left(\frac{\mathrm{dist}(y,L)}r\right)^2\,dμ(y),$ with the infimum taken over all affine $n$-planes $L\subset\mathbb R^{n+1}$. As a corollary, we obtain a characterization of the removable sets for Lipschitz harmonic functions in terms of a metric-geometric potential and we deduce that the class of removable sets for Lipschitz harmonic functions is invariant by bilipschitz mappings. |
| title | The measures with $L^2$-bounded Riesz transform and the Painlevé problem |
| topic | Classical Analysis and ODEs Analysis of PDEs 42B20, 28A75, 49Q15 |
| url | https://arxiv.org/abs/2402.08615 |