Sharpness of convolution bounds for measures
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2026
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866911670038691840 |
|---|---|
| author | Lee, Sanghyuk Lee, Sungchul |
| author_facet | Lee, Sanghyuk Lee, Sungchul |
| contents | In this paper, we determine the sharp \((p,q)\) range for \(L^p\)--\(L^q\) bounds of convolution operators \(f\mapsto μ*f\) associated with fractal measures \(μ\in \mathcal P_{α,β}(\mathbb R^d)\), namely, compactly supported Borel probability measures satisfying the \(α\)-Frostman condition \[ μ(B(x,ρ)) \lesssim ρ^α, \qquad \forall (x,ρ)\in \mathbb R^d\times (0,1), \] and the \(β/2\)-Fourier decay condition \[ |\widehatμ(ξ)| \lesssim |ξ|^{-β/2}, \qquad \forall ξ\in\mathbb R^d. \] Sharpness is established by constructing measures satisfying these conditions together with a suitable lower regularity condition. Modifications of the same constructions also refine previous sharpness results for the \(L^2\) restriction estimate of Mockenhaupt--Mitsis--Bak--Seeger by producing, in every dimension and in both the geometric \((α\geβ)\) and non-geometric \((β>α)\) regimes, a single measure in \(\mathcal P_{α,β}(\mathbb R^d)\) for which the corresponding threshold exponent is sharp. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_09809 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sharpness of convolution bounds for measures Lee, Sanghyuk Lee, Sungchul Classical Analysis and ODEs 42B20 (primary), 28A80 (secondary) In this paper, we determine the sharp \((p,q)\) range for \(L^p\)--\(L^q\) bounds of convolution operators \(f\mapsto μ*f\) associated with fractal measures \(μ\in \mathcal P_{α,β}(\mathbb R^d)\), namely, compactly supported Borel probability measures satisfying the \(α\)-Frostman condition \[ μ(B(x,ρ)) \lesssim ρ^α, \qquad \forall (x,ρ)\in \mathbb R^d\times (0,1), \] and the \(β/2\)-Fourier decay condition \[ |\widehatμ(ξ)| \lesssim |ξ|^{-β/2}, \qquad \forall ξ\in\mathbb R^d. \] Sharpness is established by constructing measures satisfying these conditions together with a suitable lower regularity condition. Modifications of the same constructions also refine previous sharpness results for the \(L^2\) restriction estimate of Mockenhaupt--Mitsis--Bak--Seeger by producing, in every dimension and in both the geometric \((α\geβ)\) and non-geometric \((β>α)\) regimes, a single measure in \(\mathcal P_{α,β}(\mathbb R^d)\) for which the corresponding threshold exponent is sharp. |
| title | Sharpness of convolution bounds for measures |
| topic | Classical Analysis and ODEs 42B20 (primary), 28A80 (secondary) |
| url | https://arxiv.org/abs/2605.09809 |