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Let\Sigmabe a strictly convex, compact patch of aC^{2}hypersurface in\mathbb{R}^{n}, with non-vanishing Gaussian curvature and surface measured\sigmainduced by the Lebesgue measure in\mathbb{R}^{n}. The Mizohata–Takeuchi conjecture states that \int |\widehat{g d\sigma}|^{2} w \leq C \|Xw\|_{\infty} \int |g|^{2} for allg\in L^{2}(\Sigma)and all weightsw \colon \mathbb{R}^{n}\rightarrow [0,+\infty), whereXdenotes theX-ray transform. As partial progress towards the conjecture, we show, as a straightforward consequence of recently-established decoupling inequalities, that for every\varepsilon>0, there exists a positive constantC_{\varepsilon}, which depends only on\Sigmaand\varepsilon, such that for allR \geq 1and all weightsw \colon \mathbb{R}^{n}\rightarrow [0,+\infty), we have \int_{B_R}|\widehat{g d\sigma}|^{2} w \leq C_{\varepsilon} R^{\varepsilon} \sup_{T} \Big(\int_{T} w^{(n+1)/2}\Big)^{2/(n+1)}\int |g|^{2}, whereTranges over the family of tubes in\mathbb{R}^{n}of dimensionsR^{1/2}\times \cdots \times R^{1/2}\times R. From this we deduce the Mizohata–Takeuchi conjecture with anR^{(n-1)/(n+1)}-loss; i.e., that \int_{B_R}|\widehat{g d\sigma}|^{2} w \leq C_{\varepsilon} R^{\frac{n-1}{n+1}+ \varepsilon}\|Xw\|_{\infty} \int |g|^{2} for any ballB_{R}of radiusRand any\varepsilon>0. The power(n-1)/(n+1)here cannot be replaced by anything smaller unless properties of\widehat{g d\sigma}beyond ‘decoupling axioms’ are exploited. We also provide estimates which improve this inequality under various conditions on the weight, and discuss some new cases where the conjecture holds.
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This content will become publicly available on February 5, 2026
Convex cocompactness for Coxeter groups
We investigate representations of Coxeter groups into\mathrm{GL}(n,\mathbb{R})as geometric reflection groups which are convex cocompact in the projective space\mathbb{P}(\mathbb{R}^{n}). We characterize which Coxeter groups admit such representations, and we fully describe the corresponding spaces of convex cocompact representations as reflection groups, in terms of the associated Cartan matrices. The Coxeter groups that appear include all infinite word hyperbolic Coxeter groups; for such groups, the representations as reflection groups that we describe are exactly the projective Anosov ones. We also obtain a large class of nonhyperbolic Coxeter groups, thus providing many examples for the theory of nonhyperbolic convex cocompact subgroups in\mathbb{P}(\mathbb{R}^{n})developed by Danciger–Guéritaud–Kassel (2024).
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- Award ID(s):
- 1945493
- PAR ID:
- 10630905
- Publisher / Repository:
- EMS Press
- Date Published:
- Journal Name:
- Journal of the European Mathematical Society
- Volume:
- 27
- Issue:
- 1
- ISSN:
- 1435-9855
- Page Range / eLocation ID:
- 119 to 181
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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