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Register a new userA sharp stability estimate for the geodesic ray transform
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We prove a sharp $L^2to H^{1/2}$ stability estimate for the geodesic X-ray transform of tensor fields of order $0$, $1$ and $2$ on a simple Riemannian manifold with a suitable chosen $H^{1/2}$ norm. We show that such an estimate holds for a family of such $H^{1/2}$ norms, not topologically equivalent, but equivalent on the range of the transform. The reason for this is that the geodesic X-ray transform has a microlocally structured range.
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We consider the geodesic X-ray transform acting on solenoidal tensor fields on a compact simply connected manifold with strictly convex boundary and non-positive curvature. We establish a stability estimate of the form $L^2mapsto H^{1/2}_{T}$, where the $H^{1/2}_{T}$-space is defined using the natural parametrization of geodesics as initial boundary points and incoming directions (fan-beam geometry); only tangential derivatives at the boundary are used. The proof is based on the Pestov identity with boundary term localized in frequency.
This paper is connected with the problem of describing path metric spaces that are homeomorphic to manifolds and biLipschitz homogeneous, i.e., whose biLipschitz homeomorphism group acts transitively. Our main result is the following. Let $X = G/H$ be a homogeneous manifold of a Lie group $G$ and let $d$ be a geodesic distance on $X$ inducing the same topology. Suppose there exists a subgroup $G_S$ of $G$ that acts transitively on $X$, such that each element $g in G_S$ induces a locally biLipschitz homeomorphism of the metric space $(X,d)$. Then the metric is locally biLipschitz equivalent to a sub-Riemannian metric. Any such metric is defined by a bracket generating $G_S$-invariant sub-bundle of the tangent bundle. The result is a consequence of a more general fact that requires a transitive family of uniformly biLipschitz diffeomorphisms with a control on their differentials. It will be relevant that the group acting transitively on the space is a Lie group and so it is locally compact, since, in general, the whole group of biLipschitz maps, unlikely the isometry group, is not locally compact. Our method also gives an elementary proof of the following fact. Given a Lipschitz sub-bundle of the tangent bundle of a Finsler manifold, then both the class of piecewise differentiable curves tangent to the sub-bundle and the class of Lipschitz curves almost everywhere tangent to the sub-bundle give rise to the same Finsler-Carnot-Caratheodory metric, under the condition that the topologies induced by these distances coincide with the manifold topology.
In this work we study the issue of geodesic extendibility on complete and locally compact metric length spaces. We focus on the geometric structure of the space $(Sigma (X),d_H)$ of compact balls endowed with the Hausdorff distance and give an explicit isometry between $(Sigma (X),d_H)$ and the closed half-space $ Xtimes mathbb{R}_{ge 0}$ endowed with a taxicab metric. Among the applications we establish a group isometry between $mbox{Iso} (X,d)$ and $mbox{Iso} (Sigma (X),d_H)$ when $(X,d)$ is a Hadamard space.
In this paper, based on the local comparison principle in [12], we study the local behavior of the difference of two spacelike graphs in a neighborhood of a second contact point. Then we apply it to the constant mean curvature equation in 3-dimensional Lorentz-Minkowski space $mathbb{L}^3$ and get the uniqueness of critical point for the solution of such equation over convex domain, which is an analogue of the result in [28]. Last, by this uniqueness, we obtain a minimum principle for a functional depending on the solution and its gradient. This gives us a sharp gradient estimate for the solution, which leads to a sharp height estimate.
Let $D$ be a nonnegative integer and ${mathbf{Theta}}subset S^1$ be a lacunary set of directions of order $D$. We show that the $L^p$ norms, $1<p<infty$, of the maximal directional Hilbert transform in the plane $$ H_{{mathbf{Theta}}} f(x):= sup_{vin {mathbf{Theta}}} Big|mathrm{p.v.}int_{mathbb R }f(x+tv)frac{mathrm{d} t}{t}Big|, qquad x in {mathbb R}^2, $$ are comparable to $(log#{mathbf{Theta}})^frac{1}{2}$. For vector fields $mathsf{v}_D$ with range in a lacunary set of of order $D$ and generated using suitable combinations of truncations of Lipschitz functions, we prove that the truncated Hilbert transform along the vector field $mathsf{v}_D$, $$ H_{mathsf{v}_D,1} f(x):= mathrm{p.v.} int_{ |t| leq 1 } f(x+tmathsf{v}_D(x)) ,frac{mathrm{d} t}{t}, $$ is $L^p$-bounded for all $1<p<infty$. These results extend previous bounds of the first author with Demeter, and of Guo and Thiele.
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