Related papers: Compactness of a restricted X-ray transform

We show that the restriction and extension operators associated to the moment curve possess extremizers and that $L^p$-normalized extremizing sequences of these operators are precompact modulo symmetries.

We show how the techniques introduces by Christ can be employed to derive endpoint $L^p-L^q$ bounds for the X-ray transform associated to the line complex generated by the curve $t\to(t,t^2,...,t^{d-1}).$ Almost-sharp Lorentz space…

In this article, we establish various facts about extremizers for $L^p$-improving convolution operators $T\colon L^p \rightarrow L^q$ associated with compactly-supported probability measures on either $\mathbb{R}^d$ or $\mathbb{T}^d$ . If…

The operator $T$, defined by convolution with the affine arc length measure on the moment curve parametrized by $h(t)=(t,t^{2},...,t^{d})$ is a bounded operator from $L^{p}$ to $L^{q}$ if $(\frac{1}{p}, \frac{1}{q})$ lies on a line segment.…

In this article, we prove that all global, nonendpoint Fourier restriction inequalities for the paraboloid in $\mathbb R^{1+d}$ have extremizers and that $L^p$-normalized extremizing sequences are precompact modulo symmetries. This result…

In this paper, we show that Hilbert transforms along some curves are bounded on $L^p({\mathbb R}^n;X)$ for some $1<p<\infty$ and some UMD spaces $X$. In particular, we prove that the Hilbert transform along some curves are completely…

We give a geometric characterization of self-extremal sets in $L_p(\Om)$ spaces that partially extends our previous results to the case of $L_p$\ spaces.

It is known that extremizers for the $L^2$ to $L^6$ adjoint Fourier restriction inequality on the cone in $\mathbb{R}^3$ exist. Here we show that nonnegative extremizing sequences are precompact, after the application of symmetries of the…

We establish endpoint Lebesgue space bounds for convolution and restricted X-ray transforms along curves satisfying fairly minimal differentiability hypotheses, with affine and Euclidean arclengths. We also explore the behavior of certain…

Let (\Omega,\mu) be a finite measure space, X a Banach space, and let 1\le p<\infty. The aim of this paper is to give an elementary proof of the Diaz--Mayoral theorem that a subset V of L^p(\mu;X) is relatively compact if and only if it is…

We prove a new characterization of complex projective space using lengths of extremal rays.

In this paper, for $1<p<\infty$, we obtain the $L^p$-boundedness of the Hilbert transform $H^{\gamma}$ along a variable plane curve $(t,u(x_1, x_2)\gamma(t))$, where $u$ is a Lipschitz function with small Lipschitz norm, and $\gamma$ is a…

We present a complete characterization of the metric compactification of $L_{p}$ spaces for $1\leq p < \infty$. Each element of the metric compactification of $L_{p}$ is represented by a random measure on a certain Polish space. By way of…

We prove a compactness criterion in $L^p({\mu},X)$: a subset of $L^p({\mu},X)$ is relatively norm compact iff the set of integrals of its functions over any measurable set is relatively norm compact, it satisfies the Fr\'echet oscillation…

The X-ray transform on a compact symmetric space M is here inverted by means of an explicit inversion formula. The proof uses the conjugacy of the minimal closed geodesics in M and of the maximally curved totally geodesic spheres in M,…

We study $L^p$-Sobolev regularity estimate for the restricted X-ray transforms generated by nondegenerate curves. Making use of the inductive strategy in the recent work by the authors, we establish the sharp $L^p$-regularity estimates for…

Using recent developments on locally compact groups, we are able to obtain quantitative results on embeddings into Lebesgue spaces for a large class of HNN extensions.

We establish near-optimal mixed-norm estimates for the X-ray transform restricted to polynomial curves with a weight that is a power of the affine arclength. The bounds that we establish depend only on the spatial dimension and the degree…

We establish the existence of extremizers for a Fourier restriction inequality on planar convex arcs without points with colinear tangents whose curvature satisfies a natural assumption. More generally, we prove that any extremizing…

We give a new proof of a recent characterization by Diaz and Mayoral of compactness in the Lebesgue-Bochner spaces $L_X^p$, where $X$ is a Banach space and $1\le p<\infty$, and extend the result to vector-valued Banach function spaces…