Related papers: Compactness criteria via Laguerre and Hankel trans…

We obtain criteria for integral transformations of Laplace and Stieltjes type to be compact on Lebesgue spaces of real functions on the semiaxis.

We provide a compactness principle which is applicable to different formulations of Plateau's problem in codimension one and which is exclusively based on the theory of Radon measures and elementary comparison arguments. Exploiting some…

In a previous work, "compact versions" of Rubio de Francia's weighted extrapolation theorem were proved, which allow one to extrapolate the compactness of an linear operator from just one space to the full range of weighted Lebesgue spaces,…

In this note we give a proof of the Sobolev and Morrey embedding theorems based on the representation of functions in terms of the fundamental solution of suitable partial differential operators. We also prove the compactness of the Sobolev…

In this paper we solve a long standing problem about the bilinear $T1$ theorem to characterize the (weighted) compactness of bilinear Calder\'{o}n-Zygmund operators. Let $T$ be a bilinear operator associated with a standard bilinear…

We prove a compactness result in GBD which also provides a new proof of the compactness theorem in GSBD, due to Chambolle and Crismale [5, Theorem 1.1]. Our proof is based on a Fr\'echet-Kolmogorov compactness criterion and does not rely on…

In this paper, we establish a Minkowski-type inequality for weak Lebesgue space, which allows us to obtain a characterization of relative compactness in these spaces. Furthermore, we are the first to investigate the compactness results of…

We introduce general translations as solutions to Cauchy or Dirichlet problems. This point of view allows us to handle the heat-diffusion semigroup as a translation. With the given examples Kolmogorov-Riesz characterization of compact sets…

We extend the classical Kolmogorov-Riesz compactness theorem to the setting of asymptotic $L_p$ spaces on $\mathbb{R}^n$. These are nonlocally convex $\mathrm{F}$-spaces that contain the standard $L_p$ spaces as dense subspaces and include…

We study compact embeddings of Sobolev, Besov, and Triebel-Lizorkin spaces with variable exponents on both bounded and unbounded metric measure spaces. We establish sufficient conditions for compactness, and under additional assumptions, we…

We give an abstract version of the hard Lefschetz theorem, the Lefschetz decomposition and the Hodge-Riemann theorem for compact Kaehler manifolds.

The Pego theorem characterizes the precompact subsets of the square-integrable functions on $\mathbb{R}^n$ via the Fourier transform. We prove the analogue of the Pego theorem on compact groups (not necessarily abelian).

We prove a sharp, quantitative analogue of Helgason's conjecture at the level of distributions: For a semisimple Lie group $G$ of real rank one, Poisson transforms map a Sobolev space on $P\backslash G$ boundedly with closed range to an…

Bessel potential spaces have gained renewed interest due to their robust structural properties and applications in fractional partial differential equations (PDEs). These spaces, derived through complex interpolation between Lebesgue and…

We prove the Hodge-Riemann bilinear relations, the hard Lefschetz theorem and the Lefschetz decomposition for compact Kahler manifolds in the mixed situation.

We prove global regularity, scattering and a priori bounds for the energy critical Maxwell-Klein-Gordon equation relative to the Coulomb gauge on (1+4)-dimensional Minkowski space. The proof is based upon a modified Bahouri-Gerard profile…

In this article, we give a complex-geometric proof of the Alexandrov-Fenchel inequality without using toric compactifications. The idea is to use the Legendre transform and develop the Brascamp-Lieb proof of the Pr\'ekopa theorem. New…

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…

The connections between the objects mentioned in the title are used to give a short proof of the Cartan--Helgason theorem and a natural construction of the compactifications.

We prove a compact embedding theorem in a class of spaces of piecewise H1 functions subordinated to a class of shape regular, but not necessarily quasi-uniform triangulations of a polygonal domain. This result generalizes the…