Multipliers in the Bessel potential spaces with positive smoothness indices: bilateral continuous embeddings and their exact character

Riesz's and Bessel's Operators in Bilateral Grand Lebesgue Spaces

In this paper we obtain the non - asymptotic estimations for Riesz's and Bessel's potential integral operators in the so - called Bilateral Grand Lebesgue Spaces. We also give examples to show the sharpness of these inequalities.

Multipliers in the scale of periodic Bessel potential spaces with smoothness indices of different signs

We prove a general type description result for the multipliers acting between two periodic Bessel potential spaces, defined on the $n$--dimensional torus, in a case when their smoothness indices are of different signs. This is done through the detailed examination of a periodic analogue of the linear operator $J_s$, which is employed in the definition of the scale of the Bessel potential space defined on the whole space $\mathbb{R}^n$. Our method of defining this periodic analogue of $J_s$ uses the results about an asymptotic behaviour of the generalized Fourier coefficients and existence of a…

Optimal Calderón Spaces for generalized Bessel potentials

In the paper we investigate the properties of spaces with generalized smoothness, such as Calderón spaces that include the classical Nikolskii-Besov spaces and many of their generalizations, and describe differential properties of generalized Bessel potentials that include classical Bessel potentials and Sobolev spaces. Kernels of potentials may have non-power singularity at the origin. With the help of order-sharp estimates for moduli of continuity of potentials, we establish the criteria of embeddings of potentials into Calderón spaces, and describe the optimal spaces for such embeddings.

Function spaces of logarithmic smoothness: embeddings and characterizations

In this paper we present a comprehensive treatment of function spaces with logarithmic smoothness (Besov, Sobolev, Triebel-Lizorkin). We establish the following results: Sharp embeddings between the Besov spaces defined by differences and by Fourier-analytical decompositions as well as between Besov and Sobolev/Triebel-Lizorkin spaces; Various new characterizations for Besov norms in terms of different K-functionals. For instance, we derive characterizations via ball averages, approximation methods, heat kernels, and Bianchini-type norms; Sharp estimates for Besov norms of derivatives and pote…

Norm-Controlled Inversion in Smooth Banach Algebras, II

We show that smoothness implies norm-controlled inversion: the smoothness of an element $a$ in a Banach algebra with a one-parameter automorphism group is preserved under inversion, and the norm of the inverse $a^{-1}$ is controlled by the smoothness of $a$ and by spectral data. In our context smooth subalgebras are obtained with the classical constructions of approximation theory and resemble spaces of differentiable functions, Besov spaces or Bessel potential spaces. To treat ultra-smoothness, we resort to Dales-Davie algebras. Furthermore, based on Baskakov's work, we derive explicit norm c…

Barron空间真的能克服维度灾难吗？

Does the Barron space really defy the curse of dimensionality?

针对Barron空间是否真能克服维度灾难的问题，本文从与无限宽浅层神经网络相关的非经典光滑性视角出发，通过定义ADZ空间提供证据表明其并不能克服该灾难，补充了经典光滑性视角外的理论认识。