Related papers: Interpolation of Hilbert and Sobolev Spaces: Quant…

In this paper Hilbert spaces are characterized among Banach spaces in terms of transitivity with respect to nicely behaved subgroups of the isometry group. For example, the following result is typical here: If X is a real Banach space…

We introduce two Bishop-Phelps-Bollob\'as moduli which measure, for a given Banach space, what is the best possible Bishop-Phelps-Bollob\'as theorem in this space. We show that there is a common upper bound for these moduli for all Banach…

We present a simple proof of some interpolation inequalities between H\"{o}lder and Lebesgue's spaces. As an example, to demonstrate the simplicity of their applications to nonlinear PDE, we give also a simple proof of an a-priory estimate…

We study embeddings and norm estimates for tensor products of weighted reproducing kernel Hilbert spaces. These results lead to a transfer principle that is directly applicable to tractability studies of multivariate problems as integration…

The research monograph gives the first systematic exposition of the elliptic (scalar and matrix) operators theory and elliptic boundary-value problems in the scales of Hilbert spaces of H\"ormander of the functions/distributions of…

Geometric quantiles are location parameters which extend classical univariate quantiles to normed spaces (possibly infinite-dimensional) and which include the geometric median as a special case. The infinite-dimensional setting is highly…

The classical Gagliardo-Nirenberg interpolation inequality is a well-known estimate which gives, in particular, an estimate for the Lebesgue norm of intermediate derivatives of functions in Sobolev spaces. We present an extension of this…

We study the problem of the existence of a common algebraic complement for a pair of closed subspaces of a Banach space. We prove the following two characterizations: (1) The pairs of subspaces of a Banach space with a common complement…

Reproducing kernel Hilbert spaces provide a foundational framework for kernel-based learning, where regularization and interpolation problems admit finite-dimensional solutions through classical representer theorems. Many modern learning…

We prove the stability of isomorphisms between Banach spaces generated by interpolation methods introduced by Cwikel-Kalton-Milman-Rochberg which includes, as special cases, the real and complex methods up to equivalence of norms and also…

The article deals with a simplified proof of the Sobolev embedding theorem for Lizorkin--Triebel spaces (that contain the $L_p$-Sobolev spaces $H^s_p$ as special cases). The method extends to a proof of the corresponding fact for general…

Let X be a real Banach space. We prove that the existence of an injective, positive, symmetric and not strictly singular operator from X into its dual implies that either X admits an equivalent Hilbertian norm or it contains a nontrivially…

The question is addressed of when a Sobolev type space, built upon a general rearrangement-invariant norm, on an $n$-dimensional domain, is a Banach algebra under pointwise multiplication of functions. A sharp balance condition among the…

Constructing or learning a function from a finite number of sampled data points (measurements) is a fundamental problem in science and engineering. This is often formulated as a minimum norm interpolation problem, regularized learning…

We consider weighted anchored and ANOVA spaces of functions with first order mixed derivatives bounded in $L_p$. Recently, Hefter, Ritter and Wasilkowski established conditions on the weights in the cases $p=1$ and $p=\infty$ which ensure…

We study how well a quasi-Banach space can be coarsely embedded into a Hilbert space. Given any quasi-Banach space X which coarsely embeds into a Hilbert space, we compute its Hilbert space compression exponent. We also show that the…

We obtain the inequality $$\int_{\Omega}|\nabla u(x)|^ph(u(x))dx\leq C(n,p)\int_{\Omega} \left( \sqrt{ |\nabla^{(2)} u(x)||{\cal T}_{h,C}(u(x))|}\right)^{p}h(u(x))dx,$$ where $\Omega\subseteq {\bf R}^n$ and $n\ge 2$, $u:\Omega\rightarrow…

Let $E$ be a Sidon subset of the integers and suppose $X$ is a Banach space. Then Pisier has shown that $E$-spectral polynomials with values in $X$ behave like Rademacher sums with respect to $L_p-$norms. We consider the situation when $X$…

We obtain sharp approximation results for into nearisometries between Lp spaces and nearisometries into a Hilbert space. Our main theorem is the optimal approximation result for nearsurjective nearisometries between general Banach spaces.

Integral operators of Abel type of order a > 0 arise naturally in a large spectrum of physical processes. Their inversion requires care since the resulting inverse problem is ill-posed. The purpose of this work is to devise and analyse a…