A dual frames multiplier is an operator consisting of analysis, multiplication and synthesis processes, where the analysis and the synthesis are made by two dual frames in a Hilbert space, respectively. In this paper we investigate the spectra of som
e dual frames multipliers giving, in particular, conditions to be at most countable. The contribution extends the results available in literature about the spectra of Bessel multipliers with symbol decaying to zero and of multipliers of dual Riesz bases.
The chain group $C(G)$ of a locally compact group $G$ has one generator $g_{rho}$ for each irreducible unitary $G$-representation $rho$, a relation $g_{rho}=g_{rho}g_{rho}$ whenever $rho$ is weakly contained in $rhootimes rho$, and $g_{rho^*}=g_{rho}
^{-1}$ for the representation $rho^*$ contragredient to $rho$. $G$ satisfies chain-center duality if assigning to each $g_{rho}$ the central character of $rho$ is an isomorphism of $C(G)$ onto the dual $widehat{Z(G)}$ of the center of $G$. We prove that $G$ satisfies chain-center duality if it is (a) a compact-by-abelian extension, (b) connected nilpotent, (c) countable discrete icc or (d) connected semisimple; this generalizes M. M{u}gers result compact groups satisfy chain-center duality.
We study the inequalities of the type $|int_{mathbb{R}^d} Phi(K*f)| lesssim |f|_{L_1(mathbb{R}^d)}^p$, where the kernel $K$ is homogeneous of order $alpha - d$ and possibly vector-valued, the function $Phi$ is positively $p$-homogeneous, and $p = d/(
d-alpha)$. Under mild regularity assumptions on $K$ and $Phi$, we find necessary and sufficient conditions on these functions under which the inequality holds true with a uniform constant for all sufficiently regular functions $f$.
This note is devoted to several results about frequency localized functions and associated Bernstein inequalities for higher order operators. In particular, we construct some counterexamples for the frequency-localized Bernstein inequalities for high
er order Laplacians. We show also that the heat semi-group associated to powers larger than one of the laplacian does not satisfy the strict maximum principle in general. Finally, in a suitable range we provide several positive results.
Answering a key point left open in a recent work of Bongers, Guo, Li and Wick, we provide the following lower bound $$ |b|_{text{BMO}_{gamma}(mathbb{R}^2)}lesssim |[b,H_{gamma}]|_{L^p(mathbb{R}^2)to L^p(mathbb{R}^2)}, $$ where $H_{gamma}$ is the parabolic Hilbert transform.
We study the embeddings of (homogeneous and inhomogeneous) anisotropic Besov spaces associated to an expansive matrix $A$ into Sobolev spaces, with focus on the influence of $A$ on the embedding behaviour. For a large range of parameters, we derive sharp characterizations of embeddings.
We introduced the quasicentral modulus to study normed ideal perturbations of operators. It is a limit of condenser quasicentral moduli in view of a recently noticed analogy with capacity in nonlinear potential theory. We prove here some basic proper
ties of the condenser quasicentral modulus and compute a simple example. Some of the results are in the more general setting of a semifinite von Neumann algebra.
This survey addresses sampling discretization and its connections with other areas of mathematics. We present here known results on sampling discretization of both integral norms and the uniform norm beginning with classical results and ending with v
ery recent achievements. We also show how sampling discretization connects to spectral properties and operator norms of submatrices, embedding of finite-dimensional subspaces, moments of marginals of high-dimensional distributions, and learning theory. Along with the corresponding results, important techniques for proving those results are discussed as well.
Let $S subset mathbb{R}^{n}$ be an arbitrary nonempty compact set such that the $d$-Hausdorff content $mathcal{H}^{d}_{infty}(S) > 0$ for some $d in (0,n]$. For each $p in (max{1,n-d},n]$ an almost sharp intrinsic description of the trace space $W_{p
}^{1}(mathbb{R}^{n})|_{S}$ of the Sobolev space $W_{p}^{1}(mathbb{R}^{n})$ is given. Furthermore, for each $p in (max{1,n-d},n]$ and $varepsilon in (0, min{p-(n-d),p-1})$ new bounded linear extension operators from the trace space $W_{p}^{1}(mathbb{R}^{n})|_{S}$ into the space $W_{p-varepsilon}^{1}(mathbb{R}^{n})$ are constructed.
Interpolation inequalities in Triebel-Lizorkin-Lorentz spaces and Besov-Lorentz spaces are studied for both inhomogeneous and homogeneous cases. First we establish interpolation inequalities under quite general assumptions on the parameters of the fu
nction spaces. Several results on necessary conditions are also provided. Next, utilizing the interpolation inequalities together with some embedding results, we prove Gagliardo-Nirenberg inequalities for fractional derivatives in Lorentz spaces, which do hold even for the limiting case when one of the parameters is equal to 1 or $infty$.
