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Register a new userFactorization in Denjoy-Carleman classes associated to representations of $(mathbb{R}^{d},+)$
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For two types of moderate growth representations of $(mathbb{R}^d,+)$ on sequentially complete locally convex Hausdorff spaces (including F-representations [J. Funct. Anal. 262 (2012), 667-681], we introduce Denjoy-Carleman classes of ultradifferentiable vectors and show a strong factorization theorem of Dixmier-Malliavin type for them. In particular, our factorization theorem solves [Conjecture 6.; J. Funct. Anal. 262 (2012), 667-681] for analytic vectors of representations of $G =(mathbb{R}^d,+)$. As an application, we show that various convolution algebras and modules of ultradifferentiable functions satisfy the strong factorization property.
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Wavelet set wavelets were the first examples of wavelets that may not have associated multiresolution analyses. Furthermore, they provided examples of complete orthonormal wavelet systems in $L^2(mathbb{R}^d)$ which only require a single generating wavelet. Although work had been done to smooth these wavelets, which are by definition discontinuous on the frequency domain, nothing had been explicitly done over $mathbb{R}^d$, $d >1$. This paper, along with another one cowritten by the author, finally addresses this issue. Smoothing does not work as expected in higher dimensions. For example, Bin Hans proof of existence of Schwartz class functions which are Parseval frame wavelets and approximate Parseval frame wavelet set wavelets does not easily generalize to higher dimensions. However, a construction of wavelet sets in $hat{mathbb{R}}^d$ which may be smoothed is presented. Finally, it is shown that a commonly used class of functions cannot be the result of convolutional smoothing of a wavelet set wavelet.
Let $S subset mathbb{R}^{n}$ be a~closed set such that for some $d in [0,n]$ and $varepsilon > 0$ the~$d$-Hausdorff content $mathcal{H}^{d}_{infty}(S cap Q(x,r)) geq varepsilon r^{d}$ for all cubes~$Q(x,r)$ centered in~$x in S$ with side length $2r in (0,2]$. For every $p in (1,infty)$, denote by $W_{p}^{1}(mathbb{R}^{n})$ the classical Sobolev space on $mathbb{R}^{n}$. We give an~intrinsic characterization of the restriction $W_{p}^{1}(mathbb{R}^{n})|_{S}$ of the space $W_{p}^{1}(mathbb{R}^{n})$ to~the set $S$ provided that $p > max{1,n-d}$. Furthermore, we prove the existence of a bounded linear operator $operatorname{Ext}:W_{p}^{1}(mathbb{R}^{n})|_{S} to W_{p}^{1}(mathbb{R}^{n})$ such that $operatorname{Ext}$ is right inverse for the usual trace operator. In particular, for $p > n-1$ we characterize the trace space of the Sobolev space $W_{p}^{1}(mathbb{R}^{n})$ to the closure $overline{Omega}$ of an arbitrary open path-connected set~$Omega$. Our results extend those available for $p in (1,n]$ with much more stringent restrictions on~$S$.
Let $H_V=-Delta +V$ be a Schrodinger operator on an arbitrary open set $Omega$ of $mathbb R^d$, where $d geq 3$, and $Delta$ is the Dirichlet Laplacian and the potential $V$ belongs to the Kato class on $Omega$. The purpose of this paper is to show $L^p$-boundedness of an operator $varphi(H_V)$ for any rapidly decreasing function $varphi$ on $mathbb R$. $varphi(H_V)$ is defined by the spectral theorem. As a by-product, $L^p$-$L^q$-estimates for $varphi(H_V)$ are also obtained.
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.
If F is an infinitely differentiable function whose composition with a blowing-up belongs to a Denjoy-Carleman class C_M (determined by a log convex sequence M=(M_k)), then F, in general, belongs to a larger shifted class C_N, where N_k = M_2k; i.e., there is a loss of regularity. We show that this loss of regularity is sharp. In particular, loss of regularity of Denjoy-Carleman classes is intrinsic to arguments involving resolution of singularities.
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