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Register a new userAlmost sharp descriptions of traces of Sobolev $W_{p}^{1}(mathbb{R}^{n})$-spaces to arbitrary compact subsets of $mathbb{R}^{n}$. The case $p in (1,n]$
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Authors
Alexander Tyulenev
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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.
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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~$S$.
We construct explicit examples of Frostman-type measures concentrated on arbitrary planar rectifiable curves of positive length. Based on such constructions we obtain for each $p in (1,infty)$ an exact description of the trace space of the first-order Sobolev space $W^{1}_{p}(mathbb{R}^{2})$ to an arbitrary planar rectifiable curve $Gamma subset mathbb{R}^{2}$ of positive length.
By employing the $1/N$ expansion, we compute the vacuum energy~$E(deltaepsilon)$ of the two-dimensional supersymmetric (SUSY) $mathbb{C}P^{N-1}$ model on~$mathbb{R}times S^1$ with $mathbb{Z}_N$ twisted boundary conditions to the second order in a SUSY-breaking parameter~$deltaepsilon$. This quantity was vigorously studied recently by Fujimori et al. using a semi-classical approximation based on the bion, motivated by a possible semi-classical picture on the infrared renormalon. In our calculation, we find that the parameter~$deltaepsilon$ receives renormalization and, after this renormalization, the vacuum energy becomes ultraviolet finite. To the next-to-leading order of the $1/N$ expansion, we find that the vacuum energy normalized by the radius of the~$S^1$, $R$, $RE(deltaepsilon)$ behaves as inverse powers of~$Lambda R$ for~$Lambda R$ small, where $Lambda$ is the dynamical scale. Since $Lambda$ is related to the renormalized t~Hooft coupling~$lambda_R$ as~$Lambdasim e^{-2pi/lambda_R}$, to the order of the $1/N$ expansion we work out, the vacuum energy is a purely non-perturbative quantity and has no well-defined weak coupling expansion in~$lambda_R$.
In the leading order of the large-$N$ approximation, we study the renormalon ambiguity in the gluon (or, more appropriately, photon) condensate in the 2D supersymmetric $mathbb{C}P^{N-1}$ model on~$mathbb{R}times S^1$ with the $mathbb{Z}_N$ twisted boundary conditions. In our large~$N$ limit, the combination $Lambda R$, where $Lambda$ is the dynamical scale and $R$~is the $S^1$ radius, is kept fixed (we set $Lambda Rll1$ so that the perturbative expansion with respect to the coupling constant at the mass scale~$1/R$ is meaningful). We extract the perturbative part from the large-$N$ expression of the gluon condensate and obtain the corresponding Borel transform~$B(u)$. For~$mathbb{R}times S^1$, we find that the Borel singularity at~$u=2$, which exists in the system on the uncompactified~$mathbb{R}^2$ and corresponds to twice the minimal bion action, disappears. Instead, an unfamiliar renormalon singularity emph{emerges/} at~$u=3/2$ for the compactified space~$mathbb{R}times S^1$. The semi-classical interpretation of this peculiar singularity is not clear because $u=3/2$ is not dividable by the minimal bion action. It appears that our observation for the system on~$mathbb{R}times S^1$ prompts reconsideration on the semi-classical bion picture of the infrared renormalon.
Let $p(cdot): mathbb R^nto(0,infty)$ be a variable exponent function satisfying the globally log-Holder continuous condition. In this article, the authors first obtain a decomposition for any distribution of the variable weak Hardy space into good and bad parts and then prove the following real interpolation theorem between the variable Hardy space $H^{p(cdot)}(mathbb R^n)$ and the space $L^{infty}(mathbb R^n)$: begin{equation*} (H^{p(cdot)}(mathbb R^n),L^{infty}(mathbb R^n))_{theta,infty} =W!H^{p(cdot)/(1-theta)}(mathbb R^n),quad thetain(0,1), end{equation*} where $W!H^{p(cdot)/(1-theta)}(mathbb R^n)$ denotes the variable weak Hardy space. As an application, the variable weak Hardy space $W!H^{p(cdot)}(mathbb R^n)$ with $p_-:=mathopmathrm{ess,inf}_{xinrn}p(x)in(1,infty)$ is proved to coincide with the variable Lebesgue space $W!L^{p(cdot)}(mathbb R^n)$.
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