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Title: Sharp lower bounds on the manifold widths of Sobolev and Besov spaces
We consider the problem of determining the manifold $$n$$-widths of Sobolev and Besov spaces with error measured in the $$L_p$$-norm. The manifold widths control how efficiently these spaces can be approximated by general non-linear parametric methods with the restriction that the parameter selection and parameterization maps must be continuous. Existing upper and lower bounds only match when the Sobolev or Besov smoothness index $$q$$ satisfies $$q\leq p$$ or $$1 \leq p \leq 2$$. We close this gap and obtain sharp lower bounds for all $$1 \leq p,q \leq \infty$$ for which a compact embedding holds. A key part of our analysis is to determine the exact value of the manifold widths of finite dimensional $$\ell^M_q$$-balls in the $$\ell_p$$-norm when $$p\leq q$$. Although this result is not new, we provide a new proof and apply it to lower bounding the manifold widths of Sobolev and Besov spaces. Our results show that the Bernstein widths, which are typically used to lower bound the manifold widths, decay asymptotically faster than the manifold widths in many cases. more »« less
Eriksson-Bique, Sylvester; Koskela, Pekka; Malý, Jan; Zhu, Zheng
(, International Mathematics Research Notices)
null
(Ed.)
Abstract We show that the 1st-order Sobolev spaces $$W^{1,p}(\Omega _\psi ),$$ $$1<p\leq \infty ,$$ on cuspidal symmetric domains $$\Omega _\psi $$ can be characterized via pointwise inequalities. In particular, they coincide with the Hajłasz–Sobolev spaces $$M^{1,p}(\Omega _\psi )$$.
Frank, Rupert L.; Larson, Simon
(, Advances in Calculus of Variations)
Abstract We consider Lane–Emden ground states with polytropic index 0 ≤ q - 1 ≤ 1 {0\leq q-1\leq 1} , that is, minimizers of the Dirichlet integral among L q {L^{q}} -normalized functions.Our main result is a sharp lower bound on the L 2 {L^{2}} -norm of the normal derivative in terms of the energy, which implies a corresponding isoperimetric inequality.Our bound holds for arbitrary bounded open Lipschitz sets Ω ⊂ ℝ d {\Omega\subset\mathbb{R}^{d}} , without assuming convexity.
Ivanisvili, P; Nazarov, F
(, International Mathematics Research Notices)
null
(Ed.)
Abstract Let $$1\leq p \leq q <\infty $$ and let $$w \in \mathbb{C}$$. Weissler conjectured that the Hermite operator $$e^{w\Delta }$$ is bounded as an operator from $$L^{p}$$ to $$L^{q}$$ on the Hamming cube $$\{-1,1\}^{n}$$ with the norm bound independent of $$n$$ if and only if $$\begin{align*} |p-2-e^{2w}(q-2)|\leq p-|e^{2w}|q. \end{align*}$$It was proved in [ 1], [ 2], and [ 17] in all cases except $$2<p\leq q <3$$ and $$3/2<p\leq q <2$$, which stood open until now. The goal of this paper is to give a full proof of Weissler’s conjecture in the case $p=q$. Several applications will be presented.
Ifrim, Mihaela; Pineau, Ben; Tataru, Daniel; Taylor, Mitchell A
(, Annals of PDE)
Abstract We provide a complete local well-posedness theory inHsbased Sobolev spaces for the free boundary incompressible Euler equations with zero surface tension on a connected fluid domain. Our well-posedness theory includes: (i) Local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and the first proof of continuous dependence on the data, all in low regularity Sobolev spaces; (ii) Enhanced uniqueness: Our uniqueness result holds at the level of the Lipschitz norm of the velocity and the$$C^{1,\frac{1}{2}}$$regularity of the free surface; (iii) Stability bounds: We construct a nonlinear functional which measures, in a suitable sense, the distance between two solutions (even when defined on different domains) and we show that this distance is propagated by the flow; (iv) Energy estimates: We prove refined, essentially scale invariant energy estimates for solutions, relying on a newly constructed family of elliptic estimates; (v) Continuation criterion: We give the first proof of a sharp continuation criterion in the physically relevant pointwise norms, at the level of scaling. In essence, we show that solutions can be continued as long as the velocity is in$$L_T^1W^{1,\infty}$$and the free surface is in$$L_T^1C^{1,\frac{1}{2}}$$, which is at the same level as the Beale-Kato-Majda criterion for the boundaryless case; (vi) A novel proof of the construction of regular solutions. Our entire approach is in the Eulerian framework and can be adapted to work in more general fluid domains.
Dodson, Benjamin
(, International Mathematics Research Notices)
Abstract In this note, we prove scattering for a defocusing nonlinear Schrödinger equation with initial data lying in a critical Besov space. In addition, we obtain polynomial bounds on the scattering size as a function of the critical Besov norm.
Siegel, Jonathan W. Sharp lower bounds on the manifold widths of Sobolev and Besov spaces. Retrieved from https://par.nsf.gov/biblio/10519854. Journal of Complexity 85.C Web. doi:10.1016/j.jco.2024.101884.
Siegel, Jonathan W. Sharp lower bounds on the manifold widths of Sobolev and Besov spaces. Journal of Complexity, 85 (C). Retrieved from https://par.nsf.gov/biblio/10519854. https://doi.org/10.1016/j.jco.2024.101884
@article{osti_10519854,
place = {Country unknown/Code not available},
title = {Sharp lower bounds on the manifold widths of Sobolev and Besov spaces},
url = {https://par.nsf.gov/biblio/10519854},
DOI = {10.1016/j.jco.2024.101884},
abstractNote = {We consider the problem of determining the manifold $n$-widths of Sobolev and Besov spaces with error measured in the $L_p$-norm. The manifold widths control how efficiently these spaces can be approximated by general non-linear parametric methods with the restriction that the parameter selection and parameterization maps must be continuous. Existing upper and lower bounds only match when the Sobolev or Besov smoothness index $q$ satisfies $q\leq p$ or $1 \leq p \leq 2$. We close this gap and obtain sharp lower bounds for all $1 \leq p,q \leq \infty$ for which a compact embedding holds. A key part of our analysis is to determine the exact value of the manifold widths of finite dimensional $\ell^M_q$-balls in the $\ell_p$-norm when $p\leq q$. Although this result is not new, we provide a new proof and apply it to lower bounding the manifold widths of Sobolev and Besov spaces. Our results show that the Bernstein widths, which are typically used to lower bound the manifold widths, decay asymptotically faster than the manifold widths in many cases.},
journal = {Journal of Complexity},
volume = {85},
number = {C},
publisher = {Journal of Complexity},
author = {Siegel, Jonathan W},
}
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