skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


This content will become publicly available on January 2, 2026

Title: Quantitative Sobolev regularity of quasiregular maps
We quantify the Sobolev space norm of the Beltrami resolvent \((I- \mu S)^{-1}\), where \(S\) is the Beurling–Ahlfors transform, in terms of the corresponding Sobolev space norm of the dilatation \(\mu\) in the critical and supercritical ranges. Our estimate entails as a consequence quantitative self-improvement inequalities of Caccioppoli type for quasiregular distributions with dilatations in \(W^{1,p}\), \(p \ge 2\). Our proof strategy is then adapted to yield quantitative estimates for the resolvent \((I-\mu S_\Omega)^{-1}\) of the Beltrami equation on a sufficiently regular domain \(\Omega\), with \(\mu\in W^{1,p}(\Omega)\). Here, \(S_\Omega\) is the compression of \(S\) to a domain \(\Omega\). Our proofs do not rely on the compactness or commutator arguments previously employed in related literature. Instead, they leverage the weighted Sobolev estimates for compressions of Calderón–Zygmund operators to domains, recently obtained by the authors, to extend the Astala–Iwaniec–Saksman technique to higher regularities.  more » « less
Award ID(s):
2054863 1800057 2000510
PAR ID:
10612877
Author(s) / Creator(s):
; ;
Publisher / Repository:
Finnish Mathematical Society
Date Published:
Journal Name:
Annales Fennici Mathematici
Volume:
50
Issue:
1
ISSN:
2737-0690
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. ; ; ; (, Leibniz international proceedings in informatics)
    Braverman, Mark (Ed.)
    We present a framework for speeding up the time it takes to sample from discrete distributions $$\mu$$ defined over subsets of size $$k$$ of a ground set of $$n$$ elements, in the regime where $$k$$ is much smaller than $$n$$. We show that if one has access to estimates of marginals $$\mathbb{P}_{S\sim \mu}[i\in S]$$, then the task of sampling from $$\mu$$ can be reduced to sampling from related distributions $$\nu$$ supported on size $$k$$ subsets of a ground set of only $$n^{1-\alpha}\cdot \operatorname{poly}(k)$$ elements. Here, $$1/\alpha\in [1, k]$$ is the parameter of entropic independence for $$\mu$$. Further, our algorithm only requires sparsified distributions $$\nu$$ that are obtained by applying a sparse (mostly $$0$$) external field to $$\mu$$, an operation that for many distributions $$\mu$$ of interest, retains algorithmic tractability of sampling from $$\nu$$. This phenomenon, which we dub domain sparsification, allows us to pay a one-time cost of estimating the marginals of $$\mu$$, and in return reduce the amortized cost needed to produce many samples from the distribution $$\mu$$, as is often needed in upstream tasks such as counting and inference. For a wide range of distributions where $$\alpha=\Omega(1)$$, our result reduces the domain size, and as a corollary, the cost-per-sample, by a $$\operatorname{poly}(n)$$ factor. Examples include monomers in a monomer-dimer system, non-symmetric determinantal point processes, and partition-constrained Strongly Rayleigh measures. Our work significantly extends the reach of prior work of Anari and Derezi\'nski who obtained domain sparsification for distributions with a log-concave generating polynomial (corresponding to $$\alpha=1$$). As a corollary of our new analysis techniques, we also obtain a less stringent requirement on the accuracy of marginal estimates even for the case of log-concave polynomials; roughly speaking, we show that constant-factor approximation is enough for domain sparsification, improving over $O(1/k)$ relative error established in prior work. 
    more » « less
  2. ; (, SIAM journal on numerical analysis)
    We develop discrete $$W^2_p$$-norm error estimates for the Oliker--Prussner method applied to the Monge--Ampère equation. This is obtained by extending discrete Alexandroff estimates and showing that the contact set of a nodal function contains information on its second-order difference. In addition, we show that the size of the complement of the contact set is controlled by the consistency of the method. Combining both observations, we show that the error estimate $$\| u - u_h \|_{W^2_{f,p}} (\mathcal{N}^I_h)$$ converges in order $$O(h^{1/p})$$ if $p > d$ and converges in order $$O(h^{1/d} \ln (\frac 1 h)^{1/d})$$ if $$p \leq d$$, where $$\|\cdot\|_{W^2_{f,p}(\mathcal{N}^I_h)}$$ is a weighted $$W^2_p$$-type norm, and the constant $C>0$ depends on $$\|{u}\|_{C^{3,1}(\bar\Omega)}$$, the dimension $$d$$, and the constant $$p$$. Numerical examples are given in two space dimensions and confirm that the estimate is sharp in several cases. 
    more » « less
  3. ; ; ; (, 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 $$M^{1,p}(\Omega _\psi )$$. 
    more » « less
  4. ; ; ; (, New York journal of mathematics)
    The complex Green operator $$\mathcal{G}$$ on CR manifolds is the inverse of the Kohn-Laplacian $$\square_b$$ on the orthogonal complement of its kernel. In this note, we prove Schatten and Sobolev estimates for $$\mathcal{G}$$ on the unit sphere $$\mathbb{S}^{2n-1}\subset \mathbb{C}^n$$. We obtain these estimates by using the spectrum of $$\boxb$$ and the asymptotics of the eigenvalues of the usual Laplace-Beltrami operator. 
    more » « less
  5. ; ; ; (, 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. 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email