Annales Fennici Mathematici

Quantitative Sobolev regularity of quasiregular maps

Francesco Di Plinio — 2025-01-02

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.

Tent spaces and solutions of Weinstein type equations with CMO(R_+,dm_λ) boundary values

Jorge J. Betancor — 2025-01-09

Let \(\{P_{t}^{[\lambda]}\}_{t>0}\) be the Poisson semigroup associated with the Bessel operator \(\Delta_{\lambda}\) on \(\mathbb{R}_+:=(0,\infty)\), where \(\lambda>0\) and

\(\Delta_{\lambda}:=-x^{-2\lambda}\frac{d}{dx}x^{2\lambda}\frac{d}{dx}\).

In this paper, the authors show that a function \(u(y,t)\) on \(\mathbb{R}_{+}\times\mathbb{R}_{+}\), has the form \(u(y,t)=P_{t}^{[\lambda]}f(y)\) with \(f\in\) CMO\((\mathbb{R}_{+},dm_{\lambda})\), where \(dm_{\lambda}(x):=x^{2\lambda}\,dx\), if and only if \(u\) satisfies the Weinstein type equation

\(\mathbb{L}_{\lambda}u(x,t):=\frac{\partial^{2}u(x,t)}{\partial t^{2}}-\Delta_{\lambda}u(x,t)=0\), \((x,t)\in{\mathbb{R}_{+}\times\mathbb{R}_{+}}\),

a Carleson type condition and certain limiting conditions. For this purpose, the authors first introduce the tent spaces \(T_{2}^{p}\) with \(p\in[1,\infty]\) and \(T_{2,C}^{\infty}\) in the Bessel setting and then show that CMO\((\mathbb{R}_{+},dm_{\lambda})\) has a connection with \(T_{2,C}^{\infty}\) via \(\{P_{t}^{[\lambda]}\}_{t>0}\). In addition, the authors obtain some boundedness results on the operator \(\pi_{\lambda}\) from tent spaces to some "ordinary" function spaces.

A note on summability in Banach spaces

José Rodríguez — 2025-01-30

Let \(Z\) and \(X\) be Banach spaces. Suppose that \(X\) is Asplund. Let \(\mathcal{M}\) be a bounded set of operators from \(Z\) to \(X\) with the following property: a bounded sequence \((z_n)_{n\in \mathbb{N}}\) in \(Z\) is weakly null if, for each \(M\in\mathcal{M}\), the sequence \((M(z_n))_{n\in\mathbb{N}}\) is weakly null. Let \((z_n)_{n\in\mathbb{N}}\) be a sequence in \(Z\) such that: (a) for each \(n\in\mathbb{N}\), the set \(\{M(z_n)\colon M\in \mathcal{M}\}\) is relatively norm compact; (b) for each sequence \((M_n)_{n\in\mathbb{N}}\) in \(\mathcal{M}\), the series \(\sum_{n=1}^\infty M_n(z_n)\) is weakly unconditionally Cauchy. We prove that if \(T\in \mathcal{M}\) is Dunford–Pettis and \(\inf_{n\in\mathbb{N}}\|T(z_n)\|\|z_n\|^{-1}>0\), then the series \(\sum_{n=1}^\infty T(z_n)\) is absolutely convergent. As an application, we provide another proof of the fact that a countably additive vector measure taking values in an Asplund Banach space has finite variation whenever its integration operator is Dunford–Pettis.

The dimension of projections of planar diagonal self-affine measures

Aleksi Pyörälä — 2025-02-07

We show that if \(\mu\) is a self-affine measure on the plane defined by an iterated function system of contractions with diagonal linear parts, then under an irrationality assumption on the entries of the linear parts,

 
\(\operatorname{dim}_{\rm H} \mu \circ \pi^{-1} = \min \{ 1, \operatorname{dim}_{\rm H} \mu \}\)

for any non-principal orthogonal projection \(\pi\).

Algebraic curves and meromorphic functions sharing pairs of values

Norbert Steinmetz — 2025-03-03

The 4IM+1CM-problem is to determine all pairs \((f,g)\) of meromorphic functions in the complex plane that are not Möbius transformations of each other and share five pairs of values, one of them CM (counting multiplicities). In the present paper it is shown that each such pair parameterises some algebraic curve \(K(x,y)=0\) of genus zero and low degree. Thus the search may be restricted to the pairs of meromorphic functions \((Q(e^z),\widetilde Q(e^z))\), where \(Q\) and \(\widetilde Q\) are non-constant rational functions of low degree. This leads to the paradoxical situation that the 4IM+1CM-problem could be solved by a computer algebra virtuoso rather than a complex analyst.

On the volumes of simplices determined by a subset of R^d

Pablo Shmerkin — 2025-03-13

We prove that for \(1\le k<d\), if \(E\) is a Borel subset of \(\mathbb{R}^d\) of Hausdorff dimension strictly larger than \(k\), the set of \((k+1)\)-volumes determined by \(k+2\) points in \(E\) has positive one-dimensional Lebesgue measure. In the case \(k=d-1\), we obtain an essentially sharp lower bound on the dimension of the set of tuples in \(E\) generating a given volume. We also establish a finer version of the classical slicing theorem of Marstrand–Mattila in terms of dimension functions, and use it to extend our results to sets of "dimension logarithmically larger than \(k\)".

Partition function for the 2d Coulomb gas on a Jordan curve

Klara Courteaut — 2025-03-13

We prove an asymptotic formula for the partition function of a 2d Coulomb gas at inverse temperature \(\beta>0\), confined to lie on a Jordan curve. The partition function can include a linear statistic. The asymptotic formula involves a Fredholm determinant related to the Loewner energy of the curve, and also an expression involving the sampling function, the exterior conformal map for the curve and the Grunsky operator. The asymptotic formula also gives a central limit theorem for linear statistics of the particles in the gas.

A growth estimate for the planar Mumford–Shah minimizers at a tip point: An alternative proof of David–Léger

Yi Ru-Ya Zhang — 2025-03-25

Let \(\Omega\subset \mathbb{R}^2\) be a bounded domain and \(u\in SBV(\Omega)\) be a local minimizer of the Mumford–Shah problem in the plane, with \(0\in \overline{S}_u\) being a tip point and \(B_1\subset \Omega\). Then there exist absolute constants \(C>0\) and \(0<r_0<1\) such that


\(|u(x)-u(0)|\le C r^{1/2}\) for any \(x\in B_r\) and \(0<r<r_0\).

This estimate is a local version of the original one in David–Léger (2002, Proposition 10.17). Our result is based on a dichotomy and the John structure of \(\Omega\setminus \overline{S}_u\), different from the one by David–Léger (2002) or Bonnet–David (2001, Lemma 21.3).

Carrot John domains in variational problems

Weicong Su — 2025-03-25

In this paper, we explore carrot John domains within variational problems, dividing our examination into two distinct sections. The initial part is dedicated to establishing the lower semicontinuity of the (optimal) John constant with respect to Hausdorff convergence for bounded John domains. This result holds promising implications for both shape optimization problems and Teichmüller theory. In the subsequent section, we demonstrate that an unbounded open set satisfying the carrot John condition with a center at \(\infty\), appearing in the Mumford–Shah problem, can be covered by a uniformly finite number of unbounded John domains (defined conventionally through cigars). These domains, in particular, support Sobolev–Poincaré inequalities.

On the existence of L^p-optimal transport maps for norms on R^N

Guoxi Liu — 2025-03-26

In this paper, we prove existence of \(L^p\)-optimal transport maps with \(p\in (1,\infty)\) in a class of branching metric spaces defined on \(\mathbb{R}^N\). In particular, we introduce the notion of cylinder-like convex function and we prove an existence result for the Monge problem with cost functions of the type \(c(x, y) = f(g(y - x))\), where \(f\colon [0, \infty) \to [0, \infty)\) is an increasing strictly convex function and \(g\colon \mathbb{R}^N \to [0, \infty)\) is a cylinder-like convex function. When specialised to cylinder-like norm, our results shows existence of \(L^p\)-optimal transport maps for several "branching" norms, including all norms in \(\mathbb{R}^2\) and all crystalline norms.

Continuity of solutions to complex Hessian equations via the Dinew–Kołodziej estimate

Per Åhag — 2025-03-28

This study extends the celebrated volume-capacity estimates of Dinew and Kołodziej, providing a foundation for examining the regularity of solutions to boundary value problems for complex Hessian equations. By integrating the techniques established by Dinew and Kołodziej and incorporating recent advances by Charabati and Zeriahi, we demonstrate the continuity of the solutions.