Annales Fennici Mathematici

Equality of different definitions of conformal dimension for quasiself-similar and CLP spaces

Sylvester Eriksson-Bique — Mon, 24 Jun 2024 00:00:00 +0300

We prove that for a quasiself-similar and arcwise connected compact metric space all three known versions of the conformal dimension coincide: the conformal Hausdorff dimension, conformal Assouad dimension and Ahlfors regular conformal dimension. This answers a question posed by Murugan. Quasisimilar spaces include all approximately self-similar spaces. As an example, the standard Sierpiński carpet is quasiself-similar and thus the three notions of conformal dimension coincide for it.

We also give the equality of the three dimensions for combinatorially p-Loewner (CLP) spaces. Both proofs involve using a new notion of combinatorial modulus, which lies between two notions of modulus that have appeared in the literature. The first of these is the modulus studied by Pansu and Tyson, which uses a Carathéodory construction. The second is the one used by Keith and Laakso (and later modified and used by Bourdon, Kleiner, Carrasco-Piaggio, Murugan and Shanmugalingam). By combining these approaches, we gain the flexibility of giving upper bounds for the new modulus from the Pansu–Tyson approach, and the ability of getting lower bounds using the Keith–Laakso approach. Additionally the new modulus can be iterated in self-similar spaces, which is a crucial, and novel, step in our argument.

Loomis–Whitney inequalities on corank 1 Carnot groups

Ye Zhang — Mon, 01 Jul 2024 00:00:00 +0300

In this paper we provide another way to deduce the Loomis–Whitney inequality on higher dimensional Heisenberg groups \(\mathbb{H}^n\) based on the one on the first Heisenberg group \(\mathbb{H}^1\) and the known nonlinear Loomis–Whitney inequality (which has more projections than ours). Moreover, we generalize the result to the case of corank 1 Carnot groups and products of such groups. Our main tool is the modified equivalence between the Brascamp–Lieb inequality and the subadditivity of the entropy developed in Carlen and Cordero-Erausquin (2009).

On Ramanujan's modular equations and Hecke groups

Md. Shafiul Alam — Mon, 01 Jul 2024 00:00:00 +0300

Inspired by the work of Ramanujan, many people have studied generalized modular equations and the numerous identities found by Ramanujan. These identities known as modular equations can be transformed into polynomial equations. There is no developed theory about how to find the degrees of these polynomial modular equations explicitly. In this paper, we determine the degrees of the polynomial modular equations explicitly and study the relation between Hecke groups and modular equations in Ramanujan's theories of signatures 2, 3, and 4.

Neargeodesics in Gromov hyperbolic John domains in Banach spaces

Vasudevarao Allu, Abhishek Pandey — Tue, 02 Jul 2024 00:00:00 +0300

In this paper, we prove that neargeodesics in Gromov hyperbolic John domains in Banach space are cone arcs. This result gives an improvement of a result of Li (2014).

Korevaar–Schoen–Sobolev spaces and critical exponents in metric measure spaces

Fabrice Baudoin — Mon, 26 Aug 2024 00:00:00 +0300

We survey, unify and present new developments in the theory of Korevaar–Schoen–Sobolev spaces on metric measure spaces. While this theory coincides with those of Cheeger and Shanmugalingam if the space is doubling and supports a Poincaré inequality, it offers new perspectives in the context of fractals for which the approach by weak upper gradients is inadequate.

Strong barriers for weighted quasilinear equations

Takanobu Hara — Thu, 29 Aug 2024 00:00:00 +0300

In potential theory, use of barriers is one of the most important techniques. We construct strong barriers for weighted quasilinear elliptic operators. There are two applications: (i) solvability of Poisson-type equations with boundary singular data, and (ii) a geometric version of Hardy inequality. Our construction method can be applied to a general class of divergence form elliptic operators on domains with rough boundary.

Weak limit of W^1,2 homeomorphisms in R^3 can have any degree

Ondřej Bouchala, Stanislav Hencl, Zheng Zhu — Fri, 13 Sep 2024 00:00:00 +0300

In this paper for every \(k\in\mathbb{Z}\) we construct a sequence of weakly converging homeomorphisms \(h_m\colon B(0,10)\to\mathbb{R}^3\), \(h_m\rightharpoonup h\) in \(W^{1,2}(B(0,10))\), such that \(h_m(x)=x\) on \(\partial B(0,10)\) and for every \(r\in(5/16,7/16)\) the degree of \(h\) with respect to the ball \(B(0,r)\) is equal to \(k\) on a set of positive measure.

Liouville type theorems for subelliptic systems on the Heisenberg group with general nonlinearity

Vishvesh Kumar, Michael Ruzhansky, Rong Zhang — Tue, 15 Oct 2024 00:00:00 +0300

In this paper, we establish Liouville type results for semilinear subelliptic systems associated with the sub-Laplacian on the Heisenberg group \(\mathbb{H}^{n}\) involving two different kinds of general nonlinearities. The main technique of the proof is the method of moving planes combined with some integral inequalities replacing the role of maximum principles. As a special case, we obtain the Liouville theorem for the Lane–Emden system on the Heisenberg group \(\mathbb{H}^{n}\), which also appears to be a new result in the literature.

Applications of the Stone–Weierstrass theorem in the Calderón problem

Tony Liimatainen, Mikko Salo — Tue, 22 Oct 2024 00:00:00 +0300

We give examples on the use of the Stone–Weierstrass theorem in inverse problems. We show uniqueness in the linearized Calderón problem on holomorphically separable Kähler manifolds and in the Calderón problem for nonlinear equations on conformally transversally anisotropic manifolds. We also study the holomorphic separability condition in terms of plurisubharmonic functions. The Stone–Weierstrass theorem allows us to generalize and simplify earlier results. It also makes it possible to circumvent the use of complex geometrical optics solutions and inversion of explicit transforms in certain cases.

Functional equations in formal power series

Fedor Pakovich — Thu, 31 Oct 2024 00:00:00 +0200

Let \(k\) be an algebraically closed field of characteristic zero, and \(k[[z]]\) the ring of formal power series over \(k\). In this paper, we study equations in the semigroup \(z^2k[[z]]\) with the semigroup operation being composition. We prove a number of general results about such equations and provide some applications. In particular, we answer a question of Horwitz and Rubel about decompositions of "even" formal power series. We also show that every right amenable subsemigroup of \(z^2k[[z]]\) is conjugate to a subsemigroup of the semigroup of monomials.

On semi-orthogonal matrices with row vectors of equal lengths

Kalle Leppälä — Wed, 13 Nov 2024 00:00:00 +0200

When does a rectangular matrix with an orthonormal set of column vectors have row vectors of equal lengths? The column spaces of such matrices are multidimensional generalizations of the projection plane used in isometric perspective. We show that in the absence of unexpected linear relations, any rectangular matrix can be row-scaled so that if we were to orthonormalize the column vectors, the row vectors would attain equal lengths in the process. We use Grassmann coordinates to reduce the question into an instance of the famous matrix scaling problem, and with the help of existing theory introduce simple numerical solutions.

Exceptional set estimates for radial projections in R^n

Paige Bright, Shengwen Gan — Fri, 15 Nov 2024 00:00:00 +0200

We prove two conjectures in this paper. The first conjecture is by Lund, Pham and Thu: Given a Borel set \(A\subset \mathbb{R}^n\) such that \(\dim A\in (k,k+1]\) for some \(k\in\{1,\dots,n-1\}\). For \(0<s<k\), we have

\(\text{dim}(\{y\in \mathbb{R}^n \setminus A\mid \text{dim} (\pi_y(A)) < s\})\leq \max\{k+s -\dim A,0\}.\)

The second conjecture is by Liu: Given a Borel set \(A\subset \mathbb{R}^n\), then

\(\text{dim} (\{x\in \mathbb{R}^n \setminus A \mid \text{dim}(\pi_x(A))<\text{dim} A\}) \leq \lceil \text{dim} A\rceil.\)