Annales Fennici Mathematici was founded in 1941 by P.J. Myrberg under the name Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica. The journal was owned and published by Academia Scientiarum Fennica until 2021 when the Finnish Mathematical Society took over as the owner. Annales Fennici Mathematici publishes original research papers in all fields of mathematics. Historically the emphasis has been on analysis. One volume, divided into two issues, is published annually.
Revisiting cyclic elements in growth spaces
We revisit the problem of characterizing cyclic elements for the shift operator in a broad class of radial growth spaces of holomorphic functions on the unit disk, focusing on functions of finite Nevanlinna characteristic. We provide results in the range of Dini regular weights, and in the regime of logarithmic integral divergence. Our proofs are largely constructive and allow for substantial simplifications of earlier works that previously relied on the Carleson Corona Theorem, such as the Korenblum–Roberts Theorem, as well as a more recent result of El-Fallah, Kellay and Seip.
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On \(L^p \to L^q\) infinitesimal relative boundedness of Schrödinger operators \((-\Delta)^{\alpha/2}+v\)
By analyzing the trace inequality for Bessel potentials, some Morrey-type sufficient conditions are given for which \(L^p \to L^q\), \(1<p,q<\infty\), infinitesimal relative boundedness of the Schrödinger operators \((-\Delta)^{\alpha/2}+v\) holds. These results provide new aspects of Morrey spaces and a nice application of weight theory.
A step towards the tensorization of Sobolev spaces
We prove that Sobolev spaces on Cartesian and warped products of metric spaces tensorize, only requiring that one of the factors is a doubling space supporting a Poincaré inequality.
A variant of inverse mean curvature flow for star-shaped hypersurfaces evolving in a cone
Given a smooth convex cone in the Euclidean \((n+1)\)-space (\(n\geq 2\)), we consider strictly mean convex hypersurfaces with boundary which are star-shaped with respect to the center of the cone and which meet the cone perpendicularly. If those hypersurfaces inside the cone evolve by a variant of inverse mean curvature flow, then, by using the convexity of the cone in the derivation of the gradient and Hölder estimates, we can prove that this evolution exists for all the time and the evolving hypersurfaces converge smoothly to a piece of a round sphere as time tends to infinity.
Anisotropic weighted Levin–Cochran–Lee type inequalities on homogeneous Lie groups
In this paper, we first prove the weighted Levin–Cochran–Lee type inequalities on homogeneous Lie groups for arbitrary weights, quasi-norms, and \(L^p\)- and \(L^q\)-norms. Then, we derive a sharp weighted inequality involving specific weights given in the form of quasi-balls in homogeneous Lie groups. Finally, we also calculate the sharp constants for the aforementioned inequalities.
anisotropic Levin–Cochran–Lee inequalities
Directional pliability, Whitney extension, and Lusin approximation for curves in Carnot groups
We show that, in arbitrary Carnot groups, pliability in a subset of directions is sufficient to guarantee the existence of a Whitney-type extension and a Lusin approximation for curves with tangent vectors in the same set of directions. We apply this to show that every horizontal curve in the Engel group must intersect a \(C^{1}\) horizontal curve in a set of positive measure.
Limits of manifolds with a Kato bound on the Ricci curvature. II
We prove that metric measure spaces obtained as limits of closed Riemannian manifolds with Ricci curvature satisfying a uniform Kato bound are rectifiable. In the case of a non-collapsing assumption and a strong Kato bound, we additionally show that for any \(\alpha \in (0,1)\) the regular part of the space lies in an open set with the structure of a \(C^\alpha\)-manifold.
Generalized Schwarzians and normal families
We study families of analytic and meromorphic functions with bounded generalized Schwarzian derivative \(S_k(f)\). We show that these families are quasi-normal. Further, we investigate associated families, such as those formed by derivatives and logarithmic derivatives, and prove several (quasi-)normality results. Moreover, we derive a new formula for \(S_k(f)\), which yields a result for families \(F\subseteq H\mathbb{D}\) of locally univalent functions that satisfy \(S_k(f)(z)\neq b(z)\) for some \(b\in M(\mathbb{D})\) and all \(f\in F\), \(z\in\mathbb{C}\) and for entire functions \(g\) with \(S_k(g)(z)\neq 0\) and \(S_k(g)(z)\neq\infty\) for all \(z\in\mathbb{C}\). The classical Schwarzian derivative \(S_f\) is contained as the case \(k=2\).
Off-diagonal matrix extrapolation for Muckenhoupt bases
In this paper we extend the theory of Rubio de Francia extrapolation for matrix weights, recently introduced by Bownik and the first author (2022), to off-diagonal extrapolation. We also show that the theory of matrix weighted extrapolation can be extended to matrix \(A_p\) classes defined with respect to a general basis, provided that a version of the Christ–Goldberg maximal operator is assumed to be bounded. Finally, we extend a recent result by Vuorinen (2024) and show that all of the multiparameter bases have this property.
Carathéodory convergence and the conformal type problem
We study Carathédory convergence for open, simply connected surfaces spread over the sphere and, in particular, provide examples demonstrating that in the Speiser class the conformal type can change when two singular values collide.
Greedy matching in optimal transport with concave cost
We consider the optimal transport problem between a set of \(n\) red points and a set of \(n\) blue points subject to a concave cost function such as \(c(x,y) = \|x-y\|^p\) for \(0< p < 1\). Our focus is on a particularly simple matching algorithm: match the closest red and blue point, remove them both and repeat. We prove that it provides good results in any metric space \((X,d)\) when the cost function is \(c(x,y) = d(x,y)^{p}\) with \(0 < p < 1/2\). Empirically, the algorithm produces results that are remarkably close to optimal—especially as the cost function gets more concave; this suggests that greedy matching may be a good toy model for optimal transport for very concave transport cost.
Matrix-weighted bounds in variable Lebesgue spaces
In this paper we prove boundedness of Calderón–Zygmund operators and the Christ–Goldberg maximal operator in the matrix-weighted variable Lebesgue spaces recently introduced by Cruz-Uribe and the second author. Our main tool to prove these bounds is through bounding a Goldberg auxiliary maximal operator. As an application, we obtain a quantitative extrapolation theorem for matrix-weighted variable Lebesgue spaces from the recent framework of directional Banach function spaces of the first author.
A dual simple proof of the classical Bernstein and Calabi–Bernstein theorems
In this note, we present a short and simple proof of both the Bernstein theorem and the Calabi–Bernstein theorem, which allows us to visualize both the common features and the remarkable differences between the two results. The proofs are based on the construction of conformal metrics that ensure the parabolicity of the surfaces. Consequently, they do not rely on complex analysis.
Sharp Fourier decay estimates for measures supported on the well-approximable numbers
We construct a measure on the well-approximable numbers whose Fourier transform decays at a nearly optimal rate. This gives a logarithmic improvement on a previous construction of Kaufman.
Exceptional set estimates in finite fields
We study the exceptional set estimate for projections in \(\mathbb{F}_q^n\). For each \(V\in G(k,\mathbb{F}^n_q)\), let \(\pi_V\colon \mathbb{F}_q^n\rightarrow V\) be the projection map. We prove the following result: If \(A\subset \mathbb{F}_q^n\) with \(\#A=q^a\) (\(n-1\le a\le n\)) and \(0< s<\frac{a+n-2}{2}\), then \(\# \{V\in G(n-1,\mathbb{F}^n_q)\colon \#\pi_V(A)< q^s \}\lessapprox q^{n-2}\). This improves the previous range \(0<s<\frac{n-1}{n}a\). Also, our range of \(s\) is sharp in the sense that if \(s>\frac{a+n-2}{2}\), then the right hand side above should be at least \(q^t\) for some \(t>n-2\).
Two footnotes to the F. & M. Riesz theorem
We present a new proof of the F. & M. Riesz theorem on analytic measures of the unit circle \(\mathbb{T}\) that is based the following elementary inequality: If \(f\) is analytic in the unit disc \(\mathbb{D}\) and \(0 \leq r \leq \varrho < 1\), then \(\|f_r-f_\varrho\|_1 \leq 2 \sqrt{\|f_\varrho\|_1^2-\|f_r\|_1^2}\), where \(f_r(e^{i\theta})=f(r e^{i\theta})\) and where \(\|\cdot\|_1\) denotes the norm of \(L^1(\mathbb{T})\). The proof extends to the infinite-dimensional torus \(\mathbb{T}^\infty\), where it clarifies the relationship between Hilbert's criterion for \(H^1(\mathbb{T}^\infty)\) and the F. & M. Riesz theorem.
On bounded energy of convolution of fractal measures
For all \(s\in[0,1]\) and \(t\in(0,s]\cup [2-s,2)\), we find the supremum of numbers \(\omega\in(0,2)\) such that \(\textup{I}_\omega(\mu\ast\sigma) \lesssim 1,\) where \(\mu\) is any Borel measure on \(B(1)\) with \(\textup{I}_t(\mu)\leq 1\) and \(\sigma\) is any \((s,1)\)-Frostman measure on a \(C^2\)-graph with non-zero curvature. As an application, we use this to show the sharp \(L^6\)-decay of Fourier transform of \(\sigma\) when \(s\in [\frac{2}{3}, 1]\).
The aim of this note is to give a short proof of a result of Pyörälä–Shmerkin–Suomala–Wu; the Sierpiński carpet, and generalisations, are tube-null; they can be covered with tubes of arbitrarily small total width. We remark that a more general class of sponge-like sets satisfy this property. For a given \(\epsilon > 0\) the proof is able to give an explicit description of the tubes for which the total width is less than \(\epsilon\).
Conformal and holomorphic barycenters in hyperbolic balls
We introduce the notions of conformal barycenter and holomorphic barycenter of a measurable set \(D\) in the hyperbolic ball. The two barycenters coincide in the disk, but they differ in multidimensional balls of \(\mathbb{C}^m \cong \mathbb{R}^{2m}\). These notions are counterparts of barycenters of measures on spheres, introduced by Douady and Earle in 1986.
A counting formula for extreme contractions on regular polygonal Banach spaces
A regular polygonal Banach space \(X\) is a two-dimensional real Banach space whose unit sphere is a regular polygon. We prove that the number of extreme contractions on a regular \(2n\)-gonal Banach space is \(2n^2+4n\). A characterization of extreme contractions on \(X\) is also presented.
A deterministic proof of Loewner energy reversibility via local reversals
We give a new proof of the orientation reversibility of chordal Loewner energy by reversing the orientation of a chord in partial increments. This fact was first proved by Wang (2019) using the reversibility of chordal Schramm–Loewner evolution (SLE) along with the interpretation of Loewner energy as the large deviation rate function of chordal SLE\(_\kappa\) as \(\kappa \to 0\). Our method is similar in spirit to Zhan's proof (2008) of chordal SLE\(_\kappa\) reversibility for \(\kappa \in (0,4]\), though it is purely deterministic. As a key step in our proof, we establish that a minimal energy chord among those passing through a fixed finite set of points is a piecewise hyperbolic geodesic.
Finite dimensionality of Besov spaces and potential-theoretic decomposition of metric spaces
In the context of a metric measure space \((X,d,\mu)\), we explore the potential-theoretic implications of having a finite-dimensional Besov space. We prove that if the dimension of the Besov space \(B^\theta_{p,p}(X)\) is \(k>1\), then \(X\) can be decomposed into \(k\) number of irreducible components (Theorem 1.1). Note that \(\theta\) may be bigger than \(1\), as our framework includes fractals. We also provide sufficient conditions under which the dimension of the Besov space is 1. We introduce critical exponents \(\theta_p(X)\) and \(\theta_p^{\ast}(X)\) for the Besov spaces. As examples illustrating Theorem 1.1, we compute these critical exponents for spaces \(X\) formed by glueing copies of \(n\)-dimensional cubes, the Sierpiński gaskets, and of the Sierpiński carpet.
The Stepanov theorem for Q-valued functions
In this work we prove the Stepanov differentiation theorem for multiple-valued functions. This theorem is proved in the wide generality of metric-space-multiple-valued functions without relying on a Lipschitz extension result. General definitions of differentiability and approximate differentiability for functions between suitable metric spaces are also introduced.
Normalized solutions for logarithmic Schrödinger equation with a perturbation of power law nonlinearity
We study the existence of normalized solutions to the following logarithmic Schrödinger equation \(-\Delta u+\lambda u=\alpha u\log u^2+\mu|u|^{p-2}u\), \(x\in\mathbb{R}^N\), under the mass constraint \(\int_{\mathbb{R}^N}u^2\,\mathrm{d}x=c^2\), where \(\alpha,\mu\in\mathbb{R}\), \(N\ge 2\), \(p>2\), \(c>0\) is a constant, and \(\lambda\in\mathbb{R}\) appears as Lagrange multiplier. Under different assumptions on \(\alpha\), \(\mu\), \(p\) and \(c\), we prove the existence of ground state solutions and excited state solutions. The asymptotic behavior of the ground state solution as \(\mu\to 0\) is also investigated. Our results include the case \(\alpha<0\) or \(\mu<0\), which is less studied in the literature.
Unique extremality of affine maps on plane domains
We prove that affine maps are uniquely extremal quasiconformal maps on the complement of a well distributed set in the complex plane answering a conjecture from Marković (2002). We construct the required Reich sequence using Bergman projections, and meromorphic partitions of unity.
On the sharp second order Caffarelli–Kohn–Nirenberg inequality
The purpose of this paper is twofold. By developing a new technique, we first prove some sharp Caffarelli–Kohn–Nirenberg inequalities of second order for non-radial functions. As a direct consequence, we recover the sharp second order uncertainty principle in Cazacu, Flynn and Lam (2022). For the class of radial functions, we next establish a sharp inequality which can be regarded as the second order version of Xia's inequality (2007).
Logarithmic vanishing theorems on weakly 1-complete Kähler manifolds
This paper aims to investigate the vanishing theorems of cohomology groups on weakly 1-complete Kähler manifolds. Firstly, we extend a logarithmic vanishing theorem originally proposed by Huang, Liu, Wan and Yang, which says that when \(N\otimes \mathcal{O}_X([\Delta])\) is a \(k\)-positive \(\mathbb{R}\)-line bundle, the \(q\)-th cohomology group of sheaf \(\Omega^{p}(\log D)\otimes L\otimes N\) vanishes for any \(p+q\geq n+k+1\) over a compact Kähler manifold \(X\). The proof employs an approximation theorem. In the local case, the result follows directly as a corollary of the original theorem. In the global case, we construct a solution for any positive real number \(c\), ensuring that the error term with respect to the exact solution converges to zero in the sense of norm. Finally, as applications of our results, we obtain logarithmic generalizations of several classical vanishing theorems and the corresponding corollaries.
A clustering theorem in fractional Sobolev spaces
We prove a general clustering result for the fractional Sobolev space \(W^{s,p}\): whenever the positivity set of a function \(u\) in a cube has measure bounded from below by a multiple of the cube's volume, and the \(W^{s,p}\)-seminorm of \(u\) is bounded from above by a convenient power of the cube's side, then \(u\) is positive in a universally reduced cube. Our result aims at applications in regularity theory for fractional elliptic and parabolic equations. Also, by means of suitable interpolation inequalities, we show that clustering results in \(W^{1,p}\) and \(BV\), respectively, can be deduced as special cases.
CMC hypersurfaces with polynomial volume growth in warped products and the nonexistence of entire solutions to the minimal hypersurface equation
We investigate constant mean curvature (CMC) complete two-sided hypersurfaces with polynomial volume growth in a class of warped products satisfying a suitable curvature constraint. In this setting, we establish the nonexistence of such a CMC hypersurface under mild hypotheses involving the mean curvature and the warping function. Applications to Einstein warped product, pseudo-hyperbolic, Schwarzschild and Reissner–Nordström spaces are also given. Furthermore, we present a nonparametric version of our main result which, in particular, guarantees the nonexistence of entire solutions with finite C2 norm of the the minimal hypersurface equation on a complete Riemannian manifold with polynomial volume growth.
constant mean curvature two-side hypersurfaces
The Landau–Bloch type theorems for certain class of holomorphic and pluriharmonic mappings in C^n
In this paper, we define two classes of holomorphic mappings defined on the unit ball \(B^n\) of \(n\)-dimensional complex space \(\mathbb{C}^n\) and obtain the lower estimates for Bloch's constant for these classes. Also, we derive the Landau–Bloch type theorem for some subclasses of pluriharmonic mappings defined on the unit ball \(B^n\).