You-Cheng Chou, Chin-Lung Wang, Po-Sheng Wu
We study the generalized Lamé equation on an elliptic curve $E$ with multiple singularities. By restricting to the locus admitting solutions with quasi-periodic properties, we construct two curves: (i) The generalized Lam'e curve: with $n:=\sum\nolimits_{i=1}^r n_i\in\mathbb Z_{\geq 0}$, we construct $\mathcal{Y}_{\mathbf n}(\mathbf p;τ)$, which lies in an affine bundle over $Sym^n E$ and parametrizes generalized Hermite--Halphen ansatz solutions. (ii) The log-free curve: each $n_i\in\frac12\mathbb N$ gives a polynomial equation in the accessory parameters. This leads to a non-complete intersection variety $V_{\mathbf{n}}(\mathbf{p};τ)$ when all $n_i\in\frac12\mathbb N$. We prove that it is a reduced curve. We analysis the GLC as an algebraic family over the pole configuration space $\mathbf{p}$. We study the shifted addition map \[ σ: Sym^n E\longrightarrow E, \] establishing a generically finite, degree formula. The geometry of boundary degenerations under pole collisions perfectly mirrors the tensor algebra of $\mathfrak{sl}_2(\mathbb{C})$-modules within the BGG category $\mathcal{O}$. We generalize pre-modular forms to a framework of twisted isomonodromic deformations. We construct $(\mathbf{n}, \mathbf{p})$-deformed pre-modular forms parameterized by pseudo-monodromy data $(t,s)$, whose vanishing governs these deformations and factorizes along boundary strata. Iterating these deformations through the boundary allows any arbitrary configuration to be continuously deformed down to the classical Lamé equation. Finally, we prove the Treibich conjecture stated for $r=2$ extra symmetric pairs, as well as its generalizations for $r \leq 4$.
Alberto Dayan, Adrián Llinares, Miguel Monsalve-López
We study properties of $A^p_α$ spaces in the Dirichlet range, recently defined by Brevig, Kulikov, Seip and Zlotnikov as the set of all holomorphic functions on the unit disc $\mathbb{D}$ such that \[ \int_{\mathbb{D}} |f(z)|^{p-2} |f'(z)|^2 (1 - |z|^2)^α \, dA(z) < \infty, \] when $0<α< 1$ and $p > 0$. We answer in the negative two questions posed by Brevig et al. by showing that, if $p\ne2$ and $p > \frac{1}{2}$, $A^p_α$ is not a vector space and that the norm is in general not increasing in $p$. This is achieved by means of an equivalent description for $A^p_α$ which is given in terms of the Poisson integral of the boundary function of its inhabitants. Such norm also leads to a description of $A^p_α$ functions in the Dirichlet range given in terms of their inner and outer factors. As a corollary, we show that $A^1_α$ is contained in the weak product of a Dirichlet-type space.
Yingying Cai, Xavier Tolsa
We provide quantitative estimates for the dimension drop of harmonic measure. We show that for a domain $Ω= \mathbb{R}^{n+1} \setminus E$ where $E$ is an $s$-Ahlfors regular compact set satisfying a uniform $L^2$-based non-flatness condition $β_2 \ge δ_0$, the dimension of its harmonic measure is strictly less than $s$ for $s \in (n - cδ_0^2, n]$. For planar domains, we establish an analogous quantitative threshold $s_0 = 1 - cδ_0^2$ under Azzam's uniform non-flatness condition $β_\infty + β_{\operatorname{hole}} \ge δ_0$.
Giovanni Savaré
The equilibrium shape of a thin, elastic, inextensible ribbon minimizes its bending energy. It has been shown that, as the width of the ribbon tends to zero, the bending energy Gamma-converges to the so called Sadowsky functional. In this paper we consider geometrically frustrated anisotropic ribbons with a possibly curved reference configuration. We prove that the Gamma-convergence remains valid under prescribed affine boundary conditions, including, in particular, those satisfied by a Möbius strip.
Ovidiu Costin, Gerald V. Dunne, Ali Saraeb
We develop a resurgent approach to the problem of unique continuation of mock theta functions across their natural boundary. The starting point is the representation of the associated Mordell-Appell integrals as Laplace transforms of resurgent functions, which serve as the primary analytic objects. By rotating the Laplace contour by $π$, i.e. onto the Stokes line, one obtains, in all known cases, the mock-modular relations between the Mordell-Appell integrals and the corresponding unary series in $\hat q=e^{-πi τ}$ and $\hat q_1=e^{-πi (-1/τ)}$. We then prove that these relations admit a unique solution on the $q$-side, expressed in terms of $q=e^{πi τ}$ and $q_1=e^{πi (-1/τ)}$, with coefficients determined by the corresponding Mordell-Appell integrals. This yields a canonical continuation across the natural boundary, given by a resurgent extension of the classical principle of permanence of relations, and singles out a distinguished family of mock theta functions in each group. We present a complete analysis for the order 3 and 5 cases (mf3 and mf5). The method extends naturally to higher orders; a general theory will appear in a separate paper.
Hugo Aimar, Ivana Gómez, Joaquín Toledo
We introduce a natural bilinear fractional integral type operator induced by a third order hypermetric on Ahlfors regular quasi-metric spaces. Given a quasi-metric space $(X,d)$ the function $ρ(x,y,z)$, defined as the distance, in $X^3$, of $(x,y,z)$ to the diagonal $\bigtriangleup_3=\{(x,x,x)\in X^3:x\in X\}$ is said to be a third order hypermetric in $X$. When $(X,d)$ is a Euclidean space or, more generally, when $(X,d,μ)$ is $η$-Ahlfors regular for some $η$ positive, the function $ρ(x,y,z)$ generates kernels for bilinear operators of the type $T^γ(f,g)(x)=\iint_{X\times X}ρ(x,y,z)^{-γ}f(y)g(z)dμ(y)dμ(z)$, for a given positive $γ$. In the setting of $η$-Ahlfors regular space, the power $-γ=-2η$ of $ρ(x,\cdot,\cdot)$ provides the natural singularity for this family of kernels. In this paper we consider the fractional integral rank $0<γ<2η$. We prove boundedness properties of the type $\|T^γ(f,g)\|_{p_3}\leq C\|f\|_{p_1}\|g\|_{p_2}$ for adequate values of the exponents $p_1,p_2$ and $p_3$. The proof is based on three upper bounds for $T^γ(f,g)$ in terms of the classical linear fractional Riesz operators $I_{η-\fracγ{2}}$, using the linear Hardy-Littlewood-Sobolev inequality.
Jonathan M. Fraser, Thang Pham
We study the Falconer distance set problem in Euclidean space and obtain improved dimensional estimates under natural Fourier analytic assumptions cast in terms of the Fourier dimension and spectrum. Interestingly, under reasonably mild assumptions, we are able to beat the $d/2$ dimension threshold in dimensions $d \geq 5$. For example, we show that (in any ambient spatial dimension $d$) a Borel set with Fourier dimension at least $2$ has a distance set of full Hausdorff dimension. We also show that (in any ambient spatial dimension $d$) a Borel set with Fourier spectrum at least $d/4+1$ at $θ=1/2$ has a distance set of full Hausdorff dimension. In particular, this can hold for sets with Fourier dimension zero (provided $d \geq 4$). We also consider pinned variants of these problems and construct examples that demonstrate the sharpness (or near sharpness) of our results.
Shuichi Sato
We prove pointwise relations between some multiparameter square functions on $\bold R^n$.
Juan José Marín, José María Martell, Dorina Mitrea, Marius Mitrea
We prove several characterizations of $\mathscr{C}^{1,ω}$-domains (aka Lyapunov domains), where $ω$ is a growth function satisfying natural assumptions. For example, given an Ahlfors regular domain $Ω\subseteq{\mathbb{R}}^n$, we show that the modulus of continuity of the geometric measure theoretic outward unit normal $ν$ to $Ω$ is dominated by (a multiple of) $ω$ if and only if the action of each Riesz transform $R_j$ associated with $\partialΩ$ on the constant function $1$ has a modulus of continuity dominated by (a multiple of) $ω$. The proof of this result requires that we establish a higher-dimensional generalization of the classical Plemelj-Privalov theorem, identifying a large class of singular integral operators that are bounded on generalized Hölder spaces. This class includes the Cauchy-Clifford operator and the harmonic double layer operator, among others.
Vjekoslav Kovač, Adian Anibal Santos Sepčić
We study density thresholds that force a measurable set $E\subseteq\mathbb{R}^d$ to contain all sufficiently large similar copies of every $n$-point configuration. We prove a lower bound of the form $1-O((\log n)/n)$, which matches the known upper bound up to the logarithmic factor, thus essentially resolving a problem posed by Falconer, Yavicoli, and the first author of the present paper. We also study the same problem for embeddings of $n$-point configurations into $\mathbb{R}^d$ equipped with the $\ell^p$ norm, obtaining an asymptotically sharp bound $1-1/n+o(1/n)$, as soon as $p\in(1,\infty)\setminus\{2\}$. In the proof of the former estimate we use equidistribution of polynomial sequences modulo $1$ combined with probabilistic thinning. The proof of the latter estimate relies on the geometry of the $\ell^p$ spaces for $p\neq2$.
Boon Suan Ho
We construct dyadic lacunary counterexamples for two problems of Erdős on pointwise behavior of dilates on the circle. The main device is a dyadic spike block: rare positive spikes create long positive runs in the lacunary averages, while a deterministic lower floor prevents cancellation from the remaining stages. The endpoint construction gives a mean-zero $f\in\bigcap_{1\le q<\infty}L^q(\mathbb T)$ and a sequence $n_j=2^{m_j}$, $n_{j+1}/n_j\ge2$, such that $$ \|f-S_Nf\|_2\ll (\log\log N)^{-1/2}, \qquad \limsup_{N\to\infty} \frac1N\sum_{j\le N}f(n_jx)=+\infty $$ for almost every $x$. Thus Matsuyama's positive theorem at exponent $c>1/2$ cannot be extended to the endpoint $c=1/2$, and Erdős Problem #996 has a negative answer. A second choice of parameters gives, for every $2\le p<\infty$, functions $f\in L^p(\mathbb T)$ with $$ \limsup_{N\to\infty} \frac{\sum_{j\le N}f(n_jx)} {N(\log N)^{1/p-\varepsilon}} =+\infty \qquad(\varepsilon>0) $$ almost everywhere; the case $p=2$ answers Erdős Problem #995. We also include a bounded small-set companion construction.
Sergey A. Denisov, Giorgio Young
We study the wave operators for a Jacobi matrix whose spectral measure satisfies the Szegö condition. We prove existence and completeness of wave operators under a mild additional assumption on the Verblunsky coefficients of the associated measure on the unit circle.
Nicolas Angelini
We study the packing dimension of Borel measures under orthogonal projections. We give a necessary and sufficient condition such that typical projections of Borel probability measures have full packing dimension and derive general lower bounds in the complementary case. Our approach shows that the Assouad dimension of the support influences the behavior of projected measures. The same method yields corresponding results for images under fractional Brownian motion.
Naoya Hatano, Ryota Kawasumi, Hiroki Saito, Hitoshi Tanaka
We show some simple sufficient conditions for which the multilinear embedding theorem holds for fractional sparse operators. By verifying these conditions, we establish the theorem for power weights. We also provide Morrey-type sufficient conditions for which the $L^p \to L^q$, $1<p,q<\infty$, infinitesimal relative bounds hold for Schrödinger operators of the form $(-Δ)^{α/2}+v$.
Naoya Hatano, Ryota Kawasumi, Hiroki Saito, Hitoshi Tanaka
By using parallel corona decomposition, the Kerman-Sawyer trace inequality is extended from Lebesgue spaces to product Morrey spaces.
Yunfeng Zhang
We present a basic pointwise bound for the irreducible characters of $\mathrm{SU}(3)$ and, as an application, derive new $L^p$ bounds for these characters. Our approach is based on the descent of characters to singular sets and the cancellation in this formula.
Gian Maria Dall'Ara, Adrian Dumitrescu
What is the largest constant $c\in [0,1]$ with the property that every finite collection $\mathcal{C}$ of axis-parallel squares in the plane admits a disjoint sub-collection $\mathcal{S}$ occupying at least a fraction $c$ of the area covered by $\mathcal{C}$? This problem was first raised by T.~Radó in 1928, who was motivated by a classical covering lemma in real analysis due to Vitali. R.~Rado later generalized the problem from axis-parallel squares in the plane to homothetic copies of any given convex body $K$ in $\mathbb{R}^d$, where now we are looking for an optimal constant $F(K)$. Our utmost interest is for cubes and balls in the high-dimensional regime $d\rightarrow \infty$. The estimates that we currently have for cubes are much more precise than those for balls: namely if $Q^d$ is a $d$-dimensional cube, then \[ (e^{-1}+o(1))\frac{2^{-d}}{d \log{d}} \leq F(Q^d)\leq 2^{-d}, \] while denoting $B^d$ a $d$-dimensional Euclidean ball, then \[ (1+ε_d)3^{-d}\leq F(B^d)\leq 2.447^{-d}, \] where $ε_d>0$ vanishes exponentially fast as $d\rightarrow \infty$. The latter upper bound is obtained here by using the Kabatiansky--Levenshtein bound for the sphere packing problem.
Naoya Hatano, Ryota Kawasumi, Hiroki Saito, Hitoshi Tanaka
By using a Hedberg-type inequality, the Adams trace inequality is extended from Lebesgue spaces to product Morrey spaces.
Gian Maria Dall'Ara, Roberto Frigerio, Ervin Hadziosmanovic
Let $M$ be a complete hyperbolic $n$-manifold, $n\geq 2$. Via integration over geodesic simplices, any closed bounded differential 2-form on $M$ defines a bounded cohomology class in $H^2_b(M)$. It was proved by Barge and Ghys (for $n=2$) and by Battista et al. (for $n>2$) that, if $M$ is closed, then this procedure defines an injective embedding of the (infinite-dimensional) space of closed differential $2$-forms on $M$ into $H^2_b(M)$. We extend this result to the case when the fundamental group of $M$ is of the first kind, i.e. its limit set is equal to the whole boundary at infinity of hyperbolic space (this holds, for example, when $M$ has finite volume). Our argument is different from Barge and Ghys' original one, and relies on the following fact of independent interest: an $L^\infty$ function on the hyperbolic plane is determined by its integrals over all ideal triangles. We prove this fact by way of Fourier analysis on the hyperbolic plane.
Dariusz Kosz, Jakub Niksiński, Błażej Wróbel
We prove a uniform upper and lower bound for Delannoy numbers. This is achieved by using the representation of Delannoy numbers as the number of lattice points in high-dimensional cross-polytopes (also known as hyper-octahedrons or $\ell^1$ balls) and proving a uniform (dimension-free) count for these lattice points. Using this count, we establish dimension-free estimates for discrete maximal functions over cross-polytopes. By proving a comparison principle with the continuous setting, we obtain a dimension-free estimate on all $\ell^p(\mathbb{Z}^d)$ spaces for radii $R>C d^{3/2}.$ We also treat the full maximal function on $\ell^2(\mathbb{Z}^d)$ for small radii $R\le d^{1-\varepsilon}$ and the dyadic maximal function for any radii.