25,444 papers in this slice of arXiv.
Andriy Bondarenko, Kristian Seip
For every 0≤β≤1/2, we construct a nonzero real-valued continuous function fβ
Benedikt Buchecker, Benjamin Eichinger, Olof Rubin +1
We describe the asymptotics of Chebyshev polynomials on an analytic Jordan arc in the plane. This gives an affirmative answer to a conjecture of Christiansen-Simon-Zinchenko, based on predictions of Widom from 1969. The proof combines weighted Faber polynomials with extremal signatures, discrete orthogonal polynomials and a Marcinkiewicz-Zygmund sampling inequality, and yields Szegő-Widom asymptotics for the Chebyshev polynomials themselves.
Rocío Ayala, Fabio Berra, Gladis Pradolini
We study two-weight weak-type estimates for the operator Svf=T(fv)/v, where T is the Hardy-Littlewood maximal operator or a Calderón-Zygmund operator (CZO) and v is a weight. Concretely, under certain conditions on the weights involved, we prove that Sv is bounded from L1(wv) to L1,∞(uv). We also consider the corresponding inequalities when T is a higher-order commutator of a CZO. These types of results are inspired by the article of Sawyer in [21], (see also [17]).
Weiyi Kong, Dachun Yang, Wen Yuan
We prove that, for any given time lag γ∈(0,1), itemize the one-sided parabolic BMO space BMO+(γ) coincides with the parabolic BMO space PBMO−(γ), i.e., PBMO−(γ)=BMO+(γ) with equivalent norms; the parabolic Muckenhoupt class A∞+(γ) defined via the reverse Jensen inequality can be represented as the union of the parabolic Muckenhoupt classes Ar+(γ) with r∈[1,∞), i.e., A∞+(γ)=⋃r∈[1,∞)Ar+(γ); the parabolic reverse Hölder classes ⋃q∈(1,∞]RHq+ coincide with the parabolic Muckenhoupt classes ⋃r∈[1,∞)Ar+(γ), i.e., ⋃q∈(1,∞]RHq+=⋃r∈[1,∞)Ar+(γ), itemize and hence give affirmative answers to Questions 4.5 and 4.6 posed by Kinnunen and Saari [Nonlinear Anal. 131 (2016), 289--299]. The main ingredients are a uniform parabolic space-time shifting property for parabolic reverse Hölder weights and a one-sided stopping-time argument, which yields a new parabolic John--Nirenberg inequality for BMO+(γ). As consequences, we obtain John--Nirenberg and exponential integrability characterizations of BMO+(γ), prove that BMO+(γ) is independent of the positive time lag, and identify its null space.
Xijun Hu, Yuwei Ou, Zhiwen Qiao +1
We study the relative periodic orbits in the Gutzwiller-type anisotropic Kepler problem, which is a generalized model derived from the classical Gutzwiller anisotropic Kepler problem. By reducing the rotational symmetry of the z-axis, we obtain a reduced system with two degrees of freedom and its energy surface forms a compact and regular three-sphere in a certain parameter range. Combining the estimation of the Seifert rotation number of the planar Kepler orbit and the CHHL formula introduced in CHHL23, we prove that this system admits infinitely many periodic orbits on every compact regular energy surface. This model can be applied to the (1+2n)-body problem to find infinitely many relative periodic orbits with hip-hop symmetry as well as relative periodic orbits in the n-pyramidal problem.
Hua Qiu, Qi Wang
We prove that the conformal dimension of every locally finite Borel measure is either zero or infinite. The main ingredient is a quasisymmetric dimension-reduction theorem: every full-support probability measure of finite Hausdorff dimension on a separable metric space admits quasisymmetrically equivalent metrics in which its Hausdorff dimension is arbitrarily small. In particular, every locally finite Borel measure on a doubling metric space has conformal dimension zero. We also prove that for every n≥1 and p>0 there are a metric space X, a quasisymmetric homeomorphism f:[0,1]n→X, and a Borel set E⊂[0,1]n such that dimHf(E)≤p and dimH([0,1]n∖E)≤n−1+p. The second bound is sharp up to p: if dimHf(E)<1, then dimH([0,1]n∖E)≥n−1.
Paata Ivanisvili
We show that for every conformal map φ:D→Ω⊊C from the unit disk onto a simply connected proper domain Ω, the length of φ−1(Ω∩L) is at most π2 for every line L.
Steven P. Clark
To each finite symmetric measure σ on the real line, with support a compact subset of (−2,2), we associate a measure μ on the unit circle by transporting mass at ±x to e±iθ(x), θ(x)=2arcsin(x/2). The linear prediction errors, Verblunsky coefficients, and Toeplitz determinants of μ are then expressed through the orthogonal polynomial data of σ at the single edge point x=2. In particular Em(μ)=21tm∥Pm∥σ2 with tm=Pm+1(2)/Pm(2). Under a condition on the first coefficients, decay of the recurrence coefficients of σ forces the tm to increase from t1 onward, a Turán-type monotonicity in the degree placing every prediction margin of μ past the first above the corresponding coefficient of σ. Taking σ to be the Lommel-polynomial measures, with atoms at rescaled reciprocals of the zeros of the Bessel function Jν, yields a two-parameter family of purely atomic circle measures with a normalized determinant limit equal to a value of Jν.
Adam Cushman, Ciprian Demeter, Shukun Wu
Let X⊂R be finite and let γ(t)=(t,f(t)), where f is strictly convex. We show that J3(γ(X))=#{(x1,…,x6)∈X6:i=1∑3γ(xi)=i=4∑6γ(xi)}≪ε∣X∣3+ε. As applications, we prove that ∣A−A∣≫ε∣A∣5/3−ε and ∣A+A∣≫ε∣A∣8/5−ε for any convex sequence A⊂R.
Brad Rodgers
We give a proof that the optimal constant in the weighted Hilbert inequality is strictly greater than π, answering in the negative a question asked by Montgomery and Vaughan. The proof proceeds via an explicit limiting counterexample.
Lukas Liehr, Mitchell A. Taylor, Peiyang Yu
We establish a sharp version of Grothendieck's theorem for Bessel sequences. Precisely, given a Bessel sequence {xj}j∈N with Bessel bound 1 in a Hilbert space, we show that there exists functions {fj}j∈N belonging to the unit ball of L∞([0,1]) such that for all j,k∈N one has ⟨xj,xk⟩=∫01fj(x)fk(x)dx. As an application, we give an affirmative answer to an extension problem of Olevskii: if E⊂[0,1] is a Lebesgue measurable set such that [0,1]∖E has positive measure, then every Bessel sequence in L2(E) with Bessel bound 1 extends to an orthonormal system in L2([0,1]) that is bounded by the (optimal) constant λ([0,1]∖E)−1/2 on [0,1]∖E. A formalization of our main result in Lean 4 accompanies the paper.
Fabrizio Colombo, Elodie Pozzi, Irene Sabadini +1
In this paper we prove the Corona theorem for slice hyperholomorphic functions in a general way that applies to functions with values in a Clifford algebra Rn for any n≥2. This general proof is based on a matrix version of Corona's theorem and then exploiting some symmetries of the matrices that allow to reconstruct the values of the functions. The crucial idea is to tie together the algebra of the functions, the Bezout identities and the Carleson-type conditions. The results we obtain rest on a new way of writing the slice product which improves the standard approach and which holds in general dimension.
Luis Daniel Abreu
Let PN be the (N+1)-dimensional Hilbert space of analytic polynomials of degree at most N. This is the natural environment to define SU(2) (Bloch) coherent states. Let Qρ be the Husimi function of a density operator ρ on . We prove an isospectral version of Lieb-Solovej inequality: if ρ↓ is obtained by placing the eigenvalues of ρ in decreasing order along the monomial basis, then equation* ∫_CΦ(Q_ρ(z)) dm(z)≤ ∫_CΦ(Q_ρ^(z)) dm(z) equation*% for every convex function Φ on [0,1]. Applying the corresponding reversed inequality to the concave function Φ(t)=−tlogt gives the Wehrl entropy. In the process it is show that the output state of ρ under Lieb-Solovej's channel is majorized by the output of the state ρ↓. As an application, among all measurable subsets of the sphere having a fixed area, spherical caps maximize every partial sum of the eigenvalues of Toeplitz operators with symbol 1Ω. Equivalently, caps maximize all Ky Fan norms, leading to the sharp inequalities, previously known for r=1: equation* ∑_j=1^rλ_j(Ω) ≤ r-∑_k=0^r-1(r-k)N+1k m(Ω)^k(1-m(Ω))^N+1-k. equation* This implies isoperimetric inequalities for all Schatten sums of TΩ, obtained without using the spherical isoperimetric inequality.
Yutong Zhang, Yaoran Yang
Cohn and Kumar conjectured in 2009 that there is a radial Schwartz function g:R24→R satisfying g(r)≤0 for r≥6, g(r)≥0 for r≥0, g(2)>0, and (g(0)−g(0))/g(2)=196560. We construct such functions from the radial Fourier interpolation basis in dimension 24. If a2,b2 denote the basis functions dual to value and radial-derivative interpolation at radius 2, then gC=a2−Cb2 has exactly the nodal data needed for Poisson summation over the Leech lattice. The sphere-packing magic function identifies b2 and supplies its strict signs. We prove that the removable quotients a2/b2 and a2/b2 are bounded on the required half-lines. The noncompact step follows from exact coefficient extraction in the interpolation kernel and an S-cusp expansion. Both quotients tend to (43+240log2)/15; for all sufficiently large radii they lie on opposite sides of this limit, with an explicit first exponential correction. Consequently the admissible parameters in this affine family form a nonempty closed ray, and every member proves the conjectured identity. The same interpolation basis recovers every nontrivial Leech-shell coefficient.
Dan Coroian, Peter Dragnev, Alan Legg +1
The constrained equilibrium problem for the logarithmic potential was introduced by Rakhmanov (1996) as he realized that the asymptotic distribution of the zeros of discrete orthogonal polynomials in a compact interval could be described in terms of the equilibrium measure of this interval in a class of measures subject to a certain constraint. Other authors such as Dragnev and Saff (1997), and Kuijlaars and Van Assche (1999), extended this approach to more general settings. These problems have proven to be useful for describing asymptotic distributions in different settings. In the current paper, we consider constrained equilibrium problems in the Riesz setting, that is, for s-Riesz potentials in the hyperplane Rd, with d≥1 and max(0,d−2)<s<d. Along with some general results, an illustrative example consisting of the solution of a constrained equilibrium problem in the unit ball is studied in detail. This is the main part of the paper. Finally, a number of numerical experiments show how the constrained equilibrium measure, the solution of the problem, may be discretized using the so-called 'constrained Leja points' introduced by Coroian and Dragnev (2001).
Yifan Jing, Shukun Wu
We study two Kakeya set problems in compact semisimple Lie groups. In the first, motivated by the group structure, a Kakeya set is required to contain a left coset of every closed one-dimensional torus. For a compact semisimple Lie group G of dimension d and rank r, we determine the optimal Minkowski dimension for this problem, proving that it is exactly (d+r)/2. The upper bound is obtained from a Lie-theoretic construction associated with a Chevalley involution, while the lower bound combines incidence geometry with geometry of numbers. We also consider a local Kakeya set problem, closer to the classical harmonic-analytic formulation, in which one requires a fixed-length one-parameter arc in every direction. For this problem we obtain lower bound (d+1)/2 and (d+2)/2 for odd and even r, and upper bound (d+r)/2. We conjecture that the upper bound should be sharp.
Efstathios Konstantinos Chrontsios Garitsis
The class of Weierstrass functions Wa,b is one of the first class of examples of continuous and nowhere differentiable real functions. A challenging line of research has been to determine the various dimensions of graphs G(Wa,b) of such functions. For instance, the Hausdorff dimension of G(Wa,b) was only recently determined by Shen in 2018, after a long series of partial results by many different authors. While the Assouad dimension of G(Wa,b) remains an open problem, also posed as a question by J. M. Fraser, there have been many indications that it might be equal to 2. Such indications include the graph of Wiener processes, the graphs of almost all Hölder functions in the Baire category sense, and the graphs of Weierstrass functions after a series of countably many reflections all having Assouad dimension equal to 2. In this paper we show that this is not the case, providing a quantitative upper bound on the Assouad dimension of G(Wa,b) that is strictly less than 2. In particular, we show that such a bound is true for a class of generalized Weierstrass functions Wa,bφ(x)=∑j=0∞ajφ(bjx), which includes Wa,b and the class of Takagi functions. The latter fact is used to also answer in the negative a conjecture of H. Yu on the Assouad dimension of graphs of Takagi functions for parameters a∈(0,1), b∈[3/a,∞)∩Z.
Sampad Lahiry
Following recent developments on matrix-valued orthogonal polynomials (MVOPs), we construct the matrix Szegő factorisation of the matrix weight function when it is supported on multiple intervals. We find that on the gaps the matrix function has unitary multiplicative jumps. In the next part, performing the Deift-Zhou steepest descent analysis we study the associated global parametrix which comes from the Riemann-Hilbert problem of the MVOPs. A solution is constructed visualising the rows of the parametrix as sections of vector bundles on a hyper-elliptic Riemann surface. An alternate view point to the global parametrix is also presented using a vanishing lemma. As an application we obtain strong asymptotics of MVOPs for multiple cuts which extends the works of Deaño, Kuijlaars, and Román.
Takumi Maesaka
We prove a double-sum analogue of Whipple's 3F2 summation formula. As an application, we establish a restricted sum formula for multiple binomial sums, which arise as special cases of Igarashi's parametrized multiple zeta series.
Chao Zhang
We determine the sharp constants in the one-sided John-Nirenberg inequality for functions of bounded lower oscillation on an interval. More precisely, if f∈(I₀), then _I⊆ I₀(1/|I|) |{x∈ I: f(x)-_I f>λ}| ≤ c₁(-c₂λf_(I₀)), λ>0, with the usual interpretation when ∥f∥BLO(I0)=0. The sharp constants are c1∗=e and c2∗=1. We give a proof based on the Riesz rising sun lemma and an independent Bellman function proof. Finally, we present several consequences of the sharp estimate.