Isoperimetric inequality for the polydisk

Let \({n\in\mathbb{N}}\). For \({k\in\{1,\dots,n\}}\) let \({\Omega_k\subset \mathbb{C}}\) be a simply connected domain with a rectifiable boundary. Let \({\Omega^n=\prod_{k=1}^n\Omega_k\subset \mathbb{C}^n}\) be a generalized polydisk with distinguished boundary \({\partial\Omega^n=\prod_{k=1}^n\partial\Omega_k}\). Let E ^r (Ω ⁿ ) be the holomorphic Smirnov class on Ω ⁿ with index r. We show that the generalized isoperimetric inequality

$ \int\limits_{\Omega^n} |f_1|^p|f_2|^qdV\le \frac{1}{(4\pi)^n}\int\limits_{\partial \Omega^n}|f_1|^pdS \int\limits_{\partial \Omega^n} |f_2|^qdS, $

holds for arbitrary \({f_1\in E^p(\Omega^n)}\) and \({f_2\in E^q(\Omega^n)}\), where 0 < p, q < ∞. We also determine necessary and sufficient conditions for equality.

     

Heat kernel approach for sup-norm bounds for cusp forms of integral and half-integral weight

In this article, using the heat kernel approach from Bouche (Asymptotic results for Hermitian line bundles over complex manifolds: The heat kernel approach, Higher-dimensional complex varieties, pp 67–81, de Gruyter, Berlin, 1996), we derive sup-norm bounds for cusp forms of integral and half-integral weight. Let \({\Gamma\subset \mathrm{PSL}_{2}(\mathbb{R})}\) be a cocompact Fuchsian subgroup of first kind. For \({k \in \frac{1}{2} \mathbb{Z}}\) (or \({k \in 2\mathbb{Z}}\)), let \({S^{k}_{\nu}(\Gamma)}\) denote the complex vector space of cusp forms of weight-k and nebentypus \({\nu^{2k}}\) (\({\nu^{k\slash 2}}\), if \({k \in 2\mathbb{Z}}\)) with respect to \({\Gamma}\), where \({\nu}\) is a unitary character. Let \({\lbrace f_{1},\ldots,f_{j_{k}} \rbrace}\) denote an orthonormal basis of \({S^{k}_{\nu}(\Gamma)}\). In this article, we show that as \({k \rightarrow \infty,}\) the sup-norm for \({\sum_{i=1}^{j_{k}}y^{k}|f_{i}(z)|^{2}}\) is bounded by O(k), where the implied constant is independent of \({\Gamma}\). Furthermore, using results from Berman (Math. Z. 248:325–344, 2004), we extend these results to the case when \({\Gamma}\) is cofinite.     

Goldbach’s conjecture in arithmetic progressions: number and size of exceptional prime moduli

Set \({T=N^{\frac{1}{3}-\epsilon}}\). It is proved that for all but \({\ll TL^{-H},\,H > 0}\), exceptional prime numbers \({k\leq T}\) and almost all integers b ₁, b ₂ co-prime to k, almost all integers \({n\sim N}\) satisfying \({n\equiv b_{1}+b_{2}(mod\,k)}\) can be written as the sum of two primes p ₁ and p ₂ satisfying \({p_{i}\equiv b_{i}(mod\,k),\,i=1,2}\). For prime numbers \({k\leq N^{\frac{5}{24}-\epsilon}}\), this result is even true for all but \({\ll (\log\,N)^{D}}\) primes k and all integers b ₁, b ₂ co-prime to k.     

Vertex and Edge Orbits of Fibonacci and Lucas Cubes

The Fibonacci cube \({\Gamma_{n}}\) is obtained from the n-cube Q _n by removing all the vertices that contain two consecutive 1s. If, in addition, the vertices that start and end with 1 are removed, the Lucas cube \({\Lambda_{n}}\) is obtained. The number of vertex and edge orbits, the sets of the sizes of the orbits, and the number of orbits of each size, are determined for the Fibonacci cubes and the Lucas cubes under the action of the automorphism group. In particular, the set of vertex orbit sizes of \({\Lambda_{n}}\) is \({\{k \geq 1; k |n\} \cup \{k \geq 18; k |2n\}}\), the number of vertex orbits of \({\Lambda_{n}}\) of size k, where k is odd and divides n, is equal to \({\sum_{d | k} \mu (\frac{k}{d})F_{\lfloor{\frac{d}{2}}\rfloor+2}}\), and the number of edge orbits of \({\Lambda_{n}}\) is equal to the number of vertex orbits of \({\Gamma_{n-3}}\). Dihedral transformations of strings and primitive strings are essential tools to prove these results.     

Some remarks on energy inequalities for harmonic maps with potential

We call the \({\delta}\)-vector of an integral convex polytope of dimension d flat if the \({\delta}\)-vector is of the form \({(1,0,\ldots,0,a,\ldots,a,0,\ldots,0)}\), where \({a \geq 1}\). In this paper, we give the complete characterization of possible flat \({\delta}\)-vectors. Moreover, for an integral convex polytope \({\mathcal{P}\subset \mathbb{R}^N}\) of dimension d, we let \({i(\mathcal{P},n)=|n\mathcal{P}\cap \mathbb{Z}^N|}\) and \({i^*(\mathcal{P},n)=|n(\mathcal{P} {\setminus}\partial \mathcal{P})\cap \mathbb{Z}^N|}\). By this characterization, we show that for any \({d \geq 1}\) and for any \({k,\ell \geq 0}\) with \({k+\ell \leq d-1}\), there exist integral convex polytopes \({\mathcal{P}}\) and \({\mathcal{Q}}\) of dimension d such that (i) For \({t=1,\ldots,k}\), we have \({i(\mathcal{P},t)=i(\mathcal{Q},t),}\) (ii) For \({t=1,\ldots,\ell}\), we have \({i^*(\mathcal{P},t)=i^*(\mathcal{Q},t)}\), and (iii) \({i(\mathcal{P},k+1) \neq i(\mathcal{Q},k+1)}\) and \({i^*(\mathcal{P},\ell+1)\neq i^*(\mathcal{Q},\ell+1)}\).     

On stable Baire classes

We introduce and study adhesive spaces. Using this concept we obtain a characterization of stable Baire maps \({f : X\to Y}\) of the class \({\alpha}\) for wide classes of topological spaces. In particular, we prove that for a topological space X and a contractible space Y a map \({f : X \to Y}\) belongs to the nth stable Baire class if and only if there exist a sequence \({(f_k)_{k=1}^\infty}\) of continuous maps \({f_k : {X \to Y}}\) and a sequence \({(F_k)_{k=1}^\infty}\) of functionally ambiguous sets of the nth class in X such that \({f|_{F_k}=f_k|_{F_k}}\) for every k. Moreover, we show that every monotone function \({f : \mathbb{R} \to \mathbb{R}}\) is of the \({\alpha}\) th stable Baire class if and only if it belongs to the first stable Baire class.     

Some estimates for the Schrödinger type operators with nonnegative potentials

In this paper we consider the Schrödinger operator ?Δ + V on \({\mathbb R^d}\), where the nonnegative potential V belongs to the reverse Hölder class \({B_{q_{_1}}}\) for some \({q_{_1}\geq \frac{d}{2}}\) with d ≥ 3. Let \({H^1_L(\mathbb R^d)}\) denote the Hardy space related to the Schrödinger operator L = ?Δ + V and \({BMO_L(\mathbb R^d)}\) be the dual space of \({H^1_L(\mathbb R^d)}\). We show that the Schrödinger type operator \({\nabla(-\Delta +V)^{-\beta}}\) is bounded from \({H^1_L(\mathbb R^d)}\) into \({L^p(\mathbb R^d)}\) for \({p=\frac{d}{d-(2\beta-1)}}\) with \({ \frac{1}{2}<\beta<\frac{3}{2} }\) and that it is also bounded from \({L^p(\mathbb R^d)}\) into \({BMO_L(\mathbb R^d)}\) for \({p=\frac{d}{2\beta-1}}\) with \({ \frac{1}{2}<\beta< 2}\).     

Local Inverse Scattering at a Fixed Energy for Radial Schrödinger Operators and Localization of the Regge Poles

We study inverse scattering problems at a fixed energy for radial Schrödinger operators on \({\mathbb{R}^n}\), \({n \geq 2}\). First, we consider the class \({\mathcal{A}}\) of potentials q(r) which can be extended analytically in \({\Re z \geq 0}\) such that \({\mid q(z)\mid \leq C \ (1+ \mid z \mid )^{-\rho}}\), \({\rho > \frac{3}{2}}\). If q and \({\tilde{q}}\) are two such potentials and if the corresponding phase shifts \({\delta_l}\) and \({\tilde{\delta}_l}\) are super-exponentially close, then \({q=\tilde{q}}\). Second, we study the class of potentials q(r) which can be split into q(r) = q ₁(r) + q ₂(r) such that q ₁(r) has compact support and \({q_2 (r) \in \mathcal{A}}\). If q and \({\tilde{q}}\) are two such potentials, we show that for any fixed \({a>0, {\delta_l - \tilde{\delta}_l \ = \ o \left(\frac{1}{l^{n-3}}\ \left({\frac{ae}{2l}}\right)^{2l}\right)}}\) when \({l \rightarrow +\infty}\) if and only if \({q(r)=\tilde{q}(r)}\) for almost all \({r \geq a}\). The proofs are close in spirit with the celebrated Borg–Marchenko uniqueness theorem, and rely heavily on the localization of the Regge poles that could be defined as the resonances in the complexified angular momentum plane. We show that for a non-zero super-exponentially decreasing potential, the number of Regge poles is always infinite and moreover, the Regge poles are not contained in any vertical strip in the right-half plane. For potentials with compact support, we are able to give explicitly their asymptotics. At last, for potentials which can be extended analytically in \({\Re z \geq 0}\) with \({\mid q(z)\mid \leq C (1+ \mid z \mid)^{-\rho}}\), \({\rho >1}\), we show that the Regge poles are confined in a vertical strip in the complex plane.     

On complete sequences

A sequence A of nonnegative integers is called complete if all sufficiently large integers can be represented as the sum of distinct terms taken form A. For a sequence \({S=\{s_{1}, s_{2}, \dots\}}\) of positive integers and a positive real number α, let S _α denote the sequence \({\{\lfloor\alpha s_{1}\rfloor, \lfloor\alpha s_{2}\rfloor, \dots\}}\), where \({\lfloor x \rfloor}\) denotes the greatest integer not greater than x. Let \({{U_S = \{\alpha \mid S_\alpha} \, is complete\}}\). Hegyvári [6] proved that if \({\lim_{n\to\infty} (s_{n+1}-s_{n})=+ \infty}\), \({s_{n+1} < \gamma s_{n}}\) for all integers \({n \geqq n_{0}}\), where \({1 < \gamma < 2}\), and \({U_{S}\ne\emptyset}\), then \({\mu(U_{S}) > 0}\), where \({\mu(U_{S})}\) is the Lebesgue measure of U _S . Yong-Gao Chen and the first author [4] proved that, if \({s_{n+1} < \gamma s_{n}}\) for all integers \({n \geqq n_{0}}\), where \({1 < \gamma \leqq 7/4=1.75}\), then \({\mu(U_{S}) > 0}\). In this paper, we prove that the conclusion holds for \({1 < \gamma \leqq \sqrt[4]{13}=1.898\dots\;}\).     

Rectifiability of harmonic measure

In the present paper we prove that for any open connected set \({\Omega\subset\mathbb{R}^{n+1}}\), \({n\geq 1}\), and any \({E\subset \partial \Omega}\) with \({\mathcal{H}^n(E)<\infty}\), absolute continuity of the harmonic measure \({\omega}\) with respect to the Hausdorff measure on E implies that \({\omega|_E}\) is rectifiable. This solves an open problem on harmonic measure which turns out to be an old conjecture even in the planar case \({n=1}\).     

Riesz Transform Characterizations of Hardy Spaces Associated to Degenerate Elliptic Operators

Let \({L_{w}}{:=-w^{-1}{\rm div}(A\nabla)}\) be the degenerate elliptic operator on the Euclidean space \({{\mathbb{R}^{n}}}\), where w is a Muckenhoupt \({A_{2}({\mathbb{R}^{n}})}\) weight. In this article, the authors establish the Riesz transform characterization of the Hardy space \({H^{p}_{L_{w}}({\mathbb{R}}^{n})}\) associated with L_w, for \({w \in A_{q}({\mathbb{R}}^{n}) \cap RH_{\frac{n}{n-2}}({\mathbb{R}^{n}})}\) with \({n \geq 3}\), \({q \in [1,2]}\) and \({p \in (q(\frac{1}{r}+\frac{q-1}{2}+\frac{1}{n})^{-1},1]}\) if, for some \({r \in (1,\,2]}\), \({{\{tL_w e^{-tL_w}\}}_{t\geq 0}}\) satisfies the weighted \({L^{r}-L^{2}}\) full off-diagonal estimates.     

Interlacing of zeros of Gegenbauer polynomials of non-consecutive degree from different sequences

A theorem due to Stieltjes’ states that if \({\{p_n\}_{n=0}^\infty}\) is any orthogonal sequence then, between any two consecutive zeros of p _k , there is at least one zero of p _n whenever k < n, a property called Stieltjes interlacing. We show that Stieltjes interlacing extends to the zeros of Gegenbauer polynomials \({C_{n+1}^{\lambda}}\) and \({C_{n-1}^{\lambda+t}}\), \({\lambda > -\frac 12}\), if 0 < t ≤ k + 1, and also to the zeros of \({C_{n+1}^{\lambda}}\) and \({C_{n-2}^{\lambda +k}}\) if \({k\in\{1,2,3\}}\). More generally, we prove that Stieltjes interlacing holds between the zeros of the kth derivative of \({C_{n}^{\lambda}}\) and the zeros of \({C_{n+1}^{\lambda}}\), \({k\in\{1,2,\dots,n-1\}}\) and we derive associated polynomials that play an analogous role to the de Boor–Saff polynomials in completing the interlacing process of the zeros.     

Conformal embeddings of affine vertex algebras in minimal <Emphasis Type="Italic">W</Emphasis>-algebras II: decompositions

We present methods for computing the explicit decomposition of the minimal simple affine W-algebra \({W_k(\mathfrak{g}, \theta)}\) as a module for its maximal affine subalgebra \({\mathscr{V}_k(\mathfrak{g}^{\natural})}\) at a conformal level k, that is, whenever the Virasoro vectors of \({W_k(\mathfrak{g}, \theta)}\) and \({\mathscr{V}_k(\mathfrak{g}^\natural)}\) coincide. A particular emphasis is given on the application of affine fusion rules to the determination of branching rules. In almost all cases when \({\mathfrak{g}^{\natural}}\) is a semisimple Lie algebra, we show that, for a suitable conformal level k, \({W_k(\mathfrak{g}, \theta)}\) is isomorphic to an extension of \({\mathscr{V}_k(\mathfrak{g}^{\natural})}\) by its simple module. We are able to prove that in certain cases \({W_k(\mathfrak{g}, \theta)}\) is a simple current extension of \({\mathscr{V}_k(\mathfrak{g}^{\natural})}\). In order to analyze more complicated non simple current extensions at conformal levels, we present an explicit realization of the simple W-algebra \({W_{k}(\mathit{sl}(4), \theta)}\) at k = ?8/3. We prove, as conjectured in [3], that \({W_{k}(\mathit{sl}(4), \theta)}\) is isomorphic to the vertex algebra \({\mathscr{R}^{(3)}}\), and construct infinitely many singular vectors using screening operators. We also construct a new family of simple current modules for the vertex algebra \({V_k (\mathit{sl}(n))}\) at certain admissible levels and for \({V_k (\mathit{sl}(m \vert n)), m\ne n, m,n\geq 1}\) at arbitrary levels.     

Hecke algebras for {{\rm GL}_n} over local fields

We study the local Hecke algebra \({\mathcal{H}_{G}(K)}\) for \({G = {\rm GL}_{n}}\) and K a non-archimedean local field of characteristic zero. We show that for \({G = {\rm GL}_{2}}\) and any two such fields K and L, there is a Morita equivalence \({\mathcal{H}_{G}(K) \sim_{M} \mathcal{H}_{G}(L)}\), by using the Bernstein decomposition of the Hecke algebra and determining the intertwining algebras that yield the Bernstein blocks up to Morita equivalence. By contrast, we prove that for \({G = {\rm GL}_{n}}\), there is an algebra isomorphism \({\mathcal{H}_{G}(K) \cong \mathcal{H}_{G}(L)}\) which is an isometry for the induced \({L^1}\)-norm if and only if there is a field isomorphism \({K \cong L}\).     

Generalized commutativity of lattice-ordered groups II

Commutative \({\ell}\)-groups G (in which for all \({x, y \in G, xy = yx}\)) were studied long ago. This was then generalized to the study of \({\ell}\)-groups G in which for a given integer n and for all \({x, y \in G, x^{n}y^{n} = y^{n}x^{n}}\). It was then discovered that if for all \({x, y \in G}\), both \({x^{n}y^{n} = y^{n}x^{n}}\) and \({x^{m}y^{m} = y^{m}x^{m}}\) for two different integers m, n, then also \({x^{d}y^{d} = y^{d}x^{d}}\), where d is the greatest common divisor of m, n.     

A Unified Framework for Linear Dimensionality Reduction in L1

For a family of interpolation norms \({\| \cdot \|_{1,2,s}}\) on \({\mathbb{R}^{n}}\), we provide a distribution over random matrices \({\Phi_s \in \mathbb{R}^{m \times n}}\) parametrized by sparsity level s such that for a fixed set X of K points in \({\mathbb{R}^{n}}\), if \({m \geq C s \log(K)}\) then with high probability, \({\frac{1}{2}\| \varvec{x} \|_{1,2,s} \leq \| \Phi_s (\varvec{x}) \|_1 \leq 2 \| \varvec{x} \|_{1,2,s}}\) for all \({\varvec{x} \in X}\). Several existing results in the literature roughly reduce to special cases of this result at different values of s: For s = n, \({\| \varvec{x} \|_{1,2,n}\equiv \| \varvec{x} \|_{1}}\) and we recover that dimension reducing linear maps can preserve the ?₁-norm up to a distortion proportional to the dimension reduction factor, which is known to be the best possible such result. For s = 1, \({\| \varvec{x} \|_{1,2,1}\equiv \| \varvec{x} \|_{2}}\), and we recover an ?₂/?₁ variant of the Johnson–Lindenstrauss Lemma for Gaussian random matrices. Finally, if \({\varvec{x}}\) is s- sparse, then \({\| \varvec{x} \|_{1,2,s} = \| \varvec{x} \|_1}\) and we recover that s-sparse vectors in \({\ell_1^n}\) embed into \({\ell_1^{\mathcal{O}(s \log(n))}}\) via sparse random matrix constructions.     

A remark on the values of binary linear forms at prime arguments

Let λ₁, λ₂ be positive real numbers such that \({\frac{{\lambda_1}}{{\lambda_2}}}\) is irrational and algebraic. For any (C, c) well-spaced sequence \({\mathcal {V} = \{{v_i}\}_{i = 1}^\infty}\) and δ > 0 let \({E( {\mathcal {V},X,\delta})}\) denote the number of elements \({v \in \mathcal {V}, v \le X}\) for which the inequality

$| {\lambda_1 p_1 + \lambda_2 p_2 - v} | < X^{- \delta}$

is not solvable in primes p ₁, p ₂. In this paper it is proved that

$E( {\mathcal {V},X,\delta}) \ll X^{\frac{4}{5} + \delta + \varepsilon}$

for any \({\varepsilon > 0}\). This result constitutes an improvement upon that of Brüdern, Cook, and Perelli for the range \({\frac{2}{{15}} < \delta < \frac{1}{5}}\).

     

Note on Bernoulli numbers associated to some Dirichlet character of prime conductor

We fix an integer \({n \geq 1}\) and a divisor m of n such that n/m is odd. Let p be a prime number of the form \({p=2n\ell+1}\) for some odd prime number \({\ell}\) with \({\ell \nmid m}\). Let \({S=pB_{1,2m\ell}}\) be the p times of the generalised Bernoulli number associated to an odd Dirichlet character of conductor p and order \({2m\ell}\), which is an algebraic integer of the \({2m\ell}\)th cyclotomic field. It is known that \({S \neq 0}\). More strongly, we show that when \({\ell}\) is sufficiently large, the trace of \({\zeta^{-1}S}\) to the \({2m}\)th cyclotomic field does not vanish for any\({\ell}\)th root \({\zeta}\) of unity. We also show a related result on indivisibility of relative class numbers.     

The Katchalski–Lewis transversal problem for regular polygons

If every k-membered subfamily of a family of plane convex bodies has a line transversal, then we say that this family has property T(k). We say that a family \({\mathcal{F}}\) has property \({T-m}\), if there exists a subfamily \({\mathcal{G} \subset \mathcal{F}}\) with \({|\mathcal{F} - \mathcal{G}| \le m}\) admitting a line transversal. Heppes [7] posed the problem whether there exists a convex body K in the plane such that if \({\mathcal{F}}\) is a finite T(3)-family of disjoint translates of K, then m = 3 is the smallest value for which \({\mathcal{F}}\) has property \({T-m}\). In this paper, we study this open problem in terms of finite T(3)-families of pairwise disjoint translates of a regular 2n-gon \({(n \ge 5)}\). We find out that, for \({5 \le n \le 34}\), the family has property \({T - 3}\) ; for \({n \ge 35}\), the family has property \({T - 2}\).     

Spaces with normal weights and Hadamard gap series

We give a sufficient and necessary condition for an analytic function f(z) on the unit disc \({\mathbb{D}}\) with Hadamard gaps, that is, for \({f(z)=\sum_{k=1}^{\infty}a_kz^{n_k}}\) where \({n_{k+1}/n_k\geq\lambda >1 }\) for all \({k\in \mathbb{N}}\), to belong to the weighted-type space \({ H_\mu^{\infty}}\), under some condition posed on the weight function μ. We can define the corresponding little weighted-type space \({H_{\mu,0}^{\infty}}\) and give a criterion for functions to belong to it.