Remarks on the nonexistence of biharmonic maps

In this short note we study a nonexistence result of biharmonic maps from a complete Riemannian manifold into a Riemannian manifold with nonpositive sectional curvature. Assume that \({\phi : (M, g) \to (N, h)}\) is a biharmonic map, where (M, g) is a complete Riemannian manifold and (N, h) a Riemannian manifold with nonpositive sectional curvature, we will prove that \({\phi}\) is a harmonic map if one of the following conditions holds: (i) \({|d\phi|}\) is bounded in L^q(M) and \({\int_{M}|\tau(\phi)|^{p}dv_{g} < \infty}\), for some \({1 \leq q \leq \infty}\), \({1 < p < \infty}\); or (ii) \({Vol(M) = \infty}\) and \({\int_{M}|\tau(\phi)|^{p}dv_{g} < \infty}\), for some \({1 < p < \infty}\). In addition, if N has strictly negative sectional curvature, we assume that \({rank\phi(q) \geq 2}\) for some \({q \in M}\) and \({\int_{M}|\tau(\phi)|^{p}dv_{g} < \infty}\), for some \({1 < p < \infty}\). These results improve the related theorems due to Baird et al. (cf. Ann Golb Anal Geom 34:403–414, 2008), Nakauchi et al. (cf. Geom. Dedicata 164:263–272, 2014), Maeta (cf. Ann Glob Anal Geom 46:75–85, 2014), and Luo (cf. J Geom Anal 25:2436–2449, 2015).     

A semigroup related to a convex combination of boundary conditions obtained as a result of averaging other semigroups

Let \({\alpha}\) be a bounded linear operator in a Banach space \({\mathbb{X}}\), and let A be a closed operator in this space. Suppose that for \({\Phi_1, \Phi_2}\) mapping D(A) to another Banach space \({\mathbb{Y}}\), \({A_{|{\rm ker}\, \Phi_1}}\) and \({A_{|{\rm ker}\, \Phi_2}}\) are generators of strongly continuous semigroups in \({\mathbb{X}}\). Assume finally that \({A_{|{\rm ker}\, \Phi_\text{a}}}\), where \({\Phi_\text{a} = \Phi_1 \alpha + \Phi_2 \beta}\) and \({\beta = I_\mathbb{X} - \alpha}\), is a generator also. In the case where \({\mathbb{X}}\) is an L ¹-type space, and \({\alpha}\) is an operator of multiplication by a function \({0 \le \alpha \le 1}\), it is tempting to think of the later semigroup as describing dynamics which, while at state x, is subject to the rules of \({A_{|{\rm ker}\, \Phi_1}}\) with probability \({\alpha (x)}\) and is subject to the rules of \({A_{|{\rm ker}\, \Phi_2}}\) with probability \({\beta (x)= 1 - \alpha (x)}\). We provide an approximation (a singular perturbation) of the semigroup generated by \({A_{|{\rm ker}\, \Phi_\text{a}}}\) by semigroups built from those generated by \({A_{|{\rm ker}\, \Phi_1}}\) and \({A_{|{\rm ker}\, \Phi_2}}\) that supports this intuition. This result is motivated by a model of dynamics of Solea solea (Arino et al. in SIAM J Appl Math 60(2):408–436, 1999–2000; Banasiak and Goswami in Discrete Continuous Dyn Syst Ser A 35(2):617–635, 2015; Banasiak et al. in J Evol Equ 11:121–154, 2011, Mediterr J Math 11(2):533–559, 2014; Banasiak and Lachowicz in Methods of small parameter in mathematical biology, Birkhäuser, 2014; Sanchez et al. in J Math Anal Appl 323:680–699, 2006) and is, in a sense, dual to those of Bobrowski (J Evol Equ 7(3):555–565, 2007), Bobrowski and Bogucki (Stud Math 189:287–300, 2008), where semigroups generated by convex combinations of Feller’s generators were studied.     

A Beurling-Blecher-Labuschagne Theorem for Noncommutative Hardy Spaces Associated with Semifinite von Neumann Algebras

We prove a Beurling-Blecher-Labuschagne theorem for \({H^\infty}\)-invariant spaces of \({L^p(\mathcal{M},\tau)}\) when \({0 < p \leq\infty}\), using Arveson’s non-commutative Hardy space \({H^\infty}\) in relation to a von Neumann algebra \({\mathcal{M}}\) with a semifinite, faithful, normal tracial weight \({\tau}\). Using the main result, we are able to completely characterize all \({H^\infty}\)-invariant subspaces of \({L^p(\mathcal{M} \rtimes_\alpha \mathbb{Z},\tau)}\), where \({\mathcal{M} \rtimes_\alpha \mathbb{Z} }\) is a crossed product of a semifinite von Neumann algebra \({\mathcal{M}}\) by the integer group \({\mathbb{Z}}\), and \({H^\infty}\) is a non-selfadjoint crossed product of \({\mathcal{M}}\) by \({\mathbb{Z}^+}\). As an example, we characterize all \({H^\infty}\)-invariant subspaces of the Schatten p-class \({S^p(\mathcal{H})}\), where \({H^\infty}\) is the lower triangular subalgebra of \({B(\mathcal{H})}\), for each \({0 < p \leq\infty}\).     

Asymptotic behavior of the p-torsion functions as p goes to 1

Let \({\Omega}\) be a Lipschitz bounded domain of \({\mathbb{R}^N}\), \({N\geq2}\), and let \({u_p\in W_0^{1,p}(\Omega)}\) denote the p-torsion function of \({\Omega}\), p > 1. It is observed that the value 1 for the Cheeger constant \({h(\Omega)}\) is threshold with respect to the asymptotic behavior of u_p, as \({p\rightarrow 1^+}\), in the following sense: when \({h(\Omega) > 1}\), one has \({\lim_{p\rightarrow 1^+}\left\|u_{p}\right\| _{L^\infty(\Omega)}=0}\), and when \({h(\Omega) < 1}\), one has \({\lim_{p\rightarrow 1^+}\left\|u_p\right\| _{L^\infty(\Omega)}=\infty}\). In the case \({h(\Omega)=1}\), it is proved that \({\limsup_{p\rightarrow1^+}\left\|u_p\right\|_{L^\infty(\Omega)}<\infty}\). For a radial annulus \({\Omega_{a,b}}\), with inner radius a and outer radius b, it is proved that \({\lim_{p\rightarrow 1^+}\left\|u_p\right\| _{L^\infty(\Omega_{a,b})}=0}\) when \({h(\Omega_{a,b})=1}\).     

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.     

A note on standard modules and Vogan L-packets

Let F be a non-Archimedean local field of characteristic 0, let G be the group of F-rational points of a connected reductive group defined over F and let \({G\prime}\) be the group of F-rational points of its quasi-split inner form. Given standard modules \({I(\tau, \nu )}\) and \({I(\tau\prime, \nu\prime)}\) for G and \({G\prime}\) respectively with \({\tau\prime}\) a generic tempered representation, such that the Harish-Chandra \({\mu}\)-function of a representation in the supercuspidal support of \({\tau}\) agrees with the one of a generic essentially square-integral representation in some Jacquet module of \({\tau\prime}\) (after a suitable identification of the underlying spaces under which \({\nu = \nu\prime}\)), we show that \({I(\tau, \nu)}\) is irreducible whenever \({I(\tau\prime, \nu\prime)}\) is. The conditions are satisfied if the Langlands quotients \({J(\tau, \nu})\) and \({J(\tau\prime, \nu\prime)}\) of respectively \({I(\tau, \nu)}\) and \({I(\tau\prime, \nu\prime)}\) lie in the same Vogan L-packet (whenever this Vogan L-packet is defined), proving that, for any Vogan L-packet, all the standard modules with Langlands quotient in a given Vogan L-packet are irreducible, if and only if this Vogan L-packet contains a generic representation. This result for generic Vogan L-packets was proven for quasi-split orthogonal and symplectic groups by Moeglin-Waldspurger and used in their proof of the general case of the local Gan-Gross-Prasad conjectures for these groups.     

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 \({1 < \gamma \leqq \sqrt[4]{13}=1.898\dots\;}\).     

{\mathbb{Z}_2}-bordism and the Borsuk–Ulam Theorem

The purpose of this work is to classify, for given integers \({m,\, n\geq 1}\), the bordism class of a closed smooth \({m}\)-manifold \({X^m}\) with a free smooth involution \({\tau}\) with respect to the validity of the Borsuk–Ulam property that for every continuous map \({\phi : X^m \to \mathbb{R}^n}\) there exists a point \({x\in X^m}\) such that \({\phi (x)=\phi (\tau (x))}\). We will classify a given free \({\mathbb{Z}_2}\)-bordism class \({\alpha}\) according to the three possible cases that (a) all representatives \({(X^m, \tau)}\) of \({\alpha}\) satisfy the Borsuk–Ulam property; (b) there are representatives \({({X_{1}^{m}}, \tau_1)}\) and \({({X_{2}^{m}}, \tau_2)}\) of \({\alpha}\) such that \({({X_{1}^{m}}, \tau_1)}\) satisfies the Borsuk–Ulam property but \({({X_{2}^{m}}, \tau_2)}\) does not; (c) no representative \({(X^m, \tau)}\) of \({\alpha}\) satisfies the Borsuk–Ulam property.     

Open book structures on semi-algebraic manifolds

Given a C ² semi-algebraic mapping \({F} : {\mathbb{R}^N \rightarrow \mathbb{R}^p}\), we consider its restriction to \({W \hookrightarrow \mathbb{R^{N}}}\) an embedded closed semi-algebraic manifold of dimension \({n-1 \geq p \geq 2}\) and introduce sufficient conditions for the existence of a fibration structure (generalized open book structure) induced by the projection \({\frac{F}{\Vert F \Vert}:W{\setminus} F^{-1}(0) \to S^{p-1}}\). Moreover, we show that the well known local and global Milnor fibrations, in the real and complex settings, follow as a byproduct by considering W as spheres of small and big radii, respectively. Furthermore, we consider the composition mapping of F with the canonical projection \({\pi: \mathbb{R}^{p} \to \mathbb{R}^{p-1}}\) and prove that the fibers of \({\frac{F}{\Vert F \Vert}}\) and \({\frac{\pi \circ F}{\Vert \pi \circ F \Vert}}\) are homotopy equivalent. We also show several formulae relating the Euler characteristics of the fiber of the projection \({\frac{F}{\Vert F \Vert}}\) and \({W \cap F^{-1}(0)}\). Similar formulae are proved for mappings obtained after composition of F with canonical projections.     

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.     

Minimal tiled orders of finite global dimension

In this paper, we consider the sequence of balancing and Lucas balancing numbers. The balancing numbers \({B_n}\) are given by the recurrence \({B_n = 6 B_{n-1} - B_{n-2}}\) with initial conditions \({B_0 = 0, B_1 = 1}\) and its associated Lucas balancing numbers \({C_n}\) are given by the recurrence \({C_n = 6 C_{n-1} - C_{n-2}}\) with initial conditions \({C_0 = 1, C_1 = 3}\). First we find the perfect powers in the sequence of balancing and Lucas balancing numbers. We also identify those Lucas balancing numbers which are products of a power of 3 and a perfect power. Using this property of Lucas balancing numbers, we solve a conjecture regarding the non-existence of positive integral solution (x, y) for the Diophantine equation \({2x^2 + 1 = 3^b y^m}\) for any even positive integers b and m with \({m > 2}\), given in (Int J Number Theory 11:1259–1274, 2015). Also we prove that the Diophantine equations \({B_n B_{n+d}\ldots B_{n+(k-1)d} = y^m}\) and \({C_n C_{n+d}\ldots C_{n+(k-1)d} = y^m}\) have no solution for any positive integers n, d, k, y, and m with \({m \geq 2, y \geq 2}\) and gcd\({(n,d) = 1}\).     

Reducing Subspaces of Multiplication Operators on the Dirichlet Space

In this paper, we study the reducing subspaces for the multiplication operator by a finite Blaschke product \({\phi}\) on the Dirichlet space D. We prove that any two distinct nontrivial minimal reducing subspaces of \({M_\phi}\) are orthogonal. When the order n of \({\phi}\) is 2 or 3, we show that \({M_\phi}\) is reducible on D if and only if \({\phi}\) is equivalent to \({z^n}\). When the order of \({\phi}\) is 4, we determine the reducing subspaces for \({M_\phi}\), and we see that in this case \({M_\phi}\) can be reducible on D when \({\phi}\) is not equivalent to \({z^4}\). The same phenomenon happens when the order n of \({\phi}\) is not a prime number. Furthermore, we show that \({M_\phi}\) is unitarily equivalent to \({M_{z^n} (n > 1)}\) on D if and only if \({\phi = az^n}\) for some unimodular constant a.     

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.     

Endomorphism rings of some Young modules

Let \({\Sigma_r}\) be the symmetric group acting on \({r}\) letters, \({K}\) be a field of characteristic 2, and \({\lambda}\) and \({\mu}\) be partitions of \({r}\) in at most two parts. Denote the permutation module corresponding to the Young subgroup \({\Sigma_\lambda}\), in \({\Sigma_r}\), by \({M^\lambda}\), and the indecomposable Young module by \({Y^\mu}\). We give an explicit presentation of the endomorphism algebra \({{\rm End}_{k[\Sigma_r]}(Y^\mu)}\) using the idempotents found by Doty et al. (J Algebra 307(1):377–396, 2007).     

Arens regularity of projective tensor products

For completely contractive Banach algebras A and B (respectively operator algebras A and B), the necessary and sufficient conditions for the operator space projective tensor product \({A\widehat{\otimes}B}\) (respectively the Haagerup tensor product \({A\otimes^{h}B}\)) to be Arens regular are obtained. Using the non-commutative Grothendieck inequality, we show that, for C*-algebras A and B, \({A\otimes^{\gamma} B}\) is Arens regular if \({A\widehat{\otimes}B}\) and \({A\widehat{\otimes}B^{op}}\) are Arens regular whereas \({A\widehat{\otimes}B}\) is Arens regular if and only if \({A\otimes^{h}B}\) and \({B\otimes^{h}A}\) are, where \({\otimes^h}\), \({\otimes^{\gamma}}\), and \({\widehat{\otimes}}\) are the Haagerup, the Banach space projective tensor norm, and the operator space projective tensor norm, respectively.     

Integral means of weighted Hardy–Bloch functions

Let \({\mathcal{B}^\omega(p, q, B_d)}\) denote the \({\omega}\)-weighted Hardy–Bloch space on the unit ball B _d of \({\mathbb{C}^d}\), \({d\ge 1}\). For \({2< p,q < \infty}\) and \({f\in \mathcal{B}^\omega(p, q, B_d)}\), we obtain sharp estimates on the growth of the p-integral means M _p (f, r) as \({r\to 1-}\).     

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 \({i^*(\mathcal{P},\ell+1)\neq i^*(\mathcal{Q},\ell+1)}\).     

Complex homomorphisms on self-conjugate algebras of functions

Let X be a non-void set and A be a subalgebra of \({\mathbb{C}^{X}}\) . We call a \({\mathbb{C}}\) -linear functional \({\varphi}\) on A a 1-evaluation if \({\varphi(f) \in f(X) }\) for all \({f\in A}\) . From the classical Gleason–Kahane–?elazko theorem, it follows that if X in addition is a compact Hausdorff space then a mapping \({\varphi}\) of \({C_{\mathbb{C}}(X) }\) into \({\mathbb{C}}\) is a 1-evaluation if and only if \({\varphi}\) is a \({\mathbb{C}}\) -homomorphism. In this paper, we aim to investigate the extent to which this equivalence between 1-evaluations and \({\mathbb{C}}\) -homomorphisms can be generalized to a wider class of self-conjugate subalgebras of \({\mathbb{C}^{X}}\) . In this regards, we prove that a \({\mathbb{C}}\) -linear functional on a self-conjugate subalgebra A of \({\mathbb{C}^{X}}\) is a positive \({\mathbb{C}}\) -homomorphism if and only if \({\varphi}\) is a \({\overline{1}}\) -evaluation, that is, \({\varphi(f) \in\overline{f\left(X\right)}}\) for all \({f\in A}\) . As consequences of our general study, we prove that 1-evaluations and \({\mathbb{C}}\) -homomorphisms on \({C_{\mathbb{C}}\left( X\right)}\) coincide for any topological space X and we get a new characterization of realcompact topological spaces.     

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 \({n=1}\).