Band 30, Heft 5

We prove upper bounds for geodesic periods of automorphic forms over general rank one locally symmetric spaces. Such periods are integrals of automorphic forms restricted to special totally geodesic cycles of the ambient manifold and twisted with automorphic forms on the cycles. The upper bounds are in terms of the Laplace eigenvalues of the two automorphic forms, and they generalize previous results for real hyperbolic manifolds to the context of all rank one locally symmetric spaces.

Let f⁢(z)=∑n=1∞a⁢(n)⁢e2⁢π⁢i⁢n⁢z{f(z)=\sum_{n=1}^{\infty}a(n)e^{2\pi inz}} be a normalized Hecke eigenform in S2⁢knew⁢(Γ0⁢(N)){S_{2k}^{\mathrm{new}}(\Gamma_{0}(N))} with integer Fourier coefficients. We prove that there exists a constant C⁢(f)>0{C(f\/)>0} such that any integer is a sum of at most C⁢(f){C(f\/)} coefficients a⁢(n){a(n)}. We have C⁢(f)≪ε,kN6⁢k-316+ε{C(f\/)\ll_{\varepsilon,k}N^{\frac{6k-3}{16}+\varepsilon}}.

We prove a universal characterization of Hopf algebras among cocommutative bialgebras over an algebraically closed field: a cocommutative bialgebra is a Hopf algebra precisely when every split extension over it admits a join decomposition. We also explain why this result cannot be extended to a non-cocommutative setting.

In this article, we compute the Petersson norm of a family of weight one cusp forms constructed by Hecke and express it in terms of the Rademacher symbol and the regulator of real quadratic fields.

We show that for every finite nonempty subset of ℕ≥2{\mathbb{N}_{\geq 2}} there are a numerical monoid H and a squarefree element a∈H{a\in H} whose set of lengths 𝖫⁢(a){\mathsf{L}(a)} is equal to L .

We give a self-contained proof that the Graev free abelian group over a convergent sequence has no Mackey topology. This means the supremum of all locally quasi-convex compatible topologies of this group is no longer compatible.

We apply the Davies method to prove that for any regular Dirichlet form on a metric measure space, an off-diagonal stable-like upper bound of the heat kernel is equivalent to the conjunction of the on-diagonal upper bound, a cutoff inequality on any two concentric balls, and the jump kernel upper bound, for any walk dimension. If in addition the jump kernel vanishes, that is, if the Dirichlet form is strongly local, we obtain a sub-Gaussian upper bound. This gives a unified approach to obtaining heat kernel upper bounds for both the non-local and the local Dirichlet forms.

We prove the vanishing of the cup product of the bounded cohomology classes associated to any two Brooks quasimorphisms on the free group. This is a consequence of the vanishing of the square of a universal class for tree automorphism groups.

In this paper, we establish existence and uniqueness of weak solutions to general time fractional equations and give their probabilistic representations. We then derive sharp two-sided estimates for fundamental solutions of a family of time fractional equations in metric measure spaces.

The Banach algebras of Harmonic Analysis are usually far from being Arens regular and often turn out to be as irregular as possible. This utmost irregularity has been studied by means of two notions: strong Arens irregularity, in the sense of Dales and Lau, and extreme non-Arens regularity, in the sense of Granirer. Lau and Losert proved in 1988 that the convolution algebra L1⁢(G){L^{1}(G)} is strongly Arens irregular for any infinite locally compact group. In the present paper, we prove that L1⁢(G){L^{1}(G)} is extremely non-Arens regular for any infinite locally compact group. We actually prove the stronger result that for any non-discrete locally compact group G , there is a linear isometry from L∞⁢(G){L^{\infty}(G)} into the quotient space L∞⁢(G)/ℱ⁢(G){L^{\infty}(G)/\mathcal{F}(G)}, with ℱ⁢(G){\mathcal{F}(G)} being any closed subspace of L∞⁢(G){L^{\infty}(G)} made of continuous bounded functions. This, together with the known fact that ℓ∞⁢(G)/𝒲⁢𝒜⁢𝒫⁢(G){\ell^{\infty}(G)/\mathscr{W\!A\!P}(G)} always contains a linearly isometric copy of ℓ∞⁢(G){\ell^{\infty}(G)}, proves that L1⁢(G){L^{1}(G)} is extremely non-Arens regular for every infinite locally compact group.

Arone and the second author showed that when the dimensions are in the stable range, the rational homology and homotopy of the high-dimensional analogues of spaces of long knots can be calculated as the homology of a direct sum of finite graph-complexes that they described explicitly. They also showed that these homology and homotopy groups can be interpreted as the higher-order Hochschild homology, also called Hochschild–Pirashvili homology. In this paper, we generalize all these results to high-dimensional analogues of spaces of string links. The methods of our paper are applicable in the range when the ambient dimension is at least twice the maximal dimension of a link component plus two, which in particular guarantees that the spaces under study are connected. However, we conjecture that our homotopy graph-complex computes the rational homotopy groups of link spaces always when the codimension is greater than two, i.e. always when the Goodwillie–Weiss calculus is applicable. Using Haefliger’s approach to calculate the groups of isotopy classes of higher-dimensional links, we confirm our conjecture at the level of π0{\pi_{0}}.

This paper deals with the problem u∈𝒦u*,ψ⁢(Ω),\displaystyle u\in{\cal K}_{u_{*},\psi}(\Omega),∀v∈𝒦u*,ψ⁢(Ω):∫Ω∑i=1n[ai⁢(x,D⁢u)-fi]⁢Di⁢(u-v)⁢d⁢x⩽∫Ωf⁢(u-v)⁢𝑑x,\displaystyle\forall v\in{\cal K}_{u_{*},\psi}(\Omega):\int_{\Omega}\sum_{i=1}% ^{n}[a_{i}(x,Du)-f^{i}]D_{i}(u-v)\,dx\leqslant\int_{\Omega}f(u-v)\,dx, where {𝒦u*,ψ⁢(Ω)={v∈u*+W01,(pi)⁢(Ω):∑i=1nai⁢(x,D⁢u)⁢Di⁢v∈L1⁢(Ω)⁢ and ⁢v⩾ψ, a.e. ⁢Ω},u*∈W1,(pi)⁢(Ω),θ=max⁡{u*,ψ}∈u*+W01,(pi)⁢(Ω),f∈L(p¯*)′⁢(Ω),fi∈Lpi′⁢(Ω),i=1,…,n,\left\{\begin{aligned} &\displaystyle{\cal K}_{u_{*},\psi}(\Omega)=\biggl{\{}v% \in u_{*}+W_{0}^{1,(p_{i})}(\Omega):\sum_{i=1}^{n}a_{i}(x,Du)D_{i}v\in L^{1}(% \Omega)\text{ and }v\geqslant\psi,\text{ a.e. }\Omega\biggr{\}},\\ &\displaystyle u_{*}\in W^{1,(p_{i})}(\Omega),\quad\theta=\max\{u_{*},\psi\}% \in u_{*}+W_{0}^{1,(p_{i})}(\Omega),\\ &\displaystyle f\in L^{(\bar{p}^{*})^{\prime}}(\Omega),\quad f^{i}\in L^{p_{i}% ^{\prime}}(\Omega),\,i=1,\dots,n,\end{aligned}\right. and the Carathéodory functions ai:Ω×ℝn→ℝ{a_{i}:\Omega\times{\mathbb{R}}^{n}\to{\mathbb{R}}}, i=1,…,n{i=1,\dots,n}, satisfy some coercivity condition. We assume that the function θ=max⁡{u*,ψ}{\theta=\max\{u_{*},\psi\}} makes ai⁢(x,D⁢θ){a_{i}(x,D\theta)} to be more integrable than Lpi′⁢(Ω){L^{p_{i}^{\prime}}(\Omega)}, i=1,…,n{i=1,\dots,n}, and then we prove that the solution u enjoys higher integrability.

In this paper, we study the boundedness of the Hardy operators on Musielak–Orlicz spaces.

We show that any bounded t-structure in the bounded derived category of a silting-discrete algebra is algebraic, i.e. has a length heart with finitely many simple objects. As a corollary, we obtain that the space of Bridgeland stability conditions for a silting-discrete algebra is contractible.

Given a vector bundle, its (stable) order is the smallest positive integer t such that the t -fold self-Whitney sum is (stably) trivial. So far, the order and the stable order of the canonical vector bundle over configuration spaces of Euclidean spaces have been studied in [F. R. Cohen, R. L. Cohen, N. J. Kuhn and J. A. Neisendorfer, Bundles over configuration spaces, Pacific J. Math. 104 1983, 1, 47–54], [F. R. Cohen, M. E. Mahowald and R. J. Milgram, The stable decomposition for the double loop space of a sphere, Algebraic and Geometric Topology (Stanford 1976), Proc. Sympos. Pure Math. 32 Part 2, American Mathematical Society, Providence 1978, 225–228], and [S.-W. Yang, Order of the Canonical Vector Bundle on Cn⁢(k)/Σk{C_{n}(k)/\Sigma_{k}}, ProQuest LLC, Ann Arbor, 1978]. Moreover, the order and the stable order of the canonical vector bundle over configuration spaces of closed orientable Riemann surfaces with genus greater than or equal to one have been studied in [F. R. Cohen, R. L. Cohen, N. J. Kuhn and J. A. Neisendorfer, Bundles over configuration spaces, Pacific J. Math. 104 1983, 1, 47–54]. In this paper, we mainly study the order and the stable order of the canonical vector bundle over configuration spaces of spheres and disjoint unions of spheres.

We prove an endpoint version of the uniform Sobolev inequalities in [C. E. Kenig, A. Ruiz and C. D. Sogge, Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators, Duke Math. J. 55 1987, 329–347]. It was known that strong type inequalities no longer hold at the endpoints; however, we show that restricted weak type inequalities hold there, which imply the earlier classical result by real interpolation. The key ingredient in our proof is a type of interpolation first introduced by Bourgain [J. Bourgain, Esitmations de certaines functions maximales, C. R. Acad. Sci. Paris 310 1985, 499–502]. We also prove restricted weak type Stein–Tomas restriction inequalities on some parts of the boundary of a pentagon, which completely characterizes the range of exponents for which the inequalities hold.

In this paper, we first establish a kind of weighted space-time L2{L^{2}} estimate, which belongs to Keel–Smith–Sogge-type estimates, for perturbed linear elastic wave equations. This estimate refines the corresponding one established by the second author [D. Zha, Space-time L2L^{2} estimates for elastic waves and applications, J. Differential Equations 263 2017, 4, 1947–1965] and is proved by combining the methods in the former paper, the first author, Wang and Yokoyama’s paper [K. Hidano, C. Wang and K. Yokoyama, On almost global existence and local well posedness for some 3-D quasi-linear wave equations, Adv. Differential Equations 17 2012, 3–4, 267–306] and some new ingredients. Then, together with some weighted Sobolev inequalities, this estimate is used to show a refined version of almost global existence of classical solutions for nonlinear elastic waves with small initial data. Compared with former almost global existence results for nonlinear elastic waves due to John [F. John, Almost global existence of elastic waves of finite amplitude arising from small initial disturbances, Comm. Pure Appl. Math. 41 1988, 5, 615–666] and Klainerman and Sideris [S. Klainerman and T. C. Sideris, On almost global existence for nonrelativistic wave equations in 3D, Comm. Pure Appl. Math. 49 1996, 307–321], the main innovation of our result is that it considerably improves the amount of regularity of initial data, i.e., the Sobolev regularity of initial data is assumed to be the smallest among all the admissible Sobolev spaces of integer order in the standard local existence theory. Finally, in the radially symmetric case, we establish the almost global existence of a low regularity solution for every small initial data in H3×H2{H^{3}\times H^{2}}.

The main purpose of this paper is to investigate the non-archimedean counterpart of the classical result stating that the dual of a real or complex metrizable locally convex space, equipped with the locally convex topology of uniform convergence on compact sets, belongs to the topological category formed by the k -spaces. We prove that this counterpart holds when the non-archimedean valued base field 𝕂{\mathbb{K}} is locally compact, but fails for any non-locally compact 𝕂{\mathbb{K}}. Here we deal with a topological subcategory, the one formed by the k0{k_{0}}-spaces, the adequate non-archimedean substitutes for k -spaces. As a product, we complete some of the achievements on the non-archimedean Banach–Dieudonné Theorem presented in [C. Perez-Garcia and W. H. Schikhof, The p -adic Banach–Dieudonné theorem and semi-compact inductive limits, p -adic Functional Analysis (Poznań 1998), Lecture Notes Pure Appl. Math. 207, Dekker, New York 1999, 295–307]. Also, we use our results to construct in a simple way natural examples of k -spaces (which are also k0{k_{0}}-spaces) whose products are not k0{k_{0}}-spaces. This in turn improves the, rather involved, example given in [C. Perez-Garcia and W. H. Schikhof, Locally Convex Spaces over non-Archimedean Valued Fields, Cambridge Stud. Adv. Math. 119, Cambridge University Press, Cambridge, 2010] of two k0{k_{0}}-spaces whose product is not a k0{k_{0}}-space. Our theory covers an important class of non-archimedean Fréchet spaces, the Köthe sequence spaces, which have a relevant influence on applications such as the definition of a non-archimedean Laplace and Fourier transform.

We construct a new homology theory for the categories of quivers and multigraphs and describe the basic properties of introduced homology groups. We introduce a conception of homotopy in the category of quivers and we prove the homotopy invariance of homology groups.