Volume 31, Issue 2

Let G be a compact Hausdorff group and let H be a closed subgroup of G . We introduce pseudo-differential operators with symbols on the homogeneous space G/H{G/H}. We present a necessary and sufficient condition on symbols for which these operators are in the class of Hilbert–Schmidt operators. We also give a characterization of and a trace formula for the trace class pseudo-differential operators on the homogeneous space G/H{G/H}.

The simplicial wedge construction on simplicial complexes and simple polytopes has been used by a variety of authors to study toric and related spaces, including non-singular toric varieties, toric manifolds, intersections of quadrics and more generally, polyhedral products. In this paper we extend the analysis to include toric orbifolds. Our main results yield infinite families of toric orbifolds, derived from a given one, whose integral cohomology is free of torsion and is concentrated in even degrees, a property which might be termed integrally equivariantly formal . In all cases, it is possible to give a description of the cohomology ring and to relate it to the cohomology of the original orbifold.

Let f be a Hecke–Maass cusp form for SL3⁢(ℤ){\mathrm{SL}_{3}(\mathbb{Z})} and χ a primitive Dirichlet character of prime power conductor 𝔮=pκ{\mathfrak{q}=p^{\kappa}}, with p prime. We prove the subconvexity bound L⁢(12,π⊗χ)≪p,π,ε𝔮3/4-3/40+εL\Big{(}\frac{1}{2},\pi\otimes\chi\Big{)}\ll_{p,\pi,\varepsilon}\mathfrak{q}^{% 3/4-3/40+\varepsilon} for any ε>0{\varepsilon>0}, where the dependence of the implied constant on p is explicit and polynomial.

We consider a nonlinear Dirichlet problem driven by the p -Laplace differential operator with a reaction which has a subcritical growth restriction only from above. We prove two multiplicity theorems producing three nontrivial solutions, two of constant sign and the third nodal. The two multiplicity theorems differ on the geometry near the origin. In the semilinear case (that is, p=2{p=2}), using Morse theory (critical groups), we produce a second nodal solution for a total of four nontrivial solutions. As an illustration, we show that our results incorporate and significantly extend the multiplicity results existing for a class of parametric, coercive Dirichlet problems.

Let G/H{G/H} be a reductive symmetric space of split rank one and let K be a maximal compact subgroup of G . In a previous article the first two authors introduced a notion of cusp forms for G/H{G/H}. We show that the space of cusp forms coincides with the closure of the space of K -finite generalized matrix coefficients of discrete series representations if and only if there exist no K -spherical discrete series representations. Moreover, we prove that every K -spherical discrete series representation occurs with multiplicity one in the Plancherel decomposition of G/H{G/H}.

Let G be an arbitrary topological group. We prove that the cellularity of G is equal to the cellularity of the quotient space G/K{G/K} for every compact subgroup K of G .

In this paper, we study some shifted convolution sums for higher rank groups. In particular, we establish an asymptotic formula for a GL⁢(4)×GL⁢(2){\mathrm{GL}(4)\times\mathrm{GL}(2)} shifted convolution sum ∑n≤x|λf⁢(n)|2⁢rl⁢(n+b),\sum_{n\leq x}\lvert\lambda_{f}(n)\rvert^{2}r_{l}(n+b), where λf⁢(n){\lambda_{f}(n)} are normalized Fourier coefficients of a Hecke holomorphic cusp form and rl⁢(n){r_{l}(n)} denotes the number of representations of n by the quadratic form x12+⋯+xl2{x_{1}^{2}+\cdots+x_{l}^{2}}.

This paper presents a description of the fourth dimension quotient, using the theory of limits of functors from the category of free presentations of a given group to the category of abelian groups. A functorial description of a quotient of the third Fox subgroup is given and, as a consequence, an identification (not involving an isolator) of the third Fox subgroup is obtained. It is shown that the limit over the category of free representations of the third Fox quotient represents the composite of two derived quadratic functors.

Let f be a Hecke cusp form of weight k for the full modular group, and let {λf⁢(n)}n≥1{\{\lambda_{f}(n)\}_{n\geq 1}} be the sequence of its normalized Fourier coefficients. Motivated by the problem of the first sign change of λf⁢(n){\lambda_{f}(n)}, we investigate the range of x (in terms of k ) for which there are cancellations in the sum Sf⁢(x)=∑n≤xλf⁢(n){S_{f}(x)=\sum_{n\leq x}\lambda_{f}(n)}. We first show that Sf⁢(x)=o⁢(x⁢log⁡x){S_{f}(x)=o(x\log x)} implies that λf⁢(n)<0{\lambda_{f}(n)<0} for some n≤x{n\leq x}. We also prove that Sf⁢(x)=o⁢(x⁢log⁡x){S_{f}(x)=o(x\log x)} in the range log⁡x/log⁡log⁡k→∞{\log x/\log\log k\to\infty} assuming the Riemann hypothesis for L⁢(s,f){L(s,f)}, and furthermore that this range is best possible unconditionally. More precisely, we establish the existence of many Hecke cusp forms f of large weight k , for which Sf⁢(x)≫Ax⁢log⁡x{S_{f}(x)\gg_{A}x\log x}, when x=(log⁡k)A{x=(\log k)^{A}}. Our results are GL2{\mathrm{GL}_{2}} analogues of work of Granville and Soundararajan for character sums, and could also be generalized to other families of automorphic forms.

We consider a generalization K0gr⁢(R){K_{0}^{\operatorname{gr}}(R)} of the standard Grothendieck group K0⁢(R){K_{0}(R)} of a graded ring R with involution. If Γ is an abelian group, we show that K0gr{K_{0}^{\operatorname{gr}}} completely classifies graded ultramatricial *{*}-algebras over a Γ-graded *{*}-field A such that (1) each nontrivial graded component of A has a unitary element in which case we say that A has enough unitaries, and (2) the zero-component A0{A_{0}} is 2-proper (a⁢a*+b⁢b*=0{aa^{*}+bb^{*}=0} implies a=b=0{a=b=0} for any a,b∈A0{a,b\in A_{0}}) and *{*}-pythagorean (for any a,b∈A0{a,b\in A_{0}} one has a⁢a*+b⁢b*=c⁢c*{aa^{*}+bb^{*}=cc^{*}} for some c∈A0{c\in A_{0}}). If the involutive structure is not considered, our result implies that K0gr{K_{0}^{\operatorname{gr}}} completely classifies graded ultramatricial algebras over any graded field A . If the grading is trivial and the involutive structure is not considered, we obtain some well-known results as corollaries. If R and S are graded matricial *{*}-algebras over a Γ-graded *{*}-field A with enough unitaries and f:K0gr⁢(R)→K0gr⁢(S){f:K_{0}^{\operatorname{gr}}(R)\to K_{0}^{\operatorname{gr}}(S)} is a contractive ℤ⁢[Γ]{\mathbb{Z}[\Gamma]}-module homomorphism, we present a specific formula for a graded *{*}-homomorphism ϕ:R→S{\phi:R\to S} with K0gr⁢(ϕ)=f{K_{0}^{\operatorname{gr}}(\phi)=f}. If the grading is trivial and the involutive structure is not considered, our constructive proof implies the known results with existential proofs. If A0{A_{0}} is 2-proper and *{*}-pythagorean, we also show that two graded *{*}-homomorphisms ϕ,ψ:R→S{\phi,\psi:R\to S} are such that K0gr⁢(ϕ)=K0gr⁢(ψ){K_{0}^{\operatorname{gr}}(\phi)=K_{0}^{\operatorname{gr}}(\psi)} if and only if there is a unitary element u of degree zero in S such that ϕ⁢(r)=u⁢ψ⁢(r)⁢u*{\phi(r)=u\psi(r)u^{*}} for any r∈R{r\in R}. As an application of our results, we show that the graded version of the Isomorphism Conjecture holds for a class of Leavitt path algebras: if E and F are countable, row-finite, no-exit graphs in which every infinite path ends in a sink or a cycle and K is a 2-proper and *{*}-pythagorean field, then the Leavitt path algebras LK⁢(E){L_{K}(E)} and LK⁢(F){L_{K}(F)} are isomorphic as graded rings if any only if they are isomorphic as graded *{*}-algebras. We also present examples which illustrate that K0gr{K_{0}^{\operatorname{gr}}} produces a finer invariant than K0{K_{0}}.

Let Δ be a matroid complex. In this paper, we explicitly compute the regularity of all the symbolic powers of its Stanley–Reisner ideal in terms of combinatorial data of Δ. In order to do that, we provide a sharp bound between the arboricity of Δ and the circumference of its dual Δ*{\Delta^{*}}.

In [G. Bianchi, R. J. Gardner and P. Gronchi, Symmetrization in geometry, Adv. Math. 306 2017, 51–88], a systematic study of symmetrization operators on convex sets and their properties is conducted. In the end of their article, the authors pose several open questions. The primary goal of this manuscript is to study these questions.

In this paper, we call a set of lines of a finite classical polar space an Erdős–Ko–Rado set of lines if no two lines of the polar space are opposite, which means that for any two lines l and h in such a set there exists a point on l that is collinear with all points of h . We classify all largest such sets provided the order of the underlying field of the polar space is not too small compared to the rank of the polar space. The motivation for studying these sets comes from [7], where a general Erdős–Ko–Rado problem was formulated for finite buildings. The presented result provides one solution in finite classical polar spaces.

For a p -adic local field F of characteristic 0, with residue field 𝔣{\mathfrak{f}}, we prove that the Rankin–Selberg gamma factor of a pair of level zero representations of linear general groups over F is equal to a gamma factor of a pair of corresponding cuspidal representations of linear general groups over 𝔣{\mathfrak{f}}. Our results can be used to prove a variant of Jacquet’s conjecture on the local converse theorem.

Our aim in this paper is to establish generalizations of Sobolev’s inequality for double phase functionals Φ⁢(x,t)=tp⁢(x)+a⁢(x)⁢tq⁢(x){\Phi(x,t)=t^{p(x)}+a(x)t^{q(x)}}, where p⁢(⋅){p(\,{\cdot}\,)} and q⁢(⋅){q(\,{\cdot}\,)} satisfy log-Hölder conditions and a⁢(⋅){a(\,{\cdot}\,)} is nonnegative, bounded and Hölder continuous of order θ∈(0,1]{\theta\in(0,1]}.

Let K/ℚ{K/\mathbb{Q}} be a finite Galois extension. Let χ1,…,χr{\chi_{1},\ldots,\chi_{r}} be r≥1{r\geq 1} distinct characters of the Galois group with the associated Artin L -functions L⁢(s,χ1),…,L⁢(s,χr){L(s,\chi_{1}),\ldots,L(s,\chi_{r})}. Let m≥0{m\geq 0}. We prove that the derivatives L(k)⁢(s,χj){L^{(k)}(s,\chi_{j})}, 1≤j≤r{1\leq j\leq r}, 0≤k≤m{0\leq k\leq m}, are linearly independent over the field of meromorphic functions of order <1{<1}. From this it follows that the L -functions corresponding to the irreducible characters are algebraically independent over the field of meromorphic functions of order <1{<1}.

A sharp logarithmic bound is established for the H1{H^{1}}-norm of oscillatory singular integrals with quadratic phases and Hölder class kernels. Prior results had relied on a C1{C^{1}}-assumption on the kernel.