Volume 31, Issue 4

Let G be a noncompact semi-simple Lie group with Iwasawa decomposition G=K⁢A⁢N{G=KAN}. For a semigroup S⊂G{S\subset G} with nonempty interior we find a domain of convergence of the Helgason–Laplace transform IS⁢(λ,u)=∫Seλ⁢(𝖺⁢(g,u))⁢𝑑g{I_{S}(\lambda,u)=\int_{S}e^{\lambda(\mathsf{a}(g,u))}\,dg}, where dg is the Haar measure of G , u∈K{u\in K}, λ∈𝔞∗{\lambda\in\mathfrak{a}^{\ast}}, 𝔞{\mathfrak{a}} is the Lie algebra of A and g⁢u=k⁢e𝖺⁢(g,u)⁢n∈K⁢A⁢N{gu=ke^{\mathsf{a}(g,u)}n\in KAN}. The domain is given in terms of a flag manifold of G written 𝔽Θ⁢(S){\mathbb{F}_{\Theta(S)}} called the flag type of S , where Θ⁢(S){\Theta(S)} is a subset of the simple system of roots. It is proved that IS⁢(λ,u)<∞{I_{S}(\lambda,u)<\infty} if λ belongs to a convex cone defined from Θ⁢(S){\Theta(S)} and u∈π-1⁢(𝒟Θ⁢(S)⁢(S)){u\in\pi^{-1}(\mathcal{D}_{\Theta(S)}(S))}, where 𝒟Θ⁢(S)⁢(S)⊂𝔽Θ⁢(S){\mathcal{D}_{\Theta(S)}(S)\subset\mathbb{F}_{\Theta(S)}} is a B -convex set and π:K→𝔽Θ⁢(S){\pi:K\rightarrow\mathbb{F}_{\Theta(S)}} is the natural projection. We prove differentiability of IS⁢(λ,u){I_{S}(\lambda,u)} and apply the results to construct of a Riemannian metric in 𝒟Θ⁢(S)⁢(S){\mathcal{D}_{\Theta(S)}(S)} invariant by the group S∩S-1{S\cap S^{-1}} of units of S .

In this paper, we investigate sharp damping estimates for a class of one-dimensional oscillatory integral operators with real-analytic phases. By establishing endpoint estimates for suitably damped oscillatory integral operators, we are able to give a new proof of the sharp Lp{L^{p}} estimates, which have been proved by Xiao in [Endpoint estimates for one-dimensional oscillatory integral operators, Adv. Math. 316 2017, 255–291]. The damping estimates obtained in this paper are of independent interest.

We explore the graded-formality and filtered-formality properties of finitely generated groups by studying the various Lie algebras over a field of characteristic 0 attached to such groups, including the Malcev Lie algebra, the associated graded Lie algebra, the holonomy Lie algebra, and the Chen Lie algebra. We explain how these notions behave with respect to split injections, coproducts, direct products, as well as field extensions, and how they are inherited by solvable and nilpotent quotients. A key tool in this analysis is the 1-minimal model of the group, and the way this model relates to the aforementioned Lie algebras. We illustrate our approach with examples drawn from a variety of group-theoretic and topological contexts, such as finitely generated torsion-free nilpotent groups, link groups, and fundamental groups of Seifert fibered manifolds.

We show that compact toric cosymplectic manifolds are mapping tori of equivariant symplectomorphisms of toric symplectic manifolds.

Let F be any non-Archimedean local field with residue field of cardinality qF{q_{F}}. In this article, we obtain a classification of typical representations for the Bernstein components associated to the inertial classes of the form [GLn⁡(F)×F×,σ⊗χ]{[\operatorname{GL}_{n}(F)\times F^{\times},\sigma\otimes\chi]} with qF>2{q_{F}>2}, and for the principal series components with qF>3{q_{F}>3}. With this we complete the classification of typical representations for GL3⁡(F){\operatorname{GL}_{3}(F)}, for qF>2{q_{F}>2}.

There exist three vector fields with complete polynomial flows on ℂn{\mathbb{C}^{n}}, n≥2{n\geq 2}, which generate the Lie algebra generated by all algebraic vector fields on ℂn{\mathbb{C}^{n}} with complete polynomial flows. In particular, the flows of these vector fields generate a group that acts infinitely transitively. The analogous result holds in the holomorphic setting.

In this paper, we study expanding phenomena in the setting of matrix rings. More precisely, we will prove that • if A is a set of M2⁢(𝔽q){M_{2}(\mathbb{F}_{q})} and |A|≫q7/2{\lvert A\rvert\gg q^{7/2}}, then |A⁢(A+A)|,|A+A⁢A|≫q4{\lvert A(A+A)\rvert,\lvert A+AA\rvert\gg q^{4}}, • if A is a set of SL2⁢(𝔽q){\mathrm{SL}_{2}(\mathbb{F}_{q})} and |A|≫q5/2{\lvert A\rvert\gg q^{5/2}}, then |A⁢(A+A)|,|A+A⁢A|≫q4{\lvert A(A+A)\rvert,\lvert A+AA\rvert\gg q^{4}}. We also obtain similar results for the cases of A⁢(B+C){A(B+C)} and A+B⁢C{A+BC}, where A,B,C{A,B,C} are sets in M2⁢(𝔽q){M_{2}(\mathbb{F}_{q})}.

In this paper we consider the sup-norm problem in the context of analytic Eisenstein series for GL2{\mathrm{GL}_{2}} over number fields. We prove a hybrid bound which is sharper than the corresponding bound for Maaß forms. Our results generalize those of Huang and Xu where the case of Eisenstein series of square-free levels over the base field ℚ{\operatorname{\mathbb{Q}}} had been considered.

Let ℒ{\mathcal{L}} be the Laplace operator on ℝd{\mathbb{R}^{d}}, d≥3{d\geq 3}, or the Laplace–Beltrami operator on the harmonic NA group (in particular, on a rank one noncompact symmetric space). For the equation ℒ⁢u-φ⁢(⋅,u)=0{\mathcal{L}u-\varphi(\,\cdot\,,u)=0} we give necessary and sufficient conditions for the existence of entire bounded or large solutions under the hypothesis of radiality of φ with respect to the first variable. A Harnack-type inequality for positive continuous solutions is also proved.

We prove partial and full boundary Hölder continuity, under a suitable regularity on the boundary datum, of the minimizers of non-autonomous integral functionals of the type ∫ΩΦ⁢((Ai⁢jα⁢β⁢(x,u)⁢Di⁢uα⁢Dj⁢uβ)12)⁢d⁢x,\int_{\Omega}\Phi\bigl{(}(A^{\alpha\beta}_{ij}(x,u)D_{i}u^{\alpha}D_{j}u^{% \beta})^{\frac{1}{2}}\bigr{)}\mathop{}\!dx, where Ω⊂ℝn{\Omega\subset\mathbb{R}^{n}} is a bounded domain, Φ⁢(t)=tp⁢logα⁡(e+t){\Phi(t)=t^{p}\log^{\alpha}(e+t)} with 1<p≤n{1<p\leq n} and α>0{\alpha>0}, and A⁢(x,s)=(Ai⁢jα⁢β⁢(x,s)){A(x,s)=(A^{\alpha\beta}_{ij}(x,s))} is a uniformly elliptic, bounded and continuous function.

It is shown that the radial averaging operator Tω⁢(f)⁢(z)=∫|z|1f⁢(s⁢z|z|)⁢ω⁢(s)⁢𝑑sω^⁢(z),ω^⁢(z)=∫|z|1ω⁢(s)⁢𝑑s,T_{\omega}(f)(z)=\frac{\int_{|z|}^{1}f\bigl{(}s\frac{z}{|z|}\bigr{)}\omega(s)% \,ds}{\widehat{\omega}(z)},\quad\widehat{\omega}(z)=\int_{|z|}^{1}\omega(s)\,ds, induced by a radial weight ω on the unit disc 𝔻{\mathbb{D}}, is bounded from the weighted Bergman space Aνp{A^{p}_{\nu}}, where 0<p<∞{0<p<\infty} and the radial weight ν satisfies ν^⁢(r)≤C⁢ν^⁢(1+r2){\widehat{\nu}(r)\leq C\widehat{\nu}(\frac{1+r}{2})} for all 0≤r<1{0\leq r<1}, to Lνp{L^{p}_{\nu}} if and only if the self-improving condition sup0≤r<1⁡ω^⁢(r)p∫r1s⁢ν⁢(s)⁢𝑑s⁢∫0rt⁢ν⁢(t)ω^⁢(t)p⁢𝑑t<∞\sup_{0\leq r<1}\frac{\widehat{\omega}(r)^{p}}{\int_{r}^{1}s\nu(s)\,ds}\int_{0% }^{r}\frac{t\nu(t)}{\widehat{\omega}(t)^{p}}\,dt<\infty is satisfied. Further, two characterizations of the weak-type inequality η⁢({z∈𝔻:|Tω⁢(f)⁢(z)|≥λ})≲λ-p⁢∥f∥Lνpp,λ>0,\eta(\{z\in\mathbb{D}:|T_{\omega}(f)(z)|\geq\lambda\})\lesssim\lambda^{-p}\|f% \|_{L^{p}_{\nu}}^{p},\quad\lambda>0, are established for arbitrary radial weights ω, ν and η. Moreover, differences and interrelationships between the cases Aνp→Lνp{A^{p}_{\nu}\to L^{p}_{\nu}}, Lνp→Lνp{L^{p}_{\nu}\to L^{p}_{\nu}} and Lνp→Lνp,∞{L^{p}_{\nu}\to L^{p,\infty}_{\nu}} are analyzed.

We relate p -adic families of Jacobi forms to big Heegner points constructed by B. Howard, in the spirit of the Gross–Kohnen–Zagier theorem. We view this as a GL⁢(2){\mathrm{GL}(2)} instance of a p -adic Kudla program.