Meromorphic continuation of the Eisenstein series induced from spherical, cuspidal data on parabolic subgroups is achieved via reworking Bernstein's adaptation of Selberg's third proof of meromorphic continuation.

We study when characteristic and Hölder continuous functions are traces of Sobolev functions on doubling metric measure spaces. We provide analytic and geometric conditions sufficient for extending characteristic and Hölder continuous functions into globally defined Sobolev functions.

The $H$ -space that represents Brown-Peterson cohomology $\text{B}{{\text{P}}^{k}}\left( - \right)$ was split by the second author into indecomposable factors, which all have torsion-free homotopy and homology. Here, we do the same for the related spectrum $P\left( n \right)$ , by constructing idempotent operations in $P\left( n \right)$ -cohomology $P{{(n)}^{k}}\left( - \right)$ in the style of Boardman-Johnson-Wilson; this relies heavily on the Ravenel-Wilson determination of the relevant Hopf ring. The resulting $\left( i\,-\,1 \right)$ -connected $H$ -spaces ${{Y}_{i}}$ have free connective Morava $K$ -homology $k{{(n)}_{*}}({{Y}_{i}})$ , and may be built from the spaces in the $\Omega$ -spectrum for $k\left( n \right)$ using only ${{v}_{n}}$ -torsion invariants.

We also extend Quillen's theorem on complex cobordism to show that for any space $X$ , the $P{{\left( n \right)}_{*}}$ -module $P{{(n)}^{*}}\,(X)$ is generated by elements of $P{{(n)}^{i}}(X)$ for $i\,\ge \,0$ . This result is essential for the work of Ravenel-Wilson-Yagita, which in many cases allows one to compute BP-cohomology from Morava $K$ -theory.

We consider maximal regularity in the ${{H}^{p}}$ sense for the Cauchy problem ${{u}^{\prime }}(t)+Au(t)=f(t)(t\,\in \mathbb{R})$ , where $A$ is a closed operator on a Banach space $X$ and $f$ is an $X$ -valued function defined on $\mathbb{R}$ . We prove that if $X$ is an AUMD Banach space, then $A$ satisfies ${{H}^{p}}$ -maximal regularity if and only if $A$ is Rademacher sectorial of type $<\frac{\pi }{2}.$ Moreover we find an operator $A$ with ${{H}^{p}}$ -maximal regularity that does not have the classical ${{L}^{p}}$ -maximal regularity. We prove a related Mikhlin type theorem for operator valued Fourier multipliers on Hardy spaces ${{H}^{p}}(\mathbb{R};\,X)$ , in the case when $X$ is an AUMD Banach space.

We define the higher order Riesz transforms and the Littlewood-Paley $g$ -function associated to the differential operator ${{L}_{\lambda }}f(\theta )\,=\,-{f}''(\theta )-2\lambda \cot \theta {f}'(\theta )+{{\lambda }^{2}}f(\theta )$ . We prove that these operators are Calderón–Zygmund operators in the homogeneous type space $((0,\,\pi ),\,{{(\sin t)}^{2\lambda }}dt)$ . Consequently, ${{L}^{p}}$ weighted, ${{H}^{1}}\,-\,{{L}^{1}}$ and ${{L}^{\infty }}\,-\,BMO$ inequalities are obtained.

We consider the $p$ -Yang-Mills functional $\left( p\,\ge \,2 \right)$ defined as $Y{{M}_{p}}(\nabla ):=\frac{1}{p}{{\int }_{M}}{{\left\| {{R}^{\nabla }} \right\|}^{p}}$ . We call critical points of $Y{{M}_{p}}(\cdot )$ the p-Yang–Mills connections, and the associated curvature ${{R}^{\nabla }}$ the $p$ -Yang-Mills fields. In this paper, we prove gap properties and instability theorems for $p$ -Yang-Mills fields over submanifolds in ${{\mathbb{R}}^{n+k}}$ and ${{\mathbb{S}}^{n+k}}$ .

We use the monomial basis theory developed by Deng and Du to present an elementary algebraic construction of the canonical bases for both the Ringel–Hall algebra of a cyclic quiver and the positive part ${{\mathbf{U}}^{+}}$ of the quantum affine $\mathfrak{s}{{\mathfrak{l}}_{n}}$ . This construction relies on analysis of quiver representations and the introduction of a new integral PBW-like basis for the Lusztig $\mathbb{Z}[v,\,{{v}^{-1}}]$ -form of ${{\mathbf{U}}^{+}}$ .

Let $K$ be a number field, and let $F$ be a symmetric bilinear form in $2N$ variables over $K$ . Let $Z$ be a subspace of ${{K}^{N}}$ . A classical theorem of Witt states that the bilinear space $\left( Z,\,F \right)$ can be decomposed into an orthogonal sum of hyperbolic planes and singular and anisotropic components. We prove the existence of such a decomposition of small height, where all bounds on height are explicit in terms of heights of $F$ and $Z$ . We also prove a special version of Siegel's lemma for a bilinear space, which provides a small-height orthogonal decomposition into one-dimensional subspaces. Finally, we prove an effective version of the Cartan-Dieudonné theorem. Namely, we show that every isometry $~\sigma$ of a regular bilinear space $\left( Z,\,F \right)$ can be represented as a product of reflections of bounded heights with an explicit bound on heights in terms of heights of $F$ , $Z$ , and $~\sigma$ .

We prove dispersive and Strichartz inequalities for the solution of the wave equation related to the full Laplacian on the Heisenberg group, by means of Besov spaces defined by a Littlewood–Paley decomposition related to the spectral resolution of the full Laplacian. This requires a careful analysis due also to the non-homogeneous nature of the full Laplacian. This result has to be compared to a previous one by Bahouri, Gérard and Xu concerning the solution of the wave equation related to the Kohn Laplacian.

We prove two spectral identities. The first one relates the relative trace formula for the spherical variety $\text{GSpin(4,}\,\text{3)/}{{G}_{2}}$ with a weighted trace formula for $G{{L}_{2}}$ . The second relates a spherical variety pertaining to ${{F}_{4}}$ to one of $GSp\left( 6 \right)$ . These identities are in accordance with a conjecture made by Jacquet, Lai, and Rallis, and are obtained without an appeal to a geometric comparison.