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 H\"older continuous functions
are traces of Sobolev functions on doubling metric measure spaces.
We provide analytic and geometric conditions sufficient for extending
characteristic and H\"older continuous functions into globally defined
Sobolev functions.
The $H$-space that represents Brown--Peterson cohomology
$\BP^k (-)$ 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 $P(n)$, by constructing
idempotent operations in $P(n)$-cohomology $P(n)^k ($--$)$ in the style
of Boardman--Johnson--Wilson; this relies heavily on the
Ravenel--Wilson determination of the relevant Hopf ring.
The resulting $(i- 1)$-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 $k(n)$
using only $v_n$-torsion invariants.
We also extend Quillen's theorem on complex cobordism to show that
for any space $X$, the \linebeak$P(n)_*$-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 $u'(t) + Au(t) = f(t)\ (t\in \R)$, where $A$ is a closed
operator on a Banach space $X$ and $f$ is an $X$-valued function
defined on $\R$. 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 $H^p(\R;X)$, 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 $L_\l f(\theta)=-f''(\theta)-2\l\cot\theta
f'(\theta)+\l^2f(\theta)$. We prove that these operators are
Calder\'{o}n--Zygmund operators in the homogeneous type space
$((0,\pi),(\sin t)^{2\l}\,dt)$. Consequently, $L^p$ weighted,
$H^1-L^1$ and $L^\infty-BMO$ inequalities are obtained.
We consider the
$p$-Yang--Mills functional
$(p\geq 2)$
defined as
$\YM_p(\nabla):=\frac 1 p \int_M \|\rn\|^p$.
We call critical points of $\YM_p(\cdot)$ the $p$-Yang--Mills
connections, and the associated curvature $\rn$ 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 $\bU^+$ of the quantum affine $\frak{sl}_n$. 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~$\bU^+$.
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 $(Z,F)$ 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--Dieudonn{\'e} theorem. Namely, we show
that every isometry $\sigma$ of a regular bilinear space $(Z,F)$ 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, G\'erard
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 $\GSpin(4,3)/G_2$ with a
weighted trace formula for $\GL_2$. The second relates a spherical
variety pertaining to $F_4$ to one of $\GSp(6)$. These identities are
in accordance with a conjecture made by Jacquet, Lai, and Rallis,
and are obtained without an appeal to a geometric comparison.