arXiv subjects

Detlef Mueller

Publications and source records attributed to Detlef Mueller.

6 recordsLinked to original sources

arXiv2008

Wave equation and multiplier estimates on Damek-Ricci spaces

Let S be a Damek-Ricci space and L be a distinguished left invariant Laplacian on S. We prove pointwise estimates for the convolution kernels of spectrally localized wave operators associated with L. This generalizes previous results proved by D. Mueller and C. Thiele on ax+b-groups. We also prove pointwise estimates of the gradient of these convolution kernels. As a corollary we reprove basic multiplier estimates from previous papers of W. Hebisch and T. Steger and M. Vallarino by different methods. Finally we derive Sobolev estimates for the solution to the wave equation associated with L.

math.CA↗

arXiv2006

Local solvability of linear differential operators with double characteristics I: Necessary conditions

This is a the first in a series of two articles devoted to the question of local solvability of doubly characteristic differential operators $L,$ defined, say, in an open set $\Om\subset \RR^n.$ Suppose the principal symbol $p_k$ of $L$ vanishes to second order at $(x_0,ξ_0)\in T^*\Om\setminus 0,$ and denote by $Q_\H$ the Hessian form associated to $p_k$ on $T_{(x_0,ξ_0)}T^*\Om.$ As the main result of this paper, we show (under some rank conditions and some mild additional conditions) that a necessary condition for local solvability of $L$ at $x_0$ is the existence of some $θ\in\RR$ such that $\Re (e^{iθ}Q_\H)\ge 0.$

math.AP↗

arXiv2005

Local solvability of second order differential operators with double characteristics I: Necessary conditions

This is a the first in a series of two articles devoted to the question of local solvability of doubly characteristic second order differential operators. For a large class of such operators, we show that local solvability at a given point implies "essential dissipativity" of the operator at this point. By means of Hoermander's classical necessary condition for local solvability, the proof is reduced to the following question, whose answer forms the core of the paper: Suppose that $Q_A$ and $Q_B$ are two real quadratic forms on a finite dimensional symplectic vector space, and let $Q_C:=\{Q_A,Q_B\}$ be given by the Poisson bracket of $Q_A$ and $Q_B.$ Then $Q_C$ is again a quadratic form, and we may ask: When can we find a common zero of $Q_A$ and $Q_B$ at which $Q_C$ does not vanish? The second paper, in combination with the first one, will give a fairly comprehensive picture of what rules local solvability of invariant second order operators on the Heisenberg group.

math.AP↗

arXiv2004

Wave equation and multiplier estimates on ax+b groups

Let L be the distinguished Laplacian on certain semidirect products of R by R^n which are of ax+b type. We prove pointwise estimates for the convolution kernels of spectrally localized wave operators for arbitrary time and scaling parameter.

math.CA↗

arXiv2003

Singular spherical maximal operators on a class of two step nilpotent Lie groups

Let $H^n\cong \Bbb R^{2n}\ltimes \Bbb R$ be the Heisenberg group and let $μ_t$ be the normalized surface measure for the sphere of radius $t$ in $\Bbb R^{2n}$. Consider the maximal function defined by $Mf=\sup_{t>0} |f*μ_t|$. We prove for $n\ge 2$ that $M$ defines an operator bounded on $L^p(H^n)$ provided that $p>2n/(2n-1)$. This improves an earlier result by Nevo and Thangavelu, and the range for $L^p$ boundedness is optimal. We also extend the result to a more general setting of surfaces and to groups satisfying a nondegeneracy condition; these include the groups of Heisenberg type.

math.CA↗

arXiv2003

Solvability of dissipative second order left-invariant differential operators on the Heisenberg group

We prove local solvability for large classes of operators of the form $$ L=\sum_{j,k=1}^{2n}a_{jk}V_jV_k+iαU,$$ where the $V_j$ are left-invariant vector fields on the Heisenberg group satisfying the commutation relations $[V_j,V_{j+n}]=U$ for $1\le j\le n$, and where $A=(a_{jk})$ is a complex symmetric matrix with semi-definite real part. Our results widely extend all of the results for the case of non-real, semi-definite matrices $A$ known to date, in particular those obtained recently jointly with F. Ricci under Sjöstrand's cone condition.

math.CA↗