Let L be the distinguished Laplacian on certain semidirect products of ℝ by ℝⁿ which are of ax + b type. We prove pointwise estimates for the convolution kernels of spectrally localized wave operators of the form $e^{it\surd L}\psi (\surd L/\lambda )$ for arbitrary time t and arbitrary λ > 0, where ψ is a smooth bump function supported in [-2,2] if λ ≤ 1 and in [1,2] if λ ≥ 1. As a corollary, we reprove a basic multiplier estimate of Hebisch and Steger [Math. Z. 245 (2003)] for this particular class of groups, and derive Sobolev...

Let $A$ be a $2m$ order differential operator in a hermitian vector bundle $E$ over a compact riemannian manifold $\bar{\Omega }$ with boundary $\Gamma$ ; and denote by $A_B$ the realization defined by a normal differential boundary condition $B\rho u=0$ ($u\in H^{2m}\left(E\right)$, $\rho u=$ Cauchy data). We characterize, by an explicit condition on $A$ and $B$ near $\Gamma$, the realizations $A_B$ for which there exists an integro-differential sesquilinear form $a_B(u,\nu )$ on $H^m\left(E\right)$ such that $(Au,\nu )=a_B(u,\nu )$ on $D\left(A_B\right)$; moreover we show that these are exactly the realizations satisfying a weak semiboundedness estimate:...

Let $\Omega$ be an open bounded set in ${ℝ}^n$$(n\ge 2)$, with $C^2$ boundary, and $N^{p,\lambda }\left(\Omega \right)$ ($1<p<+\infty$, $0\le \lambda <n$) be a weighted Morrey space. In this note we prove a weighted version of the Miranda-Talenti...

In this paper we mainly prove weighted Poincaré inequalities for vector fields satisfying Hörmander's condition. A crucial part here is that we are able to get a pointwise estimate for any function over any metric ball controlled by a fractional integral of certain maximal function. The Sobolev type inequalities are also derived. As applications of these weighted inequalities, we will show the local regularity of weak solutions for certain classes of strongly degenerate differential operators formed...

This work is concerned with the study of an initial boundary value problem for a non-conserved phase field system arising from the Penrose-Fife approach to the kinetics of phase transitions. The system couples a nonlinear parabolic equation for the absolute temperature with a nonlinear hyperbolic equation for the phase variable $\chi$, which is characterized by the presence of an inertial term multiplied by a small positive coefficient $\mu$. This feature is the main consequence of supposing that the response...