×

On positive Rockland operators. (English) Zbl 0856.43005

Let \(G\) be a homogeneous Lie group of homogeneous dimension \(D\) with a left Haar measure \(dg\) and \(L\) the action of \(G\) as left translations on \(L_p(G;dg)\). Further let \(H=dL(C)\) denote a homogeneous operator of homogeneous order \(m\) associated with \(L\). If \(H\) is positive and hypoellipctic on \(L_2\), i.e., \(H\) is a positive Rockland operator, we prove that it is closed on each of the \(L_p\)-spaces, \(p\in\langle 1,\infty\rangle\), and that it generates a semigroup \(S\) with a smooth kernel \(K\) which, with its derivatives, satisfies Gaussian bounds. The semigroup is holomorphic in the open right half-plane on all the \(L_p\)-spaces, \(p\in[1,\infty]\). Specifically, if \(|\cdot|\) denotes a homogeneous modulus on \(G\) and \(dL(a^\alpha)\) is a homogeneous differential operator of homogeneous order \(|\alpha|\) then for all \(\varepsilon>0\) there exist \(a,b>0\) such that \[ |(dL(a^\alpha)K_z)(g)|\leq a|z|^{-(D+|\alpha|)/m}e^{-b(|g|^m|z|^{-1})^{1/(m-1)}} \] uniformly for all \(z\in\mathbb{C}\setminus\{0\}\) with \(|\arg z|<\pi/2-\varepsilon\) and \(g\in G\). Further extensions of these results to non-homogeneous operators and general representations are also given.

MSC:

43A85 Harmonic analysis on homogeneous spaces
22E30 Analysis on real and complex Lie groups
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
35B45 A priori estimates in context of PDEs
PDFBibTeX XMLCite
Full Text: DOI EuDML