Università degli Studi di Genova


Spectral multipliers for Laplacians associated to some Dirichlet forms

Let $L^\phi =-\Delta-\phi^{-1}\nabla\phi\cdot\nabla$ be the self-adjoint operator associated to the Dirichlet form $Q^\phi(f)=\int_{R^d} |\nabla f(x)|^2 d\lambda^\phi(x)$, where $\phi$ is a positive $C^2$ function, $d \lambda^\phi = \phi d\lambda$ and $\lambda$ denotes Lebesgue measure on $R^d$. We study the boundedness on $L^p(\lambda^\phi)$ of spectral multipliers of $L^\phi $. We prove that if $\phi$ grows or decays at most exponentially at infinity and satisfies a suitable ``curvature condition", then functions which are bounded and holomorphic in the intersection of a parabolic region and a sector and satisfy Mihlin type conditions at infinity are spectral multipliers of $L^p(\lambda^\phi)$. The parabolic region depends on $\phi$, on $p$, and on the infimum of the essential spectrum of the operator $L^\phi$ on $L^2(\lamda^phi)$. The sector depends on the angle of holomorphy of the semigroup generated by $L^\phi$ on $L^p(\lambda^\phi)$.

Functional calculus, spectral multiplier, Dirichlet form

47, 42, 60

Pubblicato su: Proceedings of the Edinburgh Mathematical Society. Series II Vol. 51 (2008) N. 3 Pag. 581-607