## Proceedings of the Centre for Mathematics and its Applications

- Proc. Centre Math. Appl.
- Miniconference on Operators in Analysis. Ian Doust, Brian Jefferies, Chun Li, and Alan McIntosh, eds. Proceedings of the Centre for Mathematical Analysis, v. 24. (Canberra AUS: Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, 1990), 91 - 102

### $H_{\infty}$ functional calculus of second order elliptic partial differntial operators on $L^p$ spaces

#### Abstract

Let L be a strongly elliptic partial differential operator of second order, with real coefficients on $L^p(\Omega), 1 \lt p \lt \infty$, with either Dirichlet, or Neumann, or "oblique" boundary conditions. Assume that $\Omega$ is an open, bounded domain with $C^2$ boundary. By adding a oonstant, if necessary, we then establish an $H_\infty$, functional calculus which associates an operator m(L) to each bounded holomorphic function m so that \[ \| m (L) \| \leq M \| m \|\infty \] where M is a constmt independent of m. Under suitable asumptions on L, we can also obtain a similar result in the case of Dirichlet boundary conditions where $\Omega$ is a non-smooth domain.

#### Article information

**Dates**

First available in Project Euclid:
18 November 2014

**Permanent link to this document**

https://projecteuclid.org/
euclid.pcma/1416335063

**Mathematical Reviews number (MathSciNet)**

MR1060114

#### Citation

Duong, Xuan Thinh. $H_{\infty}$ functional calculus of second order elliptic partial differntial operators on $L^p$ spaces. Miniconference on Operators in Analysis, 91--102, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, Canberra AUS, 1990. https://projecteuclid.org/euclid.pcma/1416335063