\(H_ \infty\) functional calculus of elliptic operators on \(L^ p\) spaces. An approach using Calderón-Zygmund operator theory.

*(English)*Zbl 0786.35046From the author’s introduction: “This paper will explain how Calderón- Zygmund operator theory can be used to establish a bounded \(H_ \infty\) functional calculus of an elliptic partial differential operator \(A\) in an \(L^ p\) space. The aim is to establish the bound of functional calculus: \(\| f(A) \| \leq C_ p \| f \|_ \infty\), where \(C_ p\) is a constant and \(f\) belongs to a certain class of bounded holomorphic functions”. Here
\[
f( {\mathcal L} )={1 \over 2 \pi i} {\mathcal L}^{-1} (I+ {\mathcal L})^ 2\int_ \Gamma({\mathcal L}-\lambda I)^{- 1}{\lambda f(\lambda) \over (1+\lambda)^ 2} d\lambda,\quad 1<p<\infty.
\]
The authors uses the representation of \(f({\mathcal L})\) as the integral operator with the kernel \(K(x,y)=\int_ \Gamma G_ \lambda(x,y)f(\lambda)d \lambda\), \(G_ \lambda(x,y)\) being the Green’s function of the corresponding even-order elliptic regular boundary value problem with smooth data and coerciveness conditions. He elaborates on an essential technique to estimate derivatives \(D_ yK(x,y)\) for functions \(f\), decaying fast enough at 0 and \(\infty\), and it is based on the estimates of the Green’s functions by S. Agmon, R. Beals, H. Tanabe and some results of Calderón-Zygmund operator theory in spaces of homogeneous type.

The main theorem 3 of the paper is then proved via the convergence lemma of A. McIntosh and theorems of A. Yagi and P. Grisvard on interpolation.

In conclusion the author notes that “Calderón-Zygmund operators can also be used to obtain a bounded \(H_ \infty\) functional calculus for a second order elliptic partial differential operator with discontinuous coefficients on a Lipschitz domain under Dirichlet or Neumann boundary conditions”.

The main theorem 3 of the paper is then proved via the convergence lemma of A. McIntosh and theorems of A. Yagi and P. Grisvard on interpolation.

In conclusion the author notes that “Calderón-Zygmund operators can also be used to obtain a bounded \(H_ \infty\) functional calculus for a second order elliptic partial differential operator with discontinuous coefficients on a Lipschitz domain under Dirichlet or Neumann boundary conditions”.

Reviewer: G.Il’yushina (Moskva)

##### MSC:

35J40 | Boundary value problems for higher-order elliptic equations |

47A60 | Functional calculus for linear operators |

31B10 | Integral representations, integral operators, integral equations methods in higher dimensions |

47F05 | General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX) |

31B35 | Connections of harmonic functions with differential equations in higher dimensions |