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\(H_ \infty\) functional calculus of elliptic operators on \(L^ p\) spaces. An approach using Calderón-Zygmund operator theory. (English) Zbl 0786.35046
From 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”.

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
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