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Nonexistence of localization of spectral decompositions associated with elliptic operators. (English. Russian original) Zbl 0558.35049
Math. Notes 30, 760-764 (1982); translation from Mat. Zametki 30, 535-542 (1981).
From the author’s introduction: ”Let G be an arbitrary domain from \(R^ n\). We consider the formally self-adjoint elliptic differential operator with constant real coefficients \(A=A(D)=\sum_{| \alpha | <m}a_{\alpha}D^{\alpha}\quad in\quad L_ 2(G)\) with domain of definition \(C_ 0^{\infty}(G)\). We denote by \(\hat Aa\) self-adjoint extension of the operator A to \(L_ 2(G)\). Let \(\{E_{\lambda}\}\) be a resolution of the identity, \(\theta\) (x,y,\(\lambda)\) be the spectral function of the operator \(\hat A,\) i.e., the kernel of the integral operator \(E_{\lambda}\). By the symbols \(E_{\lambda}\) \(\theta\) (x,y,\(\lambda)\) we shall understand the respective Riesz means. In this paper we shall study for functions f(x) of the Zygmund-Hölder class \(C^ a(G)\) the localization conditions for \(E_{\lambda}f(x)\) for \(0<s<(n-1)/2.''\)
Reviewer: D.Robert
MSC:
35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs
35J30 Higher-order elliptic equations
35E20 General theory of PDEs and systems of PDEs with constant coefficients
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References:
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