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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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