Acta Mathematica Academiae Scientiarum Hungaricae 64. (1994)

1994 / 1. szám - Horváth M.: Local uniform convergence of the eigenfunction expansion associated with the Laplace operator. I

Acta Math. Hungar. 64 (1) (1994), 1-25. LOCAL UNIFORM CONVERGENCE OF THE EIGENFUNCTION EXPANSION ASSOCIATED WITH THE LAPLACE OPERATOR. I M. HORVÁTH (Budapest) 1. Introduction The convergence properties of the expansions formed by a system of eigenfunctions of the Laplace operator are studied by many authors, see e.g. [1], [5], [6], [7], [8], [9], [10]. In this paper the following notions will be used. Consider a bounded domain ft C R, N > 1. By an eigenfunction of the Laplace operator we mean a function 0^n£ G C2(ft) satisfying —А и = A и on if; A G C is called the eigenvalue of u. We consider a Riesz basis (14) C X2(ft) of the eigenfunctions of the Laplace operator and consider the biorthogonal system (ví) C L2(ft): (1) A Ui — A íUi\ Aj G C, (u;, j' We do not assume that the Vj are eigenfunctions. Introduce the notations Pi := \/Ä~ Pi := Re pi 2 0, := Im/r,. Introduce further the Bessel-Macdonald kernel where Kv(r) is the Macdonald function, see [1]. The Liouville classes X",l^p^oo,a>0 are defined as follows ([4]). L%(RN ) consists of the functions / : Rw —► C representable in the form /(*) Wa(|* - y\)h(y)dy (2) va(r) :=------N------------> 0 < a (2тг)ТГ(|) r—

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