On Some Function Spaces That Appear in Applied Mathematics

The linear spaces generated by the eigenfunctions of a diierential operator are well-known in applied mathematics. In this paper we examine their interpolation properties, connection with Sobolev spaces and apply these results to the solving of hyperbolic equation in Sobolev spaces of fractional order. 1. Interpolation and function spaces Let A 1 and A 2 be two Banach spaces, linearly and continuously embedded in a topological linear space A. Such two spaces are called interpolation pair fA 1 ; A 2 g. The space A 1 + A 2 we deene as A 1 + A 2 = f a 2 A : a = a 1 + a 2 ; a j 2 A j ; j = 1; 2 g; with the norm kak A1+A2 = inf a=a1+a2; aj2Aj (ka 1 k A1 + ka 2 k A2). Introduce the function K(t; a; A 1 ; A 2) = inf a2A1+A2 a=a1+a2; aj2Aj (ka 1 k A1 + tka 2 k A2): This function is a norm in A 1 + A 2 equivalent to the standard norm kak A1+A2. For 0 < < 1, 1 6 q < 1, the interpolation space (A 1 ; A 2) ;q obtained by K-method of real interpolation is deened as the set of all elements a 2 A 1 + A 2 with the nite norm kak (A1;A2) ;q Z 1 0 t ? K(t; a; A 1 ; A 2)] q dt t 1=q (see 1]).