Almost Periodic Differential Equations by A.M. Fink

By A.M. Fink

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In an analogous way the definition of the class Ώ* is constructed. 16, G where Φ and Φ0 are arbitrary analytic functions of the class yt*(G) and g is an arbitrary function of the class L{G). 17) *o(*) & ' Consider now the non-homogeneous Cauchy-Biemann equation and let us assume that / c 91? e. /=^ , fo^HG), 0oeW(G). 18) almost everywhere. All such solutions are obviously given by the formula where Φ is an arbitrary function of the class 3to((7). 5* 38 GENERALIZED ANALYTIC FUNCTIONS §6. Properties of the operator T#/ In this paragraph we shall investigate properties of the operator TG with respect to various classes of functions.

14 T/ e D-{G), TfeDz(G) if feL^G), and We shall also consider classes of functions possessing generalized derivatives of an order higher than one, with respect to z and z. ) dztdzk and they belong to the class LV{G), p ^ 1, including the case oi p = oo. The symbol Dm will stand for DmA. 4). ) ■ (5 8a - > 32 GENERALIZED ANALYTIC FUNCTIONS Spaces of the type Dm>p were first introduced by Sobolev in the class of functions of an arbitrary number of variables, [79a], and denoted by W{™\ Important properties of these spaces (imbedding theorems) were established in the papers of Sobolev and Kondrashev, [79a], Nikolski [61b, c], and others.

5* 38 GENERALIZED ANALYTIC FUNCTIONS §6. Properties of the operator T#/ In this paragraph we shall investigate properties of the operator TG with respect to various classes of functions. 1. 19. Let G be a bounded domain. If f eLp(G), p > 2, then the function g = TGf satisfies the conditions where zx and z% are arbitrary points of the plane, and Mu M2 are constants, Mx depending on p and G, while M2 depends on p only. P R O O F . 1). ί\ζ-*ιΓΎ-**Γβ*£*ν G > a<2, β<2. 5) About the point zx we draw a circle Gx of radius ρ = 2\zx—081 a n d a concentric circle G0 of radius 2ρ 0 , such that ÖC G0.

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