Projections on Lp-spaces of polyanalytic functions

Abstract

The fundamental result: for an arbitrary bounded, simply connected domain Ω in ℂ, the subspace Ln,m p(Ω) of the space Lp(Ω, σ) (σ is the plane Lebesgue measure, p ≥ 1), consisting of the (m, n)-analytic functions in Ω, is complemented in LP(Ω, σ) (a function f is said to be (m, n)-analytic if (∂m+n/∂¯Zm∂Zn)f=0 in Ω). Consequently, by virtue of a theorem of J. Lindenstrauss and A. Pelczyński, the space Ln,m P(Ω) is linearly homeomorphic to lP. In particular, for m=n=1 we obtain that the space of all harmonic LP-functions in Ω is complemented in LP(Ω, σ). This result has been known earlier only for smooth domains.