Abstract: | The fundamental result: for an arbitrary bounded, simply connected domain in , the subspace Ln,m
p( ) of the space Lp( , ) ( is the plane Lebesgue measure, p 1), consisting of the (m, n)-analytic functions in , is complemented in LP( , ) (a function f is said to be (m, n)-analytic if ( m+n/ ¯Zm Zn)f=0 in ). Consequently, by virtue of a theorem of J. Lindenstrauss and A. Pelczy ski, the space Ln,m
P( ) is linearly homeomorphic to lP. In particular, for m=n=1 we obtain that the space of all harmonic LP-functions in is complemented in LP( , ). This result has been known earlier only for smooth domains.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 190, pp. 15–33, 1991. |