Abstract. We define spaces $H_{p}^+(w),$ $H_{p}^-(w)$ for weight functions $w$ with singularities in finitely many points and show that the dimension of $H_{p}^+(w)\cap H_{p}^-(w)$ is finite and depends on the number of singularities. We find the codimension in $L^p(w)$ of the subspaces generated by $H_{p}^+(w)\cup H_{p}^-(w).$ Necessary and sufficient conditions on the weight function $w$ are found so that the natural projection from $L^p(w)$ onto $H_{p}^+(w)$ exists for $1< p< \infty.$ It is also shown that no natural projection from $L^1(w)$ onto $H_{1}^+(w)$ may exist for any weight function $w $ under consideration.

AMS Subject Classification (1991): 42C15, 42C30

Keyword(s): H^p, weightedspaces, projection operator

Received August 21, 2003, and in revised form September 20, 2006. (Registered under 5961/2009.)