2-WEIGHT WEAK AND EXTRA-WEAK TYPE INEQUALITIES FOR THE ONE-SIDED MAXIMAL OPERATOR

Let M(g)(+) be the one-sided maximal operator and let Phi be a convex non-decreasing function on <0, infinity), Phi(0)=O. We present necessa ry and sufficient conditions on a couple of weight functions (sigma, r ho) such that the integral inequalities of weak type rho({Mg+f>lambda} ).Phi(lambda)less than or equal to K integral(-infinity)(infinity)Phi( K\f(x)\)sigma dx, and of extra-weak type rho({Mg+f>lambda})less than o r equal to K integral(-infinity)(infinity)Phi(K \f(x)\/lambda sigma(x) dx hold. Our proofs do not refer to the theory of Orlicz spaces.