Norms of Maximal Functions Between Generalized and Classical Lorentz Spaces

The aim of the paper is to find the norm of the generalized maximal operator M.,.a(b), defined for all measurable functions f on R-n, with 0 < alpha < infinity and functions b, phi : (0, infinity) -> (0, infinity), by M(phi, Lambda alpha(b))f(x) := sup (Q(sic)x) ||f chi Q||Lambda(alpha)(b)/phi(|Q|), x is an element of R-n, from generalized Lorentz spaces G Gamma(p, m, v) into classical Lorentz spaces Lambda(q)(w). In order to achieve the goal, we reduce the problem to the solution of the inequality (integral(infinity)(0) [T(u,b)f *(y)](q) w(y) dy)(1/q) <= C (integral(infinity)(0) (integral(x)(0) [f *(s)]p ds) (m/p) v(x) dx)(1/m) where w and v are weight functions on (0,infinity). Here f* is the non-increasing rearrangement of a measurable function f defined on R-n and T-u,T- b is the iterated Hardy-type operator involving suprema, which is defined for a measurable non-negative function g on (0,infinity) by (T(u,b)g)(t) := sup(tau is an element of[t,infinity)) u(tau)/B(tau) integral(tau)(0) g(s)b(s) ds, t is an element of(0,infinity), where u and b are weight functions on (0,infinity) such that u is continuous on (0,infinity) and the function B(t) := integral(t)(0) b(s) ds satisfies 0 < B(t) < infinity for every t is an element of (0,infinity).