Hardy–Littlewood Inequalities for Functions in Hardy–Lorentz Type Spaces

Let (X,d,μ) be a set with quasimetric d and σ -finite measure μ , and let I=(0,t₀) , 0<t₀≤∞ . We consider the classes Hp,r , 0<p,r≤∞ , consisting of complex-valued measurable functions u on X× I for which the maximal function N u(x):=\|u(y,t)| d(x,y)<t\ , x∈ X , belongs to the Lorentz space Lp,r(X) . In special cases, these classes are extensions of certain Hardy and Hardy–Lorentz spaces (of analytic, harmonic, etc., functions). For functions in the classes Hp,r , we consider generalizations of the classical Hardy–Littlewood inequalities as well as fractional integration operators.