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

Let $$(X,d,\mu)$$ be a set with quasimetric $$d$$ and $$\sigma$$ -finite measure $$\mu$$ , and let $$I=(0,t_0)$$ , $$0<t_0\le\infty$$ . We consider the classes $$\mathcal H^{p,r}$$ , $$0<p,r\le\infty$$ , consisting of complex-valued measurable functions $$u$$ on $$X\times I$$ for which the maximal function $$\mathcal N u(x):=\sup\{|u(y,t)|\colon d(x,y)<t\}$$ , $$x\in X$$ , belongs to the Lorentz space $$L^{p,r}(X)$$ . In special cases, these classes are extensions of certain Hardy and Hardy–Lorentz spaces (of analytic, harmonic, etc., functions). For functions in the classes $$\mathcal H^{p,r}$$ , we consider generalizations of the classical Hardy–Littlewood inequalities as well as fractional integration operators.