What question did this study set out to answer?

To investigate generalizations of Hardy–Littlewood inequalities for functions in Hardy–Lorentz spaces.

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March 21, 2026

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

Key Points

To investigate generalizations of Hardy–Littlewood inequalities for functions in Hardy–Lorentz spaces.
Defined the classes of complex-valued measurable functions on a set with quasimetric and $ $-finite measure.
Formulated the maximal function and its membership in Lorentz space.
Explored fractional integration operators in relation to these functions.
Identified conditions under which the Hardy–Littlewood inequalities hold for these function classes.
Demonstrated that certain special cases extend known Hardy and Hardy–Lorentz spaces.
Provided examples of analytic and harmonic functions that fit within the proposed framework.

Abstract

Let (X, d, ) be a set with quasimetric d and -finite measure, and let I= (0, t₀), 0<t₀. We consider the classes H^p, 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 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 H^p, r, we consider generalizations of the classical Hardy–Littlewood inequalities as well as fractional integration operators.

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