Grand Lebesgue spaces without grandizers

Abstract We adapt the standard norm of grand Lebesgue spaces (Euclidean case, Lebesgue measure) to the case of functions defined on a domain Ω with possibly infinite measure, basically dropping the mean in the integral appearing in the norm of Lebesgue spaces. The novelty is that we also drop the notion of a grandizer: In general, we lose the property of containing standard Lebesgue spaces; however, we show that a number of important properties (including structural properties of the spaces, mapping properties of integral operators, and the validity of Hardy’s inequality even in higher dimensions, as well as Sobolev and Poincaré inequalities), which are usually proved using the norm in Lebesgue spaces, remain valid.