On generalized Lorentz-Zygmund spaces

Abstract

Abstract. We study generalized Lorentz-Zygmund spaces with broken logarithmic functions. We derive necessary and sufficient conditions for embeddings between them. We give a complete characterization of their associate spaces. We establish necessary and sufficient conditions for a generalized Lorentz-Zygmund space to be a Banach function space and to have absolutely continuous (quasi-)norm. We describe completely relations between these spaces and Orlicz spaces.Mathematics subject classification (1991): 46E30, 26D15.

Generalized n-Laplacian: boundedness of weak solutions to the Dirichlet problem with nonlinearity in the critical growth range

“…For other results concerning these spaces and their Trudinger-type embeddings we refer the reader to [6,[8][9][10][11][12][13][14]20].…”

On generalized Moser-Trudinger inequalities without boundary condition

“…coincides with the Orlicz space L Φ (Ω), where lim t→∞ Φ(t) t n log α (t) = 1, the space L n log n−1 L log α log L(Ω) coincides with L Φ (Ω) where lim t→∞ Φ(t) t n log n−1 (t) log α (log(t)) = 1, and so on (see for example [20,Lemma 8.1]). For other results concerning these spaces we refer the reader to [11], [12], [13], [14], [15] and [20].…”

“…For other results concerning these spaces we refer the reader to [11], [12], [13], [14], [15] and [20].…”

A sharp form of an embedding into multiple exponential spaces

“…Moreover, it is shown in [3] (see also [2] and [4]) that in the limiting case α = n − 1 we have the embedding into a double exponential space, i.e., the space W L n log n−1 L log α log L(Ω), α < n − 1, is continuously embedded into the Orlicz space with the Young function that behaves like exp(exp(t γ )) for large t. Further, in the limiting case α = n − 1 we have the embedding into the triple exponential space and so on. For other results concerning these spaces we refer the reader to [4], [5], [6], [7], [8] and [13]. In paper [11], the author studies the analogue of (1) for the case of an embedding into single and double exponential spaces.…”