The norm of the characteristic function of a set in the John‐Nirenberg space of exponent p

Abstract

We find the concrete value of || A || JN p (R) for any measurable set A ⊂ R of positive and finite Lebesgue measure, where JN p stands for the John-Nirenberg space of exponent 1 ≤ p ≤ ∞. In the case I 0 = [0, 1] we show that || I || JN p (I 0) = 2 (1 −) for any interval I ⊂ I 0 with |I| = and any 1 ≤ p ≤ p where p = max { 1− ,

A Survey on Function Spaces of John–Nirenberg Type

“…which implies that f JN 1 (Q 0 ) ≤ f L 1 (Q 0 )/C and hence the above claim holds true. Moreover, the relation between JN 1 (R) and L 1 (R) was studied in [33] (Proposition 2). (iii) Garsia and Rodemich in [55] (Theorem 7.4) showed that for any given…”

“…Theorem 10. Let p, q ∈ (1, ∞), α ∈ R, and M be the Hardy-Littlewood maximal operator as in (33). Then M is bounded on RM con p,q,α (R n ).…”

“…Therefore, John-Nirenberg spaces are new spaces between Lebesgue spaces and weak Lebesgue spaces, which motivates us to study the properties of JN p . Furthermore, various John-Nirenberg-type spaces have also attracted a lot of attention in recent years (see, for instance, [31][32][33][34][35][36][37] for the Euclidean space case and [30,[38][39][40] for the metric measure space case).…”

“…Blasco and Espinoza-Villalva [33] computed the concrete value of 1 A JN p (R) for any given p ∈ [1, ∞] and any measurable set A ⊂ R of positive and finite Lebesgue measure, where JN ∞ (R) := BMO (R).…”

A bridge connecting Lebesgue and Morrey spaces via Riesz norms

“…Recall that the John-Nirenberg space JN p (Q 0 ) , with p ∈ (1, ∞) and cube Q 0 ⫋ ℝ n , is defined to be the set of all f ∈ L 1 (Q 0 ) such that where the supremum is taken over all collections of subcubes {Q i } i of Q 0 with pairwise disjoint interiors. Recently, the John-Nirenberg space begins to attract more and more attention; see, for instance, [5,6,12,18,27,36,38] for the Euclidean space case, [1,15,23,24] for the metric measure space case and, particulary, [39] for a complete survey on spaces of such type. Observe that the norm appearing in (1) differs from the norm appearing in (2) only in subtracting integral means.…”