Special John-Nirenberg-Campanato spaces via congruent cubes

Abstract

and s be a non-negative integer. Inspired by the space JNp introduced by John and Nirenberg (1961) and the space B introduced by Bourgain et al. ( 2015), we introduce a special John-Nirenberg-Campanato space JN con (p,q,s)α over R n or a given cube of R n with finite side length via congruent subcubes, which are of some amalgam features. The limit space of such spaces as p → ∞ is just the Campanato space which coincides with the space BMO (the space of functions with bounded mean oscillations) when α = 0. Moreover, a vanishing subspace of this new space is introduced, and its equivalent characterization is established as well, which is a counterpart of the known characterization for the classical space VMO (the space of functions with vanishing mean oscillations) over R n or a given cube of R n with finite side length. Furthermore, some VMO-H 1 -BMO-type results for this new space are also obtained, which are based on the aforementioned vanishing subspaces and the Hardy-type space defined via congruent cubes in this article. The geometrical properties of both the Euclidean space via its dyadic system and congruent cubes play a key role in the proofs of all these results.

John--Nirenberg-$Q$ Spaces via Congruent Cubes

“…Motivated by [19,Proposition 2.2] and [18,Lemma 5.7], we establish an equivalence principle (namely, Proposition 2.5 below) which shows that, in the congruent setting (namely, all cubes {Q j } j have equal edge length), the summation is equivalent to the integral so long as the integrand Φ is 'almost increasing'. Definition 2.3.…”

“…To prove Theorem 3.8, we need the following geometrical lemma (Lemma 3.10 below) which is a refinement of both [30,Lemma 2.4] and [19,Lemma 2.5]. In what follows, for any set A of R n , we use A to denote its closure; moreover, two sets A and B are said to be mutually adjacent if A ∩ B ∅.…”

“…The following John-Nirenberg space via congruent cubes (for short, the congruent John-Nirenberg space) was introduced by Jia et al in [19] to shed some light on the mysterious space JN p and the space B, which are introduced, respectively, by John and Nirenberg [23] and Bourgain et al [7]. In what follows, for any ℓ ∈ (0, ∞), let Π ℓ denote the class of all collections {Q j } j of subcubes of R n with pairwise disjoint interiors and equal edge length ℓ.…”

“…Another well-known space appeared in John and Nirenberg [23] is BMO (R n ), the space containing functions of bounded mean oscillation, which can be regarded as the limit space of JN con p,q (R n ) as p → ∞; see [19,Proposition 2.21] and also [31,Proposition 2.6]. The space BMO (R n ) has wide applications in harmonic analysis and partial differential equations; see, for instance, [2,11,12,14,15,16,17,25,28,34].…”

“…Indeed, we obtain a more general equivalence principle in Proposition 2.5 below via introducing the 'almost increasing' set function Φ (see Definition 2.3 below). It should be pointed out that Proposition 2.5 may have independent interest because, as a special case of this proposition with Φ replaced by the mean oscillation as in (1.1), [19,Proposition 2.2] proves extremely useful when studying the boundedness of the Hardy-Littlewood maximal operator, Calderón-Zygmund operators, fractional integrals, and Littlewood-Paley operators on congruent John-Nirenberg spaces; see [20,21,22] for more details. Moreover, we show that Q α (R n ) can be regarded as the limit space of JNQ α p,2 (R n ) as p → ∞ in Proposition 2.10 below, and also obtain an extension result over cubes (with finite edge length) in Proposition 2.11 below.…”