Regularity for commutators of the local multilinear fractional maximal operators

1.1 Background

Over the last several years and recently regularity properties of maximal operators have been studied extensively. The first work was due to Kinnunen [13] who showed that the classical Hardy-Littlewood maximal operator

is bounded on the first order Sobolev spaces W^1,p(ℝⁿ) for 1 < p ≤ ∞, where B(x, r) is the open ball in ℝⁿ centered at x with radius r, and |B(x, r)| denotes the volume of B(x, r). Later on, Kinnunen’s result was extended to the local version in [14], to the fractional version in [16] and to the multisublinear version in [6, 19]. Since we do not have sublinearity for the weak derivatives of the Hardy-Littlewood maximal function, the continuity of M : W^1,p(ℝⁿ) → W^1,p(ℝⁿ) for 1 < p < ∞ is affirmatively a nontrivial issue, which was addressed by Luiro [23] and later extensions were given in [24]. We can consult [2, 4, 5, 7, 25, 26] for the endpoint Sobolev regularity of maximal operators, as well as [17, 20] for the regularity properties of maximal operators on other smooth function spaces, such as Triebel-Lizorkin spaces, fractional Sobolev spaces and Besov spaces.

It should be pointed out that the commutativity with translations for maximal operators plays a key role in deducing the boundedness of maximal operators on W^1,p(ℝⁿ). In fact, it was shown in [9, Theorme 1] that any sub-linear operator which commutes with translations and is bounded on L^p(ℝⁿ) for some p ∈ (1, ∞) forms a bounded operator on W^1,p(ℝⁿ). Unfortunately, the local maximal operator does not commute with translations, it makes the investigation on the regularity of the local maximal operators very interesting. In 1998, Kinnunen and Lindqvist [14] firstly studied the regularity of the local maximal operator

for f ∈ Lloc1 (Ω), where Ω is a subdomain in ℝⁿ and Ω^c = ℝⁿ∖Ω. They showed that

for almost every x ∈ Ω and f ∈ W^1,p(Ω) with some p ∈ (1, ∞) (see also [9]). As an application of (1.1), we can get the boundedness for M_Ω : W^1,p(Ω) → W^1,p(Ω) for 1 < p < ∞. Here W^1,p(Ω), 1 ≤ p ≤ ∞, consists of those functions f ∈ L^p(Ω) whose weak first order partial derivatives D_if, i = 1, …, n, belong to L^p(Ω). The W^1,p(Ω) norm of f is given by

where ∇f = (D₁f, …, D_nf) is the weak gradient of f and ∥f∥_L^p(Ω) = (∫_Ω|f(x)|^pdx)^1/p. Later on, the above result was extended by Heikkinen et al. [11] to the fractional case. Let 0 ≤ α < n and f ∈ Lloc1 (Ω), the local fractional maximal operator M_α,Ω is defined as

The authors in [11] showed that if |Ω| < ∞, 1 < p < n and 1 ≤ α < n/p, then the map M_α,Ω : W^1,p(Ω) → W^1,q(Ω) is bounded, where q = np/(n – (α – 1)p). Moreover,

for almost every x ∈ Ω if f ∈ W^1,p(Ω). They also showed that if f ∈ L^p(Ω) for p ∈ (n/(n – 1), ∞) and 1 ≤ α < min{(n – 1)/p, n – 2n/((n – 1)p)} + 1, then

for almost every x ∈ Ω. Here 𝓢_α,Ω is the local spherical maximal operator, i.e.gf

where d𝓗^n–1 is the normalized (n – 1)-dimensional Hausdorff measure. An immediate consequence of (1.2) shows that the map M_α,Ω : L^p(Ω) → W^1,q(Ω) is bounded for n/(n – 1) < p < ∞, 1 ≤ α < n/p and q = np/(n – (α – 1)p) if |Ω| < ∞. Recently, Ramos et al. [29] established the bounds for M_α,Ω : L^p(Ω) → W^1,p(Ω) for 1 < p ≤ n/(n – 1) and 1 ≤ α < n under certain conditions on Ω.

Recently, Hart et al. [10] investigated the local multilinear maximal operator and its fractional variant. Let m be a positive integer and 0 ≤ α < mn. For f⃗ = (f₁, …, f_m) with each f_j ∈ Lloc1 (Ω), the local multilinear fractional type maximal operator 𝔐_α,Ω is defined by

Particularly, when α = 0, the operator 𝔐_α,Ω reduces to the local multilinear maximal operator 𝔐_Ω. When m = 1, the operator 𝔐_Ω (resp., 𝔐_α,Ω) is just the local (resp., fractional) maximal operator M_Ω (resp., M_α,Ω). It was pointed out in [10] that

for all f⃗ = (f₁, …, f_m) with each f_j ∈ L^p_j(Ω), and

if each p_j > 1, 0 ≤ α < mn, 1/q = 1/p₁ + ⋯ + 1/p_m – α/n and one of the following conditions holds:

1. α = 0, 1 ≤ q ≤ ∞ and 1 < p₁, …, p_m ≤ ∞;

2. 0 < α < n, 1 ≤ q < ∞ and 1 < p₁, …, p_m ≤ ∞;

3. n ≤ α < mn, 1 ≤ q < ∞ and 1 < p₁, …, p_m < ∞.

Based on the above boundedness, the authors of [10] established the following results.

1.2 Commutators of local multilinear fractional maximal operators

An active extension of current research is to investigate the regularity properties for the commutators of the Hardy-Littlewood maximal function, which is defined in the form

where b is a locally integral function defined on ℝⁿ. The maximal commutator of M with b is defined by

The L^p (1 < p < ∞) boundedness for [b, M] was first established by Milman and Schonbek [27] by assuming that b ≥ and b ∈ BMO(ℝⁿ). The same conclusion was also obtained by Bastero et al. [3] without the assumption b ≥ 0. The boundedness of M_b has been studied intensively by many authors (see [1, 12, 30]). Liu et al. [22] proved that

provided that 1 < p₁, p₂, p < ∞, 1/p = 1/p₁ + 1/p₂, f ∈ W^1,p₁(ℝⁿ) and b ∈ W^1,p₂(ℝⁿ). Very recently, Liu and Xi [21] extended the above result to the fractional version. The authors in [21] also studied the commutators of the local fractional maximal operators. To be more precise, for 0 ≤ α < n and a locally integrable function b defined on Ω, we define the commutator of the local fractional maximal function by

The fractional maximal commutators of M_α,Ω with b is defined by

Liu and Xi established the following results.