Sobolev regularity for commutators of the fractional maximal functions
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Banach J. Math. Anal. https://doi.org/10.1007/s43037-020-00095-6 ORIGINAL PAPER
Sobolev regularity for commutators of the fractional maximal functions Feng Liu1 · Shuai Xi1 Received: 3 June 2020 / Accepted: 3 August 2020 © Tusi Mathematical Research Group (TMRG) 2020
Abstract In this paper the Sobolev regularity properties are investigated of the commutators of fractional maximal functions, both in the global and local case. Some new bounds for the derivatives of the above commutators will be established. As several applications, the boundedness for these operators in Sobolev spaces as well as the bounds of these operators on the Sobolev spaces with zero boundary values in the local setting are obtained. Keywords Commutator · Fractional maximal function · Local fractional maximal function · Sobolev space Mathematics Subject Classification 42B25 · 46E35
1 Introduction 1.1 Background The regularity theory of maximal operators has been the subject of many recent articles in harmonic analysis. In the seminal paper [18], Kinnunen firstly proved that the classical Hardy–Littlewood maximal operator
Mf (x) = sup r>0
1 |f (y)|dy, |B(x, r)| ∫B(x,r)
Communicated by Mox S. Moslehian. * Shuai Xi [email protected] Feng Liu [email protected] 1
College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao 266590, Shandong, People’s Republic of China Vol.:(0123456789)
F. Liu, S. Xi
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). Here W 1,p (ℝn ) is given by
W 1,p (ℝn ) ∶= {f ∶ ℝn → ℝ ∣ ‖f ‖W 1,p (ℝn ) ∶= ‖f ‖1,p = ‖f ‖p + ‖∇f ‖p < ∞},
where ‖f ‖p = ‖f ‖Lp (ℝn ) and ∇f = (D1 f , … , Dn f ) is the weak gradient of f. Since then, Kinnunen’s result was extended to various variants. For example, see [19] for the local case, [21] for the fractional case, [7, 24] for the multisublinear case. Due to the lack of sublinearity for the maximal operator at the derivative level, 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 [27] and later extensions were given in [28]. See [22, 25] for some results in the framework of Triebel–Lizorkin spaces, as well as [2, 5, 6, 8, 13] for the endpoint regularity of maximal operators. Recently an interesting extension of regularity theory is the investigation on the regularity properties for the commutators of the Hardy–Littlewood maximal function, which is defined in the form
[b, M](f )(x) = b(x)Mf (x) − M(bf )(x),
x ∈ ℝn ,
where M denotes the usual Hardy–Littlewood maximal operator and b is a locally integral function defined on ℝn . The maximal commutator of M with b is defined by
𝔐b f (x) = sup r>0
1 |b(x) − b(y)||f (y)|dy, |B(x, r)| ∫B(x,r)
x ∈ ℝn .
Milman and Schonbek [30] first proved the Lp (1 < p < ∞) bounds of [b, M] by assuming that b ≥ 0 and b ∈ BMO(ℝn ) , which was later improved by Bastero et al. [3] who got the Lp b
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