Commutators of integral operators with functions in Campanato spaces on Orlicz-Morrey spaces
- PDF / 3,077,941 Bytes
- 41 Pages / 439.37 x 666.142 pts Page_size
- 108 Downloads / 220 Views
Tusi Mathematical Research Group
ORIGINAL PAPER
Commutators of integral operators with functions in Campanato spaces on Orlicz‑Morrey spaces Minglei Shi1 · Ryutaro Arai1 · Eiichi Nakai1 Received: 2 June 2020 / Accepted: 27 August 2020 © Tusi Mathematical Research Group (TMRG) 2020
Abstract We consider the commutators [b, T] and [b, I𝜌 ] on Orlicz-Morrey spaces, where T is a Calderón-Zygmund operator, I𝜌 is a generalized fractional integral operator and b is a function in generalized Campanato spaces. We give a necessary and sufficient condition for the boundedness of the commutators on Orlicz-Morrey spaces. To do this we prove the boundedness of generalized fractional maximal operators on Orlicz-Morrey spaces. Moreover, we introduce Orlicz-Campanato spaces and establish their relations to Orlicz-Morrey spaces. Keywords Orlicz-Morrey space · Campanato space · Singular integral · Fractional integral · Commutator Mathematics Subject Classification 42B35 · 46E30 · 42B20 · 42B25
Communicated by Dachun Yang. This work was supported by Grant-in-Aid for Scientific Research (B), No. 15H03621, Japan Society for the Promotion of Science. * Eiichi Nakai [email protected]
Minglei Shi [email protected]; [email protected]
Ryutaro Arai [email protected]; [email protected]
1
Department of Mathematics, Ibaraki University, Mito, Ibaraki 310‑8512, Japan Vol.:(0123456789)
M. Shi et al.
1 Introduction Let ℝn be the n-dimensional Euclidean space. Let b ∈ BMO(ℝn ) and T be a Calderón-Zygmund singular integral operator. In 1976 Coifman, Rochberg and Weiss [3] proved that the commutator [b, T] = bT − Tb is bounded on Lp (ℝn ) ( 1 < p < ∞ ), that is,
‖[b, T]f ‖Lp = ‖bTf − T(bf )‖Lp ≤ C‖b‖BMO ‖f ‖Lp ,
where C is a positive constant independent of b and f. For the fractional integral operator I𝛼 , Chanillo [2] proved the boundedness of [b, I𝛼 ] in 1982. Coifman, Rochberg and Weiss [3] and Chanillo [2] also gave the necessary conditions for the boundedness. These results were extended to Orlicz spaces by Janson [13] and to Morrey spaces by Di Fazio and Ragusa [6]. For other extensions and generalization, see [1, 7–10, 12, 17, 21, 32, 41–44], etc. In this paper we investigate the commutators [b, T] and [b, I𝜌 ] on Orlicz-Morrey spaces, where T is a Calderón-Zygmund operator, I𝜌 is a generalized fractional integral operator and b is a function in generalized Campanato spaces. The Orlicz-Morrey spaces unify Orlicz and Morrey spaces, and the Campanato spaces unify BMO and Lipschitz spaces. Therefore, our results contain many previous results as corollaries. The boundedness of T and I𝜌 on the Orlicz-Morrey spaces are known by [29] and [28], respectively. To prove the boundedness of [b, T] and [b, I𝜌 ] , we need the generalized fractional maximal operator M𝜌 and the sharp maximal operator M ♯ . We show the boundedness of M𝜌 on Orlicz-Morrey spaces under weaker conditions than I𝜌 . Moreover, we introduce Orlicz-Campanato spaces and establish their relations to Orlicz-Morrey spa
Data Loading...
