Endpoint boundedness for commutators of singular integral operators on weighted generalized Morrey spaces

In this paper, we obtain the endpoint boundedness for the commutators of singular integral operators with BMO functions and the associated maximal operators on weighted generalized Morrey spaces. We also get similar results for the commutators of fractional integral operators with BMO functions and the associated maximal operators.


Introduction and main results
The Morrey spaces were introduced by Morrey in [11] to investigate the local behavior of solutions to second order elliptic partial differential equations. Chiarenza and Frasca [2] showed the boundedness of the Hardy-Littlewood maximal operator, singular integral operators, and fractional integral operators on the Morrey spaces.
Let f be a measurable function on R n . The Hardy-Littlewood maximal function is defined by where the supremum is taken over all balls B containing x. We say that T is a singular integral operator if there exists a function K which satisfies the following conditions: © The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
The BMO(R n ) space is defined by where b B = 1 |B| B b(y) dy. For the singular integral operator T and b ∈ BMO, the commutator [b, T] is defined by It is a classical result that the operators T are bounded on L p (w) whenever 1 < p < ∞ and w ∈ A p , and for p = 1 and w ∈ A 1 , we have the weak type result which can be found in [9]. Komori and Shirai extended them to the weighted Morrey spaces in [10].
Let f be a measurable function on R n and 1 ≤ p < ∞, 0 ≤ κ < 1. For two weights w and u, the weighted Morrey space is defined by , and the supremum is taken over all balls B in R n . When w = u, we write L p,κ (w, u) as L p,κ (w). Komori and Shirai in [10] proved that, for 1 < p < ∞ and w ∈ A p , T and [b, T] are bounded on L p,κ (w), and if p = 1 and w ∈ A 1 , then for all t > 0 and any ball B, Qi et al. [14] obtained the weighted endpoint estimates for the commutators of the singular integral operators with BMO functions and associated maximal operators on the weighted Morrey space L 1,κ (w). They also gave similar results for the commutators of the fractional integral operators with BMO functions and associated maximal operators. Let w and u be two weights and 1 ≤ q ≤ β ≤ p ≤ ∞. We define the generalized twoweight Morrey space (L q (w), L p (u)) β := (L q (w), L p (u)) β (R n ) as the space of all measurable for any r > 0, with the usual modification when p = ∞. In the case w = u, the spaces (L q (w), L p (u)) β are the spaces (L q (w), L p ) β defined by Feuto in [7]. In the case w = u ≡ 1, the spaces (L q (w), L p (u)) β are the spaces (L q , L p ) β defined in [8] by Fofana. For q < β and p = ∞, the space (L q (w), L p ) β is the weighted Morrey space L q,κ (w) with κ = 1 q -1 β . Feuto [7] proved that the singular integral operators, the commutators of the singular integral operators with BMO functions, and other operators were bounded on these generalized weighted Morrey spaces (L q (w), L p ) β for q > 1. Here we consider the boundedness of the commutators of the singular integral operators with BMO functions on the endpoint generalized weighted Morrey space (L 1 (w), L p ) β . The weighted endpoint estimates for the commutators of the singular integral operators with BMO functions have many applications in partial differential equations. The BMO functions and the associated maximal operators can be applied in optimization problems, see [5,6].
Let w, u be two weights, Ψ : [0, ∞) → [0, ∞) be an increasing function and 1 ≤ β ≤ p ≤ ∞. We define the generalized weak weighted Morrey space (L Ψ ,∞ (w), L p (u)) β as the space When Ψ (t) = t, w = u, the space (L Ψ ,∞ (w), L p (u)) β is the generalized weak weighted Morrey space (L 1,∞ (w), L p ) β defined in [7]. Feuto proved for the singular integral operator T, if w ∈ A 1 , then In this paper, we extend the methods used in [14] and obtain the endpoint boundedness for the commutators of the singular integral operators with BMO functions and the associated maximal operators on the generalized weighted Morrey spaces (L 1 (w), L p ) β . The results are more general than [14] and have different forms. We also give similar results for the commutators of the fractional integral operators with BMO functions and the associated maximal operators.
In order to state our results, we need to recall some notations and facts about the Young functions and Orlicz spaces; for further information, see [1]. A function Φ : [0, ∞) → [0, ∞) is a Young function if it is convex and increasing, and if Φ(0) = 0 and Φ(t) → ∞ as t → ∞.
Given a locally integrable function f and a Young function Φ, define the mean Luxemburg norm of f on a ball B by For α, 0 ≤ α < n, and a Young function Φ, we define the Orlicz maximal operator Given α, 0 < α < n, for an appropriate function f on R n , the fractional integral operator (or the Riesz potential) of order α is defined by For b ∈ BMO(R n ), we define the commutators of the operator I α and b by A weight w is said to belong to the class A p,q for 1 < p, q < ∞ if there exists a positive constant C such that, for any ball B in R n , The following theorems are our main results.

Theorem 1.3 Let T be any singular integral operator
We also study similar estimates for the commutators of the fractional integral operators with BMO functions and the associated maximal operators and get the following results. -1 , and Θ(t) = t 1/q log(e + t -1 ). Then there exists a constant C > 0 independent of f such that -1 , and Θ(t) = t 1/q log(e + t -1 ). Then there exists a constant C > 0 independent of f such that From these results, we see that the commutators of the fractional integral operators with the BMO functions and the associated maximal operators map the weighted Morrey spaces to some weighted Orlicz-Morrey spaces. Hence we can further consider the boundedness for these integral operators on general weighted Orlicz-Morrey spaces.
for every locally integrable function f .
for every locally integrable function f .

Then
(1) if q > 1 and T is bounded on L q (w), then it is also bounded on (L q (w), L p ) β for Proof of Theorem 1.1 By Lemma 2.2, Lemma 2.3, and Lemma 2.6, we obtain that the Hardy-Littlewood maximal operator M is bounded on (L q (w), L p ) β for w ∈ A q , and for w ∈ A 1 , then there exists a constant C > 0 independent of f such that Because M L(log L) ≈ M 2 , which was obtained by Perez in [12], we have M L(log L) is bounded on (L q (w), L p ) β . This ends the proof.
Proof of Theorem 1.2 Fix y ∈ R n and r > 0, let B = B(y, r) be a ball centered at y with radius r.

By Lemma 2.4, we have
To estimate the term I, since w ∈ A 1 , we have For the term II, observe that for x ∈ (3B) c , x ∈ B , B is a ball and B ∩ B = ∅. We have Therefore we obtain Since w ∈ A 1 , we get Hence, we obtain Thus, for any r > 0, by Lemma 2.1 and Lemma 2.5, we have This ends the proof.

Lemma 2.8 ([9])
Let w ∈ A 1 , then there exist a constant C > 0 and θ > 0 such that, for any ball B, Proof of Theorem 1.3 Fix y ∈ R n and r > 0, let B = B(y, r). By Lemma 2.7, we have To estimate the term I, since w ∈ A 1 , it is easy to prove that M L(log L) 1+ε (wχ B )(x) ≤ Cw(x), x ∈ 3B, we have For the term II, observe that for x ∈ (3B) c , x ∈ B , B is a ball and B ∩ B = ∅, by Lemma 2.8, for any δ : 0 < δ ≤ θ , we have Noticing the definition of the maximal function M, we obtain Hence, we obtain Thus, for any r > 0, by Lemma 2.1 and Lemma 2.5, we have in which we take δ > 0 small enough such that η( 1 β -1 p ) -δ 1+δ > 0. This ends the proof.
In this section, we set Φ(t) = t log(e + t), it is submultiplicative and so h Φ ≈ Φ. Let 0 < α < n, and q be a number 1/q = 1α/n. Denote The function Γ is invertible with

Lemma 3.2 ([14]
) Let 0 < α < n, 1/q = 1α/n. Then there exists a constant C > 0 such that, for any t > 0, for any weight w, we have Proof of Theorem 1.4 Fix y ∈ R n and r > 0, let B = B(y, r). By Lemma 3.2, we have Now we estimate the term I. Noticing that, for s > 0, we have Since w ∈ A 1 , we get For the term II, observe that for x ∈ (3B) c , x ∈ B , B is a ball and B ∩ B = ∅. As in the proof of Theorem 1.2, we have Since w ∈ A 1 , Θ is submultiplicative, we get Hence, we obtain Thus, for any r > 0, we have ≤ C Φ |f | (L 1 (Θ(w)),L p (w)) β .
This ends the proof.