Characterizations for the genuine Calderón-Zygmund operators and commutators on generalized Orlicz-Morrey spaces

V. S. Guliyev; Meriban N. Omarova; Maria Alessandra Ragusa

doi:10.1515/anona-2022-0307

Artikel Open Access

Characterizations for the genuine Calderón-Zygmund operators and commutators on generalized Orlicz-Morrey spaces

V. S. Guliyev , Meriban N. Omarova und Maria Alessandra Ragusa

Veröffentlicht/Copyright: 11. August 2023

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Aus der Zeitschrift Advances in Nonlinear Analysis Band 12 Heft 1

Abstract

In this article, we show continuity of commutators of Calderón-Zygmund operators [ b , T ] with BMO functions in generalized Orlicz-Morrey spaces M Φ , φ ( R n ) . We give necessary and sufficient conditions for the boundedness of the genuine Calderón-Zygmund operators T and for their commutators [ b , T ] on generalized Orlicz-Morrey spaces, respectively.

Keywords: generalized Orlicz-Morrey spaces; genuine Calderón-Zygmund operator; commutator; BMO

MSC 2010: 42B20; 42B35; 46E30

1 Introduction

It is well known that the commutator is an important integral operator, and it plays a key role in harmonic analysis. In 1965, Calderón [4] studied a kind of commutators, appearing in Cauchy integral problems of lip-line. Let T be a Calderón-Zygmund singular integral operator and b ∈ BMO ( R n ) . A celebrated result of Coifman et al. [7] states that the commutator operator [ b , T ] f = T ( b f ) − b T f is bounded on L p ( R n ) for 1 < p < ∞ . The commutator of Calderón-Zygmund operators plays an important role in the study of regularity of solutions of elliptic partial differential equations of second order (see, for example, [5,6,9,11,15,16,26,29]).

Let T be a singular integral Calderón-Zygmund operator, briefly a Calderón-Zygmund operator, i.e., a linear operator bounded from L 2 ( R n ) in L 2 ( R n ) taking all infinitely continuously differentiable functions f with compact support to the functions T f ∈ L 1 loc ( R n ) represented by

(1.1) T f ( x ) = ∫ R n K ( x , y ) f ( y ) d y a.e. on supp f .

Here, K ( x , y ) is a continuous function away from the diagonal, which satisfies the standard estimates: there exist c 1 > 0 and 0 < ε ≤ 1 such that

(1.2) ∣ K ( x , y ) ∣ ≤ c 1 ∣ x − y ∣ − n ,

for all x , y ∈ R n , x ≠ y , and

(1.3) ∣ K ( x , y ) − K ( x ′ , y ) ∣ + ∣ K ( y , x ) − K ( y , x ′ ) ∣ ≤ c 1 ∣ x − x ′ ∣ ∣ x − y ∣ ε ∣ x − y ∣ − n

whenever 2 ∣ x − x ′ ∣ ≤ ∣ x − y ∣ . Such operators were introduced in [8], see also [29].

In this article, we give an explicit expression for the definition of the singular integral operators on generalized Orlicz-Morrey spaces M p , φ ( R n ) . This definition is independent of the pre-dual and the second pre-dual of the generalized Orlicz-Morrey spaces. In addition, our result provides a solid foundation for the further studies of singular integral operators on generalized Orlicz-Morrey spaces. As an application of our approach, we study the commutators of singular integral operator on generalized Orlicz-Morrey spaces.

Notice that in view of the definition of generalized Orlicz-Morrey spaces, to study the action of singular integral operator T on f ∈ M p , φ ( R n ) , we need to estimate T f on a neighborhood of x ∈ R n . On the other hand, the definition of T in (1.1) is given in term of pointwise limit. Therefore, we cannot directly use it to study the boundedness of T on M p , φ ( R n ) .

Let ℬ = { B ( x , r ) : x ∈ R n , r > 0 } . The following definition overcome this difficulty by giving the definition of T f for f ∈ M p , φ ( R n ) in a neighborhood of x ∈ R n .

Definition 1.1

Let T be a singular integral operator. For any f ∈ M p , φ ( R n ) and x ∈ B ( z , r ) ∈ ℬ , we define

(1.4) ( T f ) ( x ) = ( T ( χ B ( z , 2 r ) f ) ) ( x ) + ∫ R n \ B ( z , 2 r ) K ( x , y ) f ( y ) d y .

We say that T is a genuine Calderón-Zygmund operator (see [3]) if it is a Calderón-Zygmund operator and, for n ≥ 2 , there exist c 1 , c 2 > 0 and a rotation ℛ such that

(1.5) K ( x , y ) ≥ c 1 ∣ x − y ∣ n

for all x ∈ R n and all y ∈ C x = x + ℛ ( C ) , where

C = { y = ( y 1 , … , y n ) ≡ ( y ¯ , y n ) ∈ R n : y n > c 2 ∣ y ¯ ∣ } .

If n = 1 , then we assume that there exists c 1 > 0 such that

K ( x , y ) ≥ c 1 ∣ x − y ∣

for all x ∈ R and all y > x or for all x ∈ R and all y < x .

Clearly, the Hilbert transform ℋ , in which case K ( x , y ) = 1 x − y , is a genuine Calderón-Zygmund operator because K ( x , y ) ≥ 1 ∣ x − y ∣ for all x ∈ R and for all y < x . Let T be a Calderón-Zygmund operator of the form

K ( x , y ) = Ω x − y ∣ x − y ∣ ∣ x − y ∣ n ,

where Ω is a continuous function on the unit sphere S n − 1 in R n , Ω ≢ 0 , which is homogeneous of order zero and such that ∫ S n − 1 Ω ( η ) d η = 0 . The properties of Ω imply that there exist c 3 > 0 , η 0 ∈ S n − 1 and δ > 0 such that Ω ( η ) ≥ c 3 for all η ∈ S n − 1 ∩ B ( η 0 , δ ) . Hence, condition (1.5) is satisfied.

The classical Morrey spaces, introduced by Morrey [35] in 1938, have been studied intensively by various authors and together with Lebesgue spaces play an important role in the theory of partial differential equations [5,6, 40,41]. Although such spaces allow to describe local properties of functions better than Lebesgue spaces, they have some unpleasant issues. It is well known that Morrey spaces are non separable and that the usual classes of nice functions are not dense in such spaces. Moreover, various Morrey spaces are defined in the process of study. Guliyev [21], Mizuhara [34] and Nakai [36] introduced generalized Morrey spaces M p , φ ( R n ) [22,24,43]. In [22], the generalized Morrey spaces M p , φ is defined with normalized norm

‖ f ‖ M p , φ ≡ sup x ∈ R n , r > 0 φ ( x , r ) − 1 ∣ B ( x , r ) ∣ − 1 ∕ p ‖ f ‖ L p ( B ( x , r ) ) ,

where the function φ is a positive measurable function on R n × ( 0 , ∞ ) . Here and everywhere in the sequel, B ( x , r ) is the ball in R n of radius r centered at x and ∣ B ( x , r ) ∣ = v n r n is its Lebesgue measure, where v n is the volume of the unit ball in R n .

In [12], the generalized Orlicz-Morrey space M Φ , φ ( R n ) was introduced to unify Orlicz and generalized Morrey spaces. Other definitions of generalized Orlicz-Morrey spaces can be found in [37,42]. In words of [25], our generalized Orlicz-Morrey space is the third kind, and the ones in [37,42] are the first kind and the second kind, respectively. According to the examples in [20], one can say that the generalized Orlicz-Morrey spaces of the first kind and the second kind are different and that second kind and third kind are different. However, we do not know the relation between the first and the second kind.

Note that, Orlicz-Morrey spaces unify Orlicz and generalized Morrey spaces. We extend some results on generalized Morrey spaces in the articles [1,17,22,24,32] to the case of Orlicz-Morrey space in [12,13,18,19,25,27].

As based on the results of [22], the following conditions were introduced in [12] for the boundedness of the singular integral operators on M Φ , φ ( R n ) ,

(1.6) ∫ r ∞ ess inf t < s < ∞ φ 1 ( x , s ) Φ − 1 ( s − n ) Φ − 1 ( t − n ) d t t ≤ C φ 2 ( x , r ) ,

where C does not depend on x and r .

The commutators generated by b ∈ L loc 1 ( R n ) and the operator T are defined by

[ b , T ] f ( x ) = ∫ R n b ( x ) − b ( y ) ∣ x − y ∣ n K ( x , y ) f ( y ) d y .

The operator [ b , T ] is defined by

[ b , T ] f ( x ) = ∫ R n ∣ b ( x ) − b ( y ) ∣ ∣ x − y ∣ n K ( x , y ) f ( y ) d y .

The operator T and its commutator in [5] are in connection with boundary estimates for solutions to elliptic equations.

Therefore, the purpose of this article is mainly to study the boundedness of the commutators of Calderón-Zygmund operators [ b , T ] on generalized Orlicz-Morrey spaces of the third kind M Φ , φ ( R n ) . We give necessary and sufficient conditions for the boundedness of the genuine Calderón-Zygmund operators T and its commutators [ b , T ] on generalized Orlicz-Morrey spaces M Φ , φ ( R n ) , respectively.

A function φ : ( 0 , ∞ ) → ( 0 , ∞ ) is said to be almost increasing (resp. almost decreasing) if there exists a constant C > 0 such that

φ ( r ) ≤ C φ ( s ) ( resp. φ ( r ) ≥ C φ ( s ) ) for r ≤ s .

For a Young function Φ , we denote by G Φ the set of all decreasing functions φ : ( 0 , ∞ ) → ( 0 , ∞ ) such that t ∈ ( 0 , ∞ ) ↦ Φ − 1 ( t − n ) φ ( t ) − 1 is almost decreasing.

The following results are the fundamental theorems in this article:

Theorem 1.2

Let T be a Calderón-Zygmund operator and φ 1 , φ 2 ∈ Ω Φ .

If Φ ∈ Δ 2 , then condition (1.6) is sufficient for the boundedness of T from M Φ , φ 1 ( R n ) to W M Φ , φ 2 ( R n ) . If Φ ∈ Δ 2 ∩ ∇ 2 , then the condition (1.6) is sufficient for the boundedness of T from M Φ , φ 1 ( R n ) to M Φ , φ 2 ( R n ) .
If φ 1 ∈ G Φ and T is a genuine Calderón-Zygmund operator, then condition
(1.7) φ 1 ( x , r ) ≤ C φ 2 ( x , r ) ,
where C does not depend on x and r , is necessary for the boundedness of T from M Φ , φ 1 ( R n ) to W M Φ , φ 2 ( R n ) and from M Φ , φ 1 ( R n ) to M Φ , φ 2 ( R n ) .
If Φ ∈ Δ 2 , T is a genuine Calderón-Zygmund operator and φ 1 ∈ G Φ satisfies the regularity condition
(1.8) ∫ t ∞ φ 1 ( r ) d r r ≤ C φ 1 ( t ) ,
for all t > 0 , where C > 0 does not depend on t , then condition (1.7) is necessary and sufficient for the boundedness of T from M Φ , φ 1 ( R n ) to W M Φ , φ 2 ( R n ) . If Φ ∈ Δ 2 ∩ ∇ 2 , then the condition (1.7) is necessary and sufficient for the boundedness of T from M Φ , φ 1 ( R n ) to M Φ , φ 2 ( R n ) .

If we take Φ ( t ) = t p , p ∈ [ 1 , ∞ ) at Theorem 1.2, we obtain the following new result for generalized Morrey spaces.

Corollary 1.3

Let T be a Calderón-Zygmund operator, p ∈ [ 1 , ∞ ) and φ 1 , φ 2 ∈ Ω p ≡ Ω t p .

The condition
(1.9) ∫ r ∞ ess inf t < s < ∞ φ 1 ( s ) s n p t n p + 1 d t ≤ C φ 2 ( r ) ,
for all r > 0 , where C > 0 does not depend on r, is sufficient for the boundedness of T from M p , φ 1 ( R n ) to W M p , φ 2 ( R n ) . If 1 < p < ß , then condition (1.9) is sufficient for the boundedness of T from M p , φ 1 ( R n ) to M p , φ 2 ( R n ) .
If φ 1 ∈ G p and T is a genuine Calderón-Zygmund operator, then condition (1.7) is necessary for the boundedness of T from M p , φ 1 ( R n ) to W M p , φ 2 ( R n ) and from M p , φ 1 ( R n ) to M p , φ 2 ( R n ) .
If T is a genuine Calderón-Zygmund operator and φ 1 ∈ G p satisfies the regularity condition (1.8), then the condition (1.7) is necessary and sufficient for the boundedness of T from M p , φ 1 ( R n ) to W M p , φ 2 ( R n ) . If, in addition, 1 < p < ß , then condition (1.7) is necessary and sufficient for the boundedness of T from M p , φ 1 ( R n ) to M p , φ 2 ( R n ) .

Theorem 1.4

Let T be a Calderón-Zygmund operator, b ∈ BMO ( R n ) and φ 1 , φ 2 ∈ Ω Φ .

If Φ ∈ Δ 2 ∩ ∇ 2 , then the condition
(1.10) ∫ r ∞ 1 + ln t r ess inf t < s < ∞ φ 1 ( x , s ) Φ − 1 ( s − n ) Φ − 1 ( t − n ) d t t ≤ C φ 2 ( x , r ) ,
where C does not depend on x and r, is sufficient for the boundedness of [ b , T ] from M Φ , φ 1 ( R n ) to M Φ , φ 2 ( R n ) .
If φ 1 ∈ G Φ and T is a genuine Calderón-Zygmund operator, then condition (1.7) is necessary for the boundedness of [ b , T ] from M Φ , φ 1 ( R n ) to M Φ , φ 2 ( R n ) .
If Φ ∈ Δ 2 ∩ ∇ 2 , T is a genuine Calderón-Zygmund operator and φ 1 ∈ G Φ satisfies the regularity type condition
(1.11) ∫ t ∞ 1 + ln t r φ 1 ( r ) d r r ≤ C φ 1 ( t ) ,
for all t > 0 , where C > 0 does not depend on t, then condition (1.7) is necessary and sufficient for the boundedness of [ b , T ] from M Φ , φ 1 ( R n ) to M Φ , φ 2 ( R n ) .

If we take Φ ( t ) = t p , p ∈ [ 1 , ∞ ) at Theorem 1.4, we obtain the following new result for generalized Morrey spaces.

Corollary 1.5

Let T be a Calderón-Zygmund operator, p ∈ [ 1 , ∞ ) , φ 1 , φ 2 ∈ Ω p , and b ∈ BMO ( R n ) .

If 1 < p < ß , then the condition
∫ r ∞ 1 + ln t r ess inf t < s < ∞ φ 1 ( s ) s n p t n p + 1 d t ≤ C φ 2 ( r ) ,
for all r > 0 , where C > 0 does not depend on r, is sufficient for the boundedness of [ b , T ] from M p , φ 1 ( R n ) to M p , φ 2 ( R n ) .
If φ 1 ∈ G p and T is a genuine Calderón-Zygmund operator, then condition (1.7) is necessary for the boundedness of [ b , T ] from M p , φ 1 ( R n ) to M p , φ 2 ( R n ) .
If 1 < p < ß , T is a genuine Calderón-Zygmund operator and φ 1 ∈ G p satisfies the regularity type condition (1.11), then condition (1.7) is necessary and sufficient for the boundedness of [ b , T ] from M p , φ 1 ( R n ) to M p , φ 2 ( R n ) .

By A ≲ B , we mean that A ≤ C B with some positive constant C independent of appropriate quantities. If A ≲ B and B ≲ A , we write A ≈ B and say that A and B are equivalent.

2 Definitions and preliminary results

2.1 On Young functions and Orlicz spaces

We recall the definition of Young functions.

Definition 2.1

A function Φ : [ 0 , ∞ ) → [ 0 , ∞ ] is called a Young function if Φ is convex, left-continuous, lim r → + 0 Φ ( r ) = Φ ( 0 ) = 0 and lim r → ∞ Φ ( r ) = ∞ .

From the convexity and Φ ( 0 ) = 0 , it follows that any Young function is increasing. If there exists s ∈ ( 0 , ∞ ) such that Φ ( s ) = ∞ , then Φ ( r ) = ∞ for r ≥ s . The set of Young functions such that

0 < Φ ( r ) < ∞ for 0 < r < ∞

will be denoted by Y . If Φ ∈ Y , then Φ is absolutely continuous on every closed interval in [ 0 , ∞ ) and bijective from [ 0 , ∞ ) to itself.

For a Young function Φ and 0 ≤ s ≤ ∞ , let

Φ − 1 ( s ) = inf { r ≥ 0 : Φ ( r ) > s } .

If Φ ∈ Y , then Φ − 1 is the usual inverse function of Φ . We note that

Φ ( Φ − 1 ( r ) ) ≤ r ≤ Φ − 1 ( Φ ( r ) ) for 0 ≤ r < ∞ .

It is well known that

(2.1) r ≤ Φ − 1 ( r ) Φ ˜ − 1 ( r ) ≤ 2 r for r ≥ 0 ,

where Φ ˜ ( r ) is defined by

Φ ˜ ( r ) = sup { r s − Φ ( s ) : s ∈ [ 0 , ∞ ) } , r ∈ [ 0 , ∞ ) ∞ , r = ∞ .

A Young function Φ is said to satisfy the Δ 2 -condition, denoted also as Φ ∈ Δ 2 , if

Φ ( 2 r ) ≤ k Φ ( r ) for r > 0

for some k > 1 . If Φ ∈ Δ 2 , then Φ ∈ Y . A Young function Φ is said to satisfy the ∇ 2 -condition, denoted also by Φ ∈ ∇ 2 , if

Φ ( r ) ≤ 1 2 k Φ ( k r ) , r ≥ 0

for some k > 1 .

Definition 2.2

(Orlicz space). For a Young function Φ , the set

L Φ ( R n ) = f ∈ L loc 1 ( R n ) : ∫ R n Φ ( k ∣ f ( x ) ∣ ) d x < ∞ for some k > 0

is called Orlicz space. If Φ ( r ) = r p , 1 ≤ p < ∞ , then L Φ ( R n ) = L p ( R n ) . If Φ ( r ) = 0 , ( 0 ≤ r ≤ 1 ) and Φ ( r ) = ∞ , ( r > 1 ) , then L Φ ( R n ) = L ∞ ( R n ) . The space L loc Φ ( R n ) is defined as the set of all functions f such that f χ B ∈ L Φ ( R n ) for all balls B ⊂ R n .

L Φ ( R n ) is a Banach space with respect to the norm

‖ f ‖ L Φ = inf λ > 0 : ∫ R n Φ ∣ f ( x ) ∣ λ d x ≤ 1 .

We note that

(2.2) ∫ R n Φ ∣ f ( x ) ∣ ‖ f ‖ L Φ d x ≤ 1 .

The weak Orlicz space

W L Φ ( R n ) = { f ∈ L loc 1 ( R n ) : ‖ f ‖ W L Φ < + ∞ }

is defined by the norm

‖ f ‖ W L Φ = inf λ > 0 : sup t > 0 Φ ( t ) m f λ , t ≤ 1 .

The following lemmas are valid.

Lemma 2.3

[2,33] Let Φ be a Young function and B a set in R n with finite Lebesgue measure. Then

‖ χ B ‖ W L Φ = ‖ χ B ‖ L Φ = 1 Φ − 1 ( ∣ B ∣ − 1 ) .

Lemma 2.4

For a Young function Φ and all balls B in R n , the following inequality is valid

‖ f ‖ L 1 ( B ) ≤ 2 ∣ B ∣ Φ − 1 ( ∣ B ∣ − 1 ) ‖ f ‖ L Φ ( B ) .

2.2 Generalized Orlicz-Morrey space

Orlicz-Morrey spaces are generalization of Orlicz spaces and Morrey spaces. There are three versions of Orlicz-Morrey spaces, i.e., Nakai’s [37], Gala et al. [20], and Deringoz et al. [12] versions. We used the definition of [12], which runs as follows.

We now define generalized Orlicz-Morrey spaces of the third kind. The generalized Orlicz-Morrey space M Φ , ϕ ( R n ) of the third kind is defined as the set of all measurable functions f for which the norm

‖ f ‖ M Φ , φ = sup x ∈ R n , r > 0 φ ( x , r ) − 1 Φ − 1 ( ∣ B ( x , r ) ∣ − 1 ) ‖ f ‖ L Φ ( B ( x , r ) )

is finite. Also by W M Φ , φ ( R n ) , we denote the weak generalized Orlicz-Morrey space of the third kind of all functions f ∈ W L loc Φ ( R n ) for which

‖ f ‖ W M Φ , φ = sup x ∈ R n , r > 0 φ ( x , r ) − 1 Φ − 1 ( ∣ B ( x , r ) ∣ − 1 ) ‖ f ‖ W L Φ ( B ( x , r ) ) < ∞ ,

where W L Φ ( B ( x , r ) ) denotes the weak L Φ -space of measurable functions f for which

‖ f ‖ W L Φ ( B ( x , r ) ) = ‖ f χ B ( x , r ) ‖ W L Φ .

Note that M Φ , ϕ ( R n ) covers many classical function spaces.

Example 2.5

Let 1 ≤ q ≤ p < ∞ and Φ ∈ Δ 2 ∩ ∇ 2 . From the following special cases, we see that our results will cover the Lebesgue space L p ( R n ) , the classical Morrey space M q p ( R n ) , the generalized Morrey space M p , φ ( R n ) , and the Orlicz space L Φ ( R n ) with norm coincidence:

If Φ ( t ) = t p and φ ( t ) = t − n p , then M Φ , φ ( R n ) = L p ( R n ) with norm equivalence.
If Φ ( t ) = t q and φ ( t ) = t − n p , then M Φ , φ ( R n ) , which is denoted by M q p ( R n ) , is the classical Morrey space.
If Φ ( t ) = t p , then M Φ , φ ( R n ) = M p , φ ( R n ) is the generalized Morrey space which were discussed in [22,34,36].
If φ ( t ) = Φ − 1 ( t − n ) , then M Φ , φ ( R n ) = L Φ ( R n ) , which is beyond the reach of generalized Orlicz-Morrey spaces of the second kind defined in [20] according to an example constructed in [42].

Other definitions of generalized Orlicz-Morrey spaces can be found in [20,37–39]. Therefore, our definition of generalized Orlicz-Morrey spaces here is named “third kind.”

In the case φ ( x , r ) = Φ − 1 ( ∣ B ( x , r ) ∣ − 1 ) Φ − 1 ( ∣ B ( x , r ) ∣ − λ ∕ n ) , we obtain the Orlicz-Morrey space ℳ Φ , λ ( R n ) from generalized Orlicz-Morrey space ℳ Φ , φ ( R n ) . We refer to [14, Lemmas 2.8 and 2.9] for more information about Orlicz-Morrey spaces.

Lemma 2.6

[14, Lemma 2.12] Let Φ be a Young function and φ be a positive measurable function on R n × ( 0 , ∞ ) .

If
(2.3) sup t < r < ∞ Φ − 1 ( ∣ B ( x , r ) ∣ − 1 ) φ ( x , r ) = ∞ for s o m e t > 0 and f o r a l l x ∈ R n ,
then ℳ Φ , φ ( R n ) = Θ .
If
(2.4) sup 0 < r < τ φ ( x , r ) − 1 = ∞ for s o m e τ > 0 and f o r a l l x ∈ R n ,
then ℳ Φ , φ ( R n ) = Θ .

Remark 2.7

Let Φ be a Young function. We denote by Ω Φ the sets of all positive measurable functions φ on R n × ( 0 , ∞ ) such that for all t > 0 ,

sup x ∈ R n Φ − 1 ( ∣ B ( x , r ) ∣ − 1 ) φ ( x , r ) L ∞ ( t , ∞ ) < ∞ ,

and

sup x ∈ R n ∥ φ ( x , r ) − 1 ∥ L ∞ ( 0 , t ) < ∞ ,

respectively. In what follows, keeping in mind Lemma 2.6, we always assume that φ ∈ Ω Φ .

The following lemma plays a key role in our main results.

Lemma 2.8

[13] Let B 0 ≔ B ( x 0 , r 0 ) . If φ ∈ G Φ , then there exist C > 0 such that

1 φ ( r 0 ) ≤ ‖ χ B 0 ‖ ℳ Φ , φ ≤ C φ ( r 0 ) .

We will use the following statements on the boundedness of the weighted Hardy operator

H w ∗ g ( r ) ≔ ∫ r ∞ 1 + ln t r g ( t ) w ( t ) d t , 0 < t < ∞ ,

where w is a fixed function nonnegative and measurable on ( 0 , ∞ ) .

The following theorem was proved in [23].

Theorem 2.9

[23] Let v 1 , v 2 , and w be positive almost everywhere and measurable functions on ( 0 , ∞ ) . The inequality

(2.5) ess sup r > 0 v 2 ( r ) H w ∗ g ( r ) ≤ C ess sup r > 0 v 1 ( r ) g ( r )

holds for some C > 0 for all nonnegative and nondecreasing g on ( 0 , ∞ ) if and only if

(2.6) B ≔ sup r > 0 v 2 ( r ) ∫ r ∞ 1 + ln t r w ( t ) d t sup t < s < ∞ v 1 ( s ) < ∞ .

Moreover, the value C = B is the best constant for (2.5).

Remark 2.10

In (2.5)–(2.6), it is assumed that 0 ⋅ ∞ = 0 .

3 Calderón-Zygmund operators in the space M Φ , φ ( R n )

Sufficient conditions on Φ for the boundedness of the operator T in Orlicz spaces L Φ ( R n ) have been obtained in [30, Theorem 1.4.3] and [44, Theorem 3.3].

Theorem 3.1

[30, 44] Let T be a singular integral operator. If Φ ∈ Δ 2 ⋂ ∇ 2 , then T is bounded on L Φ ( R n ) and if Φ ∈ Δ 2 , then T is bounded from L Φ ( R n ) to W L Φ ( R n ) .

Before the presentation of the main results, we recall some crucial inequalities to establish the boundedness of singular integral operator in generalized Orlicz-Morrey spaces.

Theorem 3.2

[30, Proposition 2.4] Let Φ be a Young function. Then, there is a constant C > 1 such that

Φ ( t ) m ( M f , t ) ≤ C ∫ R n Φ ( C ∣ f ( x ) ∣ ) d x

for every locally integrable f and every t > 0 .

Remark 3.3

For a sublinear operator S , weak modular inequality

(3.1) Φ ( t ) m ( S f , t ) ≤ C ∫ R n Φ ( C ∣ f ( x ) ∣ ) d x

implies the corresponding norm inequality. Indeed, let (3.1) holds. Then, we have

Φ ( t ) { x ∈ R n : S f ( x ) C 2 ‖ f ‖ L Φ > t } = Φ ( t ) { x ∈ R n : S f C 2 ‖ f ‖ L Φ ( x ) > t } ≤ C ∫ R n Φ ∣ f ( x ) ∣ C ‖ f ‖ L Φ d x ≤ 1 ,

which implies ‖ S f ‖ W L Φ ≲ ‖ f ‖ L Φ .

Sufficient conditions on ( Φ , φ 1 , φ 2 ) for the boundedness of the singular integral operators from one generalized Orlicz-Morrey space M Φ , φ 1 to another M Φ , φ 2 as stated in the following theorem are known, see [12, Theorem 5.5].

Theorem 3.4

Let φ 1 , φ 2 be positive measurable functions on R n × ( 0 , ∞ ) with satisfying the condition

(3.2) φ 1 ( x , 2 r ) ≤ C φ 2 ( x , r ) ,

where C does not depend on x ∈ R n and r > 0 . Let also T be a Calderón-Zygmund operator and the functions φ 1 , φ 2 ∈ Ω Φ satisfy condition (1.6).

If Φ ∈ Δ 2 , then the operator T is bounded from M Φ , φ 1 ( R n ) to W M Φ , φ 2 ( R n ) and
‖ T f ‖ M Φ , φ 2 ≲ ‖ f ‖ W M Φ , φ 1 ,
with constants independent of f.
If Φ ∈ Δ 2 ∩ ∇ 2 , then the operator T is bounded from M Φ , φ 1 ( R n ) in M Φ , φ 2 ( R n ) and
(3.3) ‖ T f ‖ M Φ , φ 2 ≲ ‖ f ‖ M Φ , φ 1 .

Proof

For the moment, we denote the singular integral operator on L Φ ( R n ) by T 0 to avoid confusion. Let ℬ = { B ( x , r ) : x ∈ R n , r > 0 } . For f ∈ M Φ , φ 1 ( R n ) and x ∈ R n , we choose a ball B = B ( x 0 , r ) ∈ ℬ , 2 B = B ( x 0 , 2 r ) such that x ∈ B , and let

(3.4) T f ( x ) ≔ T 0 f 1 ( x ) + ∫ R n K ( x , y ) f 2 ( y ) d y , f = f 1 + f 2 , f 1 = f χ 2 B .

First, we show that T f ( x ) is well-defined a . e . x and independent of the choice B containing x .

As T 0 is bounded on L Φ ( R n ) provided by Theorem 3.1 and f 1 ∈ L Φ ( R n ) , T 0 f 1 is well defined.

Next, we show that the second-term on the right-hand side defining T f ( x ) converges absolutely for any f ∈ M Φ , φ 1 ( R n ) and almost every x ∈ R n .

Observe that the inclusions x ∈ B , y ∈ R n \ 2 B = ∁ ( 2 B ) imply 1 2 ∣ x 0 − y ∣ ≤ ∣ x − y ∣ ≤ 3 2 ∣ x 0 − y ∣ . Then we obtain

∫ R n ∣ K ( x , y ) f 2 ( y ) ∣ d y ≲ ∫ 2 B C ∣ f ( y ) ∣ ∣ x − y ∣ n d y ≲ ∫ 2 B C ∣ f ( y ) ∣ ∣ x 0 − y ∣ n d y .

By Fubini’s theorem, we have

∫ 2 B C ∣ f ( y ) ∣ ∣ x 0 − y ∣ n d y ≈ ∫ 2 B C ∣ f ( y ) ∣ ∫ ∣ x 0 − y ∣ ∞ d t t n + 1 d y ≈ ∫ 2 r ∞ ∫ 2 r ≤ ∣ x 0 − y ∣ < t ∣ f ( y ) ∣ d y d t t n + 1 ≲ ∫ 2 r ∞ ∫ B ( x 0 , t ) ∣ f ( y ) ∣ d y d t t n + 1 .

By applying Lemma 2.4, we obtain

(3.5) ∫ 2 B C ∣ f ( y ) ∣ ∣ x 0 − y ∣ n d y ≲ ∫ 2 r ∞ ‖ f ‖ L Φ ( B ( x 0 , t ) ) Φ − 1 ( t − n ) d t t .

Finally, by condition (1.6), we obtain for all x ∈ B

(3.6) ∫ R n ∣ K ( x , y ) f 2 ( y ) d y ∣ ≲ ∫ 2 r ∞ ‖ f ‖ L Φ ( B ( x 0 , t ) ) Φ − 1 ( t − n ) d t t = ∫ 2 r ∞ ‖ f ‖ L Φ ( B ( x 0 , t ) ) ess inf t < s < ∞ φ 1 ( x , s ) Φ − 1 ( s − n ) ess inf t < s < ∞ φ 1 ( x , s ) Φ − 1 ( s − n ) Φ − 1 ( t − n ) d t t = ∫ 2 r ∞ ess sup t < s < ∞ ‖ f ‖ L Φ ( B ( x 0 , s ) ) Φ − 1 ( s − n ) φ 1 ( x , s ) ess inf t < s < ∞ φ 1 ( x , s ) Φ − 1 ( s − n ) Φ − 1 ( t − n ) d t t ≲ ‖ f ‖ M Φ , φ 1 ∫ 2 r ∞ ess inf t < s < ∞ φ 1 ( x , s ) Φ − 1 ( s − n ) Φ − 1 ( t − n ) d t t ≲ ‖ f ‖ M Φ , φ 1 φ 2 ( x 0 , r ) < ∞ .

Finally, it remains to show that the definition is independent of the choice of B . That is, if B 1 , B 2 ∈ ℬ and x ∈ B 1 ∩ B 2 , then

(3.7) T 0 ( f χ 2 B 1 ) ( x ) + ∫ R n \ 2 B 1 K ( x , y ) f ( y ) d y = T 0 ( f χ 2 B 2 ) ( x ) + ∫ R n \ 2 B 2 K ( x , y ) f ( y ) d y .

Actually, let B 3 ∈ ℬ be selected so that 2 B 1 ∪ 2 B 2 ⊂ B 3 . According to (3.6), for any x ∈ B 1 , both

χ B 3 \ 2 B 1 ∣ K ( x , y ) ∣ ∣ f ( y ) ∣ and χ B 3 \ 2 B 2 ∣ K ( x , y ) ∣ ∣ f ( y ) ∣

are integrable. Therefore, for any x ∈ B 1 ,

( T χ B 3 \ 2 B 1 f ) ( x ) = lim ε → 0 ∫ B 3 \ 2 B 1 , ∣ x − y ∣ > ε K ( x , y ) f ( y ) d y = ∫ B 3 \ 2 B 1 K ( x , y ) f ( y ) d y ,

( T χ B 3 \ 2 B 2 f ) ( x ) = lim ε → 0 ∫ B 3 \ 2 B 2 , ∣ x − y ∣ > ε K ( x , y ) f ( y ) d y = ∫ B 3 \ 2 B 2 K ( x , y ) f ( y ) d y .

Since f χ 2 B 1 , f χ B 3 \ 2 B 1 ∈ L Φ ( R n ) , the linearity of T 0 on L Φ ( R n ) yields

(3.8) T 0 ( f χ 2 B 1 ) ( x ) + ∫ R n \ 2 B 1 K ( x , y ) f ( y ) d y = T 0 ( f χ 2 B 1 ) ( x ) + ∫ B 3 \ 2 B 1 K ( x , y ) f ( y ) d y + ∫ R n \ B 3 K ( x , y ) f ( y ) d y = T 0 ( f χ 2 B 1 ) ( x ) + T 0 ( f χ B 3 \ 2 B 1 ) ( x ) + ∫ R n \ B 3 K ( x , y ) f ( y ) d y = T 0 ( f χ B 3 ) ( x ) + ∫ R n \ B 3 K ( x , y ) f ( y ) d y .

Similarly, we also have

(3.9) T 0 ( f χ 2 B 2 ) ( x ) + ∫ R n \ 2 B 2 K ( x , y ) f ( y ) d y = T 0 ( f χ B 3 ) ( x ) + ∫ R n \ B 3 K ( x , y ) f ( y ) d y .

Therefore, by combining (3.8) and (3.9), we obtain (3.7).

Now, we show the boundedness.

Since f 1 ∈ L Φ ( R n ) , by the boundedness of T 0 in L Φ ( R n ) provided by Theorem 3.1, it follows that

(3.10) ‖ T 0 f 1 ‖ L Φ ( B ) ≤ ‖ T 0 f 1 ‖ L Φ ( R n ) ≲ ‖ f 1 ‖ L Φ ( R n ) = ‖ f ‖ L Φ ( 2 B ) .

We also have

(3.11) 1 Φ − 1 ( ∣ 2 B ∣ − 1 ) = ‖ χ 2 B ‖ W L Φ ≲ ‖ M χ B ‖ W L Φ ≲ ‖ χ B ‖ L Φ = 1 Φ − 1 ( ∣ B ∣ − 1 )

from the well known pointwise estimate χ 2 B ( z ) ≲ M χ B ( z ) , for all z ∈ R n and Theorem 3.2.

By combining (3.2), (3.10), and (3.11), we obtain the estimate

(3.12) φ 2 ( B ) − 1 Φ − 1 ( ∣ B ∣ − 1 ) ‖ T 0 f 1 ‖ L Φ ( B ) ≲ φ 1 ( 2 B ) − 1 Φ − 1 ( ∣ 2 B ∣ − 1 ) ‖ f ‖ L Φ ( 2 B ) ≲ ‖ f ‖ M Φ , φ 1 .

From (3.6) for all x ∈ B , we have

(3.13) ∣ T f ( x ) ∣ ≤ ∣ T 0 f 1 ( x ) ∣ + ∫ R n ∣ K ( x , y ) f 2 ( y ) d y ∣ ≲ ∣ T 0 f 1 ( x ) ∣ + ‖ f ‖ M Φ , φ 1 φ 2 ( B ) .

Applying the norm ‖ ⋅ ‖ L Φ on both sides of (3.13), then by Lemma 2.3, we obtain

‖ T f ‖ L Φ ( B ) ≲ ‖ T 0 f 1 ‖ L Φ ( B ) + ‖ f ‖ M Φ , φ 1 φ 2 ( B ) Φ − 1 ( ∣ B ∣ − 1 ) .

Consequently, by (3.12), we have

φ 2 ( B ) − 1 Φ − 1 ( ∣ B ∣ − 1 ) ‖ T f ‖ L Φ ( B ) ≲ φ 2 ( B ) − 1 Φ − 1 ( ∣ B ∣ − 1 ) ‖ T 0 f 1 ‖ L Φ ( B ) + ‖ f ‖ M Φ , φ 1 ≲ ‖ f ‖ M Φ , φ 1

By taking supremum over B ∈ ℬ , we obtain the boundedness of T from M Φ , φ 1 ( R n ) to M Φ , φ 2 ( R n ) .□

To prove our main results, we need the following estimate.

Lemma 3.5

If T is a genuine Calderón-Zygmund operator and B 0 ≔ B ( x 0 , r 0 ) , then C ≤ T χ B 0 ∩ C x ( x ) for every x ∈ B 0 .

Proof

If x , y ∈ B 0 , then ∣ x − y ∣ ≤ ∣ x − x 0 ∣ + ∣ y − x 0 ∣ < 2 r 0 . We obtain C r 0 − n ≤ ∣ x − y ∣ − n . Therefore,

□ T χ B 0 ∩ C x ( x ) = ∫ B 0 ∩ C x K ( x , y ) d y ≳ ∫ B 0 ∩ C x ∣ x − y ∣ − n d y ≳ r 0 − n ∣ B 0 ∩ C x ∣ ≈ r 0 − n ∣ B 0 ∣ = C .

Proof of Theorem 1.2

The first part of the theorem follows from Theorem 3.4. We shall now prove the second part. Let B 0 = B ( x 0 , r 0 ) and x ∈ B 0 . It is easy to see that 1 ≲ T χ B 0 ∩ C x ( x ) for every x ∈ B 0 . Therefore, by Lemmas 2.3 and 3.5,

1 ≲ Φ − 1 ( ∣ B 0 ∣ − 1 ) ‖ T χ B 0 ∩ C x ‖ L Φ ( B 0 ) ≤ φ 2 ( B 0 ) ‖ T χ B 0 ∩ C x ‖ M Φ , φ 2 ≲ φ 2 ( B 0 ) ‖ χ B 0 ∩ C x ‖ M Φ , φ 1 ≤ φ 2 ( B 0 ) ‖ χ B 0 ‖ M Φ , φ 1 ≲ φ 2 ( B 0 ) φ 1 ( B 0 ) .

Since this is true for every B 0 , we are done.

The third statement of the theorem follows from the other statements of the theorem.□

4 Commutators of Calderón-Zygmund operators in the space M Φ , φ ( R n )

We recall the definition of the space of BMO ( R n ) .

Definition 4.1

Suppose that f ∈ L loc 1 ( R n ) , let

‖ f ‖ ∗ = sup x ∈ R n , r > 0 1 ∣ B ( x , r ) ∣ ∫ B ( x , r ) ∣ f ( y ) − f B ( x , r ) ∣ d y ,

where

f B ( x , r ) = 1 ∣ B ( x , r ) ∣ ∫ B ( x , r ) f ( y ) d y .

Define

BMO ( R n ) = { f ∈ L loc 1 ( R n ) : ‖ f ‖ ∗ < ∞ } .

Modulo constants, the space BMO ( R n ) is a Banach space with respect to the norm ‖ ⋅ ‖ ∗ .

Before proving our theorems, we need the following lemmas.

Lemma 4.2

[28] Let b ∈ BMO ( R n ) . Then, there is a constant C > 0 such that

(4.1) ∣ b B ( x , r ) − b B ( x , t ) ∣ ≤ C ‖ b ‖ ∗ ln t r f o r 0 < 2 r < t ,

where C is independent of b , x , r , and t .

Lemma 4.3

[31] Let f ∈ BMO ( R n ) and Φ be a Young function with Φ ∈ Δ 2 , then

(4.2) ‖ f ‖ ∗ ≈ sup x ∈ R n , r > 0 Φ − 1 ( ∣ B ( x , r ) ∣ − 1 ) ∥ f ( ⋅ ) − f B ( x , r ) ∥ L Φ ( B ( x , r ) ) .

For a function b ∈ B M O define the commutator [ b , T ] f = b T f − T ( b f ) . Our aim is to show boundedness of [ b , T ] in M Φ , φ ( R n ) . For this goal, we recall some well-known properties of the BMO functions.

Theorem 4.4

[10] Let Φ be a Young function with Φ ∈ Δ 2 ⋂ ∇ 2 and b ∈ BMO ( R n ) , then the commutator of Calderón-Zygmund operator [ b , T ] is bounded on L Φ ( R n ) .

Theorem 4.5

Let φ 1 , φ 2 be positive measurable functions on R n × ( 0 , ∞ ) with satisfying the condition (3.2). Let also T be a Calderón-Zygmund operator, Φ be a Young function with Φ ∈ Δ 2 ∩ ∇ 2 , b ∈ BMO ( R n ) , and the functions φ 1 , φ 2 ∈ Ω Φ satisfy condition (1.10). Then, the operator [ b , T ] is bounded from M Φ , φ 1 ( R n ) to M Φ , φ 2 ( R n ) and

(4.3) ‖ [ b , T ] f ‖ M Φ , φ 2 ≤ C ‖ b ‖ ∗ ‖ f ‖ M Φ , φ 1

with a constant independent of f .

Proof

For the moment, we denote the commutator on L Φ ( R n ) by [ b , T ] 0 to avoid confusion. For f ∈ M Φ , φ 1 ( R n ) and x ∈ R n , we choose a ball B r = B ( x 0 , r ) ∈ ℬ such that x ∈ B , and let

[ b , T ] f ( x ) ≔ [ b , T ] 0 f 1 ( x ) + ∫ R n K ( x , y ) ( b ( y ) − b ( x ) ) f 2 ( y ) d y , f = f 1 + f 2 , f 1 = f χ 2 B r .

First, we show that [ b , T ] f ( x ) is well-defined a.e. x and independent of the choice B containing x .

As [ b , T ] 0 is bounded on L Φ ( R n ) provided by Theorem 4.4 and f 1 ∈ L Φ ( R n ) , [ b , T ] 0 f 1 is well defined.

Next, we show that the second term on the right-hand side defining [ b , T ] f ( x ) converges absolutely for any f ∈ M Φ , φ 1 ( R n ) and almost every x ∈ R n .

It is easy to see that for arbitrary points x ∈ B r and y ∈ ( 2 B r ) c , it holds

(4.4) 1 2 ∣ x 0 − y ∣ ≤ ∣ x − y ∣ ≤ 3 2 ∣ x 0 − y ∣ .

Then, for all x ∈ B r ,

∫ R n ∣ K ( x , y ) ( b ( y ) − b ( x ) ) f 2 ( y ) ∣ d y ≲ ∫ 2 B C ∣ b ( y ) − b ( x ) ∣ ∣ x − y ∣ n ∣ f ( y ) ∣ d y ≲ ∫ 2 B C ∣ b ( y ) − b ( x ) ∣ ∣ x 0 − y ∣ n ∣ f ( y ) ∣ d y ≲ ∫ 2 B C ∣ b ( y ) − b B ∣ ∣ x 0 − y ∣ n ∣ f ( y ) ∣ d y + ∫ 2 B C ∣ b ( x ) − b B ∣ ∣ x 0 − y ∣ n ∣ f ( y ) ∣ d y = I 1 + I 2 .

We estimate I 1 as follows:

I 1 ≲ 1 Φ − 1 ( r − n ) ∫ ( 2 B r ) c ∣ b ( y ) − b 2 B r ∣ ∣ f ( y ) ∣ ∣ x 0 − y ∣ n d y = 1 Φ − 1 ( r − n ) ∫ ( 2 B r ) c ∣ b ( y ) − b 2 B r ∣ ∣ f ( y ) ∣ ∫ ∣ x 0 − y ∣ ∞ d t t n + 1 d y = 1 Φ − 1 ( r − n ) ∫ 2 r ∞ ∫ 2 r ≤ ∣ x 0 − y ∣ ≤ t ∣ b ( y ) − b 2 B r ∣ ∣ f ( y ) ∣ d y d t t n + 1 ≲ 1 Φ − 1 ( r − n ) ∫ 2 r ∞ ∫ B t ∣ b ( y ) − b 2 B r ∣ ∣ f ( y ) ∣ d y d t t n + 1 .

By applying Hölder’s inequality, Lemma 4.1, we obtain

I 1 ≲ 1 Φ − 1 ( r − n ) ∫ 2 r ∞ ∫ B t ∣ b ( y ) − b B t ∣ ∣ f ( y ) ∣ d y d t t n + 1 + 1 Φ − 1 ( r − n ) ∫ 2 r ∞ ∣ b 2 B r − b B t ∣ ∫ B t ∣ f ( y ) ∣ d y d t t n + 1 ≲ 1 Φ − 1 ( r − n ) ∫ 2 r ∞ ∥ b ( ⋅ ) − b B t ∥ L Φ ˜ ( B t ) ‖ f ‖ L Φ ( B t ) d t t n + 1 + 1 Φ − 1 ( r − n ) ∫ 2 r ∞ ∣ b 2 B r − b B t ∣ ‖ f ‖ L Φ ( B t ) Φ − 1 ( t − n ) d t t ≲ ‖ b ‖ ∗ Φ − 1 ( r − n ) ∫ 2 r ∞ 1 + ln t r ‖ f ‖ L Φ ( B t ) Φ − 1 ( t − n ) d t t .

To estimate I 2 , note that

I 2 = ∥ b ( ⋅ ) − b 2 B r ∥ L Φ ( 2 B r ) ∫ ( 2 B r ) c ∣ f ( y ) ∣ ∣ x 0 − y ∣ n d y .

By Lemma 4.3 and (3.5), we obtain

I 2 ≲ ‖ b ‖ ∗ Φ − 1 ( r − n ) ∫ ( 2 B r ) c ∣ f ( y ) ∣ ∣ x 0 − y ∣ n d y ≲ ‖ b ‖ ∗ Φ − 1 ( r − n ) ∫ 2 r ∞ ‖ f ‖ L Φ ( B t ) Φ − 1 ( t − n ) d t t .

By summing up I 1 and I 2 , we obtain

I 1 + I 2 ≲ ‖ b ‖ ∗ Φ − 1 ( r − n ) ∫ 2 r ∞ 1 + ln t r ‖ f ‖ L Φ ( B t ) Φ − 1 ( t − n ) d t t .

Finally, by condition (1.10), we obtain for all x ∈ B r

(4.5) ∫ R n ∣ K ( x , y ) ( b ( y ) − b ( x ) ) f 2 ( y ) d y ∣ ≲ ∫ 2 r ∞ 1 + ln t r ‖ f ‖ L Φ ( B ( x 0 , t ) ) Φ − 1 ( t − n ) d t t = ∫ 2 r ∞ 1 + ln t r ‖ f ‖ L Φ ( B ( x 0 , t ) ) ess inf t < s < ∞ φ 1 ( x , s ) Φ − 1 ( s − n ) ess inf t < s < ∞ φ 1 ( x , s ) Φ − 1 ( s − n ) Φ − 1 ( t − n ) d t t = ∫ 2 r ∞ 1 + ln t r ess sup t < s < ∞ ‖ f ‖ L Φ ( B ( x 0 , s ) ) Φ − 1 ( s − n ) φ 1 ( x , s ) ess inf t < s < ∞ φ 1 ( x , s ) Φ − 1 ( s − n ) Φ − 1 ( t − n ) d t t ≲ ‖ f ‖ M Φ , φ 1 ∫ 2 r ∞ ess inf t < s < ∞ φ 1 ( x , s ) Φ − 1 ( s − n ) Φ − 1 ( t − n ) d t t ≲ ‖ f ‖ M Φ , φ 1 φ 2 ( x 0 , r ) < ∞ .

Finally, it remains to show that the definition is independent of the choice of B . That is, if B 1 , B 2 ∈ ℬ and x ∈ B 1 ∩ B 2 , then

(4.6) [ b , T ] 0 ( f χ 2 B 1 ) ( x ) + ∫ R n \ 2 B 1 K ( x , y ) ( b ( y ) − b ( x ) ) f ( y ) d y = [ b , T ] 0 ( f χ 2 B 2 ) ( x ) + ∫ R n \ 2 B 2 K ( x , y ) ( b ( y ) − b ( x ) ) f ( y ) d y .

Actually, let B 3 ∈ ℬ be selected so that 2 B 1 ∪ 2 B 2 ⊂ B 3 . Since f χ 2 B 1 , f χ B 3 \ 2 B 1 ∈ L Φ ( R n ) , the linearity of [ b , T ] 0 on L Φ ( R n ) yields

(4.7) [ b , T ] 0 ( f χ 2 B 1 ) ( x ) + ∫ R n \ 2 B 1 K ( x , y ) ( b ( y ) − b ( x ) ) f ( y ) d y = [ b , T ] 0 ( f χ 2 B 1 ) ( x ) + ∫ B 3 \ 2 B 1 K ( x , y ) ( b ( y ) − b ( x ) ) f ( y ) d y + ∫ R n \ B 3 K ( x , y ) ( b ( y ) − b ( x ) ) f ( y ) d y = [ b , T ] 0 ( f χ 2 B 1 ) ( x ) + [ b , T ] 0 ( f χ B 3 \ 2 B 1 ) ( x ) + ∫ R n \ B 3 K ( x , y ) ( b ( y ) − b ( x ) ) f ( y ) d y = [ b , T ] 0 ( f χ B 3 ) ( x ) + ∫ R n \ B 3 K ( x , y ) ( b ( y ) − b ( x ) ) f ( y ) d y .

Similarly, we also have

(4.8) [ b , T ] 0 ( f χ 2 B 2 ) ( x ) + ∫ R n \ 2 B 2 K ( x , y ) ( b ( y ) − b ( x ) ) f ( y ) d y = [ b , T ] 0 ( f χ B 3 ) ( x ) + ∫ R n \ B 3 K ( x , y ) ( b ( y ) − b ( x ) ) f ( y ) d y .

Therefore, by combining (4.7) and (4.8), we obtain (4.6).

Now, we show the boundedness.

Since f 1 ∈ L Φ ( R n ) , by the boundedness of [ b , T ] 0 in L Φ ( R n ) provided by Theorem 4.4, it follows that

(4.9) ‖ [ b , T ] 0 f 1 ‖ L Φ ( B ) ≤ ‖ [ b , T ] 0 f 1 ‖ L Φ ( R n ) ≲ ‖ b ‖ ∗ ‖ f 1 ‖ L Φ ( R n ) = ‖ f ‖ L Φ ( 2 B ) .

We also have

(4.10) 1 Φ − 1 ( ∣ 2 B ∣ − 1 ) = ‖ χ 2 B ‖ W L Φ ≲ ‖ M χ B ‖ W L Φ ≲ ‖ χ B ‖ L Φ = 1 Φ − 1 ( ∣ B ∣ − 1 )

from the well known pointwise estimate χ 2 B ( z ) ≲ M χ B ( z ) , for all z ∈ R n and Theorem 3.2.

By combining (3.2), (4.9), and (4.10), we obtain the estimate

(4.11) φ 2 ( B ) − 1 Φ − 1 ( ∣ B ∣ − 1 ) ‖ [ b , T ] 0 f 1 ‖ L Φ ( B ) ≲ φ 1 ( 2 B ) − 1 ‖ b ‖ ∗ Φ − 1 ( ∣ 2 B ∣ − 1 ) ‖ f ‖ L Φ ( 2 B ) ≲ ‖ b ‖ ∗ ‖ f ‖ M Φ , φ 1 .

From (3.6) for all x ∈ B , we have

(4.12) ∣ [ b , T ] f ( x ) ∣ ≤ ∣ [ b , T ] 0 f 1 ( x ) ∣ + ∫ R n ∣ K ( x , y ) ( b ( y ) − b ( x ) ) f 2 ( y ) d y ∣ ≲ ∣ [ b , T ] 0 f 1 ( x ) ∣ + ‖ b ‖ ∗ ‖ f ‖ M Φ , φ 1 φ 2 ( B ) .

By applying the norm ‖ ⋅ ‖ L Φ on both sides of (4.12), then by Lemma 2.3, we obtain

‖ [ b , T ] f ‖ L Φ ( B ) ≲ ‖ [ b , T ] 0 f 1 ‖ L Φ ( B ) + ‖ b ‖ ∗ ‖ f ‖ M Φ , φ 1 φ 2 ( B ) Φ − 1 ( ∣ B ∣ − 1 ) .

Consequently, by (4.11), we have

φ 2 ( B ) − 1 Φ − 1 ( ∣ B ∣ − 1 ) ‖ [ b , T ] f ‖ L Φ ( B ) ≲ φ 2 ( B ) − 1 Φ − 1 ( ∣ B ∣ − 1 ) ‖ [ b , T ] 0 f 1 ‖ L Φ ( B ) + ‖ f ‖ M Φ , φ 1 ≲ ‖ b ‖ ∗ ‖ f ‖ M Φ , φ 1 .

By taking supremum over B ∈ ℬ , we obtain the boundedness of [ b , T ] from M Φ , φ 1 ( R n ) to M Φ , φ 2 ( R n ) .□

To prove our main results, we need the following estimate.

Lemma 4.6

If T is a genuine Calderón-Zygmund operator, b ∈ L loc 1 ( R n ) and B 0 ≔ B ( x 0 , r 0 ) , then

∣ b ( x ) − b B 0 ∩ C x ∣ ≤ C [ b , T ] χ B 0 ∩ C x ( x )

for every x ∈ B 0 , where b B 0 = 1 ∣ B 0 ∣ ∫ B 0 b ( y ) d y .

Proof

If x , y ∈ B 0 , then ∣ x − y ∣ ≤ ∣ x − x 0 ∣ + ∣ y − x 0 ∣ < 2 r 0 . We obtain C r 0 − n ≤ ∣ x − y ∣ − n . Therefore,

□ [ b , T ] χ B 0 ∩ C x ( x ) = ∫ B 0 ∩ C x ∣ b ( x ) − b ( y ) ∣ K ( x , y ) d y ≳ ∫ B 0 ∩ C x ∣ b ( x ) − b ( y ) ∣ ∣ x − y ∣ − n d y ≳ r 0 − n ∫ B 0 ∩ C x ∣ b ( x ) − b ( y ) ∣ d y ≈ ∣ B 0 ∩ C x ∣ − 1 ∫ B 0 ∣ b ( x ) − b ( y ) ∣ d y ≳ ∣ B 0 ∩ C x ∣ − 1 ∫ B 0 ( b ( x ) − b ( y ) ) d y ≳ ∣ b ( x ) − b B 0 ∩ C x ∣ .

Proof of Theorem 1.4

The first part of the theorem follows from Theorem 4.5.

We shall now prove the second part. Let B 0 = B ( x 0 , r 0 ) and x ∈ B 0 . By Lemma 4.6, we have ∣ b ( x ) − b B 0 ∣ ≤ C [ b , T ] χ B 0 ( x ) . Therefore, by Lemmas 4.3 and 2.8, we obtain

1 ≤ C ‖ [ b , T ] χ B 0 ∩ C x ‖ L Φ ( B 0 ∩ C x ) ‖ b ( ⋅ ) − b B 0 ∩ C x ‖ L Φ ( B 0 ∩ C x ) ≤ C ‖ b ‖ ∗ ‖ [ b , T ] χ B 0 ∩ C x ‖ L Φ ( B 0 ∩ C x ) Φ − 1 ( ∣ B 0 ∩ C x ∣ − 1 ) ≤ C ‖ b ‖ ∗ φ 2 ( r 0 ) ‖ [ b , T ] χ B 0 ∩ C x ‖ M Φ , φ 2 ≤ C φ 2 ( r 0 ) ‖ χ B 0 ∩ C x ‖ M Φ , φ 1 ≤ C φ 2 ( r 0 ) ‖ χ B 0 ‖ M Φ , φ 1 ≤ C φ 2 ( r 0 ) φ 1 ( r 0 ) .

Since this is true for every r 0 > 0 , we end the proof of the second part.

The third statement of the theorem follows from first and second parts of the theorem.□

Acknowledgments

The authors thank the referees for careful reading the article and useful comments.

Funding information: The research of V.S. Guliyev was partially supported by the grant of Cooperation Program 2532 TUBITAK - RFBR (RUSSIAN foundation for basic research) (Agreement number no. 119N455). V.S. Guliyev is also supported by the RUDN University Strategic Academic Leadership Program. The third author like to thank Faculty of Fundamental Science, Industrial University of Ho Chi Minh City, Vietnam, for the opportunity to work in collaboration.
Conflict of interest: Alessandra is a member of the Editorial Board of the journal, but it does not affect the peer-review process and the final decision.

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Received: 2022-04-30

Revised: 2023-01-30

Accepted: 2023-07-05

Published Online: 2023-08-11

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https://doi.org/10.1515/anona-2022-0307

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generalized Orlicz-Morrey spaces; genuine Calderón-Zygmund operator; commutator; BMO

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