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Extrapolation of compactness on weighted Morrey spaces

In a previous work,"compact versions"of Rubio de Francia's weighted extrapolation theorem were proved, which allow one to extrapolate the compactness of an linear operator from just one space to the full range of weighted Lebesgue spaces, where this operator is bounded. In this paper, we extend these results to the setting of weighted Morrey spaces. As applications, we easily obtain new results on the weighted compactness of commutators of Calder\'on--Zygmund singular integrals, rough singular integrals and Bochner--Riesz multipliers.


Introduction
We refer to a locally integrable positive almost everywhere function w on R d as a weight and we define the weighted Lebesgue and Morrey spaces as follows: Definition 1.1. Let 1 ≤ p < ∞ and w be a weight. Then a weighted Lebesgue space is defined by Definition 1.2 ( [34]). Let 1 ≤ p < ∞, 0 < λ < d and w be a weight. Then the Samko type weighted Morrey space is defined by where the supremum is taken over all cubes Q ⊂ R d .
Remark 1.3. Alternatively, we could define the Samko type weighted Morrey spaces with balls instead of cubes. If w ≡ 1, then L p,λ (w) = L p,λ (R d ), where L p,λ (R d ) is the classical Morrey space (see [31]).
As we will work with Muckenhoupt weight characteristics, we recall the following definitions: Definition 1.4 ( [22,32] where the supremum is taken over all cubes Q ⊂ R d and w Q := |Q| −1 Q w. A weight w is said to belong to the reverse Hölder class RH σ (R d ) (or w ∈ RH σ (R d )) if The classes RH σ (R d ) and A p (R d ) were introduced to study the L pintegrability of the partial derivatives of a quasiconformal mapping and the weighted norm inequalities for Hardy-Littlewood maximal function, respectively; see [22,32].
The following theorem of Rubio de Francia [33] (see also the work of Auscher-Martell [3]) on the extrapolation of boundedness on weighted spaces is one of the most important tools of modern harmonic analysis: Theorem 1.5 ( [3], Theorem 4.9 and [33]). Let 1 ≤ p − < p + ≤ ∞, and T be a linear operator simultaneously defined and bounded on L p 1 (w) for some p 1 ∈ [p − , p + ] and allw ∈ A p 1 /p − (R d ) ∩ RH (p + /p 1 ) ′ (R d ). Then T is also defined and bounded on L p (w) for all p ∈ (p − , p + ) and all w ∈ A p/p − (R d ) ∩ RH (p + /p) ′ (R d ).
Remark 1.6. We don't seem to have an analogue of Rubio de Francia's extrapolation theorem on weighted Morrey spaces.
In [14,Theorem 3.31], a version of Theorem 1.5 is stated in terms of non-negative measurable pairs of functions (f, g). This means that one does not need to work with specific operators since nothing about the operators themselves is used (like linearity or sublinearity) and they play no role. However, we work with the pairs (|f |, |T f |), where T is a linear operator, since the abstract compactness results that we will use in order to prove Theorem 1.8 below hold for linear operators (see Theorem 2.3 of Cwikel-Kalton and Theorem 2.4 of Cwikel-Rochberg).
In the recent paper [26], the authors provided the following version for extrapolation of compactness: (See also [8,27] for extensions to multilinear operators.) Theorem 1.7 ([26], Theorems 1.3 and 2.4). In the setting of Theorem 1.5, suppose in addition that T is compact on L p 1 (w 1 ) for some w 1 ∈ In this paper, we extend Theorem 1.7 to the setting of weighted Morrey space. In particular, we obtain the following: Theorem 1.8. Let 0 < λ < d, 1 ≤ p − < p + ≤ ∞ and T be a linear operator simultaneously defined and bounded on L p,λ (w) for all p ∈ (p − , p + ) and Remark 1.9. Theorem 1.8 remains true in the case p + = ∞, provided that p 1 < ∞. Thus the reverse Hölder conditions on w, w 1 are vacuous. Due to Theorem 2.8 below, it seems that our results don't apply to other type of weighted Morrey spaces such as the Komori-Shirai type weighted Morrey space considered in [28]. In addition, notice that if we let λ → 0 in Definition 1.2, then L p,0 (w) ≡ L p (w) and hence Theorem 1.8 formally recovers Theorem 1.7.
When w 1 ≡ 1, Theorem 1.8 says that we can obtain weighted compactness if the weighted boundedness and unweighted compactness are already known. This case is relevant to all our applications in Sections 4-7.
The paper is organized as follows: in Section 2 we collect some previously known results from which the proof of Theorem 1.8 will follow in Section 3. In Sections 4-7 we provide several applications of our main result. An example of these applications to commutators of Calderón-Zygmund operators is the following (we refer to Sections 4 and 5 for the notions of CMO(R d ) and Calderón-Zygmund operators, respectively): T be a Calderón-Zygmund operator that extends boundedly to L 2 (R d ) and satisfies for all f ∈ C ∞ c (R d ) the condition T f (x) = lim ε→0 |x−y|≥ε K(x, y)f (y)dy a.e. x ∈ R d . Then the commutator [b, T ] is compact on L p,λ (w) for all w ∈ A s (R d ) ∩ RH t (R d ) and all p, λ, s, t such that See also Theorems 6.4 and 7.4 for similar results on rough singular integrals and Bochner-Riesz multipliers. Although compactness of such operators on the unweighted Morrey spaces has been considered in the literature, obtaining compactness results on weighted Morrey spaces appears to be entirely new altogether.
Notation. Throughout the paper, we denote by C a positive constant which is independent of the main parameters but it may change at each occurrence, and we write f g if f ≤ Cg. The term cube will always refer to a cube Q ⊂ R d and |Q| will denote its Lebesgue measure. Recall from Definition 1.4 that w Q denotes |Q| −1 Q w, the average of w over Q, and p ′ is the conjugate exponent to p, that is p ′ := p/(p − 1).

Preliminaries
We collect the results from which the proof of Theorem 1.8 will follow in Section 3.
1. Define F (X 0 , X 1 ) as the set of all functions F :S → X 0 + X 1 such that (a) F is continuous onS and sup z∈S F (z) X 0 +X 1 < ∞, (b) F is holomorphic on S, (c) the functions t ∈ R → F (j + it) ∈ X j are bounded and continous on R for j = 0, 1.
Let Y be a Banach space. We let where Definition 2.2 (Calderón's second complex interpolation space). Suppose thatX = (X 0 , X 1 ) is a compatible couple of Banach spaces.
1. Define G(X 0 , X 1 ) as the set of all functions F :S :→ X 0 + X 1 such that (a) F is continuous onS and sup z∈S Our main abstract tools are the following theorems of Cwikel-Kalton [15] and Cwikel-Rochberg [16]: be Banach couples and T be a linear operator such that T : is compact for θ ∈ (0, 1) under any of the following four side conditions: (1) X 1 has the UMD (unconditional martingale differences) property, (2) X 1 is reflexive, and X 1 = [X 0 , E] α for some Banach space E and α ∈ (0, 1), for some Banach space F and β ∈ (0, 1), (4) X 0 and X 1 are both complexified Banach lattices of measurable functions on a common measure space.
We have swapped the roles of the indices 0 and 1 in comparison to [15]. For the UMD property, see [25,Ch. 4].
We consider a measurable function f , weights v, v 0 , v 1 and numbers p 0 , p, p 1 , R > 0 fixed. The following objects depend on these quantities, but we do not indicate this in the notation. We define for f ∈ L q,λ (v) and consider the following condition: We will use Theorems 2.3 and 2.4 in the following special setting: Then , and γ ∈ (0, 1).
We postpone the proof of Proposition 2.5 to the following section.
Lemma 2.6. If 0 < λ < d, 1 ≤ p j < ∞ and w j are weights, then the spaces X j = L p j ,λ (w j ) satisfy the condition (4) of Theorem 2.3.
Proof. It is easy to verify that X j = L p j ,λ (w j ) are complexified Banach lattices of measurable functions on the common measure space R d (see also [36]).
Remark 2.11. Lemma 2.10 remains true in the case p + = ∞, provided that q 1 < ∞. In this case the reverse Hölder conditions on w, w 0 , w 1 are vacuous and the proof is given in [26,Lemma 4.4].
3. The Proof of the key Proposition 2.5 and Theorem 1.8 To complete the proof of Theorem 1.8 it remains to verify Proposition 2.5: Proof of Proposition 2.5. We prove the proposition in the case p + < ∞. The case p + = ∞ is proved in a similar way. With some 0 < λ < d, we are given 1 and θ ∈ (0, 1) such that By Theorem 2.8, we then have as we claimed.
We can now give the proof of our main result: Proof of Theorem 1.8. Let p + < ∞ and recall that the assumptions of Theorem 1.8 are in force. The case p + = ∞ is proved in a similar way. In particular, T is a bounded linear operator on L p,λ (w) for all p ∈ (p − , p + ) and In addition, it is assumed that T is a compact operator on L p 1 ,λ (w 1 ) for some p 1 ∈ [p − , p + ] and some We need to prove that T is actually compact on L p,λ (w) for all p ∈ (p − , p + ) and , and that T : X 1 → Y 1 is compact (since this was assumed). By Lemma 2.6, the last condition (4) of Theorem 2.3 is also satisfied by these spaces X j = L p j ,λ (w j ). By Theorem 2.3, it follows that T is also compact on [X 0 , (1) holds}. Hence, by Theorem 2.4, we conclude that T is also compact from [X 0 , X 1 ] θ = L p,λ (w) to [Y 0 , Y 1 ] θ = {f ∈ L p,λ (w) : (1) holds}.
In particular, this implies that T is also compact on L p,λ (w).

Commutators with functions of bounded mean oscillation
We indicate several applications of Theorem 1.8 which deal with commutators of the form where the pointwise multiplier b belongs to the space of functions of bounded mean oscillation, or its subspace where the closure is in the BMO norm. We will need the following results of Duoandikoetxea-Rosenthal [20] (see also [19,21]) on the extrapolation of boundedness on the corresponding weighted Morrey spaces from the assumption of weighted estimates on L p (w) spaces: 20], Theorem 1.1). Let 1 ≤ κ ≤ p 1 < ∞ and T be a defined and bounded operator on L p 1 (w) for all w ∈ A p 1 /κ (R d ). Then T is bounded on L p,λ (w) for all p ∈ (κ, ∞) (and also p = κ if p 1 = κ), all 0 < λ < d − d σ and all w ∈ A p/κ (R d ) ∩ RH σ (R d ).
Theorem 4.2 ([20], Corollary 3.2). Let 1 ≤ p − ≤ p 1 ≤ p + < ∞ and T be a defined and bounded operator on Then for all p ∈ (p − , p + ) (and also p = p − if p 1 = p − ) and . This has the advantage of providing immediately several different versions. In our applications below, we will apply these with [b, T ] in place of T .
In [1,4], the following general results on weighted boundedness about commutators were obtained: ). Let 1 ≤ κ < p 1 < ∞, and T be a linear operator defined and bounded on L p 1 (w) for allw ∈ A p 1 /κ (R d ), with the operator norm dominated by some increasing function of [w] A p 1 /κ . Suppose also that b ∈ BMO(R d ). Then also [b, T ] extends to a bounded linear operator on L p 1 (w) for allw ∈ A p 1 /κ (R d ), and its operator norm is dominated by another increasing function of [w] A p 1 /κ .
The statement in [1, Theorem 2.13] is somewhat more general, but the above particular case is easily seen to be contained in it.
A combination of Theorems 1.8, 4.1, 4.2, 4.4 and 4.5 immediately gives the following two corollaries: Corollary 4.6. Let 1 ≤ κ < p + < ∞, and 1 < σ 1 < ∞, and and 0 < λ 1 < d − d σ 1 . Suppose moreover that: (1) T is a linear operator defined and bounded on Lp 1 (w) for somep 1 ∈ (κ, ∞) and for allw ∈ Ap 1 /κ (R d ), with the operator norm dominated by some increasing function of Then [b, T ] is compact on L p,λ 1 (w) for the same b, for all p ∈ (p − , p + ) and Proof. We verify the assumptions of Theorem 1.8 for the fixed numbers λ 1 , p 1 , σ 1 , κ, p + , d and the weight w 1 appearing in the statement of the Corollary, and the operator [b, T ] in place of T . By Theorem 4.4, [b, T ] is bounded on Lp 1 (w) for the samep 1 ∈ (κ, ∞) and allw ∈ A p 1 /κ (R d ). Then, by Theorem 4 . By choosing λ = λ 1 and σ = σ 1 to be the fixed numbers appearing in the statement of the Corollary, [b, T ] is bounded on L p,λ 1 (w) for all p ∈ (κ, ∞) and allw ∈ A p/κ (R d ) ∩ RH σ 1 (R d ). In particular, [b, T ] is bounded on L p,λ 1 (w) for all p ∈ (p − , p + ) and allw ∈ A p/p − (R d ) ∩ RH (p + /p) ′ (R d ) where we recall that p − = max κ, p + 1 − 1 σ 1 and 1 ≤ p − < p + < ∞. By assumption, [b, T ] is compact on L p 1 ,λ 1 (w 1 ) for some p 1 ∈ [p − , p + ] and some . Thus the assumptions of Theorem 1.8 hold for the operator [b, T ] in place of T . Hence, the conclusion of Theorem 1.8 gives the compactness of [b, T ] on L p,λ 1 (w) for all p ∈ (p − , p + ) and all Corollary 4.7. Let 1 ≤ p − < p + < ∞, and 1 < σ 1 < ∞, and 0 < λ 1 < d − d σ 1 , and Suppose moreover that: (1) T is a linear operator defined and bounded on Lp 1 (w) for somep 1 ∈ (p − , p + ) and for allw ∈ Ap the commutator [b, T ] is compact on L p 1 ,λ 1 (w 1 ) for some b ∈ BMO(R d ), Then [b, T ] is compact on L p,λ 1 (w) for the same b, for all p ∈ (q − , q + ) and Proof. We verify the assumptions of Theorem 1.8 for the fixed numbers λ 1 , p 1 , σ 1 , p − , p + , d and the weight w 1 appearing in the statement of the Corollary, and the operator [b, T ] in place of T . By Theorem 4.5, [b, T ] is bounded on Lp 1 (w) for the samep 1 ∈ (p − , p + ) and for allw ∈ . By choosing λ = λ 1 and σ = σ 1 to be the fixed numbers appearing in the statement of the Corollary, [b, T ] is bounded on L p,λ 1 (w) for all p ∈ (p − , q + ) and allw we recall that q − = max q + 1 − 1 σ 1 , p − and 1 ≤ q − < q + < ∞. By assumption, [b, T ] is compact on L p 1 ,λ 1 (w 1 ) for some p 1 ∈ [q − , q + ] and some . Thus the assumptions of Theorem 1.8 hold for the operator [b, T ] in place of T . Hence, the conclusion of Theorem 1.8 gives the compactness of [b, T ] on L p,λ 1 (w) for all p ∈ (q − , q + ) and all w ∈ A p/q − (R d ) ∩ RH (q + /p) ′ (R d ).

Commutators of Calderón-Zygmund singular integrals
In our application below, we consider Calderón-Zygmund singular integral operators which are defined as follows: T is a linear operator defined on a suitable class of test functions on R d , and it has the representation where the kernel K satisfies the size condition |K(x, y)| 1 |x − y| d and, for some δ ∈ (0, 1], the smoothness condition for all x, z, y ∈ R d such that |x − y| > 1 2 |x − z|. We will need the following classical result of Coifman-Fefferman [11]: 11]). Let T be a Calderón-Zygmund operator that extends to a bounded operator on L 2 (R d ). Then T extends to a bounded operator on L p (w) for all p ∈ (1, ∞) and all w ∈ A p (R d ).
The following result of Arai-Nakai [2], based on Sawano and Shirai's method in [35], provides a concrete condition to verify the assumptions of Corollary 4.6: ). Let 0 < λ < d, 1 < p < ∞ and T be a Calderón-Zygmund operator that extends to a bounded operator on L 2 (R d ). Assume also that for all By combining Corollary 4.6, Theorem 5.1 and Theorem 5.2 we obtain the following result which appears to be new: Lemma 5.4. Let 1 < p + < ∞, and 1 < σ < ∞, and T is a Calderón-Zygmund operator that extends boundedly to L 2 (R d ) and satisfies for all f ∈ C ∞ c (R d ) the condition T f (x) = lim ε→0 |x−y|≥ε K(x, y)f (y)dy a.e. x ∈ R d . Then the commutator [b, T ] is compact on L p,λ (w) for all p ∈ (p − , p + ) and Proof. Let us fix some σ 1 ∈ (1, ∞), λ 1 ∈ (0, d − d σ 1 ), p 1 ∈ [p − , p + ] and p 1 ∈ (1, ∞) for which we verify the assumptions of Corollary 4.6 with κ = 1. By Theorem 5.1, T extends to a bounded operator on Lp 1 (w) for all w ∈ Ap 1 (R d ). By Theorem 5.2, [b, T ] is a compact operator on L p 1 ,λ 1 (R d ) = . Thus Corollary 4.6 applies to give the compactness of [b, T ] on L p,λ 1 (w) for all p ∈ (p − , p + ) and Now, we check the conditions of Lemma 5.4 in such a way that we obtain the following: Proof of Theorem 1.10. Under the conditions appearing in the assumptions and conclusion of Theorem, we check that we can find parameters σ and p + , p − as in the Lemma 5.4. In particular, by choosing σ ∈ give the compactness of [b, T ] on L p,λ (w) for all p ∈ (p − , p + ) and all w ∈ . It remains to check that this covers all w ∈ A s (R d ) ∩ RH t (R d ) as in the statement of the Theorem. Since p/p − ≥ s and p + /p ≥ t ′ , the monotonicity of the A s (R d ) and RH t (R d ) classes implies that [b, T ] is compact on L p,λ (w) for all w ∈ A s (R d ) ∩ RH t (R d ) and all p, λ, s, t such that p ∈ (1, ∞), 0 < λ < d, s ∈ 1, min p, d λ , t ∈ d sλ ′ , ∞ .

Commutators of rough singular integrals
Let Ω be homogeneous of degree zero in R d , integrable, and have mean value zero on the unit sphere S d−1 . Define the singular integral operator T Ω by T Ω f (x) := lim ε→0 |x−y|>ε Ω(x − y) |x − y| d f (y)dy. Duoandikoetxea [17] and Watson [37] considered the following weighted estimates for T Ω : Theorem 6.1 ( [17,37]). Let r ∈ (1, ∞) and Ω ∈ L r (S d−1 ) be homogeneous of order zero with vanishing mean on S d−1 . Then T Ω extends to a bounded operator on L p (w) for all p ∈ (r ′ , ∞) and all w ∈ A p/r ′ (R d ).
The following Bochner-Riesz conjecture is well-known: (See also the works of [10,13] in two dimensions and [5,30] in the case d ≥ 3.) In [29], an equivalent form of the Bochner-Riesz Conjecture 3 is stated as follows: (See also [9,12,13] and [18,Chapter 8.5] for the connection between the Bochner-Riesz and the S τ Fourier multiplier) 1