EXTENSION AND BOUNDEDNESS OF OPERATORS

EXTENSION AND BOUNDEDNESS OF OPERATORS ON MORREY SPACES FROM EXTRAPOLATION TECHNIQUES AND EMBEDDINGS JAVIER DUOANDIKOETXEA & MARCEL ROSENTHAL Abstract. We prove that under the hypotheses of the extrapolation theorem for weights (quasi)-sublinear operators are bounded on weighted Morrey spaces. As a consequence, we obtain at once a number of results that have been proved individually for many operators. On the other hand, our theorems provide a variety of new results even for the unweighted case because we do not use any representation formula or pointwise bound of the operator as was assumed by previous authors. To extend the operators to Morrey spaces we show different (continuous) embeddings of (weighted) Morrey spaces into appropriate Muckenhoupt A1 weighted Lp spaces, which enables us to define the operators on the considered Morrey spaces by restriction. In this way, we can avoid the delicate problem of the definition of the operator, often ignored by the authors. In dealing with the extension problem through the embeddings (instead of using duality) one is neither restricted in the parameter range of the p’s (in particular p = 1 is admissible and applies to weak-type inequalities) nor the operator has to be linear.

1. Introduction An important tool of modern harmonic analysis is the extrapolation theorem due to Rubio de Francia. For a given operator T we suppose that for some p0 , 1 ≤ p0 < ∞, and for every weight belonging to the Muckenhoupt class Ap0 the inequality Z 1 p0 n p0 ≤ c k f | Lp0 ,w (Rn )k (1.1) k T f | Lp0 ,w (R )k = |T f (x)| w(x)dx Rn

holds, where the constant c is independent of f . Then for every p, 1 < p < ∞, and every w ∈ Ap there exists a constant depending on [w]Ap (see 2010 Mathematics Subject Classification. Primary 42B35, 46E30, 42B15, 42B20; Secondary 42B25. Key words and phrases. Morrey spaces, embeddings, Muckenhoupt weights, extrapolation, Calder´ on-Zygmund operators, Mihlin-H¨ ormander multipliers, commutators. The first author is supported by the grants MTM2014-53850-P of the Ministerio de Econom´ıa y Competitividad (Spain) and grant IT-641-13 of the Basque Gouvernment. The second author is supported by the German Academic Exchange Service (DAAD). 1

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JAVIER DUOANDIKOETXEA & MARCEL ROSENTHAL

Definition 2.1) such that (1.2)

k T f | Lp,w (Rn )k ≤ c k f | Lp,w (Rn )k

(cf. [Rub82, Rub84, CMP11, Duo11]). One fact which makes extrapolation theory so powerful is that one S does not need any assumption on T (beside that T is well-defined on w∈Ap Lp0 ,w (Rn )). Even if one is not in0 terested in any weighted results this is very interesting by the fact that for example the weighted L2 -boudednesses implies unweighted (and weighted) boundednesses in the complete Lp scale. Appropriate assertions hold also for subclasses of Ap0 , that is, one assumes (1.1) for a subclass (e.g. A1 ) and deduces (1.2) for appropriate subclasses (this applies to different Fourier multipliers, for instance). We generalize results of this theory to Morrey spaces and will get even in the classical Morrey spaces new results on the boundednesses of operators referring that we do not need any condition on T except that T satisfies (1.1) (for weights of Ap0 or distinguished subclasses), is well-defined on S n w∈Ap0 Lp0 ,w (R ) and is (quasi-)subadditive. There are lots of papers dealing with classes of operators in various Morrey type spaces which admit size estimates as Z |f (x)| (1.3) |(T f )(y)| ≤ c dx n Rn |y − x| for all f ∈ D(Rn ) and y ∈ / supp (f ), where D(Rn ) are the compactly supported smooth functions (cf. [Pee66, CF87, Nak94, Alv96, DYZ98, Sam09, KS09, GAKS11, Gul12, Mus12, SFZ13, RS14, KGS14, PT15, IST15, Nkm16, Wan16] and the references given there). To obtain the boundedness results they are explicitly using and requiring different representation formulas as (1.3) of the considered operators. We are completely avoiding these assumptions and encapsulating them through the consideration of weighted spaces. Hence, we get even new results on the classical Morrey spaces with respect to Mihlin-H¨ ormander type operators, rough operators, pseudodifferential operators, square functions, commutators, Fourier integral operators. . . Moreover, literature about (the non-separable) Morrey spaces (starting from Peetre [Pee66] and many following scholars) does not care about how to extend the considered operators (satisfying a required representation formula only on non-dense subsets of the considered Morrey spaces). Some forerunners dealing with the extension of the operators are [Alv96, AX12, RT13, Ros13, RT14, Tri14, RS14, Ada15, RS16]. Mainly, they apply duality (with some exceptions with respect to [Tri14, RS14] justifying (1.3) also for Morrey functions), which restricts the boundedness results to linear operators and moreover the parameter range of the integration parameter p (in particular the extension problem was not covered for p = 1 and the corresponding weak-type inequalities). We show different (continuous) embeddings of (weighted) Morrey spaces into different Muckenhoupt A1 weighted

BOUNDEDNESS ON MORREY SPACES, EXTRAPOLATION AND EMBEDDINGS

3

Lp spaces which enables us to define the operators on the considered Morrey spaces by restriction. Another possibility to overcome the extension problem is using the Fatou property of the Morrey space. Nevertheless, this method requires at least a bigger space where the operator is also bounded to extend it by restriction (cf. Triebel [Tri16]), which is exactly the type of our embeddings. However, by the fact that we do not require representation formulas of operators it is even sufficient to show a not necessarily continuous embedding of a MorS rey type space into a set of the type w∈Ap Lp,w (Rn ) (where the considered operator is required to be bounded, cf. Section 4 for examples). This can be seen as a powerful model case to extend operators which are a priori only defined on D(Rn ) —as multipliers and (maximally truncated) singular integrals— to the various generalizations of Morrey spaces given in literature and to overcome the extension problem (by the fact that we avoid the requirement of justifying (1.3) also for Morrey functions). If one wants to n deal with S other spacesnwhere D(R ) is not dense and which are contained in the set w∈Ap Lp,w (R ) and where the maximal operator is bounded (or in their appropriate associated spaces), then there are even very good chances that the presented results could be carried over partially. We also want to point out the following equality of sets, [ [ [ (1.4) Lp,w (Rn ) = Lq,w (Rn ), w∈Ap

q∈(1,∞) w∈Aq

where 1 < p < ∞. That means that the left-hand side of (1.4) does not depend on p which seems to be a new contribution to the classical Muckenhoupt weighted Lebesgue theory. We furthermore prove several assertions of this type. After introducing the notation and some preliminaries in Section 2, Section 3 is concerned with continuous embeddings of Morrey type spaces into Muckenhoupt weighted A1 Lebesgue spaces. In Section 4 we present the extrapolation theory generalized to Morrey type spaces which will be applied to far-reaching boundedness results given in Section 5. 2. Notation, Morrey spaces and preliminaries We use standard notation. Let N be the collection of all natural numbers and Rn be the Euclidean n-space. Put R = R1 . For 1 ≤ p < ∞, Lp,w (Rn ) is the complex Banach space of functions whose p-th power is integrable with respect to the weight w : Rn → [0, ∞] and which is normed by Z 1/p Z 1/p n p kf |Lp,w (R )k = |f (x)| w(x)dx = |f |p w . Rn Rn R Moreover, we write w(M ) = M w(x)dx for the measure of the subset M of Rn . We similarly define Lp,w (M ). If w(x) ≡ 1, we simply write Lp (M ), k · |Lp (M )k and |M |. Furthermore, χM denotes the characteristic function of M . For any p ∈ (1, ∞) we denote by p0 the conjugate index, namely, 1/p+

4


1/p0 = 1. If p = 1, then p0 = ∞. Moreover, L(Rn ) collects all equivalence classes of almost everywhere coinciding measurable complex functions. Then n n Lloc 1 (R ) ⊂ L(R ) collects all locally integrable functions, hence f ∈ L1 (M ) for any bounded measurable set M of Rn . Throughout the paper we consider cubes whose sides are parallel to the coordinate axes. For such a cube Q, δQ stands for the concentric cube with side-length δ times of the side-length of Q. We call an operator quasi-subadditive on a set A of functions, if |T (f + g)| ≤ c (|T (f )| + |T (g)|) almost everywhere for all f , g ∈ A, where the constant c does not depend on f and g. The concrete value of constants may vary from one formula to the next, but remains the same within one chain of (in)equalities. n Definition 2.1. Let w ∈ Lloc 1 (R ) with w > 0 almost everywhere. We say that w is a Muckenhoupt weight belonging to Ap for 1 < p < ∞ if !p−1 0 w(Q) w1−p (Q) [w]Ap ≡ sup < ∞, |Q| |Q| Q

where the supremum is taken over all cubes Q in Rn . The left hand-side is the Ap constant of w. We say that w belongs to A1 if, for any cube Q, (2.1)

w(Q) ≤ cw(x) for almost all x ∈ Q. |Q|

The A1 constant of w, denoted by [w]A1 , is the smallest constant c for which the inequality holds. Remark 2.2. The Ap weights are doubling. This means that if w ∈ Ap , there exists a constant c such that w(2Q) ≤ cw(Q),

(2.2) for every cube Q in

Rn

(cf. [Duo01, (7.3)]).

We recall that the Hardy-Littlewood maximal operator, which we denote by M , is bounded on Lp,w (Rn ) for 1 < p < ∞ if and only if w ∈ Ap and is of weak-type (1, 1) with respect to the measure w(x)dx if and only if w ∈ A1 . On the other hand, we will use the following construction of A1 weights: if M f (x) is finite a.e. and δ < 1, then M f (x)δ is an A1 weight. Moreover, the factorization theorem says that w ∈ Ap if and only if there exist w0 , w1 ∈ A1 such that w = w0 w11−p (cf. [Rub84] or [Gra09, Thm. 9.5.1]). We define some Muckenhoupt weighted Morrey spaces. Definition 2.3. For 1 ≤ p < ∞, − np ≤ r < 0 and w ∈ Ap we define

Lrp (w, Rn ) ≡ {f ∈ L(Rn ) : f |Lrp (w, Rn ) < ∞} with the norm r 1

f |Lrp (w, Rn ) ≡ sup w(Q)− p + n kf |Lp,w (Q)k , Q


5

where the supremum is taken over all cubes Q in Rn . Moreover,

Lrp (λ, w, Rn ) ≡ {f ∈ L(Rn ) : f |Lrp (λ, w, Rn ) < ∞} with the norm r 1

f |Lrp (λ, w, Rn ) ≡ sup |Q|− p + n kf |Lp,w (Q)k . Q

We also define their weak versions given by the norms 1 r

1

f |W Lrp (w, Rn ) ≡ sup sup w(Q)− p + n tw({x ∈ Q : |f (x)| > t}) p , Q

t>0

f |W Lrp (λ, w, Rn ) ≡ sup sup |Q|− Q

1 r +n p

1

tw({x ∈ Q : |f (x)| > t}) p .

t>0

−n/p

Remark 2.4. We observe that Lp (w, Rn ) = Lp,w (Rn ). If w ≡ 1, r n Lp (w, R ) coincides with the unweighted Morrey space Lrp (Rn ). The weak Morrey spaces W Lrp (w, Rn ) coincide also with the weak Lp,w (Rn ) spaces for r = −n/p. Definition 2.5. Let a non-negative locally integrable function w on Rn belong to the reverse H¨ older class RHσ for 1 < σ < ∞ if it satisfies the reverse H¨ older inequality with exponent σ, i.e. 1 Z Z σ 1 c σ ≤ w(x) dx w(x)dx, |Q| Q |Q| Q where the constant c is universal for all cubes Q ⊂ Rn . The RHσ classes are decreasing, that is, RHσ ⊂ RHτ for 1 < τ < σ < ∞. On the other hand, Gehring’s lemma ([Geh73]) says that if w ∈ RHσ , there exists > 0 such that w ∈ RHσ+ (openness of the reverse Hölder classes). Remark 2.6. Let w ∈ RHσ . For any cube Q and any measurable E ⊂ Q it holds that 0 |E| 1/σ w(E) ≤c . (2.3) w(Q) |Q| (See [Duo01, Cor. 7.6]). Since w ∈ Ap implies that w ∈ RHσ for some σ (cf. [Duo01, Thm. 7.4]), the inequality holds for each Ap weight for the appropriate σ. Another consequence of the fact that Ap weights satisfy reverse H¨ older inequalities is that if w ∈ Ap , then there exists > 0 such that w ∈ Ap− (openness of Ap classes). In the paper we use sometimes classes of the form Ap ∩ RHσ . There is a characterization of D. Cruz-Uribe and C. Neugebauer for these classes ([CUN95, Thm. 2.2]): (2.4)

Ap ∩ RHσ = {w : wσ ∈ Aσ(p−1)+1 }.

This is also valid for p = 1, that is, A1 ∩ RHσ = {w : wσ ∈ A1 }, which can be easily proved from the definition of A1 .

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Remark 2.7. Let w(x) = |x|α or w(x) = (1 + |x|)α . Then we have the following: (1) It holds that w ∈ Ap if and only if −n < α < n(p − 1) whenever p ∈ (1, ∞) and −n < α ≤ 0 whenever p = 1. (2) For α ∈ (−n, 0) it holds |x|α ∈ RHσ if and only if 1 < σ < −n/α. (3) For α ≥ 0 it holds |x|α ∈ RHσ for all 1 < σ < ∞. The first one is well known. The second one is easily obtained using (2.4) (for p = 1). The third one can be checked directly or derived from (2.4).

3. Embeddings of Muckenhoupt weighted Morrey spaces into Muckenhoupt weighted Lebesgue spaces In this section we establish different continuous embeddings between Morrey spaces of the types Lrp (w, Rn ) and Lrp (λ, w, Rn ) into some Muckenhoupt A1 weighted Lebesgue spaces which will give us the possibility to define operators on these Morrey spaces by restriction. Extension by continuity is usually not working in Morrey type spaces by their non-separability and also by the fact that the smooth functions are not dense in these spaces (cf. [RT14, Prop. 3.7] for Lrp (Rn ) and [RS16, Prop. 3.2] for Lrp (w, Rn ) for the non-separability, and cf. [Pic69, p. 22] for the non-density). In recent literature this difficulty has been treated using duality for linear operators whenever the integration parameter p satisfies 1 < p < ∞. Dealing with the extension problem through embeddings one is neither restricted in the parameter range of p (in particular, p = 1 is admissible) nor the operator has to be linear. Proposition 3.1. Let 1 < p < ∞, − np ≤ r < 0 and w ∈ Ap . Then there exists q0 ∈ (1, p) such that for each q ∈ [1, q0 ] there is 0 < α < n such that n Lrp (w, Rn ) ,→ Lq,(1+|x|)−α (Rn ) ⊂ Lloc 1 (R ).

(3.1)

n With the additional assumptions w ∈ RHσ and r ≤ − pσ , furthermore, n Lrp (λ, w, Rn ) ,→ Lq,(1+|x|)−α (Rn ) ⊂ Lloc 1 (R ).

(3.2)

Proof. We set B0 ≡ Q ≡ [−1, 1]n and Bi ≡ 2i Q \ 2i−1 Q for i ≥ 1. Since w ∈ Ap , there exists q > 1 such that w ∈ Ap/q . Set p˜ = p/q. By Hölder’s inequality it holds Z

q

−α

|f (x)| (1 + |x|)

dx ≤ c

Rn

(3.3)

≤c

∞ X

2

−iα

Z

|f (x)|q w(x)1/˜p w(x)−1/˜p dx

Bi i=0 Z ∞ X −iα

|f (x)|p w(x)dx

2

i=0

Bi

1/˜p

0

0

w1−˜p (2i Q)1/˜p .


7

If f ∈ Lrp (w, Rn ) we can bound the last term by ∞

q X c1 f | Lrp (w, Rn ) 2−iα w(2i Q)

1 r +n p

q

0

0

w1−˜p (2i Q)1/˜p

i=0 ∞

q X r ≤ c2 f | Lrp (w, Rn ) 2−i(α−n) w(2i Q) n q . i=0

In the last step we used that w ∈ Ap˜. Using now (2.3) and the fact that r < 0, we deduce that r

r

w(2i Q) n q ≤ c2iδrq w(Q) n q

(3.4)

for some δ > 0. Thus we obtain Z ∞

X r q −α r n q q n |f (x)| (1 + |x|) dx ≤c1 w(Q) f | Lp (w, R ) 2−i(α−n−δrq) Rn

i=0

≤c2 (w) f | Lrp (w, Rn ) , for α close enough to n (e.g., α = n + δr/2). Therefore (3.1) holds. In the case of the spaces Lrp (λ, w, Rn ) after (3.3) we get Z |f (x)|q (1 + |x|)−α dx Rn

(3.5)

∞

q X ≤c1 f | Lrp (λ, w, Rn ) 2−iα |2i Q|

1 r +n p

q

1/˜p0 0 w1−˜p 2i Q

i=0 ∞

q X in ≤c2 f | Lrp (λ, w, Rn ) 2−iα 2

r 1 +n p

q in

2 w(2i Q)−1/˜p ,

i=0

where we used again that w ∈ Ap˜. By the fact that w ∈ RHσ , from (2.3) we deduce 0 n |Q| 1/σ w(Q) (3.6) ≤c = c2−i σ0 . i i w(2 Q) |2 Q| Together with (3.5) we get Z |f (x)|q (1 + |x|)−α dx Rn

∞

X in r n q

≤c1 (w) f | Lp (λ, w, R ) 2−iα 2

r 1 +n p

q in

2

2−in

q pσ 0

i=0 ∞

q X

n −i(α−n−q(r+ pσ )) ≤c1 (w) f | Lrp (λ, w, Rn ) 2 . i=0 n pσ

If r + < 0, the geometric series converges for sufficiently large α < n. n = 0, notice that by the openness of the reverse To get the endpoint r + pσ H¨ older property, in (3.6) one can replace σ by σ+. Thus (3.2) is proved.

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Remark 3.2. Since w ∈ Ap implies that w ∈ RHσ for some σ, the range of values of r satisfying the embedding (3.2) is not empty. In the case of the weights |x|β and (1 + |x|)β the range of values of r satisfying (3.2) is ( − np ≤ r < 0, if 0 ≤ β < n(p − 1); β n − p ≤ r < p , if − n < β < 0. A consequence of Proposition 3.1 is the following. Proposition 3.3. Let 1 ≤ γ < p < ∞, − np ≤ r < 0 and w ∈ Ap/γ . Then there exists q0 ∈ (γ, p) such that for each q ∈ [γ, q0 ] there is 0 < α < n such that Lrp (w, Rn ) ,→ Lq,(1+|x|)−α (Rn ). n , furthermore, With the additional assumptions w ∈ RHσ and r ≤ − pσ Lrp (λ, w, Rn ) ,→ Lq,(1+|x|)−α (Rn ).

(3.7)

n Proof. Let f ∈ Lrp (w, Rn ) with w ∈ Ap/γ . Then |f |γ ∈ Lrγ p/γ (w, R ) (with w ∈ Ap/γ ) and by (3.1) we deduce

1

γ n

f | Lrp (w, Rn ) = |f |γ | Lrγ p (w, R )

γ

1

γ ≥c |f |γ | L q ,(1+|x|)−α (Rn ) ≥ c f | Lq,(1+|x|)−α (Rn ) , γ where the constant c does not depend on f . Taking into account (3.2) we obtain in the same manner (3.7). The following proposition shows that the union of the Lq,w (Rn ) spaces for fixed q and all w ∈ Aq is invariant for 1 < q < ∞. This observation seems to be new, we did not find the result in the literature. For instance, several statements in [Gra09, Section 9.5] require the definition of the operS ator on the union of the sets w∈Ap Lp,w (Rn ), while according to the next proposition one of them would be enough. Proposition 3.4. Let 1 < q < ∞. Then it holds that [ [ [ [ (3.8) Lq,w (Rn ) = Lp,w (Rn ) ( L1,w (Rn ). 1 12 . For 0 < |x| ≤

1 2

we observe that Z 1 dy 1 1 M f (x) ≥ c , =c 2 n n n |x| |y|≤|x| |y| ln |y| |x| | ln |x||

and M f is not integrable near the origin.

Corollary 3.5. Let be 1 ≤ γ < q < ∞. Then we have the following equality of sets [ [ [ [ (3.10) Lq,w (Rn ) = Lp,w (Rn ) ( Lγ,w (Rn ). w∈Aq/γ

γ 1. Then there exists some 0 < α < n/ν such that Lrp (w, Rn ) ,→ Lp,(1+|x|)−α (Rn ) h i 0 whenever w ∈ RHσ for some σ > ν and r ∈ − np , − np σν 0 . Moreover, (3.13)

Lrp (λ, w, Rn ) ,→ Lp,(1+|x|)−α (Rn ) h i whenever r ∈ − np , − np σ1 + ν10 .

(3.14)

Proof. We proceed as in the previous proposition (because w ∈ A1 ). Using that w ∈ RHσ and the openness of the reverse Hölder classes, from (3.6) we deduce r rp r p (3.15) w(2i Q) n ≤ c2i( σ0 −ε) w(Q) n p

BOUNDEDNESS ON MORREY SPACES, EXTRAPOLATION AND EMBEDDINGS 11

for some ε > 0. Then Z ∞

X r p −α r n p

|f (x)| (1 + |x|) dx ≤ c1 f | Lp (w, R ) 2−i(α−n) w(2i Q) n p Rn

i=0

p r ≤c2 f | Lrp (w, Rn ) w(Q) n p

∞ X

p rp 2−i(α−n− σ0 +ε) ≤ c3 f | Lrp (w, Rn )

i=0 n for suitable α (e.g., α = nν − 2ε , taking into account − rp σ 0 ≥ ν 0 ). Therefore (3.13) holds. For the spaces Lrp (w, Rn ) we have r Z ∞ p 2+ n i

X p −α r n p −iα |2 Q|

|f (x)| (1 + |x|) dx ≤ c f | Lp (λ, w, R ) 2 w(2i Q) Rn i=0

≤c1

r p 2+ n

|Q|

f | Lrp (λ, w, Rn ) p w(Q)

∞ X

n

2−i(α−2n−rp+ σ0 +ε) ,

i=0

and the series converges for a sufficient large α < n n ν − 2n − rp + σ 0 ≥ 0 and ε > 0 and thus (3.14).

n ν

taking into account that

Remark 3.8. Since we assume wν ∈ A1 , we automatically have w ∈ RHν . But each such w will be in a reverse Hölder class w ∈ RHσ for some σ > ν and the range of values of r obtained in the proposition is not empty. In the case w(x) = |x|β or w(x) = (1 + |x|)β , we need −n < βν ≤ 0 to satisfy the condition wν ∈ A1 and in such case the range of values of r satisfying the embeddings (3.13) and (3.14) are respectively −

n n 1 n ≤r σ(p). Let r ∈ − np , − nσb . There exists q ∈ (1, p) and α < n/σ(q) such that

Lrp (w, Rn ) ,→ Lq,(1+|x|)−α (Rn ). Furthermore, Lrp (λ, w, Rn ) ,→ Lq,(1+|x|)−α (Rn ) h i whenever r ∈ − np , − np σ1 − nb . (3.16)

Proof. Let f ∈ Lrp (w, Rn ) with w ∈ Ap ∩ RHσ . We obtain as in the proof of Proposition 3.1 the inequality Z ∞

q X rq |f (x)|q (1 + |x|)−α dx ≤ c f | Lrp (w, Rn ) 2−i(α−n) w(2i Q) n , Rn

i=0

12


The use of (3.15) (because w ∈ RHσ ) leads to f ∈ Lq,(1+|x|)−α (Rn ) whenever α − n − rq/σ 0 ≥ 0. To choose α so that α < n/σ(q) we need rq n n+ 0 ≤α< . σ (b/q)0 This range is not empty if r < −nσ 0 /b. To get the endpoint r = −nσ 0 /b, we use the openness of the reverse Hölder classes to increase σ. The proof for (3.16) is similar. Using (3.5) and (3.15) the condition on α becomes α − n − rq − nq/(σp) ≥ 0, which together with α < n/σ(q) gives the restriction on r. Proposition 3.10. Let 1 < b < ∞ and σ(p) = (b/p)0 for 1 < p < b. Let 1 < q < b. Then, [ [ [ Lq,w (Rn ) = Lp,w (Rn ). w∈Aq ∩RHσ(q)

1 1 such that for each q ∈ [1, q0 ] there is 0 < α < n such that W Lp,w (Rn ) ,→ Lq,(1+|x|)−α (Rn ). • Let 1 ≤ γ < p < ∞ and w ∈ Ap/γ . Then there exists q0 > γ such that for each q ∈ [γ, q0 ] there is 0 < α < n such that W Lp,w (Rn ) ,→ Lq,(1+|x|)−α (Rn ). • Let 1 ≤ p < q < ∞ and w ∈ A1 . Then there exists some 0 < α < n such that W Lq,w (Rn ) ,→ Lp,(1+|x|)−α (Rn ). 0

• Let 1 ≤ p < q ≤ σν 0 and w ∈ RHσ with wν ∈ A1 for some 1 < ν < σ. Then there exists some 0 < α < n/ν such that W Lq,w (Rn ) ,→ Lp,(1+|x|)−α (Rn ). • Let 1 < b < ∞ and σ(p) = (b/p)0 for 1 < p < b. Let w ∈ Ap ∩RHσ(p) . There exists q ∈ (1, p) and α < n/σ(q) such that W Lp,w (Rn ) ,→ Lq,(1+|x|)−α (Rn ).

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4. Boundedness on Morrey spaces: general results In this section we obtain the general theorems giving the boundedness on the corresponding Morrey spaces from the assumption of weighted estimates on Lp,w (Rn ) spaces. Theorem 4.1. Let 1 ≤ p0 < ∞. Assume that for a (quasi-)subadditive operator [ (4.1) T : Lp0 ,w (Rn ) → L(Rn ) w∈Ap0

and for every w ∈ Ap0 we have (4.2)

k T f | Lp0 ,w (Rn )k ≤ c1 k f | Lp0 ,w (Rn )k

for all f ∈ Lp0 ,w (Rn ).

Then for every 1 < p < ∞, every − np ≤ r < 0 and every w ∈ Ap we have that T is well-defined on Lrp (w, Rn ) by restriction by the fact that [ (4.3) Lrp (w, Rn ) ⊂ Lp0 ,w (Rn ) w∈Ap0

and, moreover,

T f | Lrp (w, Rn ) ≤ c2 f | Lrp (w, Rn )

(4.4)

for all f ∈ Lrp (w, Rn ). n Furthermore, for every 1 < p < ∞ and w ∈ Ap ∩ RHσ , if − np ≤ r ≤ − pσ we have that T is well-defined on Lrp (w, Rn ) by restriction by the fact that [ Lp0 ,w (Rn ) (4.5) Lrp (λ, w, Rn ) ⊂ w∈Ap0

and moreover (4.6)

T f | Lrp (λ, w, Rn ) ≤ c3 f | Lrp (λ, w, Rn )

for all f ∈ Lrp (λ, w, Rn ). The constants c1 , c2 and c3 in (4.2), (4.4) and (4.6) do not depend on f but may depend on [w]Ap , p0 , p, n and r. Proof. First we observe that (4.3) and (4.5) hold by (3.8) and Proposition 3.1 and therefore T is well-defined on the appropriate Morrey spaces. Let f ∈ Lrp (w, Rn ) and let Q be a cube. We decompose f by f0 ≡ χ2Q f and f1 ≡ χRn \2Q f . From the extrapolation theorem in the usual weighted Lebesgue spaces (cf. [Duo11, CMP11, Gra09]) we obtain (4.2) for any p ∈ (1, ∞) and the weight class Ap (instead of p0 and Ap0 ). Hence, (4.7)

k T (f0 )| Lp,w (Q)k ≤ c1 k f0 | Lp,w (Rn )k

1 1 +r +r ≤c1 w(2Q) p n f0 | Lrp (w, Rn ) ≤ c2 w(Q) p n f | Lrp (w, Rn ) ,


where we used (2.2) and c2 does not depend on f . Let q be such that 1 < q < p and set p˜ = p/q. Let 1 < s < p˜0 . Thus, Z 1 Z p ˜ p˜q |T (f1 )| w = |T (f1 )|q hwχQ n Q R (4.8) Z Z 1 1 ≤ |T (f1 )|q M (hs ws χQ ) s ≤ c |f1 |q M (hs ws χQ ) s , Rn

Rn (Rn )

where the equality holds with h ∈ Lp˜0 ,w with norm 1 and the last 1 s s s inequality because M (h w χQ ) ∈ A1 and T is bounded on Lq (v) for v ∈ A1 . 1 The weight M (hs ws χQ ) s is well defined because hs ws χQ ∈ L1 (Rn ) which implies M (hs ws χQ ) < ∞ almost everywhere (by the fact that M maps L1 (Rn ) to the weak L1 (Rn ) space). Indeed, Z 1 − 10 Z 1 0 s

(s−1) p˜0p˜−s +1 s p˜ n s s−1

0 w ≤ h| Lp˜ ,w (R ) h w w Q Q (4.9) Z 1 1 =

w

s(p ˜0 −1) p ˜0 −s

s

− p˜0

1

≤ c w(Q)1/˜p |Q|− s0 ,

Q

for s sufficiently close to 1 as to use the reverse Hölder inequality for w in the last step. On the other hand, we use that M (hs ws χQ )(x) for x ∈ 2i+1 Q\2i Q (i ≥ 1) is comparable to the average of hs ws χQ over 2i+1 Q. Together with (4.8) and H¨ older’s inequality we obtain (4.10) Z

p

q

p

|T (f1 )| w

≤ c1

Q

≤c2

≤c2

i=1

∞ Z X i=1

2i+1 Q\2i Q

!q

p

|f |p w

2i+1 Q\2i Q

|f1 |

!1 s

w(Q)1/˜p 1

1

|2i Q| s |Q| s0

1/˜p0 w(Q)1/˜p 0 w1−˜p 2i+1 Q 2in/s |Q|

∞

q X ≤c2 f | Lrp (w, Rn ) w 2i+1 Q i=1

s s Qh w |2i Q|

R q

2i+1 Q\2i Q

|f |q w1/˜p w−1/˜p

Z ∞ X i=1

∞ Z X

1 r +n p

q

1/˜p0 w(Q)1/˜p 0 w1−˜p 2i+1 Q . 2in/s |Q|

If we choose q such that w ∈ Ap˜, then 1/˜p 1−˜p0 i+1 1/˜p0 (4.11) w 2i+1 Q w 2 Q ≤ C|2i+1 Q|, and it is sufficient to show ∞ X 1 q rq 0 +r q w 2i+1 Q n w(Q) p 2in/s ≤ c w (Q) p n . (4.12) i=1

16


Taking into account (2.3), there exists δ > 0 such that r

r

w(2i+1 Q) n q ≤ c2iδrq w(Q) n q .

(4.13)

Therefore, (4.12) is satisfied for s sufficient small. By means of the (quasi-)subadditivity of T , (4.7), (4.10) and (4.12), finally our assertion (4.4) follows. One has to take care that T is well-defined on all appearing spaces in the proof (separating f in f0 and f1 ), which is ensured by (4.1), (3.8) and Proposition 3.1. In the case of the spaces Lrp (λ, w, Rn ), one deduces analogously to (4.7) that

1 +r k T (f0 )| Lp,w (Q)k ≤ c|Q| p n f | Lrp (λ, w, Rn ) . To work with T f1 , we first modify the last line of (4.10) and then use (4.11) to get Z q p p |T (f1 )| w Q ∞

X i+1 r n q

2 Q ≤c1 f | Lp (λ, w, R )

r 1 +n p

q

r 1 +n p

q

i=1 ∞

X i+1 r n q

2 Q ≤c1 f | Lp (λ, w, R )

1/˜p0 w(Q)1/˜p 0 w1−˜p 2i+1 Q 2in/s |Q| q 0 2in/s w(2i+1 Q)−1 w(Q) p .

i=1

By the latter we obtain (4.6) showing (4.14)

∞ X

2

r + s10 q ] inq[ p1 + n

w(2i+1 Q)−1 w(Q)

pq

≤ c.

i=1

By the fact that w ∈ RHσ and (3.6) we have 1 w(Q) −(i+1)n σ0 (4.15) ≤ c 2 . w(2i+1 Q) Inserting this into the left hand-side of (4.14) and taking into account that s 1 is as close to 1 as desired, the geometric series is convergent for nr + pσ < 0. n The endpoint r = − pσ of the statement is attained taking into account the openness property of the reverse Hölder classes (see comment after Definition 2.5). S Remark 4.2. That the operator is well-defined on w∈Ap Lp,w (Rn ) for some 1 < p < ∞ is for practical reasons not an additional condition (cf. 4.1). Usually one applies the theorem to T continuous noperators which are defined on an common dense subset of w∈Ap Lp,w (R ), for example singular integrals which are defined on D(Rn ). The singular integrals can be extended on each Lp,w (Rn ) by continuous extension with respect to (4.2). By the common dense subset D(Rn ) and the continuous extension it follows that T coincides in the intersection of some Muckenhoupt weighted


LebesgueSspaces, which implies the well-definedness of the singular integrals on 1 t}) ≤ t w x ∈ Q : |T f0 (x)| + |T f1 (x)| > c 1 1 p p t t ≤t w x ∈ Q : |T f0 (x)| > +t w x ∈ Q : |T f1 (x)| > . 2c 2c

Taking in account (4.21) we deduce 1 1 p p t t n tw x ∈ Q : |T f0 (x)| > ≤tw x ∈ R : |T f0 (x)| > 2c 2c

1 r + ≤c k f0 | Lp,w (Rn )k ≤ c2 w(Q) p n f | Lrp (w, Rn ) . By (4.21) we obtain further tw

t x ∈ Q : |T f1 (x)| > 2c

Z ≤t {x∈Rn : ≤c2 w(Q)

t |T f1 (x)|> 2c }

1 r +n p

1

!1

p

Z

p

=t

χQ w {x∈Rn : |T f1 (x)|> 2ct } !1 Z p 1 1 s s M (w χQ ) ≤ c1 |f1 |p M (ws χQ ) s Rn

f | Lrp (w, Rn ) ,

where the last inequality is the same as in the first part of the proof. This leads to (4.22). Analogously we get (4.23). Remark 4.4. Chiarenza and Frasca in [CF87] noticed that weighted A1 inequalities for singular integrals yields boundedness on unweighted Morrey spaces. With a different approach Adams and Xiao [AX12] revisited this method using weighted inequalities to obtain boundedness in the unweighted Morrey spaces. The weak-type assumption (4.21) in Theorem 4.3 is only interesting with respect to weights of A1 . Otherwise usual extrapolation techniques imply appropriate strong-type inequalities in the case of quasi-subadditive operators, which can be used with the next theorem to deduce the appropriate strong-type Morrey estimates. Indeed, let us assume that we have (4.21) for all weights w ∈ Ap/ν for fixed ν such that ν < p. Extrapolation implies that (4.21) holds then furthermore

20


for all p ∈ (ν, ∞) and all weights w ∈ Ap/ν (which can be seen using the proof of [CMP04, Cor. 3.10] and combining it with [CMP04, Cor. 3.14]). For an arbitrary q ∈ (ν, ∞) and an arbitrary w ∈ Aq/ν we find ε > 0 such that w ∈ A(q/ν)−ε . Taking in account that it holds also w ∈ A(q/ν)+ε we proceed as in Marcinkiewicz interpolation to obtain the strong boundedness of T in Lq,w (Rn ). Hereby, we mention that although it is usual to state Marcinkiewicz interpolation theorem for sublinear operators, it is clear that only quasi-subadditivity is needed. Theorem 4.5. Assume that for a (quasi-)subadditive operator [ T : Lp0 ,w (Rn ) → L(Rn ) w∈Ap0 /γ

for some γ and some p0 with 1 ≤ γ < p0 < ∞ and every w ∈ Ap0 /γ we have (4.27)

k T f | Lp0 ,w (Rn )k ≤ c1 k f | Lp0 ,w (Rn )k

for all f ∈ Lp0 ,w (Rn ).

Then for every γ < p < ∞, every − np ≤ r < 0 and every w ∈ Ap/γ we have that T is well-defined on Lrp (w, Rn ) by restriction by the fact that [ (4.28) Lrp (w, Rn ) ⊂ Lp0 ,w (Rn ) w∈Ap0 /γ

and, moreover, (4.29)

T f | Lrp (w, Rn ) ≤ c2 f | Lrp (w, Rn ) ,

for all f ∈ Lrp (w, Rn ). Furthermore, for every γ < p < ∞ and every w ∈ Ap/γ ∩ RHσ , if − np ≤ n r ≤ − pσ we have that T is well-defined on Lrp (λ, w, Rn ) by restriction by the fact that [ (4.30) Lrp (λ, w, Rn ) ⊂ Lp0 ,w (Rn ) w∈Ap0 /γ



for all f ∈ Lrp (λ, w, Rn ). The constants c1 , c2 and c3 in (4.27), (4.29) and (4.31) do not depend on f but may depend on w, p0 , γ, n, p and r. Proof. First we observe that (4.28) and (4.30) hold by (3.10) and Proposition 3.3 and therefore the well-definedness T on the appropriate Morrey spaces. By reason of extrapolation in the usual Lebesgue weighted spaces (cf. [CMP11, Cor. 3.14])) we obtain (4.27) with respect to the parameter p and the weight class Ap/γ (instead of p0 and Ap0 /γ ). The rest of the proof is almost the same as the proof of Theorem 4.1. For fixed Q, write f = f0 + f1 as before. For T f0 weproceed as usual. For T f1 we choose q such that 0

γ < q < p and 1 < s < pq and (4.8) holds because T is bounded on Lq (w) ˜ if w ˜ ∈ A1 . We continue as in the proof of Theorem 4.1.


Remark 4.6. We find in the applications several linear operators satisfying the assumptions of the previous theorem. In that case, the adjoint operator satisfies the dual estimates in the range 1 < p < γ 0 . Usually, the adjoint operator is of similar type and hence the original operator also satisfies these estimates via duality. The class of weights obtained for 1 < p < ∞ can be described as {w1−p : w ∈ Ap0 /γ }, which is the same as Ap ∩ RH γ 0 . γ 0 −p

In Theorem 4.7 we obtain partial results with respect to r in the Morrey scales. In the same situation we have sometimes the weak-type for p = 1 with weights in the class of {w : wγ ∈ A1 }. Moreover, in the range 1 < p < γ by interpolation with the results for p ≥ γ we also get the (strong) boundedness of the operator for weights in {w : wγ/p ∈ A1 }. The extension to Morrey spaces of these type of estimates are considered in Theorem 4.8. Theorem 4.7. Let 1 < p < γ 0 < ∞. Assume that for a (quasi-)subadditive operator [ T : Lp,w (Rn ) → L(Rn ) w∈Ap ∩RHγ 0 /(γ 0 −p)

and every w ∈ Ap ∩ RH (4.32)

γ0 γ 0 −p

we have

k T f | Lp,w (Rn )k ≤ c1 k f | Lp,w (Rn )k

for all f ∈ Lp,w (Rn ). 0

Let w ∈ Ap ∩ RHσ with σ 0 p ≤ γ 0 . Then for every − np ≤ r ≤ − nσ γ 0 we have that T is well-defined on Lrp (w, Rn ) by restriction by the fact that [ (4.33) Lrp (w, Rn ) ⊂ Lp,w (Rn ) w∈Ap ∩RHγ 0 /(γ 0 −p)

and, moreover,

T f | Lrp (w, Rn ) ≤ c2 f | Lrp (w, Rn ) (4.34)

for all f ∈ Lrp (w, Rn ).

Furthermore, for every − np ≤ r ≤ − np σ1 − γn0 we have that T is well-defined on Lrp (w, Rn ) by restriction by the fact that [ Lp,w (Rn ) (4.35) Lrp (λ, w, Rn ) ⊂ w∈Ap ∩RHγ 0 /(γ 0 −p)



for all f ∈ Lrp (λ, w, Rn ).

The constants c1 , c2 and c3 in (4.32), (4.34) and (4.36) do not depend on f but may depend on [w]Ap , γ, n, p and r. Proof. First we observe that (4.33) and (4.35) hold by Proposition 3.9 (with b = γ 0 ) and therefore T is well-defined on the appropriate Morrey spaces. As in the previous proofs, given f ∈ Lrp (w, Rn ) and a cube Q, we decompose f = f0 + f1 and bound k T (f0 )| Lp,w (Q)k as in (4.7).

22


Choose ε > 0 such that w ∈ RH 0

γ0 +ε γ 0 −p

. Let q > 1 be such that w ∈ Ap/q

and set p˜ = p/q. Let s = γ 0γ−q + δ for some δ > 0. Thus, Z 1 Z p ˜ p˜q |T (f1 )|q hwχQ = |T (f1 )| w n R Q Z Z 1 1 q s s s ≤ |T (f1 )| M (h w χQ ) ≤ c |f1 |q M (hs ws χQ ) s , Rn

where the equality holds with h ∈ Lp˜0 ,w equality because M (hs ws χQ ) v

γ0 γ 0 −q

∈ A1 ⊂ A

γ0 (q−1)+1 γ 0 −q

1 γ0 s γ 0 −q

Rn n (R ) with

norm 1 and the last in-

∈ A1 and T is bounded on Lq (v) for

. This boundedness holds by limited range ex-

trapolation (cf. [CMP11, Thm. 3.31]) taking into account (4.32). The weight 1

γ0

M (hs ws χQ ) s γ 0 −q is well defined because hs ws χQ ∈ L1 (Rn ) and as in (4.9) we get Z 1 Z 1 − 10 s(p ˜0 −1) s s p ˜ 1 s s−1 0 h w w ≤ w p˜ −s ≤ c w(Q)1/˜p |Q|− s0 . Q

Q

In the last step we need to be able to use the reverse Hölder inequality for w, that is, w ∈ RH s(p˜0 −1) . For this to hold we choose q near 1 and δ near 0 p ˜0 −s

such that

s(˜ p0 − 1) γ0 ≤ + ε. p˜0 − s γ0 − p Such a choice is possible because γ 0 + δ(γ 0 − q) s(˜ p0 − 1) = . p˜0 − s γ 0 − p + δ(γ 0 − q)(˜ p0 − 1)

The proof continues as in Theorem 4.1 up to inequality (4.12) so that we need the inequality ∞ X r 1 q nr q +n q i+1 in/s0 p p (4.37) w 2 Q w(Q) 2 ≤ c w (Q) . i=1

Here we use the fact that w ∈ RHσ together with (2.3) to get r

q

rq

r

w(2i+1 Q) n ≤ c2i( σ0 ) w(Q) n q . Therefore, the geometric series at the left side of (4.37) converges for r < 0 0 − nq σs0 . For a given r < − nσ γ 0 (without loss of generality by w ∈ RHσ+τ for some τ > 0), we assume that we haven chosen s close to γ (by fixing q close 0 enough to 1 and δ to 0) so that still r < − nq σs0 is fulfilled. By means of the (quasi-)subadditivity of T , (4.7), (4.10) and (4.12), finally our assertion (4.4) follows. One has to take care that T is well-defined on all appearing spaces in the proof (separating f in f0 and f1 ), which is ensured by (4.1), (3.8) and Proposition 3.1.


In the case of the spaces Lrp (λ, w, Rn ), we follow again the proof of Theorem 4.1 and we are left with (4.14). We use (4.15) and take into account that s > γ (close enough to γ) and that σ can be replaced by σ + τ to get the result. Theorem 4.8. Let 1 ≤ p < ∞ and ν > 1. Assume that for a (quasi-)subadditive operator [ Lp,w (Rn ) → L(Rn ) T : {w:wν ∈A1 }

and every w with

wν

∈ A1 we have

k T f | Lp,w (Rn )k ≤ c1 k f | Lp,w (Rn )k

(4.38)

for all f ∈ Lp,w (Rn ). Assume also that w ∈ RHσ . Then for every − np ≤ 0

r ≤ − np σν 0 we have that T is well-defined on Lrp (w, Rn ) by restriction by the fact that [ (4.39) Lrp (w, Rn ) ⊂ Lp,w (Rn ) {w:wν ∈A1 }

and, moreover,

T f | Lrp (w, Rn ) ≤ c2 f | Lrp (w, Rn ) for all f ∈ Lrp (w, Rn ). Furthermore, for every − np ≤ r ≤ − np σ1 + ν10 we have that T is welldefined on Lrp (w, Rn ) by restriction by the fact that [ (4.40) Lrp (λ, w, Rn ) ⊂ Lp,w (Rn ) {w:wν ∈A1 }

and, moreover,

T f | Lrp (λ, w, Rn ) ≤ c3 f | Lrp (λ, w, Rn ) for all f ∈ Lrp (λ, w, Rn ). If we replace the strong-type assumption (4.38) by the weak-type estimate k T f | W Lp,w (Rn )k ≤ c1 k f | Lp,w (Rn )k , similar results hold for weak-type Morrey spaces:

T f | W Lrp (w, Rn ) ≤ c2 f | Lrp (w, Rn ) and

T f | W Lrp (λ, w, Rn ) ≤ c3 f | Lrp (λ, w, Rn ) . Proof. We observe that (4.39) and (4.40) hold by Proposition 3.7 and therefore the well-definedness T on the appropriate Morrey spaces follows. The proof of this theorem is the same as that of Theorem 4.3 except for ν the fact that we need to choose s > ν to guarantee that M (ws χQ ) s is in A1 .

24


In the case f ∈ Lrp (w, Rn ), according to (4.25) we need (4.41)

∞ X

r

p

n

w(2i+1 Q) n w(Q)2i s0 < +∞.

i=1

If w ∈ RHσ , we use the estimate (4.15) and (4.41) is satisfied if rp n + 0 < 0. 0 σ s Since s is as close to ν as desired and σ can be replaced by σ + , the condition of the statement follows. For f ∈ Lrp (λ, w, Rn ), inserting (4.15) in (4.26) we get the condition 1 rp 1 1 rp 1 − − 0 = + 0 + < 0, n s σ n s σ in which again the endpoint is also reached by the same argument as before. 2+

Remark 4.9. In [CMP11] the authors got very general results with respect to extrapolation in Banach function spaces. Comparing with these results one has to mention that Morrey spaces are neither (quasi-)Banach function spaces, which is an assumption of their abstract result, nor they got results with respect to limited range extrapolation in this abstract setting (cf. Theorems (4.7) and (4.8)). They use a kind of duality (associated spaces) which is available for Morrey spaces of type Lrp (w, Rn ) (cf. [RS16]) but we completely avoid it. However, their results deliver weighted inequalities which means that for applications to operators one still has to deal with the welldefinedness of the operators (which we have done via embeddings). 5. Applications to mapping properties of operators There is a plentiful list of operators fulfilling the requirements of one or more of the theorems of the previous section. As we mentioned in the introduction several of those operators have been proved to be bounded in Morrey spaces (unweighted or weighted) using in general the particular form of the operator or some particular pointwise bound. In most cases only the size estimate with the Morrey norm has been proved, without any discussion about the possible definition of the operator in the corresponding Morrey space. All of this is avoided with our approach. In what follows, we present a collection of applications of our general results. 5.1. Calder´ on-Zygmund operators and their maximal truncations. It is by now a classical result that Calderón-Zygmund operators are bounded on Lp,w (Rn ) for w ∈ Ap (1 < p < ∞), and that they are of weak-type (1, 1) with respect to A1 weights. Here we can understand the term CalderónZygmund operator in the general sense, that is, when the operator is represented by a two-variable kernel K(x, y) with appropriate size and regularity estimates. The maximal operator associated to the Calderón-Zygmund operators taking the supremum of the truncated integrals also satisfies similar


weighted estimates. For a proof the reader can consult [Gra09, Chapter 9]. Then we can apply Theorems 4.1 for 1 < p < ∞ and 4.3 for p = 1. The regularity assumptions on K(x, y) can be weakened and stated in some integral form called Lr -Hörmander condition. In the convolution case it appears implicitely in [KW79], and in a more general setting in [RRT86], for instance. In that case the weighted inequalities hold for Lp,w (Rn ) for p ≥ γ and w ∈ Ap/γ , where γ = r0 . Then Theorem 4.5 applies. If the estimates are symmetric in the variables x and y of the kernel, duality can be used for the weighted estimates and Theorems 4.3, 4.7 or 4.8 apply. 5.2. Multipliers. Kurtz and Wheeden studied in [KW79] weighted inequalities for classes of multipliers defined as follows (see also [ST89]). Let Tcf (ξ) = m(ξ)fˆ(ξ). We say that m ∈ M (s, l) if !1/s Z sup Rs|α|−n

R>0

|Dα m(ξ)|s

< +∞ for all |α| ≤ l.

R 1. The origin of Morrey spaces is in the study of the smoothness properties of solutions of PDEs. For this reason results on the multipliers M (s, l) related to their smoothness are interesting (cf. [Tri13, Tri14, Ros12]). Another interesting case of multipliers is that of Marcinkiewicz multipliers in one dimension. They are bounded on Lp,w (R) for w ∈ Ap (see [Kur80]) and Theorem 4.1 gives all the Morrey boundedness results for 1 < p < ∞. 5.3. Rough operators. The rough singular integrals are defined in general by Z Ω(y 0 )h(|y|) f (x − y)dy TΩ,h f (x) = p.v. |y|n Rn with Ω ∈ L1 (S n−1 ) (y 0 = y|y|−1 ) and integral zero, and h defined on (0, ∞). The (weighted) boundedness of the operator is proved using additional assumptions on Ω and h. For instance, if both Ω and h are in L∞ , it was proved in [DR86] that TΩ,h is bounded on Lp,w (Rn ) for w ∈ Ap (1 < p < ∞), so that the full range is obtained for Morrey spaces from Theorem 4.1. If for fixed q > 1 we know that Ω ∈ Lq (S n−1 ) and h ∈ Lq ((0, ∞); dt/t) then the boundedness on Lp,w (Rn ) holds for w ∈ Ap/q0 (q 0 ≤ p < ∞) and also for some classes of weights obtained by duality for the range 1 < p ≤ q. These classes can further be extended by interpolation. Results of this type are in [Wat90] and [Duo93]. Depending on the situation we can apply Theorems 4.5, 4.3, 4.7 or 4.8. Due to the openness properties of Ap weights, D. Watson noticed that for Ω ∈ ∩q