Two-Weight Inequalities for Commutators with Fractional Integral ...

TWO-WEIGHT INEQUALITIES FOR COMMUTATORS WITH FRACTIONAL INTEGRAL OPERATORS

arXiv:1510.05331v1 [math.CA] 19 Oct 2015

IRINA HOLMES, ROBERT RAHM, AND SCOTT SPENCER Abstract. In this paper we investigate weighted norm inequalities for the commutator of a fractional integral operator and multiplication by a function. In particular, we show that, for µ, λ ∈ Ap,q and α/n + 1/q = 1/p, the norm k[b, Iα ] : Lp (µp ) → Lq (λq )k is equivalent to the norm of b in the weighted BMO space BM O(ν), where ν = µλ−1 . This work extends some of the results on this topic existing in the literature, and continues a line of investigation which was initiated by Bloom in 1985 and was recently developed further by the first author, Lacey, and Wick.

Contents 1. Introduction and Statement of Main Results 2. Background and Notation 3. Averaging Over Dyadic Fractional Integral Operators 4. Upper Bound 5. Lower Bound References

1 3 5 7 9 10

1. Introduction and Statement of Main Results Recall the classical fractional integral operator, or Riesz potential, on Rn : let 0 < α < n be fixed and, for a Schwartz function f define the fractional integral operator (or Riesz potential) Iα by Z f (y) Iα f (x) := dy. n−α Rn |x − y| These operators have been studied since 1949, when they were introduced by Marcel Riesz, and have since found many applications in analysis – such as Sobolev embedding theorems and PDEs. Also recall the Calderón-Zygmund operators: Z T f (x) := K(x, y)f (y) dy, x ∈ / suppf, Rn

where the kernel satisfies the standard size and smoothness estimates: C , |K(x, y)| ≤ |x − y|n |K(x + h, y) − K(x, y)| + |K(x, y + h) − K(x, y)| ≤ C

|h|δ

|x − y|n+δ

,

2010 Mathematics Subject Classification. Primary: 42A05, 42A50, 42B20 Secondary: 42A61, 42B25. Key words and phrases. Fractional Integral Operator, Commutator, Weighted Inequalities, Bloom BMO. 1

2

IRINA HOLMES, ROBERT RAHM, AND SCOTT SPENCER

for all |x − y| > 2 |h| > 0 and a fixed δ ∈ (0, 1]. To contrast the two, note for example that fractional integral operators are positive, which in many cases makes them easier to work with (as one example of this, it is almost trivial to dominate the fractional integral operators by sparse operators, though this isn’t important to us in the present setting). On the other hand, the fractional integral operators do not commute with dilations and therefore can never boundedly map Lp (dx) to itself. Additionally, the kernel of the fractional integral operator does not satisfy the standard estimates above. Therefore, the theory of fractional integral operators is not just a subset of the theory of Calderón–Zygmund operators. Because of this, results which are known for Calderón-Zygmund operators also need to be proved for the fractional integral operators. In this paper we will characterize the triples (b, µ, λ), where b is a function and µ and λ are Ap,q weights (to be defined shortly), such that the commutator [b, Iα ] is bounded from Lp (µp ) to Lq (λq ). Commutators with Riesz potentials were first studied in [3]. Our characterization will be in terms of the norm of b in a certain weighted BMO space, built from the weights µ and λ. This is an adaptation to the fractional integral setting of a viewpoint introduced by Bloom [1] in 1985, and recently investigated by the first author, Lacey and Wick in [12,13]. Specifically, Bloom characterized k[b, H] : Lp (µ) → Lp (λ)k, where H is the Hilbert transform and µ, λ are Ap weights, in terms of kbkBM O(ν) , where BMO(ν) is the weighted BMO space associated with the weight ν ··= µ1/p λ−1/p . Recall that the Hilbert transform is the one-dimensional prototype for Calderón-Zygmund operators, a role played by the Riesz transforms in Rn . A modern dyadic proof of Bloom’s result was recently given in [12], and the techniques developed were then used to extend the result to all Calderón-Zygmund operators in [13]. In particular, it was proved that (1.1)

k[b, T ] : Lp (µ) → Lp (λ)k ≤ ckbkBM O(ν) ,

for all Ap weights µ, λ, and all Calderón-Zygmund operators T on Rn , for some constant c depending on n, T , µ, λ and p. Specializing to the Riesz transforms, a lower bound was also proved. The center of the proof of (1.1) was the Hytönen Representation Theorem, which allows one to recover T from averaging over some dyadic operators, called dyadic shifts. Then the upper bound reduced to these dyadic operators. We take a similar approach in this paper, where the role of the dyadic shifts will be played by the dyadic version of the fractional integral operator Iα , given by: X (1.2) IαD f := |Q|α/n hf iQ 11Q . Q∈D

Our main result is:

Theorem 1.1. Suppose that α/n + 1/q = 1/p and µ, λ ∈ Ap,q . Let ν ··= µλ−1 . Then: k[b, Iα ] : Lp (µp ) → Lq (λq )k ≃ kbkBMO(ν) .

It is important to observe that we require that each weight belong to a certain Ap,q class and this will imply that µλ−1 is an A2 weight and in particular, an A∞ weight. Standard properties of these weight classes will be used throughout the paper, with out tracking dependencies on the particular weight characteristics. The liberal use of these properties indicates the subtleties involved in the general two–weight setting. For an excellent account of this and other topics related to fractional integral operators, see [5].

COMMUTATORS WITH FRACTIONAL INTEGRAL OPERATORS

3

The paper is organized as follows. In Section 2, we will give the requisite background material and definitions. Note, however, that most of the material not relating strictly to fractional integral operators (such as the Haar system, Ap weights, and weighted BMO) is standard and was also needed in [13] where it is discussed in more detail. In Section 3 we will briefly discuss how the fractional integral operator can be recovered as an average of dyadic operators. In Section 4 we will prove k[b, Iα ] : Lp (µp ) → Lq (λq )k . kbkBMO(ν) and in Section 5, we will prove the reverse inequality: kbkBMO(ν) . k[b, Iα ] : Lp (µp ) → Lq (λq )k. 2. Background and Notation 2.1. The Haar System. Let D be a dyadic grid on Rn and let Q ∈ D. For every ǫ ∈ {0, 1}n , let hǫQ be the usual Haar function defined R 1 on Q. For convenience, we write R ǫǫ = 1 if ǫ = (1, 1, . . . , 1). Note that, in this case, hQ = 1. Otherwise, if ǫ 6= 1, then hQ = 0. Moreover, recall that {hǫQ }Q∈D,ǫ6=1 forms an orthonormal basis for L2 (Rn ). For a function f , a cube Q ∈ D and ǫ 6= 1, we denote fb(Q, ǫ) ··= hf, hǫQ i,

where h·, ·i is the usual inner product in L2 (Rn ).

2.2. Ap Classes and Weighted BMO. Let w be a weight on Rn , that is, a locally integrable, almost everywhere positive function. For a subset Q ⊂ Rn we denote Z w(Q) w dx and hwiQ ··= w(Q) ··= . |Q| Q Given 1 < p < ∞, a weight w is said to belong to the Muckenhoupt Ap class provided that: ′

[w]Ap ··= suphwiQ hw 1−p ip−1 < ∞, Q Q

where p′ denotes the Hölder conjugate of p, and the supremum is over all cubes Q ⊂ Rn . ′ ′ p′ −1 Moreover, w ∈ Ap if and only if w 1−p ∈ Ap′ and, in this case, [w 1−p ]Ap′ = [w]A . Furtherp more, if 1 < p < q < ∞, then Ap ⊂ Aq , with [w]Aq ≤ [w]Ap for all w ∈ Ap . For a dyadic lattice D, recall the dyadic square function: 2 11 X Q (SD f )2 = . fb(Q, ǫ) |Q| P ∈D,ǫ6=1 Another property of Ap weights which will be useful for us is the following well–known weighted Littlewood–Paley Theorem: Theorem 2.1. Let w ∈ Ap , then:

kSD : Lp (w) → Lp (w)k ≃ c(n, p, [w]Ap ).

For a weight w on Rn , the weighted BMO space BMO(w) is defined to be the space of all locally integrable functions b that satisfy: Z 1 · |b − hbiQ | dx < ∞, (2.1) kbkBMO(w) ·= sup Q w(Q) Q

4


where the supremum is over all cubes Q in Rn . For a general weight, the definition of the BMO norm is highly dependent on its L1 average. But, if the weight is A∞ , one is free to replace the L1 -norm by larger averages. Namely, defining q1 Z 1 |b − hbiQ |q dw ′ , (2.2) kbkBMOq (w) ··= sup w(Q) Q Q

there holds (2.3)

kbkBMO(w) ≤ kbkBMOq (w) ≤ C(n, p, [w]A∞ )kbkBMO(w) .

The proof is similar to the proof in the unweighted case. In particular, the first inequality is a straightforward application of Hölder’s inequality and the second inequality follows from a suitable John–Nirenberg property (which requires a suitable Calderón–Zygmund decomposition). The details are in [21]. For a dyadic grid D on Rn , we define the dyadic versions of the norms above by taking supremum over Q ∈ D instead of over all cubes Q in Rn , and denote these spaces by BMOD (w) and BMOqD (w). Clearly BMO(w) ⊂ BMOD (w) for any choice of D, and the equivalence in (2.3) also holds for the dyadic versions of these spaces. A fact which will be crucial to our proof is the following: Lemma 2.2. If w ∈ A2 , there holds

(2.4)

|hb, Φi| . kbkBM OD2 (w) kSD ΦkL1 (w) .

This comes from a duality relationship between dyadic weighted BMO spaces and dyadic weighted Hardy spaces. For a more detailed discussion and a proof of this fact, see Section 2.6 of [13]. We remark here that Lemma 2.2 was also fundamental for the proof of the upper bound (1.1) in [13], essentially for the following reason: if µ, λ are Ap weights, then ν := µ1/p λ−1/p is an A2 weight. Thus the duality statement above applied to ν eventually ′ yields, through Hölder’s inequality, some bounds in terms of Lp (µ) and Lp (λ) norms. This is also the strategy we will adapt accordingly to the fractional integral case, which makes use of Ap,q classes instead. We discuss these next. 2.3. Ap,q Classes. Throughout this section, α, n, p, q are fixed and satisfy 1/p − 1/q = α/n. We recall first the fractional maximal operator, Mα f ··= sup |Q|α/n h|f |iQ 11Q , Q

with the supremum being over all cubes Q. This was first introduced in [20], where it was used to prove weighted inequalities for Iα , a result analogous to the classic result [4] of Coifman and Fefferman, relating the Hardy-Littlewood maximal operator and singular integrals. We will be working with the dyadic version of this operator, MαD , defined for a dyadic grid D just as above, but only taking supremum over Q ∈ D. Also in [20] was introduced a generalization of Ap classes for the fractional integral setting: we say that a weight w belongs to the Ap,q class provided that ′ q/

′ [w]Ap,q ··= suphw q iQ hw −p iQp < ∞.

Q

See [5–7, 25, 26] for other generalizations. We will use the following important result concerning Ap,q weights due to, for example, Sawyer and Muckenhoupt and Wheeden [20, 27, 28]:


5

Theorem 2.3. Let w be a weight. Then the following are equivalent: (i) w ∈ Ap,q ;

(ii) MαD : Lp (w p ) → Lq (w q ) ≃ C(n, α, p, [w]Ap,q );

(iii) IαD : Lp (w p ) → Lq (w q ) ≃ C(n, α, p, [w]Ap,q ).

We now make two observations about Ap,q weights which will be particularly useful to us. First, we note that: (2.5)

′

′

If w ∈ Ap,q , then: w p ∈ Ap , w −p ∈ Ap′ , w q ∈ Aq , and w −q ∈ Aq′ ,

where all weights above have Muckenhoupt characteristics bounded by powers of [w]Ap,q . To see that w p ∈ Ap , first notice w ∈ Ap,q if and only if w q ∈ Aq0 , with [w q ]Aq0 = [w]Ap,q , where

q0 ··= 1 + q/p′ = q(1 − α/n). Since the Ap classes are increasing and q0 < q, we have that w q ∈ Aq . In turn, this gives ′ ′ that w −q = (w q )1−q ∈ Aq′ . The other two statements in (2.5) follow in a similar fashion from the fact that w ∈ Ap,q if and only if w −1 ∈ Aq′ ,p′ . Second, suppose that µ, λ ∈ Ap,q and let ν ··= µλ−1 . Since µp , λp ∈ Ap , Hölder’s inequality implies ν ∈ A2 (with [ν]pA2 ≤ [µp ]Ap [λp ]Ap ), a fact which will be used in proving the upper bound. Moreover, we claim that for any cube Q: ′

′

µp (Q)1/p λ−q (Q)1/q . ν(Q)|Q|α/n ,

(2.6)

a fact which will be useful in proving the lower bound. To see this, note first that 1/p

1/p′

′

hµp iQ hµ−p iQ

′

1/q ′

1/q

. 1 and hλ−q iQ hλq iQ . 1,

which simply come from µp ∈ Ap and λq ∈ Aq . Since p′ > q ′ , Hölder implies 1/q′ Z q′ /p′ Z 1−q′ /p′ !1/q′ Z 1 1 ′ ′ ≤ µ−q dx µ−p dx dx |Q| Q |Q| Q Q 1/p′ Z 1 −p′ = , µ dx |Q| Q ′

1/q ′

and hence hµ−q iQ 1/p

′

1/p′

≤ hµ−p iQ . Combining these estimates gives: ′

1/q ′

hµp iQ hλ−q iQ

.

1

hµ

−p′

1/p′ iQ

1

1/q hλq iQ

.

1

hµ

−q ′

1/q 1/q ′ iQ hλq iQ

≤

1 hν −1 i

Q

≤ hνiQ .

The last two inequalities are more application of Hölder’s inequality and the fact that ν −1 = µ−1 λ. This proves (2.6). 3. Averaging Over Dyadic Fractional Integral Operators In this section, we show that Iα can be recovered from (1.2) by averaging over dyadic lattices. The proof here is modified (and abridged) from the proof in [24], but it is possible to modify any of the proofs in, for example, [14, 18, 23]. For the sake of clarity, we only give the proof for the one–dimensional case. Given an interval [a, b) (it is not too important that the interval be closed on the left and open on the right) of length r, we can create a dyadic lattice, Da,r in a standard way. In

6


particular, Da,r is the dyadic lattice on R with intervals of length r2−k , k ∈ Z, and the point a is not in the interior of any of the intervals in Da,r . For example, D0,1 is the standard k dyadic lattice on R. For a given lattice Da,r , we let Da,r denote the intervals in Da,r with −k length r2 . In this section we slightly abuse notation and let h1I = |I|−1/2 11I . Define: X

P0(a,r) f (x) :=

0 I∈Da,r

|I|α f, h1I h1I (x).

With r and x fixed, we can parameterize the dyadic grids by the set (−r, 0] and we can give this set the probability measure da/r. For a fixed x ∈ R, we want to compute: E(P0(a,r) f (x))

=

Z

0

−r

P0(a,r) f (x)

da . r

Let τt f (x) := f (x + t) be the translation operator and note that Pa−t τt = τt Pa . From this it easily follows that EP0(a,r) τt = τt P0(a,r) . That is, EP0(a,r) is given by convolution. Let: EP0(a,r) f (x) = F0,r ∗ f (x). We want to compute F0,r . First, note that P0a,r is convolution with the function Therefore, we have: F0,r ∗ f (x) = EP0(a,r) f (x) =

EP0(a/2,r) f (x)

Z

x+r/2

x−r/2

Z

rα 11 . r [−r/2,r/2]

dt rα f (s) 11−r/2,r/2 (t − s)ds . r r R

Using Fubini, we see that: F0,r (x) =

Z

x+r/2

x−r/2

x r α rα dt rα 11[−r/2,r/2] (t) = 11[−r/2,r/2] (x) 1 − = F0,1 (x/r). r r r r r

Now, fix an r ∈ [1, 2) and define:

Fr =

X

F0,2n r .

n∈Z

k The grids Da,r , k ∈ Z can be unioned to form a dyadic lattice (here a is fixed). Call r the calibre of the dyadic lattice. Convolution with Fr is averaging over all the dyadic lattices Da,r with fixed calibre r. That is:

Fr ∗ f = EPDa,r f.


7

R2 . Now, we want to Finally, we need to average over r ∈ [1, 2). Set F (x) := 1 Fr (x) dr r compute F (x). There holds: Z 2 dr F (x) = Fr (x) r 1 Z 2X dr = F0,2n r (x) r 1 n∈Z Z ∞ dρ = F0,ρ (x) ρ Z0 ∞ x ρα = F0,1 ( ) 2 dρ ρ ρ α Z0 ∞ x ρ x = 11−1/2,1/2 ( )(1 − ) 2 dr. ρ ρ ρ 0

Now, if x > 0, making the change of variable t = x/ρ, we see: Z dy 1 xα ∞ F0,1 (y) α = cα 1−α . F (x) = x 0 y x

1 Doing a similar computation for when x < 0, we see that F (x) = cα |x|1−α .

4. Upper Bound The decomposition in Section 3 means that the upper bound in Theorem 1.1 follows from the following, where the implied constants are independent of the dyadic lattice: Lemma 4.1. Suppose that α/n + 1/q = 1/p and µ, λ ∈ Ap,q . Let ν ··= µλ−1 . Then:

[b, IαD ] : Lp (µp ) → Lq (λq ) . kbk BMO(ν) .

Proof. We show that [b, IαD ] can be decomposed as the sum of four operators which will be fairly easy to bound. First note that for ǫ 6= 1, there holds: X X α/n ǫ α/n D ǫ |P | hQ (P )11P = Iα hQ = |P | 11P hǫQ = cα |Q|α/n hǫQ . P ∈D:P (Q

P ∈D:P (Q

Similarly,

IαD 11Q = (1 + cα ) |Q|α/n 11Q + |Q|

X

R∈D:Q(R

|R|α/n

11R . |R|

Using these computations: ( α , if P 6= Q or if P = Q and ǫ 6= η; cα |P ∩ Q| n hǫP hηQ D ǫ η P α 11 α 11Q Iα (hP hQ ) = R n n (1 + cα )|Q| |Q| + R)Q |R| |R| , if P = Q and ǫ = η. Thus:

 α α η ǫ , if P ( Q; |Q| n − |P | n  cα hQ (P )hP P α 11Q α 11 ǫ D η R [hP , Iα ]hQ = −|Q| n |Q| − R)Q |R| n |R| , if P = Q and ǫ = η;  0 , if Q ( P , or if Q = P and ǫ 6= η.

8


Expressing b and f in terms of their Haar coefficients, we obtain that X X bb(P, ǫ)fb(Q, η)[hǫ , I D ]hη . [b, IαD ]f = P α Q P,Q∈D ǫ,η6=1

Using this, there holds

(0,1,0)

[b, IαD ]f = cα T1 f − cα Πb,α

(4.1) where:

(0,1,0)

Πb,α

(0,0,1)

Πb,α

f ··= f ··=

T1 f ··= T2 f ··=

X

Q∈D,ǫ6=1

X

(0,0,1)

f − Πb,α

f − T2 f,

bb(Q, ǫ)hf iQ |Q| αn hǫ ; Q

bb(Q, ǫ)fb(Q, ǫ)|Q| αn 11Q ; |Q| Q∈D,ǫ6=1 X

P ∈D,ǫ6=1

X

P ∈D,ǫ6=1

bb(P, ǫ)

X

Q)P,η6=1

bb(P, ǫ)fb(P, ǫ)

fb(Q, η)hηQ (P )|Q|

X

Q)P

α

|Q| n

11Q |Q|

!

α n

!

hǫP ;

.

We will show that all of these operators are bounded Lp (µp ) → Lq (λq ). Below, all implied constants are allowed to depend on n, α, p, [µ]Ap,q , and [λ]Ap,q . Also all inner products below are taken with respect to dx and therefore it is enough to show: |hT f, gi| . kbkBMO(ν) kf kLp (µp ) kgkLq′ (λ−q′ ) ,

for each of the four operators above (this is because the dual of Lq (λq ) with respect to ′ ′ the unweighted inner product is Lq (λ−q )). The idea, which is taken from [12, 13], is to write the bilinear form, hT f, gi as hb, Φi and then show that kSD ΦkL1 (ν) is controlled by kf kLp (µp ) kgkLq′ (λ−q′ ) ; by the weighted H 1 − BMO duality, this is enough to prove the claim. The estimates for the two paraproducts are almost identical, and we only give the proof (0,1,0) for Πb,α . First with X α g (Q, ǫ)hǫQ , hf iQ |Q| n b Φ ··= Q∈D,ǫ6=1

there holds:

D

Then: (SD Φ)2 = Therefore,

X

Q∈D,ǫ6=1

(0,1,0) Πb,α f, g

2α

E

= hb, Φi .

|hf iQ|2 |Q| n |b g (Q, ǫ)|2

11Q ≤ (Mα f )2 (SD g)2 . |Q|

kSD ΦkL1 (ν) ≤ kMα f kLq (µq ) kSD gkLq′ (λ−q′ ) . kf kLp (µp ) kgkLq′ (λ−q′ ) ,

where the last inequality follows from Theorem 2.3 for the fractional maximal function, and ′ from Theorem 2.1 and the fact that λ−q ∈ Aq′ for the dyadic square function. The proof for (0,0,1) Πb,α is very similar, and we omit the details.


9

Now let us look at T1 . As above, we have hT1 f, gi = hb, Φi, with ! X X α fb(Q, η)hηQ (P )|Q| n hǫP , Φ := g (P, ǫ) b P ∈D,ǫ6=1

Q)P,η6=1

Then:

(SD Φ)2 ≤

X

P ∈D,ǫ6=1

|b g (P, ǫ)|2

≤ (IαD |f |)2(SD g)2 .

X

h|f |iQ|Q|

Q)P,η6=1

α n

!2

11P |P |

From Theorem 2.3 and Theorem 2.1, it follows that kSD ΦkL1 (ν) ≤ kIαD |f |kLq (µq ) kSD gkLq′ (λ−q′ ) . kf kLp (µp ) kgkLq′ (λ−q′ ) .

The estimates for T2 are similar and we omit the details.

5. Lower Bound In this section, we prove the lower bound in Theorem 1.1, which follows immediately from the Lemma below. In particular, we will show the following: Lemma 5.1. For all cubes, Q: Z 1 |b(x) − hbiQ | dx . k[b, Iα ] : Lp (µp ) → Lq (λq )k . ν(Q) Q Proof. The proof here follows along the lines of the proof in [2]. We first make some reductions. As with unweighted BMO, we can replace the hbiQ with any constant. Indeed, there holds: Z Z 1 |Q| 1 |b(x) − hbiQ | dx ≤ |b(x) − CQ | dx + |CQ − hbiQ | ν(Q) Q ν(Q) Q ν(Q) Z 2 ≤ |b(x) − CQ | dx. ν(Q) Q

Second, let P be the cube with l(P ) = 4l(Q), where l(Q) is the side length of Q, and with the same “bottom left corner” as Q. By the doubling property of A∞ weights, there holds ν(P ) ≃ ν(Q), and therefore it is enough to prove: Z 1 |b(x) − CQ | dx . k[b, Iα ] : Lp (µp ) → Lq (λq )k . ν(P ) Q Finally, let PR be the “upper right half” of P . Below, we will use CQ = hbiPR . Now, for x ∈ Q and y ∈ PR there holds: √ √ |x − y| 1 1 n |Q|1/n n |P |1/n |x − y| ≥ = ≤ ≤ . and √ √ √ √ 1/n 1/n 1/n 1/n 8 2 2 n |P | 2 n |P | 2 n |P | 2 n |P |

The point is that there is a function, K(x), that is smooth on [−1, 1]n , has a smooth periodic extension to Rn , and is equal to |x|n−α for 1/8 ≤ |x| ≤ 1/2. Therefore, for x ∈ Q and y ∈ PR there holds: x−y |x − y| √ √ . =K 2 n |J| 2 n |J|

10


Important for us is the fact that K has a Fourier expansion with summable coefficients. We are now ready to prove the main estimate. First, let σ(x) = sgn(b(x) − hbiPR ). Then: Z Z Z 1 (b(x) − b(y))σ(x)11Q(x)11PR (y)dydx |b(x) − hbiPR | dx = |PR | R R Q n−α Z Z |x − y| b(x) − b(y) 1 √ = σ(x)11Q (x)11PR (y)dydx n−α |PR | R R |x−y| 2 n |P | √ 2 n|P |

x−y b(x) − b(y) √ σ(x)11Q (x)11PR (y)dydx. ≃ |P | n−α K 2 n |P | R R |x − y| Observe that the integral above is positive, so the “≃” is not a problem. Expanding K in its Fourier series: X √ √ x−y √ = ak eikx/2 n|P | e−iky/2 n|P | , K 2 n |P | k −α/n

Z Z

and inserting this into the integral, we continue: X Z Z b(x) − b(y) X Z −α −α ikx/c|P | −iky/c|P | |P | ak e dydx = |P | ak hk (x)[b, Iα ]fk (x)dx, n−α σ(x)e |x − y| Q P R R k k where hk (x) = σ(x)eikx/c|P | 11P (x) and fk (y) = e−iky/c|P | 11PR (y). We control the integral by: Z hk (x)[b, Iα ]fk (x)dx ≤ k[b, Iα ] : Lp (µp ) → Lq (λq )k kfk kLp (µp ) khk kLq′ (λ−q′ ) R

′

= k[b, Iα ] : Lp (µp ) → Lq (λq )k µp (PR )1/p λ−q (P )1/q ′

By (2.6), this is dominated by: This completes the proof.

′

′

= k[b, Iα ] : Lp (µp ) → Lq (λq )k µp (P )1/p λ−q (P )1/q .

k[b, Iα ] : Lp (µp ) → Lq (λq )k |P |α ν(P ).

Acknowledgements. All three authors would like to thank Michael Lacey and Brett Wick for suggesting the problem, and for many useful conversations on this topic. The second author completed most of his portion of the work while still a student at Georgia Tech. References [1] Steven Bloom, A commutator theorem and weighted BMO, Trans. Amer. Math. Soc. 292 (1985), no. 1, 103–122. ↑2 [2] Lucas Chaffee, Characterizations of BMO through commutators of bilinear singular integral operators (2014), available at http://arxiv.org/pdf/1410.4587v3.pdf. ↑9 [3] S. Chanillo, A note on commutators, Indiana Univ. Math. J. 31 (1982), no. 1, 7–16. ↑2 [4] Coifman and Fefferman, Weighted norm inequalities for maximal functions and singular integrals (1974). ↑4 [5] David Cruz-Uribe, Two weight norm inequalities for fractional integral operators and commutators (2015), available at http://arxiv.org/abs/1412.4157. ↑2, 4 [6] David Cruz-Uribe and Kabe Moen, A fractional Muckenhoupt-Wheeden theorem and its consequences, Integral Equations Operator Theory 76 (2013), no. 3, 421–446. ↑4 [7] , One and two weight norm inequalities for Riesz potentials, Illinois J. Math. 57 (2013), no. 1, 295–323. ↑4


11

[8] Oliver Dragiˇcević, Loukas Grafakos, Mar´ıa Cristina Pereyra, and Stefanie Petermichl, Extrapolation and sharp norm estimates for classical operators on weighted Lebesgue spaces, Publ. Mat. 49 (2005), no. 1, 73–91. ↑ [9] J. Garc´ıa-Cuerva and J. M. Martell, Wavelet characterization of weighted spaces, J. Geom. Anal. 11 (2001), no. 2, 241–264. ↑ [10] I. M. Gel′fand and G. E. Shilov, Generalized functions. Vol. 1, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1964 [1977]. Properties and operations; Translated from the Russian by Eugene Saletan. ↑ [11] Loukas Grafakos, Classical Fourier analysis, 3rd ed., Graduate Texts in Mathematics, vol. 249, Springer, New York, 2014. ↑ [12] Irina Holmes, Michael T. Lacey, and Brett D. Wick, Bloom’s Inequality: Commutators in a Two-Weight Setting (2015), available at http://arxiv.org/abs/1505.07947. ↑2, 8 [13] I. Holmes, Michael T. Lacey, and Brett D. Wick, Commutators in the Two-Weight Setting, arXiv preprint arXiv:1506.05747 (2015). ↑2, 3, 4, 8 [14] Tuomas Hytönen, On Petermichl’s dyadic shift and the Hilbert transform, C. R. Math. Acad. Sci. Paris 346 (2008), no. 21-22, 1133–1136 (English, with English and French summaries). ↑5 [15] Tuomas P. Hytönen, The sharp weighted bound for general Calder´ on-Zygmund operators, Ann. of Math. (2) 175 (2012), no. 3, 1473–1506. ↑ [16] Joshua Isralowitz, A Matrix Weighted T 1 Theorem for Matrix Kernelled CZOs and a Matrix Weighted John–Nirinberg Theorem (2015), available at http://arxiv.org/abs/1508.02474. ↑ [17] Michael T. Lacey, Commutators with Reisz potentials in one and several parameters, Hokkaido Math. J. 36 (2007), no. 1, 175–191. ↑ [18] Michael Lacey, Haar shifts, commutators, and Hankel operators, Rev. Un. Mat. Argentina 50 (2009), no. 2, 1–13. MR2656521 (2011f:47045) ↑5 [19] Ming-Yi Lee, Chin-Cheng Lin, and Ying-Chieh Lin, A wavelet characterization for the dual of weighted Hardy spaces, Proc. Amer. Math. Soc. 137 (2009), no. 12, 4219–4225. ↑ [20] B. Muckenhoupt and R. L. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 192 (1974), 261–274. ↑4 [21] , Weighted bounded mean oscillation and the Hilbert transform, Studia Math. 54 (1975/76), no. 3, 221–237. ↑4 [22] K. B. Oldham and J. Spanier, Fractional calculus and its applications, Bul. Inst. Politehn. Ia¸si Sect¸. I 24(28) (1978), no. 3-4, 29–34 (English, with Romanian summary). MR552175 (80k:26007) ↑ [23] Stefanie Petermichl, Dyadic shifts and a logarithmic estimate for Hankel operators with matrix symbol, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 6, 455–460, DOI 10.1016/S0764-4442(00)00162-2 (English, with English and French summaries). MR1756958 (2000m:42016) ↑5 [24] S. Petermichl, S. Treil, and A. Volberg, Why the Riesz transforms are averages of the dyadic shifts?, Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations (El Escorial, 2000), 2002. ↑5 [25] Robert Rahm and Scott Spencer, Some Entropy Bump Conditions for Fractional Maximal and Integral Operators (2015), available at http://arxiv.org/abs/1504.05906v2. ↑4 [26] Richard Rochberg, NWO sequences, weighted potential operators, and Schr¨ odinger eigenvalues, Duke Math. J. 72 (1993), no. 1, 187–215. ↑4 [27] Eric T. Sawyer, A characterization of a two-weight norm inequality for maximal operators, Studia Math. 75 (1982), no. 1, 1–11. ↑4 [28] , A characterization of two weight norm inequalities for fractional and Poisson integrals, Trans. Amer. Math. Soc. 308 (1988), no. 2, 533–545. ↑4 [29] Si Jue Wu, A wavelet characterization for weighted Hardy spaces, Rev. Mat. Iberoamericana 8 (1992), no. 3, 329–349. ↑

12


Irina Holmes, School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA USA 30332-0160 E-mail address: [email protected] Robert Rahm, School of Mathematics, Washington University in St. Louis, One Brookings Drive, St. Louis, MO USA 63130 E-mail address: [email protected] Scott Spencer, School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA USA 30332-0160 E-mail address: [email protected]