Two Matrix Weighted Inequalities for Commutators with Fractional Integral Operators
aa r X i v : . [ m a t h . C A ] J a n TWO MATRIX WEIGHTED INEQUALITIES FORCOMMUTATORS WITH FRACTIONAL INTEGRALOPERATORS
ROY CARDENAS AND JOSHUA ISRALOWITZ
Abstract.
In this paper we prove two matrix weighted norm in-equalities for the commutator of a fractional integral operator andmultiplication by a matrix symbol. More precisely, we extend therecent results of the second author, Pott, and Treil on two matrixweighted norm inequalities for commutators of Calderon-Zygmundoperators and multiplication by a matrix symbol to the fractionalintegral operator setting. In particular, we completely extend thefractional Bloom theory of Holmes, Rahm, and Spencer to the twomatrix weighted setting with a matrix symbol. Introduction
Let w be a weight on R d and let L p ( w ) be the standard weightedLebesgue space with respect to the norm k f k L p ( w ) = (cid:18)Z R d | f ( x ) | p w ( x ) dx (cid:19) p . Furthermore, let A p,q for p, q > w satisfying sup Q ⊆ R d Q is a cube (cid:18) − Z Q w ( x ) dx (cid:19) (cid:18) − Z Q w − p ′ q ( x ) dx (cid:19) qp ′ < ∞ where − R Q is the unweighted average over Q (which will also occasionallybe denoted by m Q ). When p = q we write A p := A p,p as usual.Given a weight ν , we say b ∈ BMO ν if k b k BMO ν = sup Q ⊆ R d Q is a cube ν ( Q ) Z Q | b ( x ) − m Q b | dx < ∞ Mathematics Subject Classification. (where ν ( Q ) = R Q ν ) so that clearly BMO = BMO ν when ν ≡ T, define the commutator [ M b , T ] = M b T − T M b with M b being multiplication by b . In the papers [HLW16,HLW17] the authors extended earlier work of S. Bloom [Blo85] andproved that if u, v ∈ A p and T is any Calder´on-Zygmund operator(CZO) then k [ M b , T ] k L p ( u ) → L p ( v ) . k b k BMO ν (1.1)where ν = ( uv − ) p and it was proved in [HLW17] that if R s is the s th Riesz transform then k b k BMO ν . max ≤ s ≤ d k [ M b , R s ] k L p ( u ) → L p ( v ) . (1.2)Furthermore, let I α be the fractional integral operator defined by theformula I α f ( x ) = Z R d f ( y ) | x − y | d − α d y, for 0 < α < d. It was proved in [HRS16] that if 0 < α < d and α/d + 1 /q = 1 /p , if u, v ∈ A p,q , and if ν = u q v − q then k [ M b , I α ] k L p ( u pq ) → L p ( v ) ≈ k b k BMO( ν ) (1.3)On the other hand, matrix weighted extensions and generalizations of(1.1) and (1.2) that surprisingly hold for two arbitrary matrix weights(and provided new results even in the scalar p = 2 setting of a singlescalar weight) were proved in [IPT], and it is the purpose of this paperto extend the results of [IPT] to the fractional setting, providing matrixweighted extensions of (1.3) that hold for two arbitrary matrix weights.Note that for the rest of this paper we will assume that 0 < α < d and α/d + 1 /q = 1 /p .In particular, for any linear operator T acting on scalar valued functionson R d , we can canonically extend T to act on C n valued functions ~f bythe formula T ~f := P nj =1 (cid:16) T D ~f , ~e j E C n (cid:17) ~e j where { ~e j } is any orthonor-mal basis of C n (and note that this is easily seen to be independentof the orthonormal basis chosen.) Let W : R d → M n × n be an n × n matrix weight (a positive definite a.e. M n × n valued function on R d )and let L p ( W ) be the space of C n valued functions ~f such that k ~f k L p ( W ) = (cid:18)Z R d | W p ( x ) ~f ( x ) | p dx (cid:19) p < ∞ . WO WEIGHTED INEQUALITIES FOR COMMUTATORS 3
Furthermore, for p, q > W is amatrix A p,q weight (see [IM19]) if it satisfies k W k A p,q = sup Q ⊂ R d Q is a cube − Z Q (cid:18) − Z Q k W q ( x ) W − q ( y ) k p ′ dy (cid:19) qp ′ dx < ∞ (1.4)and when p = q we say W is a matrix A p weight (see [Rou03]).Now for scalar weights u and v , notice that by multiple uses of the A q property and H¨older’s inequality we have m Q ν ≈ ( m Q u ) q ( m Q v − q ′ q ) q ′ ≈ ( m Q u ) q ( m Q v ) − q ≈ ( m Q u q )( m Q v q ) − . Thus, b ∈ BMO ν when u and v are A q weights if and only ifsup Q ⊆ R Q is a cube − Z Q ( m Q v q )( m Q u q ) − | b ( x ) − m Q b | dx < ∞ , which is a condition that easily extends to the matrix weighted setting,noting that A p,q ⊂ A q when q > p , since then q ′ < p ′ and so H¨older’sinequality gives us − Z Q (cid:18) − Z Q k W q ( x ) W − q ( y ) k q ′ dy (cid:19) qq ′ dx ≤ − Z Q (cid:18) − Z Q k W q ( x ) W − q ( y ) k p ′ dy (cid:19) qp ′ dx. Namely, if
U, V are n × n matrix A p,q weights, then we define BMO p,qV,U to be the space of n × n locally integrable matrix functions B where k B k BMO p,qV,U = sup Q ⊆ R d Q is a cube (cid:18) − Z Q k ( m Q V q )( B ( x ) − m Q B )( m Q U q ) − k dx (cid:19) q < ∞ so that k b k BMO p,qV,U ≈ k b k BMO ν if U, V are scalar weights and b is a scalarfunction. Note that the BMO p,qV,U condition is much more naturallydefined in terms of reducing matrices, which will be discussed in Section3.We will need a definition before we state our first result. We say that alinear operator R acting on scalar functions is a fractional lower boundoperator if for any n ∈ N and any n × n matrix weight W we have k W k q A p,q . k T k L p ( W pq ) → L q ( W ) (1.5)with the bound independent of W (but not necessarily independent of n ), and k T k L p ( W pq ) → L q ( W ) < ∞ if W is a matrix A p,q weight. ROY CARDENAS AND JOSHUA ISRALOWITZ
Theorem 1.1.
Let T be any linear operator acting on scalar val-ued functions where its canonical C n valued extension is bounded from L p ( W pq ) to L q ( W ) for all n × n matrix A p,q weights W and all n ∈ N with bound depending on T, n, d, p , and k W k A p,q (which is known tobe true for fractional integral operators, see [IM19, Theorem 1.4] .) If U, V are m × m matrix A p weights and B is an m × m locally integrablematrix function for some m ∈ N , then k [ M B , T ] k L p ( U pq ) → L q ( V ) . k B k BMO p,qV,U (1.6) with bounds depending on
T, m, d, p, k U k A p,q and k V k A p,q .Furthermore, for any fractional lower bound operator T we have thelower bound estimate k B k BMO p,qV,U . k [ M B , T ] k L p ( U pq ) → L q ( V ) (1.7)Like in [IPT], we will use matrix weighted arguments inspired by[GPTV04] in the next section to prove Theorem 1.1 in terms of aweighted BMO quantity k B k ^ BMO p,qV,U that is equivalent to k B k BMO p,qV,U when U and V are matrix A p,q weights (see Theorem 3.1) but is muchmore natural for more arbitrary matrix weights U and V . More pre-cisely, define k B k q ^ BMO p,qV,U = sup Q ⊆ R d Q is a cube − Z Q (cid:18) − Z Q (cid:13)(cid:13)(cid:13) V q ( x )( B ( x ) − B ( y )) U − q ( y ) (cid:13)(cid:13)(cid:13) p ′ dy (cid:19) qp ′ dx. (1.8)We will then give relatively short proofs of the following two results inSection 2. Lemma 1.2.
Let T be any linear operator defined on scalar valuedfunctions where its canonical C n valued extension T for any n ∈ N satisfies k T k L p ( W pq ) → L q ( W ) ≤ φ ( k W k A p,q ) for some positive increasing function φ (possibly depending on T, d, n, p, q .)If
U, V are m × m matrix A p,q weights and B is a locally integrable m × m matrix valued function for some m ∈ N , then k [ M B , T ] k L p ( U pq ) → L q ( V ) ≤ k B k ^ BMO p,qV,U φ (cid:16) qp ′ (cid:16) k U k A p,q + k V k A p,q (cid:17) + 1 (cid:17) Lemma 1.3. If T is any fractional lower bound operator then for any m × m matrix A p,q weights U, V and an m × m matrix symbol B we WO WEIGHTED INEQUALITIES FOR COMMUTATORS 5 have k B k ^ BMO p,qV,U . k [ M B , T ] k L p ( U pq ) → L q ( V ) where the bound depends possibly on n, p, d and T but is independentof U and V . As in [IPT], we will prove that the fractional integral operator is afractional lower bound operator in Section 4 by utilizing the Schurmultiplier/Wiener algebra ideas from [LT13], and thus recover (1.7).These arguments will in fact prove the following (see [IPT] for an anal-ogous result with respect to the Riesz transforms). Here, for ease ofnotation, we set U ′ = U − p ′ q and V ′ = V − p ′ q . Theorem 1.4.
Let U and V be any (not necessarily A p ) matrix weights.If B is any locally integrable m × m matrix valued function then max (cid:26) k B k ^ BMO p,qV,U , k B k ^ BMO q ′ ,p ′ U ′ ,V ′ (cid:27) . k [ M B , I α ] k L p ( U pq ) → L q ( V ) . (1.9)Note that the two quantities k B k ^ BMO p,qV,U and k B k ^ BMO q ′ ,p ′ U ′ ,V ′ are equiv-alent when U, V ∈ A p,q (which will be proved in Section 4) and ingeneral should be thought of as “dual” matrix weighted BMO quanti-ties. Finally, we will show that an Orlicz “bumped” version of theseconditions are sufficient for the general two matrix weighted bound-edness of fractional integral operators. In particular, we will provethe following result in Section 5 (see [IPT] for an analogous result forCalderon-Zygmund operators) Proposition 1.5.
Let U and V be any m × m matrix weights, andsuppose that C and D are Young functions with ¯ D ∈ B p,q and ¯ C ∈ B q ′ .Then k [ M B , I α ] k L p ( U pq ) → L q ( V ) . min { κ , κ } where κ = sup Q kk V q ( x )( B ( x ) − B ( y )) U − q ( y ) k C x ,Q k D y ,Q κ = sup Q kk V q ( x )( B ( x ) − B ( y )) U − q ( y ) k D y ,Q k C x ,Q We refer the reader to Section 5 . ROY CARDENAS AND JOSHUA ISRALOWITZ Intermediate fractional upper and lower bounds
As stated in the introduction, we will give short proofs of Lemma 1.2and Lemma 1.3 in this section, beginning with Lemma 1.2.2.1.
Proof of Lemma 1.2.
Define the 2 × U,V,B : R d −→ M m ( C ) byΦ( x ) := V q ( x ) 00 U q ( x ) ! (cid:18) I B ( x )0 I (cid:19) = V q ( x ) V q ( x ) B ( x )0 U q ( x ) ! , so that for a.e. x ∈ R d ,Φ − ( y ) = V − q ( y ) − B ( y ) U − q ( y )0 U − q ( y ) ! . Thus, we haveΦ T Φ − = V q T V − q V q [ M B , T ] U − q U q T U − q ! . Note that W := (Φ ∗ Φ) q is a matrix weight and, by polar decom-position, there exists a unitary a.e. matrix function U such thatΦ( x ) = U ( x ) W q ( x ). This gives us that || T || L p (cid:18) W pq (cid:19) −→ L q ( W ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) W q T W − q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L p → L q = (cid:12)(cid:12)(cid:12)(cid:12) Φ T Φ − (cid:12)(cid:12)(cid:12)(cid:12) L p → L q ≈ max n(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) V q T V − q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L p → L q , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) V q [ M B , T ] U − q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L p → L q , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) U q T U − q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L p → L q o ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) V q [ M B , T ] U − q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L p → L q = || [ M B , T ] || L p (cid:18) U pq (cid:19) −→ L q ( V ) Using the assumption in Lemma 1.2 that || T || L p (cid:18) W pq (cid:19) −→ L q ( W ) . φ ( k W k A p,q )we get that || [ M B , T ] || L p (cid:18) U pq (cid:19) −→ L q ( V ) . φ ( k W k A p,q ) . (2.1)Unravelling the A p,q condition for W , we obtain k W k A p,q = sup Q − Z Q (cid:18) − Z Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) W q ( x ) W − q ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ′ d y (cid:19) qp ′ d x = sup Q − Z Q (cid:18) − Z Q (cid:12)(cid:12)(cid:12)(cid:12) Φ( x )Φ − ( y ) (cid:12)(cid:12)(cid:12)(cid:12) p ′ d y (cid:19) qp ′ d x WO WEIGHTED INEQUALITIES FOR COMMUTATORS 7 ≤ qp ′ || U || A p,q + || V || A p,q + sup Q − Z Q (cid:18) − Z Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) V q ( x ) ( B ( x ) − B ( y )) U − q ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ′ d y (cid:19) qp ′ d x ! = 3 qp ′ (cid:18) || U || A p,q + || V || A p,q + || B || q ^ BMO p,qV,U (cid:19) and thus || [ M B , I α ] || L p (cid:18) U pq (cid:19) −→ L q ( V ) . φ (cid:18) qp ′ (cid:18) || U || A p,q + || V || A p,q + || B || q ^ BMO p,qV,U (cid:19)(cid:19) . Re-scaling with B replaced by B || B || − ^ BMO p,qV,U now completes the proof.2.2.
Proof of Lemma 1.3.
We now prove Lemma 1.3. Let W and Φbe defined as in the previous subsection, so that (cid:18) k U k A p,q + k V k A p,q + k B k q ^ BMO p,qV,U (cid:19) q ≈ k W k q A p,q . || T || L p (cid:18) W pq (cid:19) −→ L q ( W ) . || [ M B , T ] || L p (cid:18) U pq (cid:19) −→ L q ( V ) + || T || L p (cid:18) U pq (cid:19) −→ L q ( U ) + || T || L p (cid:18) V pq (cid:19) −→ L q ( V ) . Clearly we may assume that || [ M B , T ] || L p (cid:18) U pq (cid:19) −→ L q ( V ) < ∞ and so byassumption all quantities above are finite. Re-scalling B rB for r >
0, dividing by r , and taking r −→ ∞ , we obtain || B || ^ BMO p,qV,U . || [ M B , I α ] || L p (cid:18) U pq (cid:19) −→ L q ( V ) . which is the desired lower bound.3. Proof of Theorem 1.1
We now prove Theorem 1.1 (assuming that I α is a fractional lowerbound operator, which will be proved in the next section) by provingthat k B k BMO p,qV,U ≈ k B k ^ BMO p,qV,U when
U, V are matrix A p,q weights (seeTheorem 3.1). To do this we need the concept of a reducing matrix. Inparticular, for any norm ρ on C n there exists a positive definite n × n matrix A where for any ~e ∈ C n we have n − | A~e | ≤ ρ ( ~e ) ≤ | A~e | (see [NT96, Lemma 11.4]). ROY CARDENAS AND JOSHUA ISRALOWITZ
In particular, for any matrix weight U and measurable 0 < | E | < ∞ there exists n × n matrices U E , U ′ E where for any ~e ∈ C n we have |U E ~e | ≈ (cid:18) − Z E (cid:12)(cid:12)(cid:12) U q ( x ) ~e (cid:12)(cid:12)(cid:12) q dx (cid:19) q , |U ′ E ~e | ≈ (cid:18) − Z E (cid:12)(cid:12)(cid:12) U − q ( x ) ~e (cid:12)(cid:12)(cid:12) p ′ dx (cid:19) p ′ . Similarly for a matrix weight V we will use the notation V E and V ′ E forthese reducing matrices. Using reducing matrices in conjunction withelementary linear algebra, it is easy to see that for a matrix weight U we have k U k q A p,q ≈ sup Q (cid:13)(cid:13) U Q U ′ Q (cid:13)(cid:13) = sup Q (cid:13)(cid:13) U ′ Q U Q (cid:13)(cid:13) ≈ sup Q − Z Q (cid:18) − Z Q k W q ( x ) W − q ( y ) k q dx (cid:19) p ′ q dy p ′ and similarly an easy application of H¨older’s inequality gives us that (cid:12)(cid:12)(cid:12)D ~e, ~f E C n (cid:12)(cid:12)(cid:12) ≤ |U Q ~e | (cid:12)(cid:12)(cid:12) U ′ Q ~f (cid:12)(cid:12)(cid:12) for ~e, ~f ∈ C n , which clearly implies that (cid:13)(cid:13) U − Q ( U ′ Q ) − (cid:13)(cid:13) ≤ . (3.1)We now prove the following matrix weighted John-Nirenberg type the-orem, which should be thought of as a fractional generalization of thematrix weighted John-Nirenberg theorem from [IPT]. Theorem 3.1. If U, V are two m × m matrix weights such that U, V ∈ A p,q and B is an m × m locally integrable matrix function, then thefollowing are equivalent (where the suprema is taken over all cubes Q ).(1) sup Q − Z Q (cid:13)(cid:13) V Q ( B ( x ) − m Q B ) U − Q (cid:13)(cid:13) dx (2) sup Q (cid:18) − Z Q (cid:13)(cid:13)(cid:13) V q ( x )( B ( x ) − m Q B ) U − Q (cid:13)(cid:13)(cid:13) q dx (cid:19) q (3) sup Q (cid:18) − Z (cid:13)(cid:13)(cid:13) U − q ( x )( B ∗ ( x ) − m Q B ∗ )( V ′ Q ) − (cid:13)(cid:13)(cid:13) p ′ dx (cid:19) p ′ (4) sup Q − Z Q (cid:18) − Z Q (cid:13)(cid:13)(cid:13) V q ( x )( B ( x ) − B ( y )) U − q ( y ) (cid:13)(cid:13)(cid:13) p ′ dy (cid:19) qp ′ dx ! q (5) sup Q − Z Q (cid:18) − Z Q (cid:13)(cid:13)(cid:13) V q ( x )( B ( x ) − B ( y )) U − q ( y ) (cid:13)(cid:13)(cid:13) q dx (cid:19) p ′ q dy p ′ WO WEIGHTED INEQUALITIES FOR COMMUTATORS 9 (6) sup Q − Z Q (cid:18) − Z Q (cid:13)(cid:13)(cid:13) V q ( x )( B ( x ) − B ( y )) U − q ( y ) (cid:13)(cid:13)(cid:13) q dx (cid:19) q ′ q dy q ′ Note that Lemma 2 . |U Q ~e | ≈ | m Q ( U q ) ~e | if U ∈ A q so that (1) ≈ k B k BMO p,qV,U for
U, V ∈ A p,q .Before we prove Theorem 3.1 we need to discuss some duality proper-ties of matrix A p,q weights. To better keep track of the exponents andmatrix weights that corresponding to a reducing matrix we temporar-ily use the notation V Q ( W, q ) , V ′ Q ( W, p, q ) to denote reducing matriceswhere | V Q ( W, q ) ~e | ≈ (cid:18) − Z Q (cid:12)(cid:12)(cid:12) W q ( x ) ~e (cid:12)(cid:12)(cid:12) q dx (cid:19) q , (cid:12)(cid:12) V ′ Q ( W, p, q ) ~e (cid:12)(cid:12) ≈ (cid:18) − Z Q (cid:12)(cid:12)(cid:12) W − q ( x ) ~e (cid:12)(cid:12)(cid:12) p ′ dx (cid:19) p ′ so that k W k A p,q ≈ sup Q (cid:13)(cid:13) V Q ( W, q ) V ′ Q ( W, p, q ) (cid:13)(cid:13) p ′ , k W k A q ≈ sup Q (cid:13)(cid:13) V Q ( W, q ) V ′ Q ( W, q, q ) (cid:13)(cid:13) q . Moreover (cid:12)(cid:12)(cid:12)(cid:12) V Q ( W − p ′ q , p ′ ) ~e (cid:12)(cid:12)(cid:12)(cid:12) ≈ (cid:18) − Z Q (cid:12)(cid:12)(cid:12) W − q ( x ) ~e (cid:12)(cid:12)(cid:12) p ′ dx (cid:19) p ′ ≈ (cid:12)(cid:12) V ′ Q ( W, p, q ) ~e (cid:12)(cid:12) (3.2)and similarly (cid:12)(cid:12)(cid:12)(cid:12) V ′ Q ( W − p ′ q , q ′ , p ′ ) ~e (cid:12)(cid:12)(cid:12)(cid:12) ≈ (cid:18) − Z Q (cid:12)(cid:12)(cid:12) W q ( x ) ~e (cid:12)(cid:12)(cid:12) q dx (cid:19) q ≈ | V Q ( W, q ) ~e | , (3.3)which (as observed in [IM19]) means that W ∈ A p,q if and only if W − p ′ q ∈ A q ′ ,p ′ . Note that with this notation we have V Q = V Q ( V, q ) , V ′ Q = V ′ Q ( V, p, q )and a similar statement holds for U . Proof of Theorem 3.1.
Recall from the introduction that A p,q ⊆ A q .Thus, we have from [IPT, Corollary 4.7] that (1) ⇐⇒ (2) ⇐⇒ (6).Moreover, since U, V ∈ A p,q if and only if U − p ′ q , V − p ′ q ∈ A q ′ ,p ′ , we havethat U − p ′ q , V − p ′ q ∈ A p ′ . The fact that V ∈ A p,q tells us that (by theA p,q property and (3.1))(1) = sup Q − Z Q (cid:13)(cid:13) U − Q ( B ∗ ( x ) − m Q B ∗ ) V Q (cid:13)(cid:13) dx ≈ sup Q − Z Q (cid:13)(cid:13) U ′ Q ( B ∗ ( x ) − m Q B ∗ )( V ′ Q ) − (cid:13)(cid:13) dx which by (3.2) is nothing but (1) with respect to matrix weights V − p ′ q , U − p ′ q , the symbol B ∗ , and the exponent p ′ , and thus (1) equivalent to (2) withrespect to V − p ′ q , U − p ′ q , B ∗ , and p ′ , which (again by (3.2)) is nothing but(3). This tells us that (3) ⇐⇒ (1) ⇐⇒ (2) ⇐⇒ (6) . Furthermore, (6) ≤ (5) by an easy application of H¨older’s inequality,since p ′ /q ′ >
1. Thus, if we can show that (5) . (2) + (3) then we willhave (3) ⇐⇒ (1) ⇐⇒ (2) ⇐⇒ (6) ⇐⇒ (5). To that end, − Z Q (cid:18) − Z Q (cid:13)(cid:13)(cid:13) V q ( x )( B ( x ) − B ( y )) U − q ( y ) (cid:13)(cid:13)(cid:13) q dx (cid:19) p ′ q dy p ′ . − Z Q (cid:18) − Z Q (cid:13)(cid:13)(cid:13) V q ( x )( B ( x ) − m Q B ) U − q ( y ) (cid:13)(cid:13)(cid:13) q dx (cid:19) p ′ q dy p ′ + − Z Q (cid:18) − Z Q (cid:13)(cid:13)(cid:13) V q ( x )( B ( y ) − m Q B ) U − q ( y ) (cid:13)(cid:13)(cid:13) q dx (cid:19) p ′ q dy p ′ = ( A ) + ( B ) . Using the matrix A p,q property we get( A ) ≤ − Z Q (cid:18) − Z Q (cid:13)(cid:13)(cid:13) V q ( x )( B ( x ) − m Q B ) U − Q (cid:13)(cid:13)(cid:13) q (cid:13)(cid:13)(cid:13) U Q U − q ( y ) (cid:13)(cid:13)(cid:13) q dx (cid:19) p ′ q dy p ′ = − Z Q (cid:18) − Z Q (cid:13)(cid:13)(cid:13) V q ( x )( B ( x ) − m Q B ) U − Q (cid:13)(cid:13)(cid:13) q dx (cid:19) p ′ q (cid:13)(cid:13)(cid:13) U Q U − q ( y ) (cid:13)(cid:13)(cid:13) p ′ dy p ′ . k U k p ′ A p,q (2) . and likewise( B ) ≤ − Z Q (cid:18) − Z Q (cid:13)(cid:13)(cid:13) V q ( x ) V ′ Q (cid:13)(cid:13)(cid:13) q (cid:13)(cid:13)(cid:13) ( V ′ Q ) − ( B ( y ) − m Q B ) U − q ( y ) (cid:13)(cid:13)(cid:13) q dx (cid:19) p ′ q dy p ′ = − Z Q (cid:18) − Z Q (cid:13)(cid:13)(cid:13) V q ( x ) V ′ Q (cid:13)(cid:13)(cid:13) q dx (cid:19) p ′ q (cid:13)(cid:13)(cid:13) ( V ′ Q ) − ( B ( y ) − m Q B ) U − q ( y ) (cid:13)(cid:13)(cid:13) p ′ dy p ′ . k V k p ′ A p,q (3) WO WEIGHTED INEQUALITIES FOR COMMUTATORS 11
Finally, as was observed already, (1) is equivalent to (1) with re-spect to V − p ′ q , U − p ′ q , B ∗ , and p ′ , which is equivalent to (5) with re-spect V − p ′ q , U − p ′ q , B ∗ , and p ′ , which is easily be seen to be nothing but(4). (cid:3) Commutator lower bound: proof of Theorem 1.4
In this section we will prove Theorem 1.4 and in the process provethat I α is a fractional lower bound operator (which will complete theproof of Theorem 1.1). As stated in the introduction, this will bedone by modifying Wiener algebra arguments from by [IPT, LT13].Let W be a matrix weight and suppose that ~f ∈ L p ∩ L p (cid:16) W pq (cid:17) and ~g ∈ L q ∩ L q ′ (cid:18) W − q ′ q (cid:19) . Let E ⊂ R d be measurable. For any t ∈ R d ,define k α,t ( x, y ) := e − πit · x k α ( x, y ) e πit · y where k α ( x, y ) = | x − y | α − d . We then have from H¨older’s inequalitythat (cid:12)(cid:12)(cid:12)(cid:12)Z R d Z R d χ E × E ( x, y ) k α,t ( x, y ) D ~f ( y ) , ~g ( x ) E C n d y d x (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z R d Z R d χ E × E ( x, y ) k α,t ( x, y ) D W q ( x ) ~f ( y ) , W − q ( x ) ~g ( x ) E C n d y d x (cid:12)(cid:12)(cid:12)(cid:12) ≤ || χ E I α χ E || L p (cid:18) W pq (cid:19) −→ L q ( W ) || ~f || L p (cid:18) W pq (cid:19) || ~g || L q ′ W − q ′ q ! Thus, if ψ = ˆ ρ for ρ ∈ L ( R d ) (whereˆdenotes Fourier transform) then, (cid:12)(cid:12)(cid:12)(cid:12)Z R d Z R d ψ (cid:18) x − yǫ (cid:19) χ E × E ( x, y ) k α ( x, y ) D ~f ( y ) , ~g ( x ) E C n d y d x (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z R d Z R d ǫ d (cid:20)Z R d ρ ( ǫt ) e − πit · ( x − y ) d t (cid:21) χ E × E ( x, y ) k α ( x, y ) D ~f ( y ) , ~g ( x ) E C n d y d x (cid:12)(cid:12)(cid:12)(cid:12) = || χ E I α χ E || L p (cid:18) W pq (cid:19) −→ L q ( W ) || ~f || L p (cid:18) W pq (cid:19) || ~g || L q ′ W − q ′ q ! || ρ || L ( R d ) , (4.1)which means that (4.1) holds if ψ is in the Wiener algebra W ( R d ) := { ˆ ρ : ρ ∈ L ( R d ) } . To prove Theorem 1.4 we will use (4.1) with ψ ( x ) = | x | d − α φ ( x ) where φ ∈ C ∞ c ( R d ). While it is likely known that such afunction lies in W ( R d ), a precise reference seems difficult to find and thus we will prove it (using a simple idea from [DT83]) for the sake ofcompleteness. Proposition 4.1.
Let < α < d . If φ ∈ C ∞ c ( R d ) , then |·| d − α φ ∈ W ( R d ) .Proof. Let F ( x ) = | x | d − α and F ( x ) = | x | d − α φ ( x ). If β ∈ { , } d , thenby an easy induction we have D β F ( x ) = Φ β ( x ) | x | d − α − | β | where Φ β ( x ) is the sum of monomials of degree | β | in d variables, whichmeans that (cid:12)(cid:12) D β F ( x ) (cid:12)(cid:12) . | x | d − α −| β | (4.2)Now let 1 < δ < min (cid:8) d − αα , (cid:9) . Using H¨older’s inequality, a stan-dard integration by parts argument, and the Hausdorff-Young inequal-ity, we then have by an argument identical to the one used in [IPT,Lemma 3 .
2] that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F V (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ( R d ) . (cid:18)Z R d (cid:12)(cid:12) D β F ( x ) (cid:12)(cid:12) δ d x (cid:19) δ < ∞ which is finite by (4.2), since δ [ | β | − ( d − α )] < (cid:18) d − αα (cid:19) [ d − ( d − α )] = (cid:18) d − αα (cid:19) α = d. Fourier inversion now immediately completes the proof. (cid:3)
Applying (4.1) with ψ ( x ) = | x | d − α φ ( x ) where φ ∈ C ∞ c ( R d ), we obtainthe inequality (cid:12)(cid:12)(cid:12)(cid:12)Z E Z E ǫ d − α φ (cid:18) x − yǫ (cid:19) D ~f ( y ) , ~g ( x ) E C n d y d x (cid:12)(cid:12)(cid:12)(cid:12) ≤ || χ E I α χ E || L p (cid:18) W pq (cid:19) −→ L q ( W ) || ~f || L p (cid:18) W pq (cid:19) || ~g || L q ′ W − q ′ q ! || ρ || L ( R d ) . (4.3)We proceed by citing a uniform boundedness result which was (implic-itly) proved in [IM19, Proposition 3.1]. Proposition 4.2.
Let E be measurable with < | E | < ∞ and definethe fractional averaging operator by A E : L p (cid:16) W pq (cid:17) −→ L q ( W ) by A E ~f := χ E | E | − αd Z E ~f ( x ) d x, WO WEIGHTED INEQUALITIES FOR COMMUTATORS 13 then ||W ′ E W E || ≈ || A E || L p (cid:18) W pq (cid:19) −→ L q ( W ) . (4.4)We can now state and prove the main technical lemma of this section,which immediately proves that I α is a fractional lower bound operator. Lemma 4.3. If B is a ball and E ⊂ B with | E | > , then ||W ′ E W E || . (cid:20) | B || E | (cid:21) − αd || χ E I α χ E || L p (cid:18) W pq (cid:19) −→ L q ( W ) . Proof.
Suppose that ~f ∈ L p ∩ L p (cid:16) W pq (cid:17) and ~g ∈ L q ′ ∩ L q ′ (cid:18) W − q ′ q (cid:19) Clearly we have (cid:20) | E || B | (cid:21) − αd D A E ~f , ~g E L ( R d ) = 1 | B | − αd Z E Z E D ~f ( y ) , ~g ( x ) E C n d y d x. Let B have radius ǫ and pick φ ∈ C ∞ c ( R d ) such that φ = 1 on the openball B (0 , x, y ∈ B then | x − y | ≤ ǫ and therefore (4.3) gives usthat (cid:12)(cid:12)(cid:12)(cid:12) | B | − αd Z E Z E D ~f ( y ) , ~g ( x ) E C n d y d x (cid:12)(cid:12)(cid:12)(cid:12) ≈ (cid:12)(cid:12)(cid:12)(cid:12) ǫ d ) − αd Z E Z E D ~f ( y ) , ~g ( x ) E C n d y d x (cid:12)(cid:12)(cid:12)(cid:12) . || χ E I α χ E || L p (cid:18) W pq (cid:19) −→ L q ( W ) || ~f || L p (cid:18) W pq (cid:19) || ~g || L q ′ W − q ′ q ! which means (cid:12)(cid:12)(cid:12)(cid:12)D A E ~f , ~g E L ( R d ) (cid:12)(cid:12)(cid:12)(cid:12) . (cid:20) | B || E | (cid:21) − αd || χ E I α χ E || L p (cid:18) W pq (cid:19) −→ L q ( W ) || ~f || L p (cid:18) W pq (cid:19) || ~g || L q ′ W − q ′ q ! . (4.5)Duality, the density of L p ∩ L p (cid:16) W pq (cid:17) in L p (cid:16) W pq (cid:17) and L q ∩ L q ′ (cid:18) W − q ′ q (cid:19) in L q ′ (cid:18) W − q ′ q (cid:19) (see [CUMR16, Proposition 3.7]), and (4.4) now com-pletes the proof. (cid:3) Proof of Theorem 1.4.
We assume that || [ M B , I α ] || L p (cid:18) U pq (cid:19) −→ L q ( V ) < ∞ ,or the lower bound holds trivially. Let B be a ball and, for each M > define E M := (cid:8) x ∈ B : max (cid:8) || U ( x ) || , (cid:12)(cid:12)(cid:12)(cid:12) U − ( x ) (cid:12)(cid:12)(cid:12)(cid:12) , || V ( x ) || , (cid:12)(cid:12)(cid:12)(cid:12) V − ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:9) < M (cid:9) . (4.6)By continuity of Lebesgue measure, there exists M such that 2 | E M | > | B | . Further, define || W || A p,q ( E M ) := − Z E M (cid:18) − Z E M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) W q ( x ) W − q ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ′ d y (cid:19) qp ′ d x (4.7)and || B || ^ BMO p,qV,U ( E M ) := − Z E M (cid:18) − Z E M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) V q ( x )( B ( x ) − B ( y )) U − q ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ′ d y (cid:19) qp ′ d x ! q . (4.8)Let W and Φ be defined as in Subsection 2.1. Using ideas from thatsubsection, we have || W || A p,q ( E M ) = − Z E M (cid:18) − Z E M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) W q ( x ) W − q ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ′ d y (cid:19) qp ′ d x = − Z E M (cid:18) − Z E M (cid:12)(cid:12)(cid:12)(cid:12) Φ( x )Φ − ( y ) (cid:12)(cid:12)(cid:12)(cid:12) p ′ d y (cid:19) qp ′ d x ≈ || U || A p,q ( E M ) + || V || A p,q ( E M ) + || B || q ^ BMO p,qV,U ( E M ) , so that (cid:18) || U || A p,q ( E M ) + || V || A p,q ( E M ) + || B || q ^ BMO p,qV,U ( E M ) (cid:19) q ≈ || W || q A p,q ( E M ) ≈ (cid:12)(cid:12)(cid:12)(cid:12) W ′ E M W E M (cid:12)(cid:12)(cid:12)(cid:12) . (cid:20) | B || E M | (cid:21) − αd || χ E M I α χ E M || L p (cid:18) W pq (cid:19) −→ L q ( W ) . || [ M B , I α ] || L p (cid:18) U pq (cid:19) −→ L q ( V ) + || χ E M I α χ E M || L p (cid:18) U pq (cid:19) −→ L q ( U ) + || χ E M I α χ E M || L p (cid:18) V pq (cid:19) −→ L q ( V ) . By assumption, all quantities above are finite so we can rescale withthe replacement B rB , for r >
0. Upon dividing by r and taking r −→ ∞ , we obtain || B || ^ BMO p,qV,U ( E M ) . || [ M B , I α ] || L p (cid:18) U pq (cid:19) −→ L q ( V ) . Applying Fatou’s lemma with M −→ ∞ and taking the supremumover all balls B in R d , we obtain || B || ^ BMO p,qV,U . || [ M B , I α ] || L p (cid:18) U pq (cid:19) −→ L q ( V ) , WO WEIGHTED INEQUALITIES FOR COMMUTATORS 15 and a slight modification to the arguments above proves that k B k ^ BMO q ′ ,p ′ U ′ ,V ′ . || [ M B , I α ] || L p (cid:18) U pq (cid:19) −→ L q ( V ) which proves Theorem 1.4. (cid:3) Proof of theorem 1.5
Our proof is a combination and modification of the arguments in [IPT,CUIM18, Li06]. For additional information on Orlicz spaces, see e.g.,[BL12].
Proposition 5.1.
There exists d dyadic grids D t , t ∈ { , } d where D V q [ M B , I α ] U − q ~f , ~g E L . X t ∈{ , } d X Q ∈D t | Q | − αd Z Q Z Q (cid:12)(cid:12)(cid:12)D V q ( x )( B ( x ) − B ( y )) U − q ( y ) ~f ( y ) , ~g ( x ) E C n (cid:12)(cid:12)(cid:12) dx dy. Proof.
Noting that V q [ M B , I α ] U − q is an integral operator with kernel V q ( x )( B ( x ) − B ( y )) U − q ( y ) | x − y | α − d , the proof is almost identical tothe proof of [IM19, Lemma 3 . (cid:3) As in [Li06], we will make use of the following well known fact aboutOrlicz spaces
Proposition 5.2.
For an increasing, convex function Φ , if k f k Φ ,Q = inf (cid:26) λ > − Z Q Φ (cid:18) | f ( y ) | λ (cid:19) dy ≤ (cid:27) and k f k ∗ Φ ,Q = inf s> (cid:26) s + s − Z Q Φ (cid:18) | f ( y ) | s (cid:19) dy (cid:27) then k f k Φ ,Q ≤ k f k ∗ Φ ,Q ≤ k f k Φ ,Q Proof of Proposition 1.5.
Clearly it is enough to prove for a dyadic grid D that X Q ∈D | Q | αd − Z Q Z Q (cid:12)(cid:12)(cid:12)D V q ( x )( B ( x ) − B ( y )) U − q ( y ) ~f ( y ) , ~g ( x ) E C n (cid:12)(cid:12)(cid:12) dx dy . min { κ , κ }k ~f k L p k ~g k L q ′ . Also, by Fatou’s lemma, it is enough to assume that ~f , ~g are boundedwith compact support. For that matter, the generalized H¨older in-equality gives X Q ∈D | Q | αd − Z Q Z Q (cid:12)(cid:12)(cid:12)D V q ( x )( B ( x ) − B ( y )) U − p ( y ) ~f ( y ) , ~g ( x ) E C n (cid:12)(cid:12)(cid:12) dx dy ≤ min { κ , κ } X Q ∈D | Q | αd k ~f k D,Q k ~g k C,Q
Fix a > d +1 and define the collection of cubes Q k = { Q ∈ D : a k < k ~f k ¯ D,Q ≤ a k +1 } , and let S k be the disjoint collection of Q ∈ D that are maximal withrespect to the inequality k f k ¯ D,Q > a k . Note that since ~f is boundedwith compact support, such maximal cubes are guaranteed to exist.Set S = S k S k . We now continue the above estimate: X k X Q ∈Q k | Q | αd k ~f k D,Q k ~g k C,Q ≤ X k a k +1 X Q ∈Q k | Q | αd k ~g k C,Q = X k a k +1 X P ∈S k X Q ∈Q k Q ⊂ P | Q | αd k ~g k C,Q . (5.1)We now estimate the inner sum. Let ℓ ( P ) = 2 − m , so X Q ⊂ P | Q | αd k ~g k C,Q = ∞ X m = m − mα X ℓ ( Q )=2 − m Q ⊂ P | Q |k ~g k C,Q (5.2)However, Proposition 5.2 gives us that X ℓ ( Q )=2 − m Q ⊂ P | Q |k g k C,Q ≤ X ℓ ( Q )=2 − m Q ⊂ P | Q | inf s> ( s + s − Z Q C | ~f ( y ) | s ! dy ) ≤ inf s> X ℓ ( Q )=2 − m Q ⊂ P " | Q | s + s Z Q C | ~f ( y ) | s ! dy = inf s> ( | P | s + s Z P C | ~f ( y ) | s ! dy ) = | P | inf s> ( s + s − Z P C | ~f ( y ) | s ! dy ) ≤ | P |k g k C,P
Plugging this into (5.2) we get
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University at Albany, SUNY, Department of Mathematics, 1400 Wash-ington Ave, Albany, NY 12222
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