A multilinear reverse H\"older inequality with applications to multilinear weighted norm inequalities

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by  David Cruz-Uribe

A MULTILINEAR REVERSE HO¨LDER INEQUALITY WITH
APPLICATIONS TO MULTILINEAR WEIGHTED NORM
7
1 INEQUALITIES
0
2
DAVID CRUZ-URIBE, OFS AND KABE MOEN
n
a
J
Abstract. In this note we prove a multilinear version of the reverse Ho¨lder in-

6 equalityinthetheoryofMuckenhouptA weights. Wegivetwoapplicationsofthis
p
2
inequality to the study of multilinear weighted norm inequalities. First, we prove

] a structure theorem for multilinear Ap~ weights; second, we give a new sufficient
A condition for multilinear, two-weight norm inequalities for the maximal operator.
C
.
h
t
a 1. Introduction
m
[ The purpose of this note is to prove a multilinear version of the reverse H¨older

inequality in the theory of the Muckenhoupt A classes. We briefly recall the defini-
1 p
v tions; for complete details, see [5]. For 1 < p < ∞, a weight w ∈ Ap if
0
p−1
80 [w]Ap = sup(cid:18)ˆ− wdx(cid:19)(cid:18)ˆ− w1−p′dx(cid:19) < ∞;
7 Q Q Q
0 here and below, the supremum is taken over all cubes in Rn with sides parallel to the
.
1
coordinate axes. We denote the union of all the A classes by A . If w ∈ A , then
0 p ∞ ∞
it satisfies the reverse H¨older inequality: there exists s > 1 such that
7
1 1 −1
: s
v [w] = sup − wsdx − wdx < ∞.
i RHs (cid:18)ˆ (cid:19) (cid:18)ˆ (cid:19)
X Q Q Q
r Our main result extends this condition to m-tuples of A∞ weights.
a
Theorem 1.1. Given 1 < s ,...,s < ∞ such that 1 = 1, suppose w ∈ RH .
1 m si i si
Then for all cubes Q, P
m 1 m
(1.1) − wsidx si ≤ C− w dx.
(cid:18)ˆ i (cid:19) ˆ i
Yi=1 Q QYi=1
Date: 1/26/2017.
2000 Mathematics Subject Classification. 42B20, 42B25.
The first author is supported by NSF Grant DMS-1362425and researchfunds from the Dean of
the College of Arts & Sciences, the University of Alabama. The second author is supported by the
Simons Foundation.
1
2 DAVIDCRUZ-URIBE,OFS ANDKABE MOEN
The proof requires the rescaling properties of Muckenhoupt weights from [4, The-
orems 2.1, 2.2] which we state here as a lemma. (Also see [9].)
Lemma 1.2. Given a weight w:
(a) w ∈ RH , 1 < s < ∞, if and only if ws ∈ A ;
s ∞
(b) w ∈ A ∩RH , 1 < p, s < ∞ if and only if ws ∈ A , q = s(p−1)+1.
p s q
Proof. By Lemma 1.2, for each i, since w ∈ RH , wsi ∈ A . Moreover, there exists
i si i ∞
0 < r < 1 such that wirsi ∈ A2 ∩ RH1; since the Ap classes are nested (Ap ⊂ Aq if
r
p < q) we can find a single value of r that works for all i. Therefore, if we apply
the reverse H¨older inequality, the A condition, multilinear H¨older’s inequality with
2
exponents s , and then H¨older’s inequality twice more, we get that for every cube Q,
i
m 1 m 1 m − 1
− wsidx si ≤ C − wrsidx rsi ≤ C − w−rsidx rsi
(cid:18)ˆ i (cid:19) (cid:18)ˆ i (cid:19) (cid:18)ˆ i (cid:19)
Yi=1 Q Yi=1 Q Yi=1 Q
m −1 m 1 m
r r
≤ C − w−rdx ≤ C − wrdx ≤ C− w dx.
(cid:18)ˆ i (cid:19) (cid:18)ˆ i (cid:19) ˆ i
QYi=1 QYi=1 QYi=1
(cid:3)
Remark 1.3. In the bilinear case, a version of this result was proved in [4, Theo-
rem 2.6]. However, our proof here is considerably simpler.
Theorem 1.6 has two corollaries. The first when p = 2 is part of the folklore of
harmonic analysis, but we have not been able to find a proof in the literature. We
include this result since it bears a resemblance to Theorem 1.6 below.
Corollary 1.4. For 1 < p < ∞, if a weight w is such that w, w1−p′ ∈ A , then
∞
w ∈ A .
p
Proof. Let w1 = wp1, w2 = w−p1, and s1 = p, s2 = p′. Then by Lemma 1.2, wi ∈ RHsi,
and inequality (1.1) becomes the A condition. (cid:3)
p
Our second corollary is a version of Theorem 1.1 that is particularly useful for
applications to multilinear weights.
Corollary 1.5. Let 1 < p ,...,p < ∞, 0 < p < ∞ be such that m 1 = 1, and
1 m i=1 pi p
let w~ = (w1,...,wm) be a vector of weights. If wi ∈ A∞, 1 ≤ i ≤Pm, then for all
cubes Q,
m p m
pi p
− w dx ≤ C− wpi dx.
(cid:18)ˆ i (cid:19) ˆ i
Yi=1 Q QYi=1
A MULTILINEAR REVERSE HO¨LDER INEQUALITY 3
p
Proof. By Lemma 1.2, for each i, since wi ∈ A∞ we have wipi ∈ RHpi. The desired
p
inequality now follows from Theorem 1.1 with s = pi. (cid:3)
i p
We now give two applications of Theorem 1.1 (or more properly, Corollary 1.5)
to the theory of multilinear weights. We first consider the structure of multilinear
A weights. To put our result into context, we briefly sketch the history of multilin-
p~
ear weighted norm inequalities. A multilinear Caldero´n-Zygmund singular integral
operator T is an m-linear operator with kernel K such that
m
~
T(f)(x) = K(x,y ,...,y ) f (y )dy ...dy ,
ˆ 1 m i i 1 m
Rmn Yi=1
where f~ = (f ,...,f ), f ∈ C∞(Rn) and x 6∈ m supp(f ), and the kernel satisfies
i m i c i=1 i
certain regularity estimates. (Precise definitionTs are not necessary for our purposes;
seeGrafakosandTorres[8]fordetailsandfurtherreferences.) These operatorssatisfy
the following Lp space estimates: if 1 < p ,...,p < ∞ and 1 = 1, then
1 m p pi
P
m
~
kT(f)k ≤ C kf k .
Lp i Lpi
Yi=1
Weighted norm inequalities for multilinear singular integral operators were first
considered by Grafakos and Torres [7]. They showed that if p = min(p ,...,p )
0 1 m
and w ∈ A , then
p0
m
~
kT(f)k ≤ C kf k .
Lp(w) i Lpi(w)
Yi=1
This result was generalized by Grafakos and Martell [6], who showed that if w ∈ A
i pi
p
and w = wpi, then
i
Q
m
~
(1.2) kT(f)k ≤ C kf k .
Lp(w) i Lpi(wi)
Yi=1
The optimal class of weights for which (1.2) holds was found by Lerner, et al. [10].
p
Given p~ = (p ,...,p ), w~ = (w ,...,w ) and w = wpi, w~ ∈ A if
1 m 1 m i p~
Q
1 m 1
p 1−p′ p′i
[w~] = sup − wdx − w idx < ∞.
Ap~ (cid:18)ˆ (cid:19) (cid:18)ˆ i (cid:19)
Q Q Yi=1 Q
4 DAVIDCRUZ-URIBE,OFS ANDKABE MOEN
It follows at once by H¨older’s inequality that if w ∈ A , then w~ ∈ A . Such weights
i pi p~
are referred to as product A weights.1 However, in [10] they constructed examples
p~
of weights w~ ∈ A such that the weights w 6∈ L1 and so not in A .
p~ i Loc pi
Here we prove that if we impose some additional regularity on the weights w , then
i
the weight w~ is a product weight.
Theorem1.6. Given 1 < p ,...,p < ∞ and 1 = 1, supposew~ = (w ,...,w ) ∈
1 m p pi 1 m
Ap~. If wi ∈ A∞, 1 ≤ i ≤ m, then wi ∈ Api. P
Proof of Theorem 1.6. By H¨older’s inequality, for every i and every cube Q,
p p
pi 1−p′ p′i
1 ≤ − w dx − w idx .
(cid:18)ˆ i (cid:19) (cid:18)ˆ i (cid:19)
Q Q
Thus, if we fix a value of i,
− w dx − w1−p′idx pi−1 pp′i = − w dx ppi − w1−p′idx pp′i
(cid:20)ˆ i (cid:18)ˆ i (cid:19) (cid:21) (cid:18)ˆ i (cid:19) (cid:18)ˆ i (cid:19)
Q Q Q Q
m p p
pi 1−p′ p′i
≤ − w dx − w idx .
(cid:18)ˆ i (cid:19) (cid:18)ˆ i (cid:19)
Yi=1 Q Q
By Corollary 1.5,
m p p
− w dx pi − w1−p′idx p′i
(cid:18)ˆ i (cid:19) (cid:18)ˆ i (cid:19)
Yi=1 Q Q
m m p
≤ C − wppi dx − w1−p′idx p′i ≤ C[w~] .
(cid:18)ˆ i (cid:19) (cid:18)ˆ i (cid:19) Ap~
QYi=1 Yi=1 Q
If we combine these two inequalities, we see that w ∈ A . (cid:3)
i pi
It is tempting to speculate that the condition w ∈ A in Theorem 1.6 can be
i ∞
weakened to w ∈ L1 . This, however, is false, and we construct a counter-example.
i Loc
We note in passing that our example also gives an example of a weight w~ ∈ A that
p~
is not a product weight; our construction is somewhat simpler than that given in [10,
Remark 7.2].
For simplicity we consider the bilinear case m = 2 on the real line and we will use
the characterization w~ ∈ Ap~ if and only if w ∈ A2p and σi = w1−p′i ∈ A2p′ (see [10,
i
1If the weights w are taken as prior, then this name makes sense. If the weight w is taken
i
as prior, then it makes more sense to refer to these weights as “factored” A weights. However,
p~
“product weights” has become the standard terminology.
A MULTILINEAR REVERSE HO¨LDER INEQUALITY 5
Theorem 3.6]). Let
χ (x)
[−1,1]
w (x) = +χ (x) and w (x) = 1.
1 |x|log(e/|x|)2 [−1,1]c 2
p
First, we have that w = wp1 belongs to A ⊂ A . To see this, notethat for0 < a < 1
1 1 2p
and b > 0, if we define
χ (x)
[−1,1]
v(x) = +χ (x),
|x|alog(e/|x|)b [−1,1]c
then v ∈ A . To show this, it suffices to check the A condition on intervals [0,t],
1 1
0 < t < 1 (cf. [3]), but in this case it is straightforward to show that
t ds 1
(1.3) ≈ .
ˆ salog(e/s)b ta−1log(e/t)b
0
(Just compare the integrand to the derivative of the righthand side.) Since p/p < 1
1
we have that w ∈ A1. Next we will show that the weight σ1 = w11−p′1 belongs to A2p′1.
To see this, consider the dual weight (dual in the class A(2p1)′) σ11−(2p′1)′ which is given
by
σ1(x)1−(2p′1)′ = w1(x)(p′1−1)((2p′1)′−1) = w1(x)2pp′1′1−−11.
Again, since 2pp′1′1−−11 < 1 the calculation (1.3) shows that σ11−(2p′1)′ ∈ A1 ⊂ A(2p′1)′ which
impliesσ1 ∈ A2p′1. Thisshowsw~ ∈ Ap~. However, wi 6∈ Api: sincew1 ∈ L1Loc\ p>1LpLoc
it cannot satisfy a reverse H¨older inequality and so is not in A∞. S
Remark 1.7. It would be interesting to have an intrinsic characterization of those A
p~
weights that can be factored as product weights.
Our second application of Corollary 1.5 is to two-weight multilinear norm inequal-
ities. In [10], the authors introduced a new multilinear maximal function as a tool to
prove the estimates (1.2):
m
~
M(f)(x) = sup − |f(y )|dy .
Q∋xYi=1(cid:18)ˆQ i i(cid:19)
They proved that
m
~
kM(f)k ≤ C kf k
Lp(w) i Lpi(wi)
Yi=1
6 DAVIDCRUZ-URIBE,OFS ANDKABE MOEN
if and only if w~ ∈ A . This result led naturally to the question of proving two-weight
p~
inequalities for M: more precisely, inequalities of the form
m
~
(1.4) kM(fσ)k ≤ C kf k ,
Lp(u) i Lpi(σi)
Yi=1
~
where fσ = (f σ ,...,f σ ). Chen and Damia´n [2] defined a multilinear analog of
1 1 m m
the Sawyer testing condition for the Hardy-Littlewood maximal operator: (u,~σ) ∈ S
p~
if there exists a constant C such that for all cubes Q,
1 m
p 1
(cid:18)ˆ M(~σχQ)pudx(cid:19) ≤ C σi(Q)pi.
Q Yi=1
The S condition is necessary for the two weight norm inequality (1.4), but it is not
p~
known if it is also sufficient. Chen and Damia´n [2, Theorem 3.2] proved that it is
sufficient if they also assume that the weights satisfy a multilinear reverse H¨older
condition: for all cubes Q,
m p m
pi p
(1.5) − σ dx ≤ C− σpi dx.
(cid:18)ˆ i (cid:19) ˆ i
Yi=1 Q QYi=1
Therefore, by Corollary 1.5, if we assume the weights σ are in A , then we get that
i ∞
the condition S is sufficient the two weight norm inequality (1.4).
p~
Corollary 1.8. Suppose u,σ ,...,σ are weights, 1 < p ,...,p < ∞, and p is
1 m 1 m
defined by 1 = m 1. If σ ∈ A and (u,~σ) ∈ S , then the two-weight norm
p i=1 pi i ∞ p~
inequality (1.4) hPolds.
Remark 1.9. It is an open question to characterize the m-tuples of weights for
which (1.5) holds. A version of this question was posed in [4, Theorem 2.7]; they
showed that when m = 2, if (1.1) holds and w ∈ RH , then w ∈ RH .
1 s1 2 s2
Remark 1.10. For an additional application of Corollary 1.5, see [1], where it is
used to prove necessary conditions on b for the bilinear commutators [b,T] to satisfy
i
weighted norminequalities, where T isa bilinear Caldero´n-Zygmund singular integral
or a bilinear fractional integral operator.
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DavidCruz-Uribe,OFS,DepartmentofMathematics,UniversityofAlabama,Tusca-
loosa, AL 35487, USA
E-mail address: [email protected]
Kabe Moen, Department of Mathematics, University of Alabama, Tuscaloosa, AL
35487, USA
E-mail address: [email protected]