Kato-Ponce inequalities on weighted and variable Lebesgue spaces

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by  David Cruz-Uribe

KATO-PONCE INEQUALITIES ON WEIGHTED AND VARIABLE
6 LEBESGUE SPACES
1
0
2 DAVID CRUZ-URIBE, OFS AND VIRGINIA NAIBO

y
a
M Abstract. WeprovefractionalLeibnizrulesandrelatedcommutatorestimatesin
the settings of weightedand variableLebesgue spaces. Our main tools are uniform

3 weighted estimates for sequences of square-function-type operators and a bilinear
2 extrapolation theorem. We also give applications of the extrapolation theorem to
theboundednessonvariableLebesguespacesofcertainbilinearmultiplieroperators
]
P and singular integrals.
A
.
h
t
a 1. Introduction and main results
m
For s ≥ 0, the inhomogeneous and homogeneous s-th differentiation operators Js
[
and Ds, respectively, are defined via the Fourier transform as
2
v J\s(f)(ξ) = (1+|ξ|2)2sf(ξ) and D\s(f)(ξ) = |ξ|sf(ξ).
4
5
The following inequalities, known as Kato-Ponce inequalities or fractional Leibniz
8
6 rules, hold for such operators and forbf,g ∈ S(Rn): b
0
(1.1) kDs(fg)k . (kDsfk kgk +kfk kDsgk ),
1. Lr Lp1 Lq1 Lp2 Lq2
0 (1.2) kJs(fg)k . (kJsfk kgk +kfk kJsgk ),
Lr Lp1 Lq1 Lp2 Lq2
6
1 where 1 < p ,p ,q ,q ≤ ∞, 1 < r ≤ ∞, 1 = 1 + 1 = 1 + 1, s > max{0,n(1 −1)}
: 1 2 1 2 2 r p1 q1 p2 q2 r
v or s is a non-negative even integer, and the implicit constants depend only on the
i
X parameters involved. The cases 1 < r < ∞, 1 < q1,p2 ≤ ∞, 1 < p1,q2 < ∞
r and s > 0 for such estimates have been known for a long time and go back to the
a
pioneering work in [5, 27, 28] for the study of the Euler, Navier-Stokes and Korteweg-
de Vries equations. Very recently in[21] a different approach was used to extend (1.1)
and (1.2) to the range 1 < r ≤ 1 with s > max{0,n(1 −1)} or s a non-negative even
2 r
Date: May 24, 2016.
2010 Mathematics Subject Classification. Primary: 42B25, 42B35. Secondary: 42B20, 46E35,
47G40.
Key words and phrases. Kato-Ponce inequalities, fractional Leibniz rules, weights, variable
Lebesgue spaces, bilinear extrapolation.
The firstauthoris supportedbythe NSFunder grantDMS 1362425,andby researchfunds from
the Dean of the College of Arts & Sciences. The second author is supported by the NSF under
grant DMS 1500381.
1
2 DAVID CRUZ-URIBE,OFS ANDVIRGINIANAIBO
integer. (Also see [3] for (1.2) with 1 < r ≤ 1 and s > n, and [34] for (1.1).) The
2
case r = ∞ was settled in [4] (see also [19]).
In [27], an important tool in the study of the Cauchy problem for the Euler and
Navier-Stokes equations in the setting of Lp-based Sobolev spaces was the following
commutator estimate closely related to (1.2):
kJs(fg)−fJs(g)k . kJsfk kgk +k∇fk Js 1g ,
Lp Lp L∞ L∞ − Lp
where 1 < p < ∞ and s ≥ 0. For other commutator estimates(cid:13)of the (cid:13)sort, see [4] and
(cid:13) (cid:13)
the references it contains.
ThegoalofthispaperistoproveKato-Ponceinequalitiesinthesettingsofweighted
Lebesgue spaces and variable Lebesgue spaces. The techniques employed in the
context of weighted Lebesgue spaces also let us obtain fractional Leibniz rules in
weighted Lorentz spaces and Morrey spaces. To state our two main results, let
S(Rn) denote the Schwartz class of rapidly decreasing functions defined on Rn and
let A denote the Muckenhoupt class of weights. For brevity we will use the notation
p
A . B to mean that A ≤ cB, where c is a constant that may only depend on some of
the parameters and weights used but not on the functions involved. For the notation
used in the statement of Theorem 1.1, we refer the reader to Section 2.
Theorem 1.1. Let 1 < p,q < ∞ and 1 < r < ∞ be such that 1 = 1 + 1. If v ∈ A ,
2 r p q p
w ∈ A , and s > max{0,n(1 −1)} or s is a non-negative even integer, then for all
q r
f,g ∈ S(Rn),
(1.3) kDs(fg)kLr(vprwqr) . kDsfkLp(v)kgkLq(w) +kfkLp(v)kDsgkLq(w),
(1.4) kJs(fg)kLr(vprwqr) . kJsfkLp(v)kgkLq(w) +kfkLp(v)kJsgkLq(w),
and
(1.5) kDs(fg)−fDs(g)k r r
Lr(vpwq)
. kDsfk kgk +k∇fk Ds 1g ,
Lp(v) Lq(w) Lp(v) − Lq(w)
(cid:13) (cid:13)
(1.6) kJs(fg)−fJs(g)k r r (cid:13) (cid:13)
Lr(vpwq)
. kJsfk kgk +k∇fk Js 1g .
Lp(v) Lq(w) Lp(v) − Lq(w)
The implicit constants depend on p, q, s, [v] and [w] . (cid:13) (cid:13)
Ap Aq (cid:13) (cid:13)
The “factored” bilinear weights in Theorem 1.1 were introduced in [22] and further
studied in [20]. In [31] a more general case of bilinear A weights was introduced.
p
We do not know if our result can be proved for these weights. However, the factored
weights are sufficient to apply extrapolation and prove our second main result. For
the notation used in the statement of Theorem 1.2, we refer the reader to Section 3.
KATO-PONCE INEQUALITIES 3
Theorem 1.2. Let p(·),q(·),r(·) ∈ P be exponent functions such that 1 = 1 +
0 r() p()
1 . Suppose further that there exist 1 < p < p and 1 < q < q ·such t·hat
q()
· − −
p(·) ′, q(·) ′ ∈ B. If s > max{0,n(1−1)}, where 1 = 1+1, or s is a non-negative
p q r r p q
(cid:16)even(cid:17)inte(cid:16)ger,(cid:17)then for all f,g ∈ S(Rn),
(1.7) kDs(fg)k . kDsfk kgk +kfk kDsgk ,
Lr(·) Lp(·) Lq(·) Lp(·) Lq(·)
(1.8) kJs(fg)k . kJsfk kgk +kfk kJsgk ,
Lr(·) Lp(·) Lq(·) Lp(·) Lq(·)
(1.9) kDs(fg)−fDs(g)k . kDsfk kgk +k∇fk Ds 1g ,
Lr(·) Lp(·) Lq(·) Lp(·) − Lq(·)
(1.10) kJs(fg)−fJs(g)kLr(·) . kJsfkLp(·) kgkLq(·) +k∇fkLp(·) J(cid:13)(cid:13)s−1g L(cid:13)(cid:13)q(·) .
Remark 1.1. Implicit in the proof of Theorem 1.1 is the fact that,(cid:13)in the(cid:13)case w = v,
(cid:13) (cid:13)
it is possible to have different pairs p, q for each term on the righthand side of (1.3),
(1.4), (1.5) and (1.6). Also, the proof of Theorem 1.2 allows for different pairs of
exponents p(·),q(·) for each term on the righthand side of the inequalities for variable
Lebesgue spaces.
Remark 1.2. As we will explain in Section 3, the hypotheses of Theorem 1.2 imply
that p(·) and q(·) are both bounded, and this restriction is intrinsic to our proof.
However, in the scale of variable Lebesgue spaces it is possible to have unbounded
exponents oreven exponents thatareequal toinfinityonsetsofpositive measure. We
conjecture that there is a version of Theorem 1.2 that allows unbounded exponents
and that includes the endpoint inequality in [4] as a special case.
Our proofs of (1.3) and (1.4) in Theorem 1.1 exploit the ideas used in [21] to
prove their unweighted counterparts (1.1) and (1.2). This approach requires us to use
weighted estimates for a certain family of square-functions; moreover we need to have
good control on their norms. The central step in this argument is Theorem 2.1 where
we obtain uniform estimates for such operators. We note that Theorem 2.1 improves
theestimategottenin[21, Corollary1]; theirproofreliesonaweak(1,1)estimateand
interpolation. Thisisenoughfortheproofoftheunweightedestimates(1.1)and(1.2),
but not for their weighted analogs of Theorem 1.1. We instead prove the necessary
weighted strong type estimates directly. A novel feature of our approach is that we
avoid using the maximal operator to estimate convolution operators and instead use
an argument based on averaging operators that was developed in a different context
in [11]. We do not know if the weak type estimates in weighted or variable Lebesgue
spaces that correspond to the weak-type estimates proved in [21] hold.
The proof of the weighted commutator estimates (1.5) and (1.6) in Theorem 1.1
relies on a decomposition of the operators Ds(fg) and Js(fg) given in [27] for the
case of Js as well as weighted estimates for certain bilinear operators from [20, 22].
4 DAVID CRUZ-URIBE,OFS ANDVIRGINIANAIBO
The variable Lebesgue space estimates in Theorem 1.2 are a consequence of Theo-
rem 1.1 and a bilinear extrapolation theorem (Theorem 3.1) that allows us to obtain
bilinear estimates in variable Lebesgue spaces from bilinear estimates in weighted
Lebesgue spaces. This result generalizes both the extrapolation theorem for variable
Lebesgue spaces in [9] (see also [8]) and the bilinear extrapolation theorem in [20].
Our result is interesting in its own right as it lets us easily prove a number of other
bilinear estimates in the variable Lebesgue space setting. To illustrate this, we use
Theorem 3.1 to prove estimates in variable Lebesgue spaces for several types of bi-
linear operators, including bilinear Caldero´n-Zygmund operators, bilinear multiplier
operators with symbols that have limited smoothness, and certain rough bilinear
singular integrals.
The remainder of this paper is organized as follows. In Section 2 we present def-
initions and basic results about weights and weighted norm inequalities, and then
state and prove Theorem 2.1. In Section 3 we give the necessary definitions and
background information on variable Lebesgue spaces, and prove Theorem 3.1. In
Section 4 we state and prove the applications of Theorem 3.1 to other bilinear oper-
ators. In Section 5 we prove Theorems 1.1 and 1.2. Finally, in Section 6 we present
Kato-Ponce inequalites in the settings of weighted Lorentz spaces and Morrey spaces
and explain how they follow from tools developed for the proof of Theorem 1.1.
Throughout this paper, all notation will be standard or will be defined as needed.
We will let Z be the set of integers, N the set of natural numbers, and N = N∪{0}.
0
Unless explicitly indicated otherwise, all function spaces that appear will consist of
complex-valued functions defined on Rn.
Acknowledgements: The authors thank the referee for the careful reading of the
manuscript.
2. Weighted square function estimates
The main result in this section is Theorem 2.1, which gives a uniform weighted
estimate for certain families of square-functions. This estimate is in turn used in the
proofs of our main results. We divide this section into three parts. In the first we
give some definitions and state some known results. In the second we state and prove
Theorem 2.1. The proof appears relatively short, but it depends on several technical
propositions which we prove in the final part.
Preliminary results. By a weight w we mean a non-negative, locally integrable
function defined on Rn. Given 0 < p < ∞, let Lp(w) denote the class of complex-
valued, measurable functions f defined on Rn such that
1
p
kfk = |f(x)|pw(x)dx < ∞.
Lp(w)
(cid:18)ZRn (cid:19)
KATO-PONCE INEQUALITIES 5
For brevity, when w = 1 we write Lp and kfk instead of Lp(w) and kfk .
Lp Lp(w)
Given a locally integrable function f on Rn and a set E ⊂ Rn of positive measure,
define
1
− f dy = f(y)dy.
|E|
ZE ZE
The Hardy-Littlewood maximal operator M is defined as follows: for each x ∈ Rn,
let
M(f)(x) = sup− |f(y)| dyχ (x),
B
B ZB
where the supremum is taken over all Euclidean balls in Rn.
Given 1 < p < ∞, the Muckenhoupt class A consists of all weights w such that
p
p 1
1 −
[w]p = sup − w(y)dy − w−p−1 < ∞,
Q (cid:18)ZQ (cid:19)(cid:18)ZQ (cid:19)
where the supremum is taken over all cubes Q ⊂ Rn. The class A consists of all
1
weights w such that M(w)(x) . w(x) for almost every x ∈ Rn; we define
[w] = sup − w(y)dy w 1 ,
A1 − L∞(Q)
Q (cid:18)ZQ (cid:19)
(cid:13) (cid:13)
where the supremum is taken over all cubes Q ⊂(cid:13)Rn. N(cid:13)ote that A ⊂ A for 1 < p <
1 p
∞ and [w] ≤ [w] .
Ap A1
It is well known that for 1 < p < ∞, the Muckenhoupt condition characterizes the
weights w such that M is bounded on Lp(w) (e.g., see [15]). Below, we will need the
vector-valued version of this result, also referred to as the weighted Fefferman-Stein
inequality. For a proof, see [1, 10].
Lemma 2.A. Given 1 < p,q < ∞ and w ∈ A , for all sequences {f } of locally
p k
integrable functions defined on Rn,
1 1
q q
(2.1) |M(f )|q . |f |q ,
k k
(cid:13)(cid:18)k Z (cid:19) (cid:13)Lp(w) (cid:13)(cid:18)k Z (cid:19) (cid:13)Lp(w)
(cid:13) X∈ (cid:13) (cid:13) X∈ (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
where the implicit(cid:13)constant depends o(cid:13)n p, q, a(cid:13)nd [w]Ap. (cid:13)
The next result we recall is a weighted Littlewood-Paley estimate. To state it, let
f denote the Fourier transform of a tempered distribution on Rn: more precisely, for
all ξ ∈ Rn and f ∈ S(Rn),
b
f(ξ) = f(x)e 2πixξdx.
− ·
Rn
Z
b
6 DAVID CRUZ-URIBE,OFS ANDVIRGINIANAIBO
Lemma 2.B. Let ϕ ∈ S(Rn) be such that supp(ϕ) ⊂ {ξ ∈ Rn : c < |ξ| < c } for
1 2
some 0 < c < c < ∞. Set ϕ (x) = 2knϕ(2kx) and suppose further that for some
1 2 k
constant c > 0 and for all ξ 6= 0,
ϕ b
ϕ(2 kξ) 2 = c .
− ϕ
k Z
X∈ (cid:12) (cid:12)
Then for every 1 < p < ∞, w ∈ A ,(cid:12)abnd f ∈(cid:12)Lp(w),
p
1
2
(2.2) |ϕ ∗f|2 ∼ kfk ,
k Lp(w)
(cid:13)(cid:18)k Z (cid:19) (cid:13)Lp(w)
(cid:13) X∈ (cid:13)
where the implicit const(cid:13)ants depend on ϕ a(cid:13)nd [w] but do not depend on f.
(cid:13) (cid:13) Ap
For a proof, see [37, Proposition 1.9] and the comment that follows; see also [29].
Statement and proof of Theorem 2.1. In Lemma 2.B the implicit constant in
(2.2)depends onthefunctionϕ; inparticular, ifwereplaceϕbyatranslationϕ(·+z),
z ∈ Rn, then we do not know a priori whether the constants will depend on z. Our
main result in this section shows that in this case they do not.
Theorem 2.1. Let Ψ ∈ S(Rn) be such that supp(Ψ) ⊂ {ξ ∈ Rn : c < |ξ| < c }
1 2
for some 0 < c1 < c2 < ∞. Given a sequence z¯ = {zk,m}k Z,m Zn ⊂ Rn, define
Ψz¯ (x) = 2knΨ(2k(x + z )) for x ∈ Rn, m ∈ Zbn and k ∈∈ Z∈. Then for every
k,m k,m
1 < p < ∞, w ∈ A , and f ∈ Lp(w),
p
1
Ψz¯ ∗f 2 2 . kfk ,
k,m Lp(w)
(cid:13)(cid:18)k Z (cid:19) (cid:13)Lp(w)
(cid:13) X∈ (cid:12) (cid:12) (cid:13)
where the implicit cons(cid:13)tants d(cid:12)epend on(cid:12)Ψ an(cid:13)d [w] but are independent of m and z¯.
(cid:13) (cid:13) Ap
Remark 2.1. As we noted in the Introduction, Theorem 2.1 improves the correspond-
ing unweighted estimate from [21, Corollary 1]. There, the authors showed that for
the particular sequence z = 2 km, k ∈ Z and m ∈ Zn, we have that
k,m −
1
Ψz¯ ∗f 2 2 . log(1+|m|)kfk .
k,m Lp
(cid:13)(cid:18)k Z (cid:19) (cid:13)Lp
(cid:13) X∈ (cid:12) (cid:12) (cid:13)
They proved this i(cid:13)nequali(cid:12)ty by me(cid:12)ans o(cid:13)f an unweighted, weak (1,1) inequality and
(cid:13) (cid:13)
then interpolating with the unweighted L2 estimate. They prove the weak (1,1)
estimate by showing that the operator satisfies a vector-valued H¨ormander condition;
such a condition is not sufficient for proving weighted norm inequalities: see [33]. Our
proof of Theorem 2.1 makes use of Rubio de Francia extrapolation, and so does not
yield a weighted endpoint estimate. We do not know if such an estimate holds, either
with a constant independent of m or with a constant of order log(1+|m|).
KATO-PONCE INEQUALITIES 7
Remark 2.2. We refer the reader to Section 6 for versions of Theorem 2.1 in the
settings of weighted Lorentz spaces and Morrey spaces.
The proof of Theorem 2.1 requires two propositions which we state here; their
proofs are given in the final part of this section.
Proposition 2.2. Let ϕ ∈ S(Rn) be such that supp(ϕ) ⊂ {ξ ∈ Rn : c < |ξ| < c }
1 2
for some 0 < c < c < ∞ and ϕ(2 kξ) 2 = c for some c > 0 and for all
1 2 k Z − ϕ ϕ
ξ 6= 0. Set ϕ (x) = 2knϕ(2kx) for k ∈∈ Z. If 1 < p < ∞band w ∈ A , then
k P (cid:12) (cid:12) p
(cid:12)b (cid:12) 1
2
ϕ ∗f . |f |2
k k k
(cid:13)k Z (cid:13)Lp(w) (cid:13)(cid:18)k Z (cid:19) (cid:13)Lp(w)
(cid:13)X∈ (cid:13) (cid:13) X∈ (cid:13)
for all sequences {fk}(cid:13)(cid:13)k Z ⊂ Lp(w)(cid:13)(cid:13)such tha(cid:13)(cid:13)t the righthand(cid:13)(cid:13)side is finite. The implicit
constant depends only∈on ϕ and [w] .
Ap
Proposition 2.3. Given Ψ ∈ S(Rn) and a sequence z¯= {zk,m}k Z,m Zn ⊂ Rn, define
Ψz¯ (x) = 2knΨ(2k(x + z )) for x ∈ Rn, m ∈ Zn and k ∈ Z.∈If 1∈ < p < ∞ and
k,m k,m
w ∈ A , then for all f ∈ Lp(w),
p
1
sup Ψz¯ ∗f . [w]p kfk ,
k,m Z k,m Lp(w) Ap Lp(w)
∈
(cid:13) (cid:13)
where the implicit constant d(cid:13)epends on(cid:13) Ψ and is independent of z¯ and w.
Proof of Theorem 2.1. Our proof is inspired by the argument in [16], but it is simpler
becauseofthestructureofouroperator. ByRubiodeFranciaextrapolation(see[10]),
it is enough to prove the desired inequality when p = 2.
Set Tmz¯,ε(f) = k ZεkΨzk¯,m ∗ f, where m ∈ Zn and ε = {εk}k Z with εk = ±1
for each k ∈ Z. Wit∈hout loss of generality we may assume c =∈ 1 and c = 2.
P 1 2 2
Fix ϕ ∈ S(Rn) such that ϕ ≡ 1 on {ξ ∈ Rn : 1 < |ξ| < 2}, ϕ is supported on
2
an annulus, and ϕ(2 kξ) 2 = c for all ξ 6= 0 for some constant c . Define
k Z − ϕ ϕ
ϕ (x) = 2knϕ(2kx). T∈hen εbΨz¯ ∗f = ϕ ∗ε Ψz¯ ∗ϕ ∗f pointwbise for all k ∈ Z,
k P (cid:12) k k,m(cid:12) k k k,m k
m ∈ Zn, and f ∈ S′(Rn)(cid:12).bBy Pro(cid:12)positions 2.2 and 2.3, and by Lemma 2.B, it follows
that for w ∈ A , f ∈ L2(w), and all z¯ and ε,
2
2
Tz¯ f 2 = ϕ ∗ε Ψz¯ ∗ϕ ∗f
m,ε L2(w) k k k,m k
(cid:13)k Z (cid:13)L2(w)
(cid:13) (cid:13) (cid:13)X∈ (cid:13)
(cid:13) (cid:13) (cid:13) . Ψz¯ ∗ϕ ∗f(cid:13) 2 . kϕ ∗fk2 . kfk2 ;
(cid:13) k,m k (cid:13) L2(w) k L2(w) L2(w)
k Z k Z
X∈ (cid:13) (cid:13) X∈
(cid:13) (cid:13)
8 DAVID CRUZ-URIBE,OFS ANDVIRGINIANAIBO
theimplicit constants dependonlyonϕ, Ψ, and[w] . Inotherwords, wehave shown
A2
that Tz¯ is bounded on L2(w) for every w ∈ A with the operator norm controlled
m,ε 2
by a constant independent of m, ε and z¯. Clearly, the same argument shows that the
operators Tz¯,+ and Tz¯, defined as Tz¯ but with k ∈ N and −k ∈ N, respectively,
m,ε m,−ε m,ε 0
instead of k ∈ Z, are bounded on L2(w) with constants independent of the same
quantities.
We can now argue as in [15, p. 177]. Let {rk}k N0 be the system of Rademacher
functions. That is, define r (t) = −1 for 0 ≤ t <∈1, r (t) = 1 for 1 ≤ t < 1, and
0 2 0 2
extend it as a periodic function on R. Then define r (t) = r (2kt) for 0 ≤ t < 1.
k 0
Recall that {rk}k N0 is an orthonormal system in L2([0,1]) and we have that for all
{a } ∈ ℓ2, ∈
k
1
2
∞ ∞
a r = |a |2 .
k k k
(cid:13)k=0 (cid:13)L2([0,1]) k=0 !
(cid:13)X (cid:13) X
Therefore, if we set εt =(cid:13)(cid:13){rk(t)}k∈N(cid:13)(cid:13)0 for 0 ≤ t < 1, then we have that
1 2 1 1 2
Ψz¯ ∗f(x) 2 = ∞ r (t)Ψz¯ ∗f(x) dt+ r (t)Ψz¯ ∗f(x) dt
k,m k k,m k k,m
Xk∈Z(cid:12) (cid:12) Z0 (cid:12)(cid:12)Xk=0 (cid:12)(cid:12) Z0 (cid:12)(cid:12)kX=−∞ − (cid:12)(cid:12)
(cid:12) (cid:12) 1(cid:12) 1(cid:12) (cid:12) (cid:12)
= (cid:12)(cid:12)Tmz¯,,+εtf(x) 2 dt+ (cid:12)(cid:12)Tmz¯,,−εtf(x)(cid:12)(cid:12)2 dt. (cid:12)(cid:12)
Z0 Z0
If we fix w ∈ A2 and com(cid:12)(cid:12)pute the (cid:12)(cid:12)L2(w) norm(cid:12)(cid:12), then we(cid:12)(cid:12)get
1 2
Ψz¯ ∗f(x) 2 2
k,m
(cid:13)(cid:18)k Z (cid:19) (cid:13)L2(w)
(cid:13) X∈ (cid:12) (cid:12) (cid:13)
(cid:13) (cid:12) (cid:12) (cid:13) 1 1
(cid:13) =(cid:13) Tz¯,+f 2 dt+ Tz¯, f 2 dt . kfk2 ,
m,εt L2(w) m,−εt L2(w) L2(w)
Z0 Z0
with the implicit constant indep(cid:13)endent(cid:13)of m, z¯ and f.(cid:13) (cid:13) (cid:3)
(cid:13) (cid:13) (cid:13) (cid:13)
Proof of Propositions 2.2 and 2.3.
Proof of Proposition 2.2. Fix w, ϕ and {fk}k Z as in the hypotheses. As before, we
can assume without loss of generality that in∈defining the support of ϕ, c = 1 and
1 2
c = 2. For all N ∈ N , define
2 0
F = ϕ ∗f . b
N ℓ ℓ
ℓ N
|X|≤
Then for all N > N and all k ∈ Z we have that
2 1
ϕ ∗(F −F ) = ϕ ∗ϕ ∗f .
k N2 N1 k ℓ ℓ
N1<X|ℓ|≤N2
KATO-PONCE INEQUALITIES 9
Comparing supports on the Fourier transform side, it follows that
0 if |k| ≤ N −1 or |k| ≥ N +2,
1 2
1 ϕ ∗ϕ ∗f if N +2 ≤ |k| ≤ N −1,
 ℓ= 1 k k ℓ k ℓ 1 2
 0 − ϕ ∗ϕ − ∗f − if |k| = N +1,
ϕk ∗(FN2 −FN1) = PP1ℓℓ==0−1ϕkk∗ϕkk−ℓ ∗ℓ fkkℓ−ℓ if |k| = N21,
 − −
ϕ ∗ϕ ∗f if |k| = N ,
Pk k+1 k+1 1
ϕ ∗ϕ ∗f if |k| = N +1.
 k k 1 k 1 2
 − −

Recall that for all k∈ Z, |ϕk ∗g| . M(g) with a constant that depends only on
ϕ. (See [15].) Since M is bounded on Lp(w), it follows that F ∈ Lp(w) for all N.
N
Moreover, by Lemma 2.B, this maximal operator inequality and Lemma 2.A, we get
that
1 1
2
kF −F k . |ϕ ∗ϕ ∗f |2
N1 N2 Lp(w) k k ℓ k ℓ
− −
ℓX=−1(cid:13)(cid:13)(cid:18)N1≤|Xk|≤N2+1 (cid:19) (cid:13)(cid:13)Lp(w)
1 (cid:13) 1 (cid:13) 1
(cid:13) 2 (cid:13) 2
. |M(M(f ))|2 . |f |2 .
k ℓ k
−
ℓX=−1(cid:13)(cid:13)(cid:18)N1≤|Xk|≤N2+1 (cid:19) (cid:13)(cid:13)Lp(w) (cid:13)(cid:13)(cid:18)N1−1≤X|k|≤N2+2 (cid:19) (cid:13)(cid:13)Lp(w)
(cid:13) (cid:13) (cid:13) (cid:13)
The last(cid:13)term in the above chain of ineq(cid:13)ualities c(cid:13)onverges to 0 as N ,N (cid:13)tend to
1 2
infinity; therefore, we have that {FN}N N0 is a Cauchy sequence in Lp(w). Hence,
ϕ ∗f converges in Lp(w). Moreov∈er, the same argument as before also shows
k Z k k
that∈
P
1
2
ϕ ∗f . |f |2 .
k k k
(cid:13) k N (cid:13)Lp(w) (cid:13)(cid:18) k N+2 (cid:19) (cid:13)Lp(w)
(cid:13)|X|≤ (cid:13) (cid:13) | |X≤ (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
with the implicit c(cid:13)onstant indep(cid:13)endent o(cid:13)f the sequence {fk(cid:13)}k Z. If we let N tend to
∈
infinity we get the desired inequality. (cid:3)
If we did not care about the size of the constant, we could easily prove a version of
Proposition 2.3 using the pointwise inequality Ψz¯ ∗f(x) ≤ CM(f)(x). However,
k,m
the constant may depend on z¯ and blow up with k and m. Moreover, the uniform
(cid:12) (cid:12)
pointwise estimate Ψz¯ ∗f(x) ≤ C M(f)(x(cid:12)−z ) onl(cid:12)y gives the desired result
k,m Ψ k,m
if w is constant since otherwise Lp(w) is not translation invariant.
(cid:12) (cid:12)
Therefore, to prov(cid:12)e the desire(cid:12)d uniform estimate in weighted spaces we will use an
argument developed in [11] for matrix weights. The proof requires two lemmas. The
first was proved in [11]; for the convenience of the reader we give the proof for the
scalar case here.
10 DAVID CRUZ-URIBE,OFS ANDVIRGINIANAIBO
Lemma 2.4. If 1 < p < ∞ and w ∈ A , then for every ball B and f ∈ Lp(w),
p
1
|B|−1(χB ∗f) Lp(w) . [w]App kfkLp(w),
with the implicit consta(cid:13)nt independent(cid:13)of w.
(cid:13) (cid:13)
Proof. Let p, w and f be as in the hypotheses. It is enough to prove this estimate for
cubes with sides parallel to the coordinate axes instead of balls: given any ball B, if
Q is the smallest cube that contains B, then |B|−1(χB ∗f) ≤ Cn|Q|−1(χQ ∗|f|).
Fix a cube Q, denote its side length by ℓ(Q) and define the cubes Q = Q+ℓ(Q)m
m
(cid:12) (cid:12)
for m ∈ Zn. Then {Qm}m Zn is a pairwise dis(cid:12)joint partition(cid:12)of Rn. Further, we can
dividethefamily{3Qm}m ∈Zn into3n familiesQJ,J ∈ {1,··· ,3n},ofpairwisedisjoint
∈
cubes (3Q denotes the cube with the same center as Q and side length 3ℓ(Q )).
m m m
Then for all x ∈ Q ,
m
|Q|−1|(χQ ∗f)(x)| ≤ |Q|−1 |f(y)χQ(x−y)| dy ≤ 3n− |f(y)| dy.
ZRn Z3Qm
Hence,
|Q|−1(χQ ∗f)(x) pw(x)dx
Rn
Z
(cid:12) (cid:12)
(cid:12) = (cid:12)|Q|−1(χQ ∗f)(x) pw(x)dx
m ZnZQm
X∈ (cid:12) (cid:12)
(cid:12) p(cid:12)
. − |f(y)| dy w(x)dx
m ZnZQm(cid:18)Z3Qm (cid:19)
X∈
3n p
1 1
. w(x)dx − |f(y)|w(y)pw(y)−pdy .
XJ=13QXm∈QJ Z3Qm (cid:18)Z3Qm (cid:19)
If we apply H¨older’s inequality, and use the facts that w ∈ A and the cubes in Q
p J
are pairwise disjoint, we get
3n p
. |f(x)|pw(x)dx − w(x)dx − w(x)−pp′ dx p′
XJ=13QXm∈QJ (cid:18)Z3Qm (cid:19)(cid:18)Z3Qm (cid:19)(cid:18)Z3Qm (cid:19)
. [w] kfkp .
Ap Lp(w)
The implicit constants are independent of w and f, and the proof is complete. (cid:3)
The next lemma is also from [11] where it was given implicitly and without proof.