Smoothing Properties of Bilinear Operators and Leibniz-Type Rules in Lebesgue and Mixed Lebesgue Spaces

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by  Jarod Hart

SMOOTHING PROPERTIES OF BILINEAR OPERATORS
AND LEIBNIZ-TYPE RULES IN LEBESGUE AND MIXED
LEBESGUE SPACES
7
1
0
2 JARODHART,RODOLFOH.TORRES,XINFENGWU

n
a Abstract. We prove that bilinear fractional integral operators and
J
similar multipliers are smoothing in the sense that they improve the

0 regularity of functions. We also treat bilinear singular multiplier oper-
1
ators which preserve regularity and obtain several Leibniz-typerules in

the contextsof Lebesgue and mixed Lebesgue spaces.
]
A
C
.
h
1. Introduction
t
a
m Let K be an integral operator of order −ν. That is, let K be of the
ν ν
[ form

1 (1) K f(x)= k (x,y)f(y)dy,
v ν ˆ ν
Rn
1
3 where the kernel satisfies the estimate |k (x,y)| . 1 , for some 0 <
ν |x−y|n−ν
6
ν < n. It is easy to see that K is smoothing, or rather improving, in the
2 ν
0 scale of Lebesgue spaces, in the sense that it maps a Lebesgue space into
. another one with a larger exponent. More precisely,
1
0 K : Lp → Lq
7 ν
1
provided 0 < 1/q = 1/p −ν/n < 1. Under suitable additional regularity
:
v and cancellation conditions (see e.g. [56]), such K is also smoothing in the
ν
i
X Sobolev scale. Namely
r K :Lp → W˙ ν,p
ν
a
where W˙ ν,p is the homogeneous Sobolev space of functions with their deriv-
ative of order ν in Lp (the precise definitions of all function spaces used in
thisarticle aregiven in Section 2below). Thisis astronger smoothingprop-
erty, since by Sobolev embedding W˙ ν,p ⊂ Lq, when p and q are related as
above. Of course, the most classical situation is that of the Riesz potential
operators
1
I f(x)= c f(y)dy,
ν νˆRn |x−y|n−ν
2010 Mathematics Subject Classification. Primary: 42B20; Secondary: 42B15, 47G99.
Key words and phrases. Bilinear operators, multipliers, maximal function, smoothing
properties, fractional derivatives, Leibniz rule, mixed Lebesgue spaces.
1
2 J.HART,R.H.TORRES,X.WU
where the constant c is selected so that the Fourier transform of I f is
ν ν
given by
I f(ξ) = |ξ|−νf(ξ).
ν
It is immediate that by defining Dsf(ξ) = |ξ|sf(ξ) we have for s < ν,
d b
(2) DsI = I .
ν ν−s
d b
Formally, the case ν = 0 in (1) corresponds to Calder´on-Zygmund opera-
tors which are no longer smoothing, but a slight modification of this simple
calculus in (2) still holds for convolution operators. For example for n > 1,
(3) ∂ I f = R f
j 1 j
where for j = 1,...,n, R are the Riesz transforms in Rn given by the
j
multiplier R f(ξ) = −iξ |ξ|−1f(ξ). As operators of order zero, the Riesz
j j
transforms R are not smoothing, but since they commute with derivative,
j
(4) d Ds(Rb f)= R (Dsf),
j j
they preserve both Lebesgue and Sobolev spaces for 1< p < ∞.
Properly interpreted the calculus in (2)–(4) extends not only to other
multiplier operators, but also beyond the convolution case to several classes
of pesodifferential operators and even more general non-convolution opera-
tors of Calder´on-Zygmund type (see e.g the book by Stein [53] for several
results and references to the vast literature in the subject).
In this article, we are interested in stating and proving analogous ver-
sions of (2)–(4) for bilinear multiplier operators, improving and extending
numerous results already in the literature in the subject and uncovering
several completely new ones. The prototypes for our work for 0 < ν < 2n
will be bilinear fractional integral operators, while for ν = 0 they will be
Coifman-Meyer multipliers. We will obtain, however, results for more gen-
eraloperatorsunderminimalregularityassumptionsonthemultiplierwhich
do not allow for pointwise smooth estimates on their corresponding kernels.
The bilinear fractional integral operators are defined for 0 < ν < 2n by
1
I (f,g)(x) = C f(y)g(z)dydz.
ν νˆR2n (|x−y|2+|x−z|2)(2n−ν)/2
The constant C is chosen again so that, using the Fourier transform, we
ν
have the representation
1
I (f,g)(x)= e−ix(ξ+η)f(ξ)g(η)dξdη.
ν ˆR2n (|ξ|2 +|η|2)ν/2
More generally we can consider for 0 ≤ ν < 2n bilbineabr multipliers of the
form
T (f,g)(x) = m (ξ,η)e−ix(ξ+η)f(ξ)g(η)dξdη,
mν ˆ ν
R2n
where
b b
(5) |∂β∂γm (ξ,η)| . (|ξ|+|η|)−ν−|β|−|γ|.
ξ η ν βγ
SMOOTHING BILINEAR OPERATORS AND LEIBNIZ-TYPE RULES 3
Note that we are allowing now ν = 0, which corresponds to the case of
the nowadays classical Coifman-Meyer multipliers. We will actually treat
multipliers where the pointwise regularity estimates in (5) are replaced by
H¨ormander-type ones using only appropriate Sobolev space regularity.
Roughlyspeaking,ifT isabilinearoperatoroforder−ν describedabove,
ν
we will show that T (f,g) has ν more derivatives than f and g (hence it is
ν
smoothing if ν > 0). Our main results could be interpreted by saying that
(6) DsT (f,g) ∼ T (Ds−νf,g)+T (f,Ds−νg),
ν 0 0
where T is an operator of order 0.
0
In making these informal statements precise, we need to review some of
the existing literature alluded to before. Our recount is not intended to be
exhaustive, but we shall rather point out some of the results most closely
related to ours. As it will be clear from our narrative below, there is a
high level of interest in the subject and a very active community working
on similar problems. Several overlapping recent results have been obtained
independently by different authors.
As already mentioned, for ν = 0 the operators in (5) are Coifman-Meyer
multipliers as studied by those authors in [22]-[25]. They are examples of
operators within the multilinear Calder´on-Zygmund theory further devel-
oped by Christ-Journ´e [19], Kenig-Stein [44] and Grafakos-Torres [40]. In
particular, bilinear Calder´on-Zygmund operators are operators of the form
(7) K(f,g)(x) = k(x,y,z)f(y)g(z)dydz
ˆ
R2n
for x ∈/ suppf ∩suppg, where the kernel satisfies the estimates
(8) |DαDβDγk(x,y,z)| . (|x−y|+|x−z|)−2n−|α|−|β|−|γ|,
x y z
and such that they act as the product of functions on Lebesgue spaces, i.e,
K :Lp1 ×Lp2 → Lq
for 1 < p ,p < ∞, 1 + 1 = 1 (appropriate end-point results hold too).
1 2 p1 p2 q
Other examples of these operators are provided by bilinear pesuodifferential
operators of order zero. For m ∈ R, a bilinear pseudodifferential operator
of order m is given by
P (f,g)(x) = a (x,ξ,η)e−ix(ξ+η)f(ξ)g(η)dξdη,
am ˆ m
R2n
where b b
(9) |∂ρ∂β∂γa (x,ξ,η)| .(1+|ξ|+|η|)m−|β|−|γ|.
x ξ η m
B´enyi-Torres [11] showed that for m = 0 these bilinear Calder´on-Zygmund
operators also satisfy for s > 0 and 1/p +1/p = 1/q < 1 the estimate
1 2
kJsPa0(f,g)kLq . kJsfkLp1kgkLp2 +kfkLp1kJsgkLp2,
4 J.HART,R.H.TORRES,X.WU
where Js is the inhomogeneous derivative operator
\
(Jsf)(ξ) = (1+|ξ|2)s/2f(ξ).
Moreover, along the lines of (6), B´enyi-Nahmod-Torres [10] showed that for
b
a symbol of order m > 0,
(10) P (f,g) = P (Jmf,g)+P (f,Jmg),
am b0 c0
for some symbols of order zero b and c , which gives then
0 0
kPam(f,g)kLq . kJmfkLp1kgkLp2 +kfkLp1kJmgkLp2,
for all 1 < p ,p < ∞ and 1 + 1 = 1. This idea goes back to the work of
1 2 p1 p2 q
Kato-Ponce [42]. Similar estimates for more general classes of symbols were
given by B´enyi et al [8] and [9] and Naibo [52]. Several classes of operators
in the H¨ormander classes BSm given by symbols satisfying the differential
ρ,δ
inequalities
(11) |∂α∂β∂γa (x,ξ,η)| . (1+|ξ|+|η|)m+ρ|α|−δ(|β|+|γ|),
x ξ η m
for 0 ≤ ρ,δ ≤ 1 were considered in those works. In particular, it was shown
in [52] that the boundedness Lp1 × Lp1 → Lp with 1/p + 1/p = 1/p,
1 2
1 < p ,p < ∞ of an operator with symbol in a class BSm , automatically
1 2 ρ,δ
impliesitsboundednessonBesov spaceswithpositivesmoothnessandbased
on the same Lp exponents. It was also proved in [52] that the same result is
truefor any bilinear multiplier operator mappingLp1×Lp1 → Lp. A similar
result for multipliers was obtained in [10] in the scale of Sobolev spaces but
with p > 1.
The boundedness properties of the operators I in the scale of Lebesgue
ν
spaces were studied by Kenig-Stein [44]. They showed that
(12) I : Lp1 ×Lp2 → Lq
ν
for 1 < p ,p < ∞, 0 < 1 + 1 − ν = 1, and 0 < ν < 2n. Bernicot et
1 2 p1 p2 n q
al [12] looked at bilinear pseudodifferential operators P with m < 0 and
am
also homogeneous version P˙ , where the estimates in (9) are modified by
am
replacing(1+|ξ|+|η|)with(|ξ|+|η|). Inparticular,theoperatorsI (ormore
ν
generally T satisfying (5)) are homogeneous bilinear pseudodifferential
mν
operators of order m = −ν. The authors in [12] showed, using a calculus
similar to (10), that
(13) kIν(f,g)kLq . kDs−νfkLp1kgkLp2 +kfkLp1kDs−νgkLp2
if 1 < p ,p < ∞, 0 < 1 + 1 − s = 1, and 0 < s < 2n and ν ≤ s. We will
1 2 p1 p2 n q
show that actually
(14) kDsIν(f,g)kLp . kDs−νfkLp1kgkLp2 +kfkLp1kDs−νgkLp2,
if 1 < p ,p < ∞, 1 + 1 = 1, 0 < ν < 2n and s > max(0, n −n). This is
1 2 p1 p2 p p
now a smoothing property on the Sobolev scale and by Sobolev embedding
SMOOTHING BILINEAR OPERATORS AND LEIBNIZ-TYPE RULES 5
an improvement of (13) for some range of the exponents. In particular,
(15) I :Lp1 ×Lp2 → W˙ ν,p
ν
for 1< p ,p < ∞, 0 < 1 + 1 = 1, and 0< ν < 2n, improving (12).
1 2 p1 p2 p
We point out that other smoothing-type estimates have been proved for
the bilinear fractional integral operators before. For example, in [1], Aimar
et al proved that the I maps from products of Lebesgue spaces with appro-
ν
priate indices into certain Campanato-BMO type spaces when 1 + 1 ≤ ν.
p1 p2 n
Such spaces provide the right setting when working on spaces of homoge-
neoustype. More recently, Chaffee-Hart-Oliveira [15] showed usingdifferent
methods that
(16) I : Lp1 ×Lp2 → I (BMO),
ν s
for certain ranges of 1 < p ,p < ∞ and 0≤ s < ν satisfying 1 + 1 = ν−s.
1 2 p1 p2 n
Note that (15) is also an improvement of (16) whenever ν − s < n since
W˙ ν,p ⊂ I (BMO) if 1 = ν−s < 1. The results in [15], however, apply to a
s p n
larger rangeof exponents and also to more general operators that we cannot
cover with our techniques.
The estimate (14), and hence (15), hold for the multipliers T as well,
mν
(17) kDsTmν(f,g)kLp . kDs−νfkLp1kgkLp2 +kfkLp1kDs−νgkLp2,
and we can allow the Coifman-Meyer case ν = 0 too. We note that after our
work was completed we received an independent preprint from Brummer-
Naibo [14] dealing with homogeneous pseudodifferential operators of differ-
ent orders. Their results can be applied to smooth multipliers too, obtain-
ing estimates similar to (17). The techniques employed by these authors,
however, are very different from ours. They rely on smooth molecular de-
compositions. To some extent, they are a bilinear counterpart of the results
by Torres [55] and Grafakos-Torres [39] in the linear case. The results in
[14] apply also to x-dependent smooth symbols, which cannot be treated
by our methods, but the multipliers we study have very limited amount of
regularity and, as far as we know, estimates involving smooth molecular
decompositions require pointwise smoothness on the symbols.
Taking m = 1, (17) leads to the already known Leibniz rule
0
(18) kDs(fg)kLp . kDsfkLp1kgkLp2 +kfkLp1kDsgkLp2.
This estimate also has a long history starting with works of Kato-Ponce [42]
and Christ-Weinstein [20]. The validity of the rule for the optimal range of
exponents 1/2 < p < ∞, 1 < p ,p ≤ ∞, 1 = 1 + 1 and s > max(0, n−n)
1 2 p p1 p2 p
or s a positive even integer, was finally settled by Muscalu-Schlag [51] and
Grafakos-Oh [37]. We refer to [37] for previous works, additional weak-
type estimates, and counterexamples for the limitations on s. The case
p = p = p = ∞ was then further considered by Grafakos-Maldonado-
1 2
Naibo [35] and completely resolved by Bourgain-Li [13].
6 J.HART,R.H.TORRES,X.WU
Motivated by applications in time-dependent partial differential equa-
tions, there has also been some interest in obtaining Leibniz rules in the
mixed Lebesgue spaces LpLq(Rn+1). The first such result involved a com-
t x
mutator estimates with fractional derivatives only in the space variable x
and was obtain by Kenig-Ponce-Vega [43]. Torres-Ward [57] obtain then a
result with the full derivatives in all variables. Denoting by Ds the frac-
t,x
tional derivatives in Rn+1, it was a shown in [57] that
(19) kDts,x(fg)kLpLq . kfkLp1Lq1kDts,xgkLp2Lq2 +kDts,xfkLp1Lq1kgkLp2Lq2,
for 1 < p,q,p ,q ,p ,q < ∞, 1 = 1 + 1 , 1 = 1 + 1, and s > 0. Notice
1 1 2 2 p p1 p2 q q1 q2
that this restricts the target indices p,q to be larger than 1. In this article,
we adapt the arguments in [37] to mixed Lebesgue spaces and obtain
kDs T (f,g)k
t,x mν Lp,q
(20) . kDts,−xµfkLp1Lq1kgkLp2Lq2 +kfkLp1Lq1kDts,−xµgkLp2Lq2
for 0 ≤ ν < 2n + 2, 1 < p ,q < ∞, i = 1,2, 1/p = 1/p + 1/p , 1/q =
i i 1 2
1/q +1/q , and s ∈ 2N or s > max(0, n+1 −(n+1), n+1 −(n+1)). In
1 2 p q
particular the case ν = 0 in (20) can beused to extend (19) to the full range
1/2 < p,q < ∞ for the appropriate values of s.
We mention that other authors have considered mixed derivatives varia-
tionsof (19)too. Whenn = 1, letDs andDs bethefractionalderivatives in
x t
the respective one-dimensional variables x and t. Benea-Muscalu [2] showed
first that in R1+1,
kDβDα(fg)k
t x LpLq
(21) . kfkLp1Lq1kDtβDxαgkLp2Lq2 +kDtβfkLp1Lq1kDxαgkLp2Lq2
+kDxαfkLp1Lq1kDtβgkLp2Lq2 +kDtβDxαfkLp1Lq1kgkLp2Lq2.
forα,β > 0,1< p ,q ≤ ∞,1 ≤ p,q < ∞, 1 = 1 + 1 ,and 1 = 1 + 1 . The
j j q q1 q2 p p1 p2
authors also state that the result hold in higher dimensions. In the case of
Lebesgue spaces the analog mixed derivative version of (18) was previously
studied by Muscalu et al [50].
Using different methods, Di Plinio and Ou [27] prove some multiplier
resultswhichimplicitlyallowthemtoextended(21)tothecase1/2 < p <∞
provided α > max(0, 1 −1) and q ≥ 1. Finally, we recently became aware
p
of a new version [3] of the work of Benea-Muscalu [2], and another preprint
[4] by the same authors treating (21) in the full quasi-Banach space case.
The combined results of [3] and [4] allow for 1/2 < p < ∞ and 1/2 < q <∞
under the condition α,β > max(0, 1 − 1,1 − 1). Moreover, in an even
p q
more recent version [5] the same authors reduced the condition on β to
β > max(0, 1 −1).
q
Wepointoutthatneither(21)implies(20)northeotherwayaround. Our
proofof (19)iscarriedoutinalldimensionsnandallowsalso1/2 <p,q ≤ 1.
In the context of mixed Lebesgue spaces, the version using full derivatives
SMOOTHING BILINEAR OPERATORS AND LEIBNIZ-TYPE RULES 7
faces a new technical difficulty that forces us to consider versions of Hardy
spaces in the mixed-norm setting. This does not seem to be the case in
the mixed derivative situation, where one can iterate some vector valued
estimates in x and t in some computations. We believe our arguments could
be modified to give the mixed derivatives version (21) of Benea-Muscalu for
thefull range of exponents too, butwe will not carry out such computations
here.
We are able to treat multipliers T with limited amount of regularity
mν
by applying some of the tools introduced by Tomita [54], and further de-
veloped by Fujita-Tomita [32], Grafakos-Si [38], Grafakos-Miyachi-Tomita
[36], Miyachi-Tomita [49], and Li-Sun[46] for ν = 0, and Chaffee-Torres-Wu
[16] for ν > 0. The techniques for the boundedness results of multipliers
(or rather paraproducts) in [27], [3], and [4] are then substantially different
from ours. Once the boundedness of certain multiplier operators is estab-
lished, the Leibniz rules follow by what are now familiar arguments, which
also work on mixed Lebesgue spaces. As already mentioned, we follow the
proof of the Leibniz rules in [37], which also share some features with the
ones used in [51], [3], and [4], and the ones alluded to in [27]. One common
ingredient is the important log estimate for the translated square function.
The arguments given in [37] for such estimates immediately extend to the
mixed-norm situation.
AfterthedefinitionsinSection 2,alloftheresultsinvolvingmultiplierop-
erators in Lebesguespaces are presented in Section 3. Ourmain resultthere
isTheorem3.3. Wethenextend inSection 4thesmoothingandLeibnizrule
estimates for T to mixed Lebesgue spaces, proving in Theorem 4.4 the
mν
analogous of Theorem 3.3 in this context. The Appendix at the end of this
article has a technical estimate involving Hardy spaces in the context of
mixed norms, which appears to be new.
2. Function spaces
Let S(Rn) denote the Schwartz class of smooth, rapidly decreasing func-
tions, with the its standard topology, and S′(Rn) be the topological dual
of S(Rn). For a function f ∈ S(Rn), we take for definition of the Fourier
transform the expression given by
f(ξ) = f(x)e−ix·ξdx,
ˆ
Rn
and, as usual, extend thisbdefinition to S′(Rn) by duality. Let S0(Rn) be
the subspace of all f ∈ S(Rn) such that
(22) f(x)xαdx = 0
ˆ
Rn
for all α ∈ Nn.
0
We have already defined in the introduction I f(ξ) = |ξ|−sf(ξ), for 0 <
s
s < n; and Dsf(ξ) = |ξ|sf(ξ), for any s > 0. These definitions certainly
c b
d b
8 J.HART,R.H.TORRES,X.WU
make sense for any function in S(Rn). We can extend them to all s ∈ R in
the same way, Dsf(ξ) = |ξ|sf(ξ), butrestricting f to S (Rn) when s < −n.
0
Note that since in such a case f(ξ) vanishes to infinite order at the origin
and so Ds nowdmaps S (Rn)bcontinuously into S (Rn). Hence we can also
0 0
extend the definition of Ds to tbhe dual of S (Rn), which can be identified
0
as the class of distributions S′(Rn) modulo polynomials.
Fixafunctionψ ∈ S (Rn)whoseFouriertransformissupportedin1/2 <
0
|ξ| < 2 and ψ(ξ) > c for 3/5 < |ξ| < 5/3 and for k ∈ Z define the
0
Littlewood-Paley operator
b
∆ f = ψ ∗f,
k 2−k
where ψ (x) = 2knψ(2kx). We will call such function a Littlewood-Paley
2−k
function.
For 0 < p,q < ∞ and s ∈ R, we recall that the homogeneous Triebel-
Lizorkin spaceF˙s,q is thecollection of all f ∈ S′(Rn)(modulopolynomials)
p
such that
1
q
kfk = (2sk|∆ f(x)|)q <∞.
F˙ps,q (cid:13)(cid:13) k∈Z k ! (cid:13)(cid:13)
(cid:13) X (cid:13)Lp
(cid:13) (cid:13)
When this is taken modul(cid:13)o polynomials, it is a B(cid:13)anach space norm if 1 ≤
(cid:13) (cid:13)
p,q < ∞ and a Banach quasi-norm if either p or q are less than 1. A choice
of a different function ψ with the same properties stated above produce
equivalent (quasi-)norms. Furthermore, we define W˙ s,p for s ∈ R and 0 <
p < ∞tobethesetofallf ∈ S′(Rn)suchthatDsf ∈ Lp with(quasi-)norm
kDsfk , and note that for 1 < p < ∞ and s ∈ R, one has W˙ s,p = F˙s,2
Lp p
with comparable norms. In particular, F˙0,2 = Lp for that range of p. On
p
the other hand for 0 < p ≤ 1, F˙0,2 coincides with the Hardy space Hp. The
p
inhomogeneous versions aregiven by Fs,q = F˙s,q∩Lp andWs,p = W˙ s,p∩Lp.
p p
Similarly the homogeneous Besov spaces defined by the (quasi-)norms
1
q q
kfk = 2skk∆ f(x)k
B˙ps,q Xk∈Z(cid:16) k Lp(cid:17) !
and their inhomogeneous counterparts are given by Bs,q = B˙s,q ∩Lp.
p p
For the purposes of the article, we will only consider the mixed Lebesgue
spaces LpLq(R×Rn), or simply LpLq(Rn+1), or LpLq(Rn+1), for 0< p,q <
t x t x
∞, which for us will be defined by the (quasi-)norms
1/p
p/q
kfkLptLqx(Rn+1) = ˆR(cid:18)ˆRn|f(t,x)|q dx(cid:19) dt! .
We could obtain, of course, versions of our results in mixed Lebesgue spaces
defined by a different ordering of the variables, but we just consider the
SMOOTHING BILINEAR OPERATORS AND LEIBNIZ-TYPE RULES 9
above one because of the significance in applications in partial differential
equations.
If ψ has the same properties as before but in Rn+1 and 1 < p,q < ∞, it
also holds (see [57]) that
1
2
(23) kfkLptLqx(Rn+1) ≈ (cid:13)(cid:13)(cid:13) Xk∈Z|∆kf|2! (cid:13)(cid:13)(cid:13)LptLqx(Rn+1).
(cid:13) (cid:13)
Finally, we will need a mixed-n(cid:13)orm version of t(cid:13)he Hardy spaces. For 0 <
(cid:13) (cid:13)
p,q < ∞, the mixed Hardy space Hp,q(Rn+1) is defined to be the collection
of all f ∈ S′(Rn+1) (modulo polynomials) such that
1
2
kfkHp,q(Rn+1) = (cid:13) |∆kf|2 (cid:13) < ∞.
(cid:13)(cid:13) Xk∈Z ! (cid:13)(cid:13)LptLqx(Rn+1)
(cid:13) (cid:13)
Clearly, by definition and (2(cid:13)3), Hp,q(Rn+1) =(cid:13) LpLq(Rn+1) whenever 1 <
(cid:13) (cid:13)
p,q < ∞. When either p or q is less than or equal to one, there appears to
be much less known about other properties of these spaces. We do mention
that a different definition was given by Cleanthous-Georgiadis-Nielsen [21]
using non-tangential maximal functions. They showed that their mixed
Hardy spaces also coincide with mixed Lebesgue spaces when both indices
are larger than one. We do not know if such mixed Hardy spaces coincide
with the Hp,q(Rn+1) above for other values of p and q, but it is likely.
Also, a wavelet characterization of Hp,q as defined above was obtained in
Georgiadis-Johsen-Nielsen [33]. In any case, for our purposes,what we need
is the following estimate. If 0 < q,p < ∞ and f ∈ Hp,q(Rn+1)∩L2(Rn+1),
then
(24) kfkLpLq(Rn+1) ≤ Cp,qkfkHp,q(Rn+1).
Although the case p = q is well known, we could not locate in the literature
the case p 6= q when either exponent is smaller or equal to one. We find this
case to be rather non-trivial and we provide a proof in the Appendix.
3. Bilinear multipliers on Lebesgue spaces
Our first result is concerned with the bilinear Fourier multipliers of the
form
T (f,g)(x) = m (ξ,η)ei(ξ+η)xf(ξ)g(η)dξdη
mν ˆ ν
R2n
for 0≤ ν < 2n, and f,g ∈ S(Rn) where m satisfies the size condition
ν b b
(25) |m (ξ,η)| . (|ξ|+|η|)−ν.
ν
Note that the size condition (25) guarantees that the operators are well-
defined and the integral is absolutely convergent. However, the multipliers
T are not a priori boundedon Lebesgue spaces without regularity on m .
mν ν
10 J.HART,R.H.TORRES,X.WU
We will need the following auxiliary functions. Let M (Rn) be the col-
ν
lection of all sequences of functions {Φkν}k∈Z satisfying suppΦkν ⊂ {|(ξ,η)| ≈
2k} and
(26) |∂β∂γΦk(ξ,η)| ≤ C (|ξ|+|η|)ν−|β|−|γ|,
ξ η ν β,γ
for all (ξ,η) 6= 0 and all multi-indices β,γ ∈ Nn, whereC is aconstant in-
0 α,β
dependentofk. Atypicalexampleis{Φk} := {(|ξ|2+|η|2)ν/2φ(2−kξ,2−kη)},
ν
where φ is a Schwartz function supported in {|(ξ,η)| ≈ 1}.
Thefollowing resultprovides a sufficient condition for T to besmooth-
mν
ing. In the case ν = 0 and m = 1 it is just the Leibniz rule (18) with the
0
same range of exponents in [37] and [51].
Theorem 3.1. Let m be a multipliers satisfying (25) for some 0 ≤ ν <2n,
ν
and let 1 < p ,p < ∞ and p be such that 1/p +1/p =1/p. Suppose that
1 2 1 2
(i) for any Φ ∈ C∞(R2n\{0}) satisfying (26), T is bounded from
ν mνΦν
Lp1 ×Lp2 to Lp with norm C , and
1
(ii) for any {Φk} ∈ M (Rn), {f } ∈ Lp1(ℓ2), and {g }∈ Lp2(ℓ2)
ν ν k k
k{TmνΦkν(fk,gk)}k∈ZkLp(ℓ1) ≤ C2k{fk}k∈ZkLp1(ℓ2)k{gk}k∈ZkLp2(ℓ2).
Then for s ∈2N or s > max(0, n −n), and f,g ∈ S(Rn),
p
(27) kTmν(f,g)kW˙ s,p ≤ C′(kfkW˙ s−ν,p1kgkLp2 +kfkLp1kgkW˙ s−ν,p2).
Moreover, if C ,C . A for some quantity A depending on m , then
1 2 mν mν ν
C′ . A .
mν
The proof of Theorem 3.1 needs the following version of the Littlewood-
Paley estimate that we take from [37].
Lemma 3.2 ([37]). Let m ∈ Zn\{0} and ψm(x) = ψ(x + m) for some
Schwartz function ψ whose Fourier transform is supported in the annulus
m m
1/2 ≤ |ξ| ≤ 2. Let ∆ (f) = Ψ ∗ f. Then for 1 < p < ∞ there is a
j 2−j
constant C = C(n,p) such that
1/2
(28) (cid:13)(cid:13) |∆mj (f)|2 (cid:13)(cid:13) ≤ Cln(1+|m|)kfkLp(Rn).
(cid:13)(cid:13) Xj∈Z (cid:13)(cid:13)Lp(Rn)
(cid:13)  (cid:13)
We note(cid:13)for further use that t(cid:13)he proof of this result in [37] is based, as in
(cid:13) (cid:13)
the classical case, on the vector valued singular integral
Tf(x)= K(x−y)f(y)dy = Ψ (x−y)f(y)dy
ˆ ˆ 2−j
Rn (cid:26) Rn (cid:27)j
as an operator from Lp(Rn,C) → Lp(Rn,ℓ2) and showing that the kernel
satisfies the H¨ormander integral condition
ˆ kK(x−y)−K(x)kC→ℓ2dx ≤ ˆ |Ψ2−j(x−y)−Ψ2−j(x)|dx
|x|>2|y| j∈Z |x|>2|y|
X