Young type inequalities for weighted spaces

0.23 MB·English

by  Joachim Toft

YOUNG TYPE INEQUALITIES FOR WEIGHTED
SPACES
JOACHIM TOFT, KAROLINE JOHANSSON, STEVAN PILIPOVIC´,
AND NENAD TEOFANOV
3 Abstract. Weestablishsharpconvolutionandmultiplicationes-
1 timates in weighted Lebesgue, Fourier Lebesgue and modulation
0 spaces. Especially we recover some results in [2, 6].
2

n
a
J
0. Introduction
5
2
The aim of the paper is to establish H¨older-Young type proper-

] ties for convolution and multiplications on weighted Lebesgue, Fourier
A
Lebesgue and modulation spaces.
F
A frequently used convolution property concerns Young’s inequality,
.
h
which in terms of the Young functional
t
a
1 1 1
m R(p) ≡ 2− − − , p = (p ,p ,p ) ∈ [1,∞]3, (0.1)
0 1 2
[ p0 p1 p2

1 asserts that
v Lp1 ∗Lp2 ⊆ Lp′0 (0.2)
8 t1 t2 −t0
7 when R(p) = 0,
9
5 t +t ≥ 0, t +t ≥ 0, and t +t ≥ 0. (0.3)
0 1 0 2 1 2
.
1
We note that the latter inequalities imply
0
3
t +t +t ≥ 0. (0.4)
1 0 1 2
v: Here Lp is the weighted Lebesgue space with parameters p and t, and
t
Xi consists of all measurable functions f on Rd such that f · h·it ∈ Lp,
r where hxi = (1+ |x|2)1/2. Furthermore, if p ∈ [1,∞], then p′ ∈ [1,∞]
a
is the conjugate exponent for p, i.e. 1/p+1/p′ = 1.
Especially we are interested to find conditions on t , j = 0,1,2, such
j
that (0.2) holds when R(p) in (0.1) stays somewhere in the interval
[0,2−1].
AroughestimateisobtainedbyanappropriateapplicationofH¨older’s
inequality on Young’s inequality, here above. More precisely, by rewrit-
ing f (x) · hxitj into (f (x) · hxiσj) · hxitj−σj, it follows by applying
j j
H¨older’s inequality on the Lpj-norms in Young’s inequality that the
tj
following result holds true.
2000 Mathematics Subject Classification. 44A35,42B05,46E35,46F99.
Key words and phrases. Fourier, Lebesgue, modulation, sharpness.
1
Proposition 0.1. Let t ∈ R, p ∈ [1,∞], j = 0,1,2 and let R(p) be
j j
given by (0.1). Also assume that 0 < R(p) ≤ 1/2, (0.3) holds true with
at least two inequalities strict, and that
t +t +t > d·R(p) (0.4)′
0 1 2
holds. Then Lp1 ∗Lp2 ⊆ Lp′0 .
t1 t2 −t0
We remark that Proposition 0.1 holds true also after removing the
condition R(q) ≤ 1/2.
In contrast to Young’s inequality here above, the most of the in-
equalities in (0.3) and (0.4)′ in Proposition 0.1 are strict, and if it is
possible to replace any such strict inequality by a non-strict one, then
the situation is improved. On the other hand, it seems not to be possi-
ble to perform such improvement by only using H¨older’s inequality in
such simple way as described here above.
In this paper we use the framework in Chapter 8 in [2] and Section
3 in [6] to decompose the involved functions in the convolutions in
convenient ways. These investigations lead to Theorem 2.2 in Section
2, which in particular gives the following improvement of Proposition
0.1.
Proposition 0.1′. Let t ∈ R, p ∈ [1,∞], j = 0,1,2 and let R(p) be
j j
given by (0.1). Also assume that 0 < R(p) ≤ 1/2, (0.3) holds true, and
that
t +t +t ≥ d·R(p) (0.4)′′
0 1 2
holds, with strict inequality in (0.4)′′ when t = d·R(p) for some j =
j
0,1,2. Then Lp1 ∗Lp2 ⊆ Lp′0 .
t1 t2 −t0
Furthermore, if t 6= d·R(p), then we prove that Proposition 0.1′ is
j
optimal in the sense that if (0.3) or (0.4)′′ are violated, then Lp1 ∗Lp2
is not continuously embedded in Lp′0 . t1 t2
−t0
Obviously, except for a few cases, the strict inequalities in Proposi-
tion 0.1 have been replaced by non-strict ones in Proposition 0.1′. The
H¨ormander theorem [2, Theorem 8.3.1] on microlocal regularity of a
product is obtained by choosing p = p = p = 2 in Proposition 0.1′,
0 1 2
and note that in contrast to Proposition 0.1′, the latter theorem is not
covered by Proposition 0.1. We remark that the results in [2] are given
in the framework of weighted Sobolev spaces of the form H2, and the
s
analysis is based on an intensive use of their Hilbert space structure.
On the other hand, here (as well as in [6]) our result considerations
include Banach spaces which might not be Hilbert spaces, and thereby
use a more sophisticated techniques in the proofs are needed.
Finally we remark that Theorem 2.2 leads to Theorem 2.4 in Sec-
tion 2, which concerns convolution properties for modulation spaces.
In particular, if t , p and R(p) are the same as in Proposition 0.1′, and
j j
2
that
1 1 1
+ + = 1, and 0 ≤ s +s +s ,
0 1 2
q q q
0 1 2
then it follows from Theorem 2.4 Mp1,q1 ∗Mp2,q2 ⊆ Mp′0,q0′.
s1,t1 s2,t2 s0,t0
Acknowledgment
This research is supported by Ministry of Education, Science and
Technological Development of Serbia through the Project no. 174024.
1. Preliminaries
In this section we review notions and notation, and discuss basic
preliminary results. We put N = {0,1,2,...}, and A . B to indicate
A ≤ cB for a suitable constant c > 0. Any extension of the L2-scalar
product on C∞(Rd) is denoted by (·, ·) = (·, ·).
0 L2
The scalar product of x and ξ in Rd is denoted by hx,ξi. For p ∈
[1,∞]weletp′ ∈ [1,∞]denotetheconjugateexponent (1/p+1/p′ = 1).
The Fourier transform F is the operator on S′(Rd) which takes the
form
(Ff)(ξ) = f(ξ) ≡ (2π)−d/2 f(x)e−ihx,ξidx, ξ ∈ Rd,
Z
when f ∈ L1(Rd).b
The (weighted) Fourier Lebesgue space FLq(Rd), s ∈ R is the Ba-
s
nach space which consists of all f ∈ S′(Rd) such that
kfk ≡ kf ·h·isk (1.1)
FLqs Lq
is finite. Here and in what follows, hξi = (1+|ξ|2)1/2.
b
Let X be an open set in Rd. Then the local Fourier Lebesgue space
FLq (X) consists of all f ∈ D′(X) such that ϕf ∈ FLq(Rd) for
s,loc s
every ϕ ∈ C∞(X). The topology in FLq (X) is defined by the family
0 s,loc
of seminorms f 7→ kϕfkFLq , where ϕ ∈ C0∞(X).
(ω)
We note that
FLq(Rd) ⊆ FLq (X). (1.2)
s s,loc
X
(cid:12)
and (cid:12)
(cid:12)
FLq1 (X) ⊆ FLq2 (X), when q ≤ q and s ≤ s . (1.3)
s1,loc s2,loc 1 2 2 1
(See e.g. [6].)
Next we define modulationspaces. Let φ ∈ S′(Rd)\0 be fixed. Then
the short-time Fourier transform of f ∈ S′(Rd) with respect to φ is
defined by
(V f)(x,ξ) = F(f ·φ(· −x))(ξ).
φ
3
Heretheleft-handside makes sense, since it isthepartialFourier trans-
form of the tempered distribution F(x,y) = (f ⊗φ)(y,y−x) with re-
spect to the y-variable. We also note that if f,φ ∈ S(Rd), then V f
φ
takes the form
V f(x,ξ) = (2π)−d/2 f(y)φ(y−x)e−ihy,ξidy. (1.4)
φ
Z
Let s,t ∈ R and p,q ∈ [1,∞] be fixed. Then the modulation space
Mp,q(Rd) consists of all f ∈ S′(Rd) such that
s,t
1/q
q/p
kfkMp,q ≡ |Vφf(x,ξ)hxithξis|pdx dξ
s,t ZRd (cid:18)ZRd (cid:19) !
is finite (with obvious interpretation of the integrals when p = ∞ or
q = ∞). In the same way, the modulation space Wp,q(Rd) consists of
s,t
all f ∈ S′(Rd) such that
1/p
p/q
kfkWp,q ≡ |Vφf(x,ξ)hxithξis|qdξ dx
s,t ZRd(cid:18)ZRd (cid:19) !
is finite.
2. Multiplication and convolution properties
In this section we derive multiplication and convolution results on
Lebesgue, Fourier Lebesgue and modulation spaces. In particular, we
extend some results in [6]. The proofs of the theorems are postponed to
Section 3. Our main results are Theorems 2.2 and 2.4. Here we present
sufficient conditions on t ∈ R and p ∈ [1,∞], j = 0,1,2, to en-
j j
sure that v ∗ v ∈ Lp′0 when v ∈ Lpj, j = 1,2, and similarly when
1 2 −t0 j tj
the convolution product and Lebesgue spaces are replaced by multi-
plication and Fourier-Lebesgue spaces. The results also include related
multiplication and convolution properties for modulation spaces.
Certain parts of the analysis concerns reformulation of 0 ≤ R(p) ≤
1/2 into equivalent statements. For convenience we let
2
G(x) = G(x ,x ,x ) = 2− x , (2.1)
0 1 2 j
j=0
X
and note that R(p) is equal to G(x) when x = 1/p . We also let
j j
1
H (x) = max min x ,max ,min(x ,x ) . (2.2)
0 π(0) π(1) π(2)
π∈S3 (cid:18) (cid:18) (cid:18)2 (cid:19)(cid:19)(cid:19)
max(x ,x ,x ), x ,x ,x < 1,
0 1 2 0 1 2 2
H (x) = min(x ,x ,x ), x ,x ,x > 1, (2.3)
1  0 1 2 0 1 2 2
1, otherwise,
2
4


and
1
H (x) = max ,min(x ,x ,x ) , (2.4)
2 0 1 2
2
(cid:18) (cid:19)
Here S is the permutations of {0,1,2}. The following lemma justifies
3
the introduction of the functions H (x), j = 0,1,2, in (2.2)–(2.4).
j
Lemma 2.1. Let x = (x ,x ,x ), and let G(x) and H (x), l = 0,1,2,
0 1 2 l
be given by (2.1)–(2.4). Then H (x) = H (x). Furthermore, if l ∈
0 1
{0,1,2}, then the following conditions are equivalent.
1
(1) 0 ≤ G(x) ≤ ;
2
(2) 0 ≤ G(x) ≤ H (x).
l
Proof. WebegintoproveH (x) = H (x).WehaveH (x) = max(y ,y ,y ),
0 1 0 0 1 2
where
1
y = min x ,max ,min(x ,x ) ,
0 0 1 2
2
(cid:18) (cid:18) (cid:19)(cid:19)
1
y = min x ,max ,min(x ,x ) ,
1 1 0 2
2
(cid:18) (cid:18) (cid:19)(cid:19)
1
y = min x ,max ,min(x ,x ) .
2 2 0 1
2
(cid:18) (cid:18) (cid:19)(cid:19)
If x ≤ 1/2, then y = x , j = 0,1,2, giving that
j j j
H (x) = max(x ,x ,x ) = H (x)
0 0 1 2 1
in this case.
If instead x ≥ 1/2, then y = min(x ,x ,x ), j = 0,1,2, giving that
j j 0 1 2
H (x) = min(x ,x ,x ) = H (x),
0 0 1 2 1
in this case as well.
Next assume that x > 1/2 and x < 1/2, for some choices of
j k
j,k ∈ {0,1,2}. By reasons of symmetry we may assume that x =
0
min(x ,x ,x ) < 1/2 and x = max(x ,x ,x ) > 1/2. Then
0 1 2 1 0 1 2
1 1 1 1
H (x) = , y = x , y = and y = min x , ≤ .
1 0 0 1 2 2
2 2 2 2
(cid:18) (cid:19)
Hence,
1
H (x) = max(y ,y ,y ) = = H (x).
0 0 1 2 1
2
which shows that H (x) = H (x) for all x.
0 1
It remains to prove the equivalence between (1) and (2). It is obvious
that (2) with l = 1 or (1) implies (2) with l = 2. Next assume that
5
(2) with l = 2 holds but not (1). Then G(x) > 1/2 and H (x) > 1/2,
2
which implies that min{x ,x ,x } > 1/2. This gives
0 1 2
2
3 1
G(x) = 2− x < 2− = ,
j
2 2
j=0
X
which is a contradiction. Hence (2) with l = 2 implies (1), and we have
proved the equivalence between (1) and (2) when l = 2.
Since H (x) = H (x) = H (x) when x ≥ 1/2 for some j = 0,1,2,
0 1 2 j
it suffices to consider the case x < 1/2, j = 0,1,2, when proving that
j
(2) is invariant under the choice of l = 0,1,2. Then by the first part of
the proof we have H (x) = H (x) < 1/2, H (x) = 1/2 and
0 1 2
2
G(x) = 2− x > 1/2.
j
j=0
X
Hence (2) is violated in this case for any j ∈ {0,1,2}. This proves the
invariance of (2) under the choice of j, and the proof is complete. (cid:3)
In the main results here below we consider convolutions between
elements in weighted Lebesgue and modulation spaces, and multiplica-
tions between elements in (weighted) Fourier-Lebesgue spaces. For the
convolution results, the parameters on the weights should satisfy
0 ≤ t +t , j,k = 0,1,2, j 6= k, (2.5)
j k
0 ≤ t +t +t −d·R(p), (2.6)
0 1 2
and
0 ≤ s +s +s . (2.7)
0 1 2
(Cf. (0.3) and (0.4)′.) If the convolution is replaced by multiplication,
then the roles for p and q , and for s and t are interchanged. There-
j j j j
fore, (2.5)–(2.7) should be replaced by
0 ≤ s +s , j,k = 0,1,2, j 6= k, (2.5)′
j k
0 ≤ s +s +s −d·R(q), (2.6)′
0 1 2
and
0 ≤ t +t +t , (2.7)′
0 1 2
when the Lebesgue parameters are q and q instead of p and p , respec-
j j
tively.
Theorem 2.2. Let s ,t ∈ R, p ,q ∈ [1,∞], j = 0,1,2 and let R be
j j j j
the functional in (0.1). Then the following is true:
6
(1) Assume that 0 ≤ R(p) ≤ 1/2, and that (2.5) and (2.6) hold true
with strict inequality in (2.6) when R(p) > 0 and s = d·R(p)
j
for some j = 0,1,2. Then the map (f ,f ) 7→ f ∗f on C∞(Rd)
1 2 1 2 0
extends uniquely to a continuous map from Lq1(Rd)×Lq2(Rd)
to Lq0′ (Rd); t1 t2
−t0
(2) Assume that 0 ≤ R(q) ≤ 1/2, and that (2.5)′ and (2.6)′ hold
true with strict inequality in (2.6)′ when R(q) > 0 and s =
j
d·R(q) for some j = 0,1,2. Then the map (f ,f ) 7→ f ·f on
1 2 1 2
C∞(Rd) extendsuniquelyto a continuous mapfrom FLq1(Rd)×
F0Lq2(Rd) to FLq0′ (Rd). s1
s2 −s0
The following corollary follows immediately from (1.2) and Theorem
2.2.
Corollary 2.3. Let the hypothesis in Theorem 2.2 hold true, and let
X ⊆ Rr be open. Then the map (f ,f ) 7→ f · f on C∞(X) extends
1 2 1 2 0
uniquely to a continuous map from (FLq1) (X) × (FLq2) (X) to
(FLq0′ ) (X) s1 loc s2 loc
−s0 loc
The next result concerns corresponding properties for modulation
spaces.
Theorem 2.4. Let the hypothesis in Theorem 2.2 hold true and Then
the following is true:
(1) Assume that 0 ≤ R(p) ≤ 1/2, R(q) ≤ 1 and (2.5)–(2.7) hold
true, with strict inequality in (2.6) when R(p) > 0 and t =
j
d · R(p) for some j = 0,1,2. Then the map (f ,f ) 7→ f ∗ f
1 2 1 2
on C∞(Rd) extends to a continuous map from Mp1,q1(Rd) ×
0 s1,t1
Mp2,q2(Rd) to Mp′0,q0′ (Rd);
s2,t2 −s0,−t0
(2) Assume that R(p) ≤ 1 and 0 ≤ R(q) ≤ 1/2, and (2.5)′–(2.7)′
hold true, with strict inequality in (2.6)′ when R(q) > 0 and
s = d·R(q) for some j = 0,1,2. Then the map (f ,f ) 7→ f ·f
j 1 2 1 2
on C∞(Rd) extends to a continuous map from Mp1,q1(Rd) ×
0 s1,t1
Mp2,q2(Rd) to Mp′0,q0′ (Rd).
s2,t2 −s0,−t0
The same is true afterMpj,qj have been replaced by Wpj,qj, j = 0,1,2.
sj,tj sj,tj
Furthermore, the extensions of these mappings are unique, except
when p or q are equal to ∞ for more than one choice of j = 0,1,2.
j j
Remark 2.5. By letting x = 1/p in Lemma 2.1, we may replace the
j j
condition 0 ≤ R(p) ≤ 1/2 in Theorems 2.2–2.4 with
1 1 1 1
0 ≤ R(p) ≤ max ,min , , .
2 p p p
(cid:18) (cid:18) 0 1 2(cid:19)(cid:19)
The following result shows that the conditions (2.5) and (2.6) are
also necessary in order for the continuity in Theorem 2.2 should hold
true.
7
Proposition 2.6. Let p ,q ∈ [1,∞] and s ,t ∈ R, j = 0,1,2. As-
j j j j
sume that at least one of the following statements hold true:
(1) the map (f ,f ) 7→ f ∗f on S(Rd) is continuously extendable
1 2 1 2
to a map from Lp1(Rd)×Lp2(Rd) to Lp′0 (Rd);
t1 t2 −t0
(2) the map (f ,f ) 7→ f ∗f on S(Rd) is continuously extendable
1 2 1 2
to a map from Mp1,q1(Rd)×Mp2,q2(Rd) to Mp′0,q0′ (Rd);
s1,t1 s2,t2 −s0,−t0
(3) the map (f ,f ) 7→ f ∗f from S(Rd)×S(Rd) to S(Rd) is
1 2 1 2
continuously extendable to a map from Wp1,q1(Rd)×Wp2,q2(Rd)
s1,t1 s2,t2
to Wp′0,q0′ (Rd).
−s0,−t0
Then (2.5) and (2.6) hold true.
By Fourier transformation, it follows that Proposition 2.6 is equiva-
lent to the following result.
Proposition 2.7. Let p ,q ∈ [1,∞] and s ,t ∈ R, j = 0,1,2. As-
j j j j
sume that at least one of the following statements hold true:
(1) the map (f ,f ) 7→ f ·f on S(Rd) is continuously extendable
1 2 1 2
to a map from FLq1(Rd)×FLq2(Rd) to FLq0′ (Rd);
s1 s2 −s0
(2) the map (f ,f ) 7→ f ·f on S(Rd) is continuously extendable
1 2 1 2
to a map from Mp1,q1(Rd)×Mp2,q2(Rd) to Mp′0,q0′ (Rd);
s1,t1 s2,t2 −s0,−t0
(3) the map (f ,f ) 7→ f ·f from S(Rd)×S(Rd) to S(Rd) is
1 2 1 2
continuously extendable to a map from Wp1,q1(Rd)×Wp2,q2(Rd)
s1,t1 s2,t2
to Wp′0,q0′ (Rd).
−s0,−t0
Then (2.5)′ and (2.6)′ hold true.
Remark 2.8. Intheliterature,thereareseveral resultswhicharerelated
to Theorems 2.2 and 2.4 (cf. e.g. the first part of Proposition 2.3 in [4]
and the references therein). It seems that some of these results contain
some mistakes.
More precisely, let p ,q ∈ [1,∞], j = 0,1,2, and s ≥ 0 be such
j j
that R(p) = 1 and R(q) = 0. Then it is remarked in [3] that the
map (f ,f ) 7→ f · f on S is extendable to a continuous map from
1 2 1 2
Mp1,q1 ×Mp2,q2 to Mp′0,q0′. (Cf. Remark 2.4 in [3].) We claim that this
s,0 0,0 s,0
is not correct when s > 0.
In fact, by applying the Fourier transform and using duality, the
statement is equivalent to the following statement:
Let p ,q ∈ [1,∞] and t ∈ R, j = 0,1,2, be such that R(p) = 0,
j j j
R(q) = 1, t = −t ≥ 0 and t = 0. Then the map (f ,f ) 7→ f ·f on
1 2 0 1 2 1 2
S is extendable to a continuous map from Wp1,q1 ×Wp2,q2 to Wp′0,q0′.
0,t1 0,t2 0,t0
The hypothesis in Proposition 2.6 is therefore fulfilled, but (2.5) is
violated. This contradicts Proposition 2.6, and the claim follows.
8
3. Proofs
In this section we present proofs of the results in Section 2. In Sub-
section 3.1 we study in details the problem of extensions of an auxiliary
three-linear map. In Subsection 3.2 we use the results from Subsection
3.1 to prove Lebesgue norm estimates of the three-linear form on dif-
ferent regions. Finally, in Subsection 3.3 we prove the main results.
3.1. The map T (f,g). In this subsection we introduce and study a
F
convenient bilinear map (denoted by T here below when F ∈ L1 is
F loc
appropriate). We refer to [2] and [6] for similar construction.
For F ∈ L1 (R2d) and p,q ∈ [1,∞], we set
loc
q/p 1/q
kFk ≡ |F(x,y)|pdx dy
Lp,q
1
(cid:16)Z (cid:16)Z (cid:17) (cid:17)
and
p/q 1/p
kFkLp,q ≡ |F(x,y)|qdy dx ,
2
(cid:16)Z (cid:16)Z (cid:17) (cid:17)
and we let Lp1,q(R2d) be the set of all F ∈ L1loc(R2d) such that kFkLp,q
is finite. The space Lp,q is defined analogously. (Cf. [5, 6].) We also l1et
2
Θ be defined as
(ΘF)(x,y) = F(x,x−y), F ∈ L1 (R2d). (3.1)
loc
If F ∈ L1 (R2d) is fixed, then we are especially concerned about
loc
extensions of the mappings
(F,f,g) 7→ T (f,g) ≡ F(·,y)f(y)g(· −y)dy (3.2)
F
Z
and
(F,f,g) 7→T (f,g) ≡ F(·,y)f(· −y)g(y)dy. (3.3)
ΘF
Z
from C∞(Rd)×C∞(Rd) to S′(Rd).
0 0
The following extend [2, Lemma 8.3.2] and [6, Proposition 3.2].
Proposition 3.1. Let F ∈ L1 (R2d), p ∈ [1,∞], j = 0,1,2. Also
loc j
assume that R(p) in (0.1) is non-negative, and let r = 1/R(p) ∈ (0,∞].
Then the following is true:
(1) if R(p) ≤ 1/p , then the mappings (3.2) and (3.3) are continu-
0
ous from L∞,r(R2d)×Lp1(Rd)×Lp2(Rd) to Lp′0(Rd). Further-
2
more,
kTF(f,g)kLq0′ . kFkL∞2 ,rkfkLp1kgkLp2 (3.4)
and
kTΘF(f,g)kLp′0 . kFkL∞2 ,rkfkLp1kgkLp2. (3.5)
9
(2) if in addition R(p) ≤ max(1/2,1/p ), then the map (3.2) is
1
continuous from Lr,∞(R2d) × Lp1(Rd) × Lp2(Rd) to Lp′0(Rd).
1
Furthermore,
kTF(f,g)kLp′0 . kFkLr1,∞kfkLp1kgkLp2.
(3) if in addition R(p) ≤ max(1/2,1/p ), then the map (3.3) is
2
continuous from Lr,∞(R2d) × Lp1(Rd) × Lp2(Rd) to Lp′0(Rd).
1
Furthermore,
kTΘF(f,g)kLp′0 . kFkLr1,∞kfkLp1kgkLp2.
We note that Proposition 3.1 agrees with [2, Lemma 8.3.2] when
p = p = 2 and with [6, Proposition 3.2] when p = p ∈ [1,∞].
1 2 1 2
Proof. (1) We only prove (3.4) and leave (3.5) for the reader.
First, assume that p ,p < ∞, and let f,g ∈ C∞(Rd). By H¨older’s
1 2 0
inequality we get
|TF(f,g)(x)|q0′ dx 1/q0′
(cid:16)Z (cid:17)
≤ |F(x,y)|rdy 1/r |f(y)|r′|g(x−y)|r′dy 1/r′ q0′ dx 1/q0′.
(cid:16)Z h(cid:16)Z (cid:17) (cid:16)Z (cid:17) i (cid:17) (3.6)
Next we use the assumption R(q) ≤ 1/p , that is r ≥ p and Young’s
0 0
inequality to obtain
|TF(f,g)(x)|q0′ dx 1/q0′
(cid:16)Z (cid:17)
≤ kFkL∞2 ,r k|f|r′ ∗|g|r′kLq0′/r′ 1/r′ ≤ kFkL∞2 ,r k|f|r′kLr1k|g|r′kLr2 1/r′
(cid:16) (cid:17) (cid:16) (cid:17)
= kFkL∞,rkfkLp1kgkLp2, (3.7)
2
where r = p /r′ and r = p /r′. The result now follows from the fact
1 1 2 2
that C∞ is dense in Lp1 and Lp2 when p ,p < ∞.
0 1 2
Next, assume that p = ∞ and p < ∞, andlet f ∈ L∞ and g ∈ C∞.
1 2 0
Then, it follows that T (f,g) is well-defined, and that (3.7) still holds.
F
The result now follows from the fact that C∞ is dense in Lp2. The case
0
p < ∞ and p = ∞ follows analogously.
1 2
Finally, if p = p = ∞, then the assumptions implies that r = 1
1 2
and p′ = ∞. The inequalities (3.4) and (3.5) then follow by H¨older’s
0
inequality.
(2) First we consider the case r ≥ p . Let h ∈ C (Rd) when r < ∞
1 0
and h ∈ L1(Rd) if r = ∞. Also let F ∈ Lr,∞(R2d) and F (y,x) =
1 0
F(x,y) and gˇ(x) = g(−x). By [6, page 354], we have |hT (f,g),hi| =
F
10