Lp estimates for the bilinear Hilbert transform y - CiteSeerX
Lp estimates for the bilinear Hilbert transform y Michael Lacey z Christoph Thiele x November 18, 1996 Abstract
For the bilinear Hilbert transform given by ( ) = p. v.
H fg x
we announce the inequality k 1, 1 3 = 1 1 + 1 2 and 1
H fg
=p
=p
=p
Z
( ? )( + )
f x
kp3
< p
3 < 2.
K
y g x
y
dy y
p1 ;p2 kf kp1 kgkp2 , provided 2 < p1 ; p2
y This operation is initially de ned only for certain functions f and g, for instance those in the Schwartz class on R. The conjectures concern the extension of H to a bounded operator on Lp spaces. We have proved
Theorem 1 H extends to a bounded operator on Lp1 Lp2 into Lp3 , provided 2 < p1; p2 < 1 and 1 < p3 < 2, where 1=p3 = 1=p1 + 1=p2. Some thirty years ago, in connection with the Cauchy integral on Lipschitz curves [1] A. P. Calderon raised the question of H mapping L2 L2 into L1 . This inequality is true and will be addressed in a subsequent paper. Indeed, the bilinear Hilbert transform maps into Lp3 provided only that p3 > 2=3. Study of the bilinear Hilbert transform is intimately related to L. Carleson's Theorem [2] asserting the pointwise convergence of Fourier series. A seminal result, it has The rst author has been supported by the NSF. Both authors acknowledge the support of a NATO travel grant. y To appear in Proc. Nat. Acad. Sci., USA z School of Mathematics, Georgia Institute of Technology, Atlanta GA 30332, USA x Department of Mathematics, Universit at Kiel, Kiel, Germany
1
2 received two proofs, with the alternative proof provided by C. Fe erman [3]. These proofs have provided us with ingenious and complementary methods of time frequency analysis. A similar analysis seems necessary to understand H , and so our proof entails signi cant aspects of both Carleson and Fe erman's proofs. We give a description of our proof, with details presented in their most concrete form. Complete proofs, which will appear in [6], require de nitions and constructions somewhat more general than those presented here. The bilinear Hilbert transform must be broken into scales and theP frequency be1 2j (2j y ) havior of each scale understood. Hence we replace the kernel 1=y with j =?1 R where is a Schwartz function with Fourier transform ^( ) = e?2ix (x) dx supported on [1=2; 2). For each j , consider Z
Hj fg(x) = f (x ? y)g(x + y)2j (2j y) dy; which has bilinear symbol ^(2?j ( ? )). More speci cally ZZ Hj fg(x) = f^( )^g()e2i(+)x^(2?j ( ? )) dd: Therefore, if g is supported in frequency on the interval [n2j ; (n +1)2j ], then Hj fg(x) acts on the inverse Fourier transform of f^( )1[(n + 1=2)2j ; (n + 3)2j ]( ), and is supported in frequency on the interval [(2n + 1=2)2j ; (2n + 4)2j ]. The di ering rates of translation make these three intervals distinct. It is important to note that the location of the intervals is arbitrary, and therefore, for all j and j 0, the inner product of Hj fg and Hj0 fg need not tend to zero as jj ? j 0j tends to in nity. The analysis of H must be done in terms of both time and frequency. Instead of proceeding with a decomposition of H , we de ne a model of it adapted to the combinatorics of the time{frequency plane. Let D be a dyadic grid in R. Call I ! 2 DD a tile if jI jj!j = 1. The interval ! is a union of four dyadic subintervals of equal length, !1; !2; !3 and !4, which we list in ascending order. Thus, i < j for all 1 i < j 4 and j 2 !j . (We will only use !j for j = 1; 2; 3.) We adopt the notation t = It !t and tj = It !tj for j = 1; 2; 3. RFix a Schwartz function with ^ supported on [?1=8; 1=8], in addition require that (x ? 16n) (x) dx = 0 for all integers n. Set for all tiles t, and j = 1; 2; 3, ?2ic(! )x x ? c(It ) e jI j ; tj (x) = q t jIt j tj
where c(J ) denotes the center of the interval J . Then our model of the bilinear Hilbert transform is X M f1f2(x) = hfq1 ; t1i hf2; t2i t3 (x); jIt j t which is initially de ned only for Schwartz functions f1 and f2. We emphasize that the sum extends over all tiles, and hence all scales. The analogue of Theorem 1 is
3
Theorem 2 M extends to a bounded operator on Lp1 Lp2 into Lp03 , provided 2 < p1; p2 < 1 and 1 < p03 = (1=p1 + 1=p2)?1 < 2. With more liberal notions of \grid," \tile" and \ tj " the bilinear Hilbert transform is in the convex hull of terms like our model M. In the present situation we can give a proof by way of duality. Thus we take f3 2 Lp3 , where 2 < p3 < 1, and show that X
t
hf1; t1 i hf ; ihf ; i K 3 kf k : j p 2 t2 3 t3 jIt j j =1
q
Y
j
The sum is over positive quantities, namely the decomposition above already captures all of the cancellation necessary for convergence of the sums. It also shows that the sum de ning M is unconditionally convergent in t. And as each fj 2 Lp , where pj > 2, it follows that each function is locally square integrable. As it turns out, L2 arguments are decisive in proving Theorem 2. We localize the sum above in the x variable by setting j
Ft (x) =
jhfj ; tj ij 1[I ](x): t jItj j =1 3
Y
q
Certainly Ft (x) dx = jIt j?1=2 Q3j=1 jhfj ; tj ij. And so we show that F(x) = Pt Ft (x) is integrable. This follows from a weak{type result: For pj as above, there is a > 0 so that for all jrj ? pj j < , 1 j 3, the operator F(x) maps Lr1 Lr2 Lr3 into Lr;1, where 1=r = 1=r1 + 1=r2 + 1=r3. Then a variant of the Marcinkiewicz Interpolation Theorem due to Janson [4] implies the strong type inequality. A single instance of the weak type inequality is R
(1)
jfx j F(x) > K
3
Y
j =1
kfj kr gj K: j
for some constant K . But this inequality implies the weak type result because F commutes with dilations by powers of 2, and so it suces to establish this last inequality. These observations are useful since some of our estimates begin to break down on exceptional sets of small measure. Due the localization of Ft in the time variable, and that we only aim for a distributional inequality, we can delete tiles t whose time coordinate falls in a set of bounded measure. The combinatorics of the time frequency plane enter in by way of the partial order on the tiles given by t < t0 if It It0 and ! !0. Note that t and t0 are not comparable with respect to < if and only if t \ t0 = ;. Being disjoint suggests orthogonality for the functions tj and t0 j0 , the dominant theme of the Lemmas we state below. Call a collection of tiles T a Carleson{Fe erman (CF) set with top q if t < q for all t 2 T. Thus !q \ !t 6= ; for t 2 T. Call T a j {CF set if T is a CF{set for which the intervals !tj intersect for all t 2 T. Notice that if T is a 1{CF set say, then the
4 intervals f!tj j t 2 Tg are pairwise disjoint for j = 2; 3. Therefore, by application of Cauchy{Schwartz
1=2 3 X Y
X
jh f ; ij 1 t 1 ? 1 ? 1 2
jIq j Ft (x) sup q jIq j jhfj ; tj ij : (2) 1 t2T jIt j j=2 t2T t2T Notice that the last two square functions are Littlewood{Paley g{functions, albeit conjugated by an exponential to account for the location of the CF set in frequency. This estimate forms the motivation for the Lemma below, which formalizes a decomposition of the set of tiles that is fundamental to our argument. Lemma 3 Fix pi > 2. There is a > 0 and an 0 > 0 and a constant K so that for all jri ? pi j < and 0 < < 0 , the following holds. The collection of all tiles S is a union 1 [ 3 [ 0 Sn;i;j S=S [ with these properties. First, (4)
S0
n=0 i;j =1
is trivial in that [
s2S0
Is K:
The collection Sn;i;j is a union of disjoint i{CF sets Tq with tops q 2 Sn;i;j , and X
(5)
t2T
Ft (x) 2?n(1=r?) for all x, q 2 Sn;i;j .
q
Here, recall that 1=r = i 1=ri, which can be taken arbitrairly close to 1. And, most signi cantly, for t = mini fpi =2g ? , P
kNn;i;j kt =
(6)
X
q2S
q
1I K 2n(1=t+K): t
n;i;j
With the Lemma in place, we estimate 1
3
X X
X
n=0 i;j =1 t2S
Ft (x)
t
n;i;j
1
X
n=0
K
2?n(1=r?) 1
X
n=0
3
X
i;j =1
kNn;i;j kt
2?n(1=r?1=t?C) K 0:
The last sum is nite as r is arbitrairly close to one, while t + = minfpi=2g > 1 is a xed distance from one. Therefore, with (4), (1) holds. We cannot give the complete construction of the Sn;i;j , but rather the initial steps, in which the nearly orthogonal classes of ti are identi ed. First we make an important comparison to a maximal function. If Tq is an i{CF set with top q, we have for j 6= i, "
X 2 (Tq ; j ) = 1 jIq j t2T jhfj ; tj ij q
#
1=2
C xinf M f (x): 2I 2 j q
5 Here M2 g is the maximal function (M jgj2)1=2 . Thus the set F = SifM2 fi > C ?1g has bounded measure and we de ne S0 = fs j Is F g, making (4) trivial. For all i{CF sets T with top q, and T SnS0 , we have (T; j ) 1, for j 6= i. The remaining construction is inductive. Assume Sthat the Sm;i;j are de ned for S r all m < n and all i; j , in such a way that for S = Sn( m
For the bilinear Hilbert transform given by ( ) = p. v.
H fg x
we announce the inequality k 1, 1 3 = 1 1 + 1 2 and 1
H fg
=p
=p
=p
Z
( ? )( + )
f x
kp3
< p
3 < 2.
K
y g x
y
dy y
p1 ;p2 kf kp1 kgkp2 , provided 2 < p1 ; p2
y This operation is initially de ned only for certain functions f and g, for instance those in the Schwartz class on R. The conjectures concern the extension of H to a bounded operator on Lp spaces. We have proved
Theorem 1 H extends to a bounded operator on Lp1 Lp2 into Lp3 , provided 2 < p1; p2 < 1 and 1 < p3 < 2, where 1=p3 = 1=p1 + 1=p2. Some thirty years ago, in connection with the Cauchy integral on Lipschitz curves [1] A. P. Calderon raised the question of H mapping L2 L2 into L1 . This inequality is true and will be addressed in a subsequent paper. Indeed, the bilinear Hilbert transform maps into Lp3 provided only that p3 > 2=3. Study of the bilinear Hilbert transform is intimately related to L. Carleson's Theorem [2] asserting the pointwise convergence of Fourier series. A seminal result, it has The rst author has been supported by the NSF. Both authors acknowledge the support of a NATO travel grant. y To appear in Proc. Nat. Acad. Sci., USA z School of Mathematics, Georgia Institute of Technology, Atlanta GA 30332, USA x Department of Mathematics, Universit at Kiel, Kiel, Germany
1
2 received two proofs, with the alternative proof provided by C. Fe erman [3]. These proofs have provided us with ingenious and complementary methods of time frequency analysis. A similar analysis seems necessary to understand H , and so our proof entails signi cant aspects of both Carleson and Fe erman's proofs. We give a description of our proof, with details presented in their most concrete form. Complete proofs, which will appear in [6], require de nitions and constructions somewhat more general than those presented here. The bilinear Hilbert transform must be broken into scales and theP frequency be1 2j (2j y ) havior of each scale understood. Hence we replace the kernel 1=y with j =?1 R where is a Schwartz function with Fourier transform ^( ) = e?2ix (x) dx supported on [1=2; 2). For each j , consider Z
Hj fg(x) = f (x ? y)g(x + y)2j (2j y) dy; which has bilinear symbol ^(2?j ( ? )). More speci cally ZZ Hj fg(x) = f^( )^g()e2i(+)x^(2?j ( ? )) dd: Therefore, if g is supported in frequency on the interval [n2j ; (n +1)2j ], then Hj fg(x) acts on the inverse Fourier transform of f^( )1[(n + 1=2)2j ; (n + 3)2j ]( ), and is supported in frequency on the interval [(2n + 1=2)2j ; (2n + 4)2j ]. The di ering rates of translation make these three intervals distinct. It is important to note that the location of the intervals is arbitrary, and therefore, for all j and j 0, the inner product of Hj fg and Hj0 fg need not tend to zero as jj ? j 0j tends to in nity. The analysis of H must be done in terms of both time and frequency. Instead of proceeding with a decomposition of H , we de ne a model of it adapted to the combinatorics of the time{frequency plane. Let D be a dyadic grid in R. Call I ! 2 DD a tile if jI jj!j = 1. The interval ! is a union of four dyadic subintervals of equal length, !1; !2; !3 and !4, which we list in ascending order. Thus, i < j for all 1 i < j 4 and j 2 !j . (We will only use !j for j = 1; 2; 3.) We adopt the notation t = It !t and tj = It !tj for j = 1; 2; 3. RFix a Schwartz function with ^ supported on [?1=8; 1=8], in addition require that (x ? 16n) (x) dx = 0 for all integers n. Set for all tiles t, and j = 1; 2; 3, ?2ic(! )x x ? c(It ) e jI j ; tj (x) = q t jIt j tj
where c(J ) denotes the center of the interval J . Then our model of the bilinear Hilbert transform is X M f1f2(x) = hfq1 ; t1i hf2; t2i t3 (x); jIt j t which is initially de ned only for Schwartz functions f1 and f2. We emphasize that the sum extends over all tiles, and hence all scales. The analogue of Theorem 1 is
3
Theorem 2 M extends to a bounded operator on Lp1 Lp2 into Lp03 , provided 2 < p1; p2 < 1 and 1 < p03 = (1=p1 + 1=p2)?1 < 2. With more liberal notions of \grid," \tile" and \ tj " the bilinear Hilbert transform is in the convex hull of terms like our model M. In the present situation we can give a proof by way of duality. Thus we take f3 2 Lp3 , where 2 < p3 < 1, and show that X
t
hf1; t1 i hf ; ihf ; i K 3 kf k : j p 2 t2 3 t3 jIt j j =1
q
Y
j
The sum is over positive quantities, namely the decomposition above already captures all of the cancellation necessary for convergence of the sums. It also shows that the sum de ning M is unconditionally convergent in t. And as each fj 2 Lp , where pj > 2, it follows that each function is locally square integrable. As it turns out, L2 arguments are decisive in proving Theorem 2. We localize the sum above in the x variable by setting j
Ft (x) =
jhfj ; tj ij 1[I ](x): t jItj j =1 3
Y
q
Certainly Ft (x) dx = jIt j?1=2 Q3j=1 jhfj ; tj ij. And so we show that F(x) = Pt Ft (x) is integrable. This follows from a weak{type result: For pj as above, there is a > 0 so that for all jrj ? pj j < , 1 j 3, the operator F(x) maps Lr1 Lr2 Lr3 into Lr;1, where 1=r = 1=r1 + 1=r2 + 1=r3. Then a variant of the Marcinkiewicz Interpolation Theorem due to Janson [4] implies the strong type inequality. A single instance of the weak type inequality is R
(1)
jfx j F(x) > K
3
Y
j =1
kfj kr gj K: j
for some constant K . But this inequality implies the weak type result because F commutes with dilations by powers of 2, and so it suces to establish this last inequality. These observations are useful since some of our estimates begin to break down on exceptional sets of small measure. Due the localization of Ft in the time variable, and that we only aim for a distributional inequality, we can delete tiles t whose time coordinate falls in a set of bounded measure. The combinatorics of the time frequency plane enter in by way of the partial order on the tiles given by t < t0 if It It0 and ! !0. Note that t and t0 are not comparable with respect to < if and only if t \ t0 = ;. Being disjoint suggests orthogonality for the functions tj and t0 j0 , the dominant theme of the Lemmas we state below. Call a collection of tiles T a Carleson{Fe erman (CF) set with top q if t < q for all t 2 T. Thus !q \ !t 6= ; for t 2 T. Call T a j {CF set if T is a CF{set for which the intervals !tj intersect for all t 2 T. Notice that if T is a 1{CF set say, then the
4 intervals f!tj j t 2 Tg are pairwise disjoint for j = 2; 3. Therefore, by application of Cauchy{Schwartz
1=2 3 X Y
X
jh f ; ij 1 t 1 ? 1 ? 1 2
jIq j Ft (x) sup q jIq j jhfj ; tj ij : (2) 1 t2T jIt j j=2 t2T t2T Notice that the last two square functions are Littlewood{Paley g{functions, albeit conjugated by an exponential to account for the location of the CF set in frequency. This estimate forms the motivation for the Lemma below, which formalizes a decomposition of the set of tiles that is fundamental to our argument. Lemma 3 Fix pi > 2. There is a > 0 and an 0 > 0 and a constant K so that for all jri ? pi j < and 0 < < 0 , the following holds. The collection of all tiles S is a union 1 [ 3 [ 0 Sn;i;j S=S [ with these properties. First, (4)
S0
n=0 i;j =1
is trivial in that [
s2S0
Is K:
The collection Sn;i;j is a union of disjoint i{CF sets Tq with tops q 2 Sn;i;j , and X
(5)
t2T
Ft (x) 2?n(1=r?) for all x, q 2 Sn;i;j .
q
Here, recall that 1=r = i 1=ri, which can be taken arbitrairly close to 1. And, most signi cantly, for t = mini fpi =2g ? , P
kNn;i;j kt =
(6)
X
q2S
q
1I K 2n(1=t+K): t
n;i;j
With the Lemma in place, we estimate 1
3
X X
X
n=0 i;j =1 t2S
Ft (x)
t
n;i;j
1
X
n=0
K
2?n(1=r?) 1
X
n=0
3
X
i;j =1
kNn;i;j kt
2?n(1=r?1=t?C) K 0:
The last sum is nite as r is arbitrairly close to one, while t + = minfpi=2g > 1 is a xed distance from one. Therefore, with (4), (1) holds. We cannot give the complete construction of the Sn;i;j , but rather the initial steps, in which the nearly orthogonal classes of ti are identi ed. First we make an important comparison to a maximal function. If Tq is an i{CF set with top q, we have for j 6= i, "
X 2 (Tq ; j ) = 1 jIq j t2T jhfj ; tj ij q
#
1=2
C xinf M f (x): 2I 2 j q
5 Here M2 g is the maximal function (M jgj2)1=2 . Thus the set F = SifM2 fi > C ?1g has bounded measure and we de ne S0 = fs j Is F g, making (4) trivial. For all i{CF sets T with top q, and T SnS0 , we have (T; j ) 1, for j 6= i. The remaining construction is inductive. Assume Sthat the Sm;i;j are de ned for S r all m < n and all i; j , in such a way that for S = Sn( m
