On the Boundedness of The Bilinear Hilbert Transform along “nonflat” smooth curves.
The Banach triangle case ().
Abstract.
We show that the bilinear Hilbert transform along curves with is bounded from where are Hölder indices, i.e. , with , and . Here stands for a wide class of smooth “nonflat” curves near zero and infinity whose precise definition is given in Section 2. This continues author’s earlier work in [13], extending the boundedness range of to any triple of indices within the Banach triangle. Our result is optimal up to endpoints.
1. Introduction
This paper, building upon the ideas in [13], continues the investigation of the boundedness properties of the bilinear Hilbert transform along curves. More precisely, if where here is a suitable^{1}^{1}1In a sense that will be specified later. smooth, nonflat curve near zero and infinity, we want to understand the behavior of the bilinear Hilbert transform along defined as
(1) 
One can easily notice that taking we obtain the standard bilinear Hilbert transform. Thus, the problem considered in this paper is, in fact, the “curved” analogue of the celebrated problem of providing bounds to the classical (“flat”) bilinear Hilbert transform  the latter being solved in the seminal work of M. Lacey and C. Thiele ([9], [10]).
It is worth noticing that similar studies regarding curved analogues of classical “flat” objects arise naturally in harmonic analysis in various contexts. Indeed, recalling the discussion in the introduction of [13], a prominent such example is given by the study of the boundedness of the linear Hilbert transform along curves given by
where here , , is a suitable curve. This latter problem first appeared in the work of Jones ([8]) and Fabes and Riviere ([5]) in connection with the analysis of the constant coefficient parabolic differential operators. The study of was later extended to cover more general and diverse situations ([16] [19], [4],[1], [2]).
In the bilinear setting, the work on the “curved” model referring to (1) was initiated by X. Li in [12]. There, he showed that, for the particular case with , , one has that
(2) 
His proof relies on the concept of uniformity, previously used in [3] and originating in the work of T. Gowers ([6]).
In [13], the author proved that (2) remains true for any curve that belongs to  a suitable class^{2}^{2}2For its definition the reader is invited to consult Section 2. It is worth saying that contains in particular the class of all polynomials without linear term  for more on this see Observation 1. of smooth nonflat curves near zero and infinity. Our work greatly extended Li’s result both qualitatively by significantly widening the class of curves, and quantitatively by revealing the scale type decay relative to the level sets of the multiplier’s phase. Our proof is based on completely different methods, involving in a first instance a delicate analysis of the multiplier followed then by a special wavepacket discretization adapted to the twodirectional oscillatory behavior of the phase. In the Appendix of the same paper, we also explained the main idea of how to upgrade the methods employed in [12] in order to be able to obtain the scale type decay.
Later, X. Li and L. Xiao, ([11]), relying heavily on [13] in both the analysis treatment^{3}^{3}3See for example the decomposition of pp. 16 in [11] versus the one in Section 5.3. of [13], the treatment at pp. 29 in [11] versus the corresponding one at pp. 323 in [13], the perturbative strategy applied in analyzing the phase of the multiplier etc. of the multiplier and the “upgraded” uniformity approach required for capturing the key necessary scale type decay revealed in [13], proved that if is any polynomial of degree , having no constant and no linear term ^{4}^{4}4Any such polynomial is just a particular example of an element . then taking one has that maps boundedly where obey , , and . (The bounds here depend only on the degree but not on the coefficients of ). Finally, more recently, in [7], the authors take the proof in [13] and translate it in the uniformity language used initially in [12].
The present paper should be regarded as a natural continuation of [13]. Relying on the author’s previous methods, we extend the earlier results by showing that for any the bounds on can be extended to cover the natural Banach triangle case. Our result is optimal up to end points. For a precise statement of our result as well as for a reminder of the definition of the class of “smooth nonflat” curves near zero and infinity one is invited to consult the next section. Acknowledgement. I would like to thank my wife Anca for drawing Figure 1 in this paper.
2. Main results
We start by recalling from [13] the definition of the set of all curves which are smooth nonflat functions near the origin:

smoothness, no critical points, variation (near origin)
(3) (possibly depending on ) and such that and on ; moreover
(4) where here .

asymptotic behavior (near origin)
There exists with such that:
For any and we have
(5) with and .
For we require
(6) where with
(The existence of , the inverse of , will be a consequence of the next hypothesis.)

nonflatness (near origin)
The main terms in the asymptotic expansion obey
(7) and
(8)
In a similar fashion one can define  the class of smooth, nonflat near infinity functions having the following properties:

smoothness, no critical points, variation (near infinity)
(9) (possibly depending on ) and such that and on ; moreover
(10) where here .

asymptotic behavior (near infinity)
There exists with such that:
For any and we have
(11) with and .
For we require
(12) where with

nonflatness (near infinity)
The main terms in the asymptotic expansion obey
(13) and
(14)
With this done, we set
and .
Observation 1.
Following [13], we list here some interesting features of the class :

Any real polynomial of degree with no constant and no linear term belongs to ;

More generally, any finite linear combination over of terms of the from with belongs to ;

Even more, any finite linear combination over of terms of the form with and is in ;

If then there exist and (all constants are allowed to depend on ) such that for any or , respectively
(15) 
either ;

or .
Thus, if , one has:
(16) 
In [13], we proved the following result
Theorem. Let be a curve such that . Recall the definition of the bilinear Hilbert transform along the curve :
Then extends boundedly from to .
In the present paper, we extend the boundedness range of the above theorem to the Banach triangle (see Figure 1):
Main Theorem. If and defined as above, we have that
(17) 
where the indices obey
(18) 
with
(19) 
Observation 2.
The Banach range in our Main Theorem is optimal up to endpoints. Indeed, let us define be the class of all real polynomials of degree with no constant and no linear terms. Then, in [12] (pp. 9), it is shown that for any , , there exists a polynomial such that for one has that
(20) 
whenever obeys (18) with .
3. Preparatives; isolating the main component of our operator
In Section 3 of [13], after elaborated technicalities, we proved that the study of our bilinear operator , can be reduced to the corresponding study of the bilinear operator defined as follows:
(21) 
with
(22) 
and
(23) 
where we have

the function is smooth, compactly supported with
(24) 
the phase of the multiplier is defined as
(25) 
for fixed (and based on the properties of ) there exists exactly one critical point defined by
(26) where here is an integer depending only on .

the function obeys
(27)
(28) 
The other cases can be treated similarly and we will not discuss them in detail here.
Setting
(29) 
and following [13], we used the scaling symmetry in order to define the following operators:

For (thus )
Remark that
(30) 
For (thus )
As before, notice that
(31)
Observation 3.
In what follows we will focus on the case as the reasonings for the other case can be treated in a similar fashion.
Main reductions
For , using (28) and , one follows the reasonings from Section 5 in [13] and successively simplifies the structure of as follows:

for , recalling the definition of and in (6), we set
(32) and first notice that
(33) Based on the properties of and on the assumptions on it is enough to treat the term
(34) and thus
(35) 
Because the phase of the multiplier has roughly the size with the dominant factor in the variable , we further divide the support in intervals of equal size. Thus, after running the same approximation algorithm as in [13], we deduce that the main component of is given by
(36)
For notational simplicity, we will allow a small abuse, and redenote as just simply .
Let us introduce now several notations:
(37) 
(38) 
(39) 
It will be useful to record the following identity
(40) 
where here
(41) 
is a smooth function supported away from the origin with and finally, the Error represents a smooth, fast decaying term relative to the parameter.
Also, throughout the paper, we have , with and .
Now for , and (recall ) we set
(42) 
From the above considerations, using relation (28) and Observation 3, we deduce that^{5}^{5}5Throughout this paper we will ignore the contribution of the error term in (40) as its treatment was carefully presented in [13]. Also, for notational convenience, we make a small abuse by letting the same function represent the Fourier localization of both and ; in reality the support of should be slightly (e.g. twice) larger than the support of .
(43) 
We now define
(44) 
A similar decomposition can be done for treating the case . In this situation the analogue of (44) will read
(45) 
where for this case is an appropriate adaptation of (43) to the context of (31) instead of (30). For the brevity of the exposition, we won’t insist more on this.
With all these done, our main focus will be on understanding the main properties of the trilinear form
(46) 
with
(47) 
Based on the reductions presented above, we deduce that for proving our Main Theorem it is enough to show
Theorem 1.
Give relation (46), we further notice that Theorem 1 follows from the theorem below^{6}^{6}6For a helpful geometric perspective, the reader is invited to consult Figure 1. after applying real interpolation methods and a telescoping sum argument (see Section 5):
Theorem 2.
Let . Then the following estimates hold ^{7}^{7}7Throughout the paper :
(49) 
and
(50) 
4. Controlled bounds for on the edge .
In this section, we focus on providing “slowly increasing” bounds^{8}^{8}8Relative to the parameter. for our form on the edge (see Figure 1). More precisely, our goal is to get the following tamer result:
Theorem 3.
Let . Then the following estimate holds:
(51) 
4.1. Estimates for the edge .
The main result of this subsection is
Proposition 1.
If , the following holds:
(52) 
Proof.
Now, letting stand for the standard HardyLittlewood maximal function, we deduce the key relation
(54) 
Now, from (54), we deduce that
where for the last relation we used standard LittelwoodPaley theory (for providing bounds on the square function for ) and FeffermanStein’s inequality ([FS]) (for the term involving the functions and ).
Now we make use of the following observation:
(55) 
Indeed, this is a simple consequence of the fact that
(56) 
Thus applying CauchySchwarz inequality we get
(57) 
Inserting (57) in the last estimate on we get:
(58) 
Define now