The Boundedness of the (Sub)Bilinear Maximal Function along "non-flat" smooth curves
Alejandra Gaitan Victor Lie




 













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    Boundedness of the (Sub)Bilinear maximal function]The Boundedness of the (Sub)Bilinear Maximal Function along “non-flat" smooth curvesAlejandra Gaitan: Department of Mathematics
    Purdue University
    
    West Lafayette, Indiana
    47906ygaitanm@purdue.eduVictor Lie: Department of Mathematics
    Purdue University
    
    West Lafayette, Indiana
    47906vlie@purdue.eduAnd Institute of Mathematics of the
    Romanian Academy, Bucharest, RO 70700, P.O. Box 1-764, Romania.The second author was supported by the National Science Foundation under Grant No. DMS-1500958. The final revision of the paper before publication was performed while the second author was supported by the National Science Foundation under Grant No. DMS-1900801.
    
    
    
    
    Let  be the class of smooth non-flat curves near the origin and near infinity introduced in <cit.> and let γ∈. We show - via a unifying approach relative to the corresponding bilinear Hilbert transform H_Γ - that the (sub)bilinear maximal function along curves Γ=(t,-γ(t)) defined as 
        M_Γ(f,g)(x):=sup_>01/2∫_-^ |f(x-t)g(x+γ(t))|dt
    
    is bounded from L^p()× L^q()→ L^r() for all p, q and r Hölder indices, 1/p+1/q=1/r, with 1<p, q≤∞ and 1≤ r≤∞. This is the maximal boundedness range for M_Γ, that is, our result is sharp.
    [
        Alejandra Gaitan and Victor Lie
        December 30, 2023
    ===================================
    
    
    
    
    
    
    
    § INTRODUCTION
    
    
    
    In this paper we study the boundedness properties of the bilinear[Throughout the paper we allow this slight abuse by referring to our maximal operator as “bilinear", though of course, strictly speaking, this is a sub-linear operator in each of the two inputs.] maximal function along a properly chosen class of curves. That is, given Γ=(t,-γ(t)) where γ is a suitable smooth, non-flat curve near zero and near infinity, we ask about the boundedness properties of the bilinear maximal function along Γ defined by
    
        M_Γ   :()×()→ L^∞() 
        
         M_Γ(f,g)(x)   :=sup_>01/2∫_-^ |f(x-t)g(x+γ(t))|dt.
    
    
    Seen as a forerunner to <cit.>, this paper presents a short proof of the maximal boundedness range for M_Γ. This proof along with the corresponding approach to the singular integral version H_Γ in <cit.> and <cit.> emerge as constituent parts of a whole. Together with the work in <cit.> and its follow-up <cit.>, this should be regarded as part of a larger enterprise to provide a unified perspective on several central themes in harmonic analysis that deal with the boundedeness properties of various classes of singular integral and maximal operators.
    
    
    
    
     §.§ Historical background and motivation
    
    
    The problem regarding the boundedness properties of the maximal bilinear operator M_ and of its singular integral analogue defined by
    
    H_Γ(f,g)(x):= p.v.∫_ f(x-t) g(x+(t))dt/t
    has a long history being initially motivated in areas such as ergodic theory - in relation with almost everywhere convergence of bilinear averages (<cit.>) or the L^p-norm convergence of (non-)conventional bilinear averages (see e.g. <cit.>, <cit.>) as well as PDE area - in relation to commutators involving differential operators (<cit.>, <cit.>).
    
    A brief account[Our bibliography here is not exhaustive, listing only the references that are closest and most relevant to our concise study.] of the development of this subject within harmonic analysis is given by
    
      *  in the zero-curvature (or flat) case, i.e. when (t)=a t for some a∈∖{-1}, the problem of providing bounds for the bilinear Hilbert transform H_Γ was raised by A. Calderón in his study of the Cauchy transform along Lipschitz curves, <cit.>. This was approached by M. Lacey and C. Thiele in  <cit.> and <cit.> where they proved that H_Γ obeys L^p× L^q → L^r bounds[It is still an open problem if this range is optimal.] for 1/p+1/q=1/r with 1<p, q≤∞ and 2/3<r<∞. The analogous result for the operator M_Γ was proved by Lacey in <cit.>.
    
    
      *   in the nonzero-curvature (or non-flat) case, e.g. when ” is nonzero near origin and infinity,  the problem of providing bounds for H_Γ was first addressed by X. Li (<cit.>), in the special case Γ(t) = (t, t^d), 2 ≤ d ∈ℕ, by proving that H_Γ obeys L^2× L^2→ L^1 bounds. His proof relies on the concept of σ-uniformity introduced in <cit.> and is inspired by  Gowers's work in <cit.>.
    
        In <cit.> and <cit.> the second author of the present paper proved the maximal range up to end-points for H_ where here  belongs to a suitable class of curves  that includes in particular any generalized (Laurent) real polynomial with no linear term.[I.e., any expression of the form (t):=∑_j=1^d a_j t^_j with {a_j}_j=1^d⊂, {_j}_j=1^d⊂∖{1} and d∈. Here, we use the following convention: if , t∈ we let t^ stand for either |t|^ or sgn  (t)  |t|^. In a follow up paper, <cit.>, the first author extends the present result (and its singular integral analogue in <cit.> and <cit.>) to the case in which one allows the curve  to be a generalized polynomial but with the linear term included.] Besides the general character of the class of curves , a novelty of <cit.>  is that it correctly identifies a scale-type decay relative to the size of the phase of the multiplier. The proof of this result resides on wave-packet analysis (Gabor frames), time-frequency discretization techniques, and orthogonality methods.
    
        Later, elaborating on the ideas and techniques in both <cit.> and <cit.>, the authors in <cit.> prove for both H_Γ  and M_Γ the expected Hölder range in the case (t) a standard polynomial with no linear term with bounds that are uniform in the polynomial's coefficients.
    
    
    For more on the historical evolution of our problem and further connections with other mathematical subjects one is invited to consult <cit.>.
    
    
    
     §.§ The main result
    
     In what follows, we refer to <cit.> for the definition of the class  of smooth “non-flat” curves near zero and infinity.
    
    With this we can state the main theorem of our paper:
    
    
     Let Γ=(t,-γ(t)) be a curve such that[It is easy to notice that our main result extends to the class of curves ^C that is defined to be the set of all curves + C with ∈ and C∈, i.e., our result is closed under translation by constants of  with ∈.]γ∈. Consider the bilinear maximal function defined by (<ref>).
    
     Then M_Γ extends boundedly from L^p()× L^q() into L^r() where the indices p, q, r obey
    	
        1/p+1/q=1/r,
    
    	and[Observe that one gets trivially the desired bound for the triple of indices (p,q,r)=(∞,∞,∞) corresponding to the point C in Figure <ref>. That is why we will exclude this case from all our future reasonings.]
        1<p≤∞ ,  1<q≤∞ 1≤ r≤∞.
    
    This result is sharp.
    
    Let 𝒫_d be the class of all real polynomials of degree d with no linear and constant terms. In <cit.>, the authors show that for any d∈, d≥2, there exists a polynomial
    P_d∈𝒫_d such that for Γ=(t,P(t)) one has that M_Γ is unbounded whenever p, q, r obey (<ref>) with 1<p, q<∞ and r<d-1/d.
    
    However, for any d∈, d≥2 we have that 𝒫_d⊂. Thus, in order for (<ref>) and (<ref>) to hold for any γ∈ we must have r≥ 1, hence the claimed optimality of the range in our theorem above.
    
    
     §.§ Main ideas; relevance
    
    
    
    
    As already mentioned earlier, our approach of the maximal operator M_ is developed along a natural correspondence with the singular integral approach associated to the bilinear Hilbert transform H_. For the remaining part of this section, in order to make our reasoning transparent, we will keep our presentation at an informal level.
    
    The philosophy behind our approach relies on the following observation:
    
    The bilinear Hilbert transform H_ can be written as
    
        H_(f,g)=∑_j∈ H_j(f,g):=∑_j∈∫_ f(x-t) g(x+(t)) _̊j(t) dt ,
    
    where  here ρ_j(t)=2^j ρ(2^j t) for a suitable ρ∈ C^∞_0() with ρ⊆{t∈|1/4<|t|<1} and obeying the mean zero condition
        ∫_ρ(t)dt=0 .
    
    
    In contrast with this, assuming from now on wlog that f, g≥ 0, the bilinear maximal function M_ can be expressed as
    
        M_(f,g)=sup_j∈ M_j(f,g):=sup_j∈∫_ f(x-t) g(x+(t))_j(t) dt ,
    
    where  as before ρ_j(t)=2^j ρ(2^j t) for a suitable ρ∈ C^∞_0()positive with ρ⊆{t∈|1/4<|t|<1} and obeying
    
        ∫_ρ(t)dt=1 .
    Thus, in a nutshell, one has
    
      * H_ is a conditional l^1-sum of pieces {H_j}_j∈ with the associated kernels {_̊j} having mean zero;
    
      * M_ is an l^∞-sum of pieces {M_j}_j∈ with the associated kernels {_j} integrating to one.
    
    
    Once at this point, one can isolate the corresponding components M_j (and H_j) and regard them as bilinear Fourier multiplier operators. In doing so, it becomes transparent that the phase oscillation in the integral definition of the multiplier will play a key role in the proof. Thus, following the strategy designed for H_ in <cit.>, we analyze the stationary points of the phase and decompose accordingly[Same type of decomposition holds for H_j.]
        M_j=M_j^L+M_j^HΔ̸+M_j^HΔ ,
    
    where
    
      * M_j^L is the j-low frequency component (essentially no phase oscillation);
    
    
      * M_j^HΔ̸ is the j-high frequency component away from the stationary points region;
    
    
      * M_j^HΔ is the j-high frequency component along the stationary points region.
    As it turns out, only the first components M_j^L and its Hilbert analogue H_j^L are sensitive to the existent distinction
    between (<ref>) and (<ref>). Consequently, for the last two components one will be able to identify M_j^HΔ̸ and M_j^HΔ with their analogue H_j^HΔ̸ and H_j^HΔ, respectively.
    
    From here on, the strategy follows the dichotomy present in <cit.>, <cit.> and <cit.>:
    
    
      *  the control over the low frequency component ^L_Γ:=sup_j |M_j^L| is obtained via Taylor series expansions exploiting the lack of oscillation on the multiplier side; indeed, Theorem <ref> states that[Here M stands for the standard Hardy-Littlewood maximal function.]
        |_Γ^L(f,g)(x)|≲_γ Mf(x)Mg(x).
    
      *  the bounds for the high frequency component away from the stationary points region ^HΔ̸_Γ:=sup_j |M_j^HΔ̸| rely on a further discretization combined with (non)-stationary principle in disguise - essentially a careful integration by parts procedure - and a novel shifted square/maximal function argument. As a byproduct of the latter, each of the elementary building blocks in the decomposition of ^HΔ̸_Γ will be pointwise bounded by a product of shifted maximal functions (multiplied by a suitable decaying factor) thus mirroring (<ref>) (see relation (<ref>)). The superposition of these pointwise estimates will provide us with the global control over ^HΔ̸_Γ.
         This is the content of Theorem <ref>.
    
    
      *  the high frequency component along the stationary points region defined as ^HΔ_Γ:=sup_j| M_j^HΔ| is of course the main term of our operator. After the linearization of the supremum, one decomposes the main term as
     
        M_j(x)^HΔ(f,g)(x)=∑_m∈  M_j(x),m^HΔ(f,g)(x) ,
    
    where each M_j(x),m^HΔ(f,g)(x) has the phase of the multiplier oscillating at height ≈2^m.
    
    Applying now Remark 2, for j(x)=j, one can identify M_j,m^HΔ(f,g) with the analogue H_j,m^HΔ(f,g) and thus use the key estimate obtained in <cit.> to get that there exists ϵ∈(0,1) such that, for any j∈ and m∈, one has
    
        M^HΔ_j,m(f,g)_L^1≲_γ 2^-ϵ mf_2 g_2.
    
    
    At this point hinted by Remark 1 and the approach in <cit.>, one makes the simple observation
    
        M^HΔ_j(x),m(f,g)(x)_L^1(dx)≤∑_jM^HΔ_j,m(f,g)_L^1≲_γ 2^-ϵ mf_2 g_2 ,
    
    where in the last line we used Cauchy-Schwarz and the almost orthogonality of the inputs along the sequence {M^HΔ_j,m(f,g)}_j∈. This takes care of the bound L^2× L^2 → L^1.
    
    The complete boundedness range, stated in Theorem <ref> relies in part on the techniques developed in <cit.> along with (shifted or generalized) square function arguments.
    
    
    
    The above description is part of a more general, philosophical approach - see <cit.> - of treating simultaneously and in a unitary fashion both the singular - here H_Γ - and its maximal variant - here M_.
    
    Finally, this paper is meant as a preface to the significantly more complex study in <cit.>, in which, completing the unification of the three themes introduced in <cit.>, we will develop a unified approach for the boundedness of H_ and M_ in the case in which Γ=(t,-γ(x,t)) - thus allowing an x-dependence of  - where here  is a suitable non-degenerate curve that is smooth and doubling in t but only measurable in x.
    
    
    
    
    
    § NOTATION
    
    
    Without lost of generality, from now on throughout the paper, we will assume that f and g are non-negative functions. For transparency, we will mostly follow the notations and conventions as in <cit.>.
    
    For example, given any ϕ∈(), j, l∈ and k∈, we set
    
        ϕ_j(x)   :=ϕ(x/2^j),
        ϕ̃_l(x)   :=x^lϕ(x),
        ϕ^(k)(x)   :=(d/dx)^kϕ(x),
        ϕ̃_l,j(x)   :=ϕ̃_l(x/2^j).
    
    
    If f∈() we denote the Fourier transform of f with f̂, where[In our later reasonings we will often ignore the constant 1/√(2π).]
        f̂(ξ):=1/√(2π)∫_ f(x) e^-i x ξ d x ,      ξ∈
    
    and the inverse Fourier transform of f with f̌, where
    
        f̌(ξ):=1/√(2π)∫_ f(x) e^i x ξ d x ,      ξ∈ .
    
    
    We denote by M the standard Hardy-Littlewood maximal operator defined on L^1_loc() as
    
        Mf(x):=sup_I∋ x
         I interval1/|I|∫_I |f(x)|dx.
    
    
    For t∈ and ∈̌C^∞_0 with (̌0)=0 we define the t-shifted square functions with respect to $̌ by
    
        S_t^f(x):=(∑_j∈|(f* (2^j(2^j·)))(x-t/2^j)|^2)^1/2 ,
    
    and thet-shifted maximal function by[As expected, in the maximal function case, the mean zero condition of $̌ becomes irrelevant and one can replace the original conditions imposed on$̌ by merely ∈() any (normalized) function with ≥ 0 and _L^1()=1. Given this, in what follows we no longer specify the dependence of our maximal function on $̌.
        M^(t)f(x):=sup_j∈|(f* (2^j(2^j·)))(x-t/2^j)| .
    
    
    If1/4 <|t|<1,∈̌C^∞_0with(̌0)=0,n∈and∈we define the(n,_t)-shifted square functions with respect to$̌ by
    
    
        S_n,_t^f(x):=(∑_j∈|(f*{1/2^-jγ'(2^-j)2^n (·/2^-jγ'(2^-j)2^n)}) (x+γ(2^-jt))|^2)^1/2 ,
    
    and the (n,_t)-shifted maximal function by
    
        M^(2^n)__tf(x):=sup_j∈|(f*{1/2^-jγ'(2^-j)2^n (·/2^-jγ'(2^-j)2^n)}) (x+γ(2^-jt))| .
    
    
    
    Throughout the paper p^*=min{p,p'} where p' denotes de Hölder conjugate of p.
    
    Also we set a_+:=max{a,0}.
    
    For A, B>0, we say that A≲ B if there is C>0 such that A≤ CB.  We say that A≈ B if A≲ B and B≲ A. Finally, if d is a real parameter, we write A≲_d B if there exists C=C(d)>0 such that A≤ CB with the obvious correspondence for A≈_d B.
    
    
    
    
    § PRELIMINARIES: THE BILINEAR MAXIMAL FUNCTION AS A MULTIPLIER OPERATOR; REDUCTION TO THE MAIN COMPONENT
    
    
    In this section, we first reshape the maximal operator in a convenient form adapted to Fourier analytic methods followed by a careful analysis of the associated multiplier. As a result, we will be able to decompose our initial operator in three components: the first two of them - considered as “error terms" - will be treated in the present section, while the remaining one - i.e the main component - will be left for the next sections.
    
    
    
    Focusing now on the main subject, we record the following simple observation: since we deal with a positive integral operator, it is enough to study our maximal function with the supremum ranging over dyadic numbers, i.e., letting  ∼ 2^j+1 with j∈, we have that
    
        M_Γ(f,g)(x)≈sup_j∈1/2^j+1∫_2^j≤|t|≤2^j+1 f(x-t) g(x+γ(t)) dt.
    
    Based on the properties of the curves ∈ we have that[The condition discussed below will be needed when discussing the boundedness of our operator M_Γ(f,g) in the regime L^∞× L^q for 1<q≤∞ - see Subsection 5.2.]
        ∃  (possibly large)      j_0∈  depending only on  such that
        for any nonzero    |t|≤ 2^-j_0 or |t|≥ 2^j_0 one has
        
        |(t)|>0,     |'(t)|>0,  and  |”(t)|>0 .
    
    With this, based on (<ref>), we notice that
    
        M_Γ(f,g)(x)   ≲sup_|j|>j_01/2^j+1∫_2^j≤|t|≤2^j+1 f(x-t) g(x+γ(t)) dt 
           + ∑_|j|≤ j_01/2^j+1∫_2^j≤|t|≤2^j+1 f(x-t) g(x+γ(t)) dt .
    
    
    Now for the latter term, using triangle and Minkowski's inequality, we trivially have
    
        ∑_|j|≤ j_01/2^j+1∫_2^j≤|t|≤2^j+1 f(x-t) g(x+γ(t)) dt_L^r(dx)
           ≤∑_|j|≤ j_0∫_2^j≤|t|≤2^j+1f(x-t) g(x+γ(t))_L^r(dx) dt/2^j+1
           ≲_p,r j_0 f_p g_q ,
    
    with p, q, r satisfying (<ref>) and (<ref>).
    
    Thus, with a slight notational abuse, we will assume from now on that
    
        M_Γ(f,g)(x)=sup_j∈|j|>j_01/2^j+1∫_2^j≤|t|≤2^j+1 f(x-t) g(x+γ(t)) dt.
    
    
    
    Let now ρ be a nonnegative, even, C^∞_0() function with ρ⊆{t∈|1/4<|t|<1} and
    
        ∫ρ(t)dt=1 .
    
    
    Set ρ_j(t)=2^jρ(2^j t) (with j∈).
    
    With this, standard reasonings show that M_Γ≈_Γ, where here
    
        _Γ(f,g)(x):=sup_j∈|j|>j_0_Γ,j(f,g)(x):=sup_j∈|j|>j_0∫_ f(x-t)g(x+γ(t))ρ_j(t)dt .
    
    
    Turning our attention on the Fourier side, we have that
    
        _Γ(f,g)(x)=sup_j∈|j|>j_0∫_∫_f̂(ξ)ĝ(η)m_j(ξ,η)e^iξ xe^iη xdξ dη ,
    
    where the multiplier is given by
    
        m_j(ξ,η):=∫_ e^-iξ te^iηγ(t)ρ_j(t)dt
        =∫_ e^-iξ/2^j te^iηγ(t/2^j)ρ(t)dt.
    
    
    
    Since the integrand in m_j is highly oscillatory, the analysis of our multiplier relies fundamentally
    on understanding the stationary points of the phase function
    
        φ_ξ,η(t):=-ξ/2^jt+ηγ(t/2^j) .
    
    
    Thus, as in <cit.> and <cit.>, it is natural to decompose the multiplier based on the size of the terms ξ/2^j and γ'(2^-j)η/2^j.
    Let now ϕ be a positive even Schwartz function supported in {t:1/2≤|t|≤ 2} with
    ∑_n∈ϕ_n(t)=1   for all   t≠0 .
    
    Then, for every j∈∖ [-j_0, j_0], we write
    
        m_j=∑_m,n∈ m_j,m,n,
    
    where
    
        m_j,m,n(ξ,η):=m_j(ξ,η) ϕ(ξ/2^m+j) ϕ(γ'(2^-j)η/2^n+j) .
    
    
    From the stationary phase principle  we have that the main contribution for the integrand in m_j comes from the values near the stationary point(s). With this, we follow the approach in <cit.>, and split the analysis of our multiplier into three components, corresponding to the behavior of the phase in (<ref>), as follows:
    
    
      I)Low frequency case - the phase function is essentially constant:
    
        m_j^L=∑_(m,n)∈(_-)^2 m_j,m,n.
    
      II)High frequency far from diagonal - high oscillation without stationary points:
    
        m_j^HΔ̸=∑_(m,n)∈^2∖((_-)^2∪Δ)m_j,m,n.
    
      III)High frequency close to diagonal - high oscillation with present stationary points:
    
        m_j^HΔ=∑_(m,n)∈Δm_j,m,n.
    
    Here Δ={(m,n)∈^2:m,n≥0,|m-n|≤ C(γ)} with C(γ)≥ 1 a large constant depending only on γ.
    
    With this, we have
    
        m_j=m_j^L+m_j^HΔ̸+m_j^HΔ.
    
    
    We end our preliminaries by transforming the maximal nature of our operator via a linearization procedure
    
        _Γ(f,g)(x)≈∫_∫_f̂(ξ)ĝ(η)m_j(x)(ξ,η)e^iξ xe^iη xdξ dη
    
    where j(x):→∖ [-j_0, j_0] is a measurable function who assigns for each point x∈ a value j∈∖ [-j_0, j_0] for which _Γ,j(f,g)(x) is at least half of the value of _Γ(f,g)(x).
    
    
    
    
    
    
    
    
    
     §.§ Low frequency term
    
    As discussed in <cit.>, the dichotomy between the singular integral behavior and its maximal version manifests precisely in the low frequency case: indeed, this is the only situation that requires the mean zero condition of the kernel 1/t in the bilinear Hilbert transform case which translates, after the standard discretization procedure, into a condition of the from (<ref>) as opposed to (<ref>) in the maximal case. Based on this, the bilinear Hilbert transform required similar techniques with those used to prove the Coifman-Meyer theorem in order to obtain the necessary bounds for the corresponding low frequency component (see <cit.>).
    
    In the bilinear maximal function case, our function ρ only satisfies ∫ρ(t)dt=1 but the good news is that - recall Remark 1 in Section 1.3. -  we only need to control an l^∞()-sum/norm (in j) as opposed to a conditional l^1()-sum in the bilinear Hilbert transform case.
    
    With this being said, we have
    
    
    Set
    
        ^L_Γ(f,g)(x):=∫_∫_f̂(ξ)ĝ(η)m_j(x)^L(ξ,η)e^iξ xe^iη xdξ dη.
    
    Then, the following holds
    
        |_Γ^L(f,g)(x)|≲_γ Mf(x)Mg(x).
    
    Furthermore, for any p, q and r satisfying (<ref>) and (<ref>) we have
    
        _Γ^L(f,g)_r≲_γ,p,qf_pg_q.
    
    
    Fix m,n ∈_-. From the definition of ϕ we have that |ξ/2^j(x)|, |γ'(2^-j(x))η/2^j(x)|≲ 1, and by property (3)  as part of the definition of  in <cit.>, we have
    
        sup_t∈ρ|ηγ(t/2^j(x))|<C_γ,
    
    where C_γ is a positive constant that is allowed to change from line to line.
    
    Recalling now (<ref>) - (<ref>), we develop the phase in (<ref>) in a Taylor series
    
        m_j(x),m,n(ξ,η)   =(∫_∑_k∈(-iξ/2^j(x)t)^k/k!∑_l∈(iηγ(t/2^j(x)))^l/l !ρ(t)dt)ϕ(ξ/2^m+j(x))ϕ(γ'(2^-j(x))η/2^n+j(x))
        
           =∑_k,l≥0(-1)^k i^k+l/k! l !C_k,l(ξ/2^j(x))^kϕ(ξ/2^m+j(x)) (γ'(2^-j(x))η/2^j(x))^lϕ(γ'(2^-j(x))η/2^n+j(x)),
    
    where
    
        C_k,l=C_j(x),k,l:=∫_ t^k(γ(2^-j(x)t)/2^-j(x)γ'(2^-j(x)))^l ρ(t)dt.
    
    Since
    
        |C_k,l|≲ C_γ^k+l
    
    uniformly in x, the sum (<ref>) converges absolutely.
    
    Set now
    
        ψ(x):=∑_m<0ϕ(x/2^m) ,
    
    and notice that ψ∈() since ψ=1-∑_m≥0ϕ_m.
    
    Inserting (<ref>) in (<ref>) and recalling (<ref>), we have
    
        m^L_j(x)(ξ,η)   =∑_k,l≥0(-1)^k i^k+l/k! l !C_k,l(ξ/2^j(x))^kψ(ξ/2^j(x)) (γ'(2^-j(x))η/2^j(x))^l
        ψ(γ'(2^-j(x))η/2^j(x)),
           =∑_k,l≥0(-1)^k i^k+l/k! l !C_k,l ψ̃_k(ξ/2^j(x)) ψ̃_l(γ'(2^-j(x))η/2^j(x)).
    
    
    
    
    Thus, putting together (<ref>) and (<ref>) and recalling (<ref>), we have
    
        _Γ^L(f,g)(x)   =∑_k,l≥0(-1)^k i^k+l/k! l !C_k,l(∫_f̂(ξ)ψ̃_k(ξ/2^j(x))e^iξ xdξ) (∫_ĝ(η)ψ̃_l(γ'(2^-j(x))η/2^j(x)) e^iη xdη)
           ≈∑_k,l≥0(-1)^k /k! l !C_k,l(f*{1/2^-j(x)ψ̌^(k)(·/2^-j(x))})(x)
           ×(g*{1/2^-j(x)γ'(2^-j(x))ψ̌^(l)(·/2^-j(x)γ'(2^-j(x)))})(x).
    
    
    We can construct the function ϕ such that for every k∈, the function ψ̌^(k) has an integrable radially decreasing majorant Φ_k with Φ_k_1≲ C^k for some constant C>0. Thus, using a classical result in <cit.>, one has
    
        |(f*{1/2^-j(x)ψ̌^(k)(·/2^-j(x))})(x)|   ≤sup_t>0|(f*{1/tψ̌^(k)(·/t)})(x)|
           ≤Φ_k_1 Mf(x)
           ≲ C^k Mf(x).
    
    
    Analogously,
    
        |(g*{1/2^-j(x)γ'(2^-j(x))ψ̌^(l)(·/2^-j(x)γ'(2^-j(x)))})(x)|≲ C^l Mg(x).
    
    Putting together (<ref>), (<ref>) and (<ref>) we have
    
        |_Γ^L(f,g)(x)|   ≲∑_k,l≥01/k! l !|C_k,l|C^k+lMf(x)Mg(x)
           ≲ Mf(x)Mg(x)∑_k,l≥01/k! l !C_γ^k+l C^k+l
           ≲_γ Mf(x)Mg(x),
    
    where here we used (<ref>).
    
    Therefore, (<ref>) holds.
    
    Taking now p,  q,  r satisfying (<ref>) and (<ref>), Hölder’s inequality and the standard L^p strong type estimate for the classical Hardy-Littlewood maximal function imply
     
        _Γ^L(f,g)_r   ≲_γMf Mg_r≲_p,qMf_pMg_q≲_p,qf_p g_q,
    
    which concludes our proof.
    
    
    
    
    
     §.§ High frequency term far from diagonal
    
    
    In this section we discuss the second (off-diagonal) term in our decomposition of m_j(x), that is, m_j(x)^HΔ̸.
    
    Our main focus, will be to prove the following
    
    
     Set
     
        ^HΔ̸(f,g)(x):=∫_∫_f̂(ξ)ĝ(η)m_j(x)^HΔ̸(ξ,η)e^iξ xe^iη
        xdξ dη,
    
     and for (m,n)∈^2∖((_-)^2∪Δ) let
      
        ^HΔ̸_m,n(f,g)(x):=∫_∫_f̂(ξ)ĝ(η)m_j(x),m,n(ξ,η)e^iξ xe^iη
         	xdξ dη.
    
     Then, for any p, q, r satisfying (<ref>) and  (<ref>), the following holds
     
      *  If (m,n)∈^2∖((_-)^2∪Δ) then, recalling (<ref>) and (<ref>), one has the pointwise estimate
      
        |^HΔ̸_m,n(f,g)(x)|≲1/2^max{m,n} ∫_{1/4 <|t|<1 } M^(2^m t)f(x) M^(2^n)__t g(x) dt        ∀ x∈ ,
    
    which further implies
     	
        ^HΔ̸_m,n(f,g)_r≲_γ,p,q1/2^max{m,n}(m_++10) (n_++10)f_pg_q .
    
      *  Deduce thus that the global component obeys
     	
        _Γ^HΔ̸(f,g)_r≲_γ,p,qf_pg_q.
    
    
    We first state two lemmas which will be used in the proof of the above statement.
    
    The first lemma contains two items: point i) represents Lemma 4.8 proved in <cit.> and can be found in the math literature under various modified forms - see e.g.<cit.> and <cit.>. The second point ii) can be proved with similar techniques to those used for i) and relies in a key fashion on the properties of the class of curves . The second lemma mirrors the first one and addresses the maximal function analogue. The case of M^(2^n t) was also referred to and proved within Proposition 42 in <cit.>.
    
    
    Let 1/4 <|t|<1, ∈̌C^∞_0 with (̌0)=0, n∈ and ∈.
    
     i) Recall the definition of S_t^ given by (<ref>) in the Notation section.
    
    Then, for any 1<p<∞, one has – uniformly in t– that
    
        S_2^n t^f_p≲_,̌p (1+n_+)^2/p^*-1f_p.
     ii) Similarly, if S_n,_t^ stands for the shifted square functions defined in (<ref>) then, for any 1<p<∞, one has – uniformly in t– that
    
        S_n,_t^f_p≲_,,̌p(1+n_+)^2/p^*-1f_p .
    
    With the same assumptions as above, and appealing to (<ref>) and (<ref>) in the Notation section we have
    that
    
        M^(2^n t)f_p, M^(2^n)__tf_p≲_,,̌p(1+n_+)^1/pf_p ,
    
    holds uniformly in t and for all 1<p≤∞.
    
    
    With these we are now ready to present
    
    
    
    Fix a pair (m,n)∈^2∖((_-)^2∪Δ). As in the proof of <cit.>, integrating by parts, we write
    
        m_j(x),m,n=A_j(x),m,n(ξ,η)+B_j(x),m,n(ξ,η),
    
    where
    
        A_j(x),m,n(ξ,η):=(∫_ e^-iξ/2^j(x)t e^iηγ(t/2^j(x))iρ'(t)/-ξ/2^j(x)+η/2^j(x)γ'(t/2^j(x))dt)
        ϕ(ξ/2^m+j(x)) ϕ(γ'(2^-j(x))η/2^n+j(x)),
    
    and
    
        B_j(x),m,n(ξ,η):= (∫_ e^-iξ/2^j(x)te^iηγ(t/2^j(x))-iη/2^2j(x)γ”(t/2^j(x))/(-ξ/2^j(x)+η/2^j(x)γ'(t/2^j(x)))^2ρ(t)dt)
        ϕ(ξ/2^m+j(x))ϕ(γ'(2^-j(x))η/2^n+j(x)).
    
    
      * : m-n≥ C_γ>>1
    
    By (3) in the definition of , we have that
    
        1-ξ2^j(x)+η2^j(x)γ'(t/2^j(x))=-12^m1/ξ/2^m+j(x)∑_l∈12^l(m-n)(γ'(2^-j(x))η/2^n+j(x)/ξ/2^m+j(x))^l (Q'(t)+Q'_j(x)(t))^l.
    
    Set
    
        A_j(x),m,n=∑_l∈ A_j(x),m,n,l,
    
    with
    
        A_j(x),m,n,l(ξ,η)   =-1/2^m1/2^l(m-n)(∫_ e^-iξ/2^j(x)t e^iηγ(t/2^j(x)) i ρ'(t) (Q'(t)+Q_j(x)(t))^l dt)
           ×ϕ̃_-l-1(ξ/2^m+j(x))ϕ̃_l(γ'(2^-j(x))η/2^n+j(x))
    
    where here ϕ̃_l, l∈, is smooth, compactly supported away from the origin, and obeying
    
        ϕ̃_l_C^β≲β! (|l|+)̱^ 
        C^|l|+
    
    for ∈̱ and some fixed C>0.
    
    We recall now the fact that, from the properties of ∈ one has that Q_j_C^N≤ a_j with a_j→ 0 as
    j→∞ and thus, since |j(x)|≥ j_0, one can choose j_0∈ large enough so that Q_j(x)_C^N≤Q_C^N  a.e. x∈.
    
     Defining the operator Λ^A as
    
        Λ^A_m,n,l(f,g)(x):=∫_∫_f̂(ξ)ĝ(η) A_j(x),m,n,l(ξ,η)e^iξ xe^iη xdξ d η,
    
    we have
    
        |Λ^A_m,n,l(f,g)(x)|   =|1/2^m1/2^l(m-n)∫_ (f*ϕ̌̃̌_-l-1,m+j(x))(x-2^mt/2^m+j(x))
           ×(g*{ϕ̃_l(2^-j(x)γ'(2^-j(x))2^n ·)}1̌)(x+γ(2^-j(x)t))  ρ'(t) (Q'(t)+Q'_j(x)(t))^l  dt|
           ≲1/2^m(2Q'_∞/2^m-n)^l ∫_{t∈:1/4 <|t|<1 }sup_j∈|(f*ϕ̌̃̌_-l-1,j)(x-2^mt/2^j)| 
           ×sup_j∈|(g*{1/2^-jγ'(2^-j)2^nϕ̌̃̌_l (·/2^-jγ'(2^-j)2^n)}) (x+γ(2^-jt)) | dt .
    
    Once at this point one can proceed in two ways:
    
      *  the simplest route is to apply the estimate
    
        |Λ^A_m,n,l(f,g)(x)|≲1/2^m(C_/2^m-n)^l ∫_{1/4 <|t|<1 } M^(2^m t)f(x) M^(2^n)__t g(x) dt .
    
      *  alternatively, one can appeal to a shifted square function argument, and write
    
        |Λ^A_m,n,l(f,g)(x)|≲1/2^m(C_/2^m-n)^l (∫_{1/4 <|t|<1 } S_2^mt^ϕ̃_-l-1f(x) S_n,_t^ϕ̃_lg(x) dt) .
    
    
    
    
    
    Let Λ^A_m,n:=∑_l∈Λ^A_m,n,l. Using now Fubini, Hölder, Lemma <ref> and (<ref>) (or alternatively, Lemma <ref> and (<ref>)), one deduces
    
        Λ^A_m,n(f,g)_r≲_γ,p,q1/2^m(m_++10)(n_++10) f_p g_q .
    
    
    
    
    Similarly, one gets the analogues of (<ref>) for the multiplier B_j,m,n, where in this latter case one uses that
    1/2^2jγ”(2^-jt)=2^-jγ'(2^-j)(Q”(t)+Q_j”(t)) .
    
      * : n-m>C_>>1
    
    As we did in Case 1, we write
    
        1/-ξ/2^j(x)+η/2^j(x)γ'(t/2^j(x))
        =1/2^n1/γ'(2^-j(x))η/2^n+j(x)∑_l∈1/2^l(n-m)(ξ/2^m+j(x)/γ'(2^-j(x))η/2^n+j(x))^l1/(Q'(t)+Q'_j(x)
        	(t))^l+1.
    
    Write
    
        Ã_j(x),m,n=∑_l∈Ã_j(x),m,n,l,
    
    where
    
        Ã_j(x),m,n,l(ξ,η)   =1/2^n1/2^l(n-m)(∫_
        e^-iξ/2^j(x)te^iηγ(t/2^j(x))i
        ρ'(t)/(Q'(t)+Q'_j(x)(t))^l+1dt) 
           ×ϕ̃_l(ξ/2^m+j(x)) ϕ̃_-l-1(γ'(2^-j(x)) η/2^n+j(x)).
    
    
    Letting Λ^Ã_m,n,l be the operator with symbol
    Ã_j(·),m,n,l one proceeds as before in order to get
    
        |Λ^Ã_m,n,l(f,g)(x)|≲1/2^n (C_/2^m-n)^l (∫_{1/4 <|t|<1 } M^(2^m t)f(x) M^(2^n)__t g(x) dt)  ,
    
    and respectively
    
        Λ^Ã_m,n(f,g)_r≲_γ,p,q1/2^n(m_++10) (n_++10) f_p g_q.
    
    Again, a similar reasoning applies to the multiplier B̃_j(·),m,n.
    
    Putting together all of the above, one concludes
    
        M_Γ^HΔ̸(f,g)_r    ≲_γ,p,q∑_(m,n)∈^2∖((_-)^2∪Δ)
         m-n≥ C_1/2^m(m_++10) (n_++10) f_p g_q
            +
        ∑_(m,n)∈^2∖((_-)^2∪Δ)
         n-m≥ C_1/2^n(m_++10) (n_++10) f_p g_q
           ≲_γ,p,qf_p g_q.
    
    
    
    
    
     §.§ High frequency term close to diagonal
    
    
    In this section we consider the last and most relevant component of m_j(x), that is, the term m_j(x)^HΔ.
    
    Our goal for the remaining part of the paper will be to prove the following
    
    
    Set
     
        ^HΔ(f,g)(x):=∫_∫_f̂(ξ)ĝ(η)m_j(x)^HΔ(ξ,η)e^iξ xe^iη
        xdξ dη.
    
    Then for any p, q,  r satisfying (<ref>) and (<ref>) we have
    
        _Γ^HΔ(f,g)_r≲_γ,p,qf_pg_q.
    
    
    
    In this subsection we will describe the strategy of reducing our Theorem <ref> to two intermediate results.
    
    We start by noticing that in (<ref>) it is enough to only consider the case m=n. Also, wlog we assume that our t-integration encoded in the expression of m_j(x)^HΔ is performed over _+.
    
     Recalling Remark 2 in Section <ref>, since the mean zero condition in (<ref>) plays no role in the regions where the phase of the multiplier is highly oscillatory, we are justified from now on to identify our operators T_j,m and B_j,m introduced below with the corresponding ones defined in <cit.>. Consequently, one can transfer with no modifications the theorems regarding T_j,m and B_j,m in <cit.> to our current setting.
    
    
    Using now the notation from <cit.> and after some elaborate technicalities, one can prove that the study of the operator with multiplier m_j(x)^HΔ can be reduced to the study of the bilinear operator T(f,g) defined by
    
        T(f,g)(x):=∑_m∈T_j(x),m(f,g)(x) .
    
    Here, for each j∈ and m∈, we define
    
        T_j,m(f,g)(x):=∫_∫_f̂(ξ)ĝ(η)v_j,m(ξ,η)e^iξ xe^iη x dξ dη
    
    and
    
        v_j,m(ξ,η):=2^-m/2e^i φ_ξ,η(t_c)ζ(γ'(2^-j)η/2^m+j,
        ξ/2^m+j/γ'(2^-j)η/2^m+j) ϕ(ξ/2^m+j) ϕ(γ(2^-j)η/2^m+j)
    
    where we have
    
      *  the phase of the multiplier - recall (<ref>) - is defined as
     
        φ_ξ,η(t):=-ξ/2^jt+ηγ(t/2^j).
    
      *  For ξ,η fixed, φ_ξ,η has a unique critical point t_c=t_c(j,ξ,η) ∈ [2^-k(γ),2^k(γ)], where k(γ)∈ depends only on γ.
     
      * ζ:[1/10,10]×[2^-k(γ),2^k(γ)]→ satisfies ζ_N-3≲  1 for some N≥7.[The regularity index N here can be lowered but we will not detail this fact here.]
    
    Also, from the properties of the class , wlog we can assume that
    
        lim_t→0
         t≠0γ'(t)=0   and   lim_t→∞γ'(t)=∞.
    
    
    Now it turns out that in formulas (<ref>) and (<ref>) one can replace the function ζ by 1. In order to clarify this point and make transparent the parallelism with the reasonings in <cit.>, we first need to recall some of the notations that we used in <cit.>. Thus, letting Ψ_η(ξ)=-φ_ξ,η(t_c), we have
    
    
    
    
    
      *  For j>0 one sets
    
        B_j,m(f(·),g(·))(x):=[γ'(2^-j)]^1/2T_j,m(f(2^m+j·),g(2^m+j/γ'(2^-j)·))(xγ'(2^-j)/2^m+j),
    
    that is
    
        B_j,m(f,g)(x)=2^-m/2[γ'(2^-j)]^1/2∫_∫_f̂(ξ)ĝ(η)e^i(γ'(2^-j)ξ+η)xe^-i 2^m 2^jΨ_η/γ'(2^-j)(ξ)ζ(η,ξ/η)ϕ(ξ)ϕ(η)dξ dη.
    
      *  For j≤ 0 one sets
    
        B_j,m(f(·),g(·))(x):=[γ'(2^-j)]^-1/2T_j,m(f(2^m+j·),g(2^m+j/γ'(2^-j)·))(x/2^m+j),
    
    that is
    
        B_j,m(f,g)(x)=2^-m/2[γ'(2^-j)]^-1/2∫_∫_f̂(ξ)ĝ(η)e^i(ξ+η/γ'(2^-j))x e^-i 2^m 2^jΨ_η/γ'(2^-j)(ξ)ζ(η,ξ/η)ϕ(ξ)ϕ(η)dξ dη.
    
    In what follows we will assume j>0, as the other case j < 0 can be treated in a similar way.
    
    From the definition of γ∈ we define for x∈
        R(u):=∫_1^u r(s)ds     R_j(u)=∫_1^u r_j(s)ds,
    
    then
    
        2^jΨ_η/γ'(2^-j)(ξ)=η R(ξ/η)+η R_j(ξ/η).
    
    From the properties of ∈ we know that lim_|j|→∞R_j_C^N=0. Thus, by properly choosing j_0∈ in (<ref>) (based on the properties of ∈), one can assume wlog that one has the pointwise estimate |R_j|≤1/C_|R| for some large C_>>1. Consequently, R_j behaves as an error term relative to R, and thus, for notational simplicity,  we will discard R_j in what follows.
    
    
    Thus, we have
    
        B_j,m(f,g)(x):=2^-m/2[γ'(2^-j)]^1/2∫_∫_f̂(ξ)ĝ(η)e^i(γ'(2^-j)ξ+η)xe^-i 2^m η
        R(ξ/η)ζ(η,ξ/η
        )ϕ(ξ)ϕ(η)dξ dη.
    
    
    
    With these we are now ready to state the following
    
    
    As in <cit.>, one can show that the function
    ζ above can be replaced by the constant function 1. This brings a series of simplifications especially when dealing later with the situation p≠2. Instead of following the argument in  <cit.>, we present here a much simpler approach: the secret lies in changing the perspective and focusing on the function
    
        ϱ(ξ,η):=ζ(η,ξ/η) .
    
    Indeed, by doing this, one can perform a double Fourier series development on ϱ and notice that the linear complex exponentials will preserve the curvature of the phase given by 2^mη R(ξ/η); in contrast with this, in the original argument focusing on ζ(η,ξ/η), after the double Fourier series argument one had to work extra in order to deal with expressions of the form {ξ/ηn_2}_n_2∈.
    
    Returning now to the above definition of ϱ we notice that ϱ:[2^-k(γ)/10,10· 2^k(γ)]
    ×[1/10,10] → satisfies ϱ_C^N-3≲_γ
    1. This last property follows from the fact that both ξ
    and η are away from 0.
    
    We can now assume without loss of generality that ϱ is compactly supported on [2π,4π]×[2π,4π]. Regarding now ϱ as a 2π-periodic function on ^2, we represent it as a multiple Fourier series:
    
        ϱ(ξ,η)=∑_n_1,n_2∈c_n_1,n_2e^i n_1 ξe^i n_2 η.
    
    
    From the hypothesis that ϱ_C^N-3≲_γ 1 with N≥ 7, we have
    
        |c_n_1,n_2|≲_γ1/(1+|n_1|+|n_2|)^4.
    
    Thus, for j>0, it follows that
    
        B_j,m(f,g)(x)=∑_n_1,n_2∈c_n_1,n_2B_j,m^n_1,n_2(f,g)(x)
    
    with
    
        B_j,m^n_1,n_2(f,g)(x):=2^-m/2[γ'(2^-j)]^1/2∫_∫_f̂(ξ)ĝ(η)e^i(γ'(2^-j)ξ+η)xe^-i
        	2^m η R(ξ/η)e^iξ n_1e^iη n_2ϕ(ξ)ϕ(η)dξ dη.
    
    At this point we make the following simple observation: B_j,m^0,0_L^2× L^2 → L^1=B_j,m^n_1,n_2_L^2× L^2 → L^1 for any n_1, n_2∈ since
    the factors e^iξ n_1 and e^iη n_2 can be absorbed into the functions f̂ and ĝ in (<ref>) without changing their corresponding L^2-norms.
    
    Consequently, since (<ref>) implies the absolute convergence of the sum ∑_n_1,n_2|c_n_1,n_2| B_j,m^n_1,n_2_L^2× L^2 → L^1=B_j,m^0,0_L^2× L^2 → L^1 ∑_n_1,n_2|c_n_1,n_2| ,
    one realizes that the boundedness of each of T_j,m can be thought as equivalent with the corresponding boundedness of B_j,m^0,0. Therefore, for notational simplicity, we redenote B_j,m as B_j,m^0,0 and set T_j,m as the correspondent operator associated with the newly defined B_j,m.
    
    
    Given the observation above, we will only focus our attention on
    
        B_j,m(f,g)(x)=2^-m/2[γ'(2^-j)]^1/2∫_∫_f̂(ξ)ĝ(η)e^i(γ'(2^-j)ξ+η)x e^-i 2^mη R(ξ/η)ϕ(ξ)ϕ(η)dξ dη ,
    
    or equivalently, on the corresponding operator T_j,m obtained from B_j,m via (<ref>) (and (<ref>) respectively).
    
    Finally, we record the following key relation:
    
        B_j,m_L^2× L^2 → L^1=T_j,m_L^2× L^2 → L^1 .
    Inspired by <cit.>, our intention is to show that even in the variable case, the operator
    T_j(x),m  (or equivalently  B_j(x),m) 
    obeys similar decay bounds with T_j,m which can be extracted from the corresponding bounds for the j(x)=j constant case. In other words, one can identify a unified approach that deals simultaneously with both the bilinear Hilbert transform and the maximal operator along non-flat curves.
        < g r a p h i c s >
    Maximal boundedness range for the Bilinear Maximal Function M_Γ. Our Main Theorem states that our operator  M_Γ:L^p()× L^q()→ L^r() is bounded for all triples (1/p,1/q,1/r') in the region int( ABC)∖ ({A}∪{B}). This result is sharp.
    
    Recall now the definition of our main operator T(f,g) in (<ref>). Our proof will be decomposed into two main parts each split in several stages - see Figure <ref>:
    
      *  First part - boundedness properties of T on int( ABC)∪ (AB) - via m-decay bounds for T_j(x),m:
    
      *  in the first stage we provide m-decay bounds for T_j(x),m on the edge (AB);
    
    
      *  in the second stage we provide m-decay bounds for T_j(x),m on the edge (CD);
    
    
    Then our result holds in int( ABC) by applying interpolation.
    
      *  Second part - boundedness properties of T on ∂( ABC):
    
    
      *  for this situation, in the first stage we provide bounds on the edge (AC];
    
    
      *  in the second stage we provide bounds on the edge (BC);
    
    
      *  in the third stage we prove the unboundedness of our operator at the remaining vertices {A} and {B}.
    
    
    § BOUNDEDNESS PROPERTIES OF T ON INT( ABC)∪ (AB)
    
    
    In this section, we focus on the boundedness of our operator T_j(x),m(f,g) on the edges (AB) and (CD). This together with interpolation imply the boundedness of the our main component operator T for all triples (1/p,1/q,1/r') within the interior of ABC together with the edge (AB), i.e., for all p,  q,  r satisfying 1/p+1/q=1/r with 1<p,  q<∞, and 1≤ r<∞.
    
    
    
     §.§ Bounds on the edge (AB): 1/p+1/q=1 with 1<p<∞
    
    
    Appealing to Observation <ref> and (<ref>), we will transfer in our context the key result in <cit.>:
    
    <cit.>
     There exists ϵ∈(0,1) such that, for any j∈ and m∈, one has
    
        T_j,m(f,g)_L^1≲_γ 2^-ϵ mf_2 g_2.
    
    
    We claim that one can get an extension of the above result to the variable case, that is
    
    
     There exists ϵ∈(0,1) such that, for any m∈, one has
    
        T_j(x),m(f,g)(x)_L^1(dx)≲_γ 2^-ϵ mf_2 g_2.
    
    
    For the case when input functions are not in L^2, we get inspired by the route presented in <cit.>, and prove (at first) the following tame bounds
    
    
    For any m∈ and [Throughout this subsection q=p'.]p, p' satisfying 1/p+1/p'=1 with 1<p<∞, one has
    
        T_j(x),m(f,g)(x)_L^1(dx)≲_γ,p m f_p g_p'.
    
    
    Notice now that for the case r=1, our Theorem <ref> follows from Theorems <ref> and <ref> above via interpolation and geometric summation in the m-parameter.
    
    
    
      §.§.§ The proofs of Theorems <ref> and <ref>
    
    
    Before starting our journey, for m∈, we introduce the following operator:
    
        T_m(f,g)(x):=∑_j∈|T_j,m(f,g)(x)| .
    
    We observe that one trivially has
    
        |T_j(x),m(f,g)(x)|≤ T_m(f,g)(x) .
    
    
    We thus deduce that both Theorems <ref> and <ref> will be now direct consequences of the following
    
    
    
    With the notations in (<ref>), (<ref>) and (<ref>), the following hold:
    
    
      *  There exists ϵ∈(0,1) such that
    
        T_m(f,g)_L^1≲_γ 2^-ϵ mf_2 g_2.
    
      *  For p, p' satisfying 1/p+1/p'=1 with 1<p<∞, one has
    
        T_m(f,g)_L^1≲_γ,p m f_p g_p'.
    
    
      §.§.§ The proof of (<ref>)
    
    
    Using the following standard notation
    ϕ_m+j(ξ):=ϕ(ξ/2^m+j) and ϕ_γ,m,j(η):= ϕ(γ'(2^-j)η/2^m+j) we rewrite (<ref>) in  Theorem <ref> as
    
        T_j,m(f,g)_L^1≲_γ 2^-ϵ m f*ϕ̌_m+j_2 g*ϕ̌_γ,m,j_2.
    
    Using this together with (<ref>) we deduce that
    T_m(f,g)_L^1≤∑_j∈T_j,m(f,g)_L^1≲_γ 2^-ϵ
    m ∑_j∈f*ϕ̌_m+j_2 g*ϕ̌_γ,m,j_2
    which, via an application of Cauchy-Schwarz, further gives
    ≲ 2^-ϵ
    m (∑_j∈f*ϕ̌_m+j_2^2)^1/2 (∑_j∈g*ϕ̌_γ,m,j_2^2)^1/2≲ 2^-ϵ m f_2 g_2 ,
    where in the last inequality we used that the supports of the functions {ϕ_m+j}_{j∈} and  respectively {ϕ_γ,m,j}_{j∈} are almost disjoint, with the latter being a direct consequence of the properties of the curve γ (see the property “smoothness, no critical points, variation near 0 and ∞" in the definition of  in <cit.>).
    
    
    
    
      §.§.§ The proof of (<ref>)
     For this point, of fundamental importance is the approach in <cit.>.
    
    Fix p, p' as in our hypothesis and take f∈ L^p and g∈ L^p'. In the spirit of <cit.>, for h∈ L^∞ (we will assume from now on wlog that h_∞=1 and h is positive), we define[It is important to notice here that as opposed to the similar object defined in <cit.>, in our context, in definition (<ref>) below, the operator T_j,m(f,g) is taken with absolute values thus making Λ_j,m(f,g,h) a sublinear “form".] for each j∈ and m∈
        Λ_j,m(f,g,h):=∫_ |T_j,m(f,g)(x)|  h(x) dx.
    
    
    For m∈, we define
    
        Λ_m(f,g,h):=Λ_m^+(f,g,h)+Λ_m^-(f,g,h),
    
    where
    
        Λ_m^+(f,g,h):=∑_j∈Λ_j,m(f,g,h)   and   Λ_m^-(f,g,h):=∑_j∈∖Λ_j,m(f,g,h).
    
    Since our Λ's involve absolute values inside the integral expressions, in order to be able to use the techniques in <cit.>, we will first apply a linearization procedure. Thus, for a suitable sequence of L^∞-functions {_j,m} with the property |_j,m(x)|=1 a.e x∈, we will re-write
    
        Λ_m^+(f,g,h):=∑_j∈∫_ T_j,m(f,g)(x) _j,m(x) h(x) dx,
    
    with the obvious correspondence for Λ_m^-(f,g,h).
    
    
    Deduce now that (<ref>) is in fact equivalent with
    
        |Λ_m^±(f,g,h)|≲_γ,p m f_p g_p' h_∞.
    
    In what follows we will only focus on the + component since the - component can be treated in a similar fashion.
    
    We will split our discussion in two sub-cases Case 1.1<p≤ 2
    
    With the notations[We will only recall here that ϕ_j,p_0(η):= ϕ('(2^-j) η/2^j-p_0) and ψ_m,p_0,j(ξ):= 2^-m/2 e^- i p_0 R(ξ/2^j p_0) ϕ(ξ/2^m+j) where j∈ and p_0∈ [2^m,  2^m+1)∩ with m∈.] in <cit.>, following the first part of the argument provided for the proof of Proposition 4.2 in Section 4 of <cit.>, we have
    |≪_m^+(f,g,h)|≲∫_( ∑_j∈|(f*ϕ̌_m+j)(y)|^2)^1/2 ( ∑_j∈|M(∑_p_0=2^m^2^m+1|(g*ϕ̌_j,p_0) ((_j,m h)*ϕ̌_j,p_0)|)(y)|^2)^1/2 dy≲( ∑_j∈|f*ϕ̌_m+j|^2)^1/2_p ( ∑_j∈|M(∑_p_0=2^m^2^m+1|(g*ϕ̌_j,p_0) ((_j,m  h)*ϕ̌_j,p_0)|)|^2)^1/2_p'≲f_p ( ∑_j∈(∑_p_0=2^m^2^m+1|(g*ϕ̌_j,p_0) ((_j,m  h)*ϕ̌_j,p_0)|)^2)^1/2_p' ,
    where for the last relation we used standard Littlewood-Paley theory (for providing bounds on the square function for f) and Fefferman-Stein's inequality (<cit.>) (for the term involving the functions g and h).
    
    Using now that
    
        ((_j,m  h)*ϕ̌_j,p_0)(x)=∫_ (_j,m  h)(x-'(2^-j)/2^j y) ϕ̌(y) e^i p_0 y dy ,
    
    we deduce that
    
        ∑_p_0=2^m^2^m+1|((_j,m  h)*ϕ̌_j,p_0)|^2≲_h_∞^2 .
    
    
    Thus, applying Cauchy-Schwarz inequality, we get
    
        (∑_p_0=2^m^2^m+1|(g*ϕ̌_j,p_0) ((_j,m  h)*ϕ̌_j,p_0)|)^2≲_h_∞^2 ∑_p_0=2^m^2^m+1|(g*ϕ̌_j,p_0)|^2 ,
    
    from which we deduce that
    
        |≪_m^+(f,g,h)|≲_f_p h_∞ (∑_j∈∑_p_0=2^m^2^m+1|g*ϕ̌_j,p_0|^2)^1/2_p' .
    
    
    Finally, since p'≥2, we are allowed to apply Rubio de Francia's inequality (<cit.>):
    
        ( ∑_j∈∑_p_0=2^m^2^m+1|g*ϕ̌_j,p_0|^2)^1/2_p'≲_,p'g_p' .
    
    Putting now together (<ref>), (<ref>), (<ref>) we conclude that (<ref>) holds.
    Case 2.2<p<∞
    
    In this second case, we follow part of the argument inside the proof of Proposition 4.3 in Section 4 of <cit.>.
    
    Indeed, by applying Cauchy-Schwarz and then Hölder's inequality, we have
    |≪_m^+(f,g,h)|≤∑_j∈∫_ (∑_p_0=2^m^2^m+1|(f*ϕ̌_m+j*ψ̌_m,p_0,j)(x)|^2)^1/2×(∑_p_0=2^m^2^m+1|(g*ϕ̌_j,p_0)(x) ((_j,m  h)*ϕ̌_j,p_0)(x)|^2)^1/2 dx≤ (∑_j∈∑_p_0=2^m^2^m+1|(f*ϕ̌_m+j*ψ̌_m,p_0,j)|^2)^1/2_p(∑_j∈∑_p_0=2^m^2^m+1|(g*ϕ̌_j,p_0) ((_j,m  h)*ϕ̌_j,p_0)|^2)^1/2_p' .
    
    Now the content of Proposition 4.4 in <cit.> is the statement that for p≥2 one has
    
        (∑_j∈∑_p_0=2^m^2^m+1|f*ϕ̌_m+j*ψ̌_m,p_0,j|^2)^1/2_p≲_,p m^2/p'-1 f_p ,
    
    while from the Lemma 4.9 in <cit.> we deduce that
    
        ∑_p_0=2^m^2^m+1|(g*ϕ̌_j,p_0)(x) ((_j,m  h)*ϕ̌_j,p_0)(x)|^2≲_h_∞^2  M(g*ϕ̌_,m,j)^2(x) ,
    
    where here we recall that ϕ_,m,j(η):=ϕ('(2^-j)/2^j+m η) .
    
    Inserting (<ref>) and (<ref>) in (<ref>), we conclude that
    
    |≪_m^+(f,g,h)|≤_,p  m^2/p'-1 f_p h_∞ (∑_j∈ M(g*ϕ̌_,m,j)^2)^1/2_p'≲_p m^2/p'-1 f_p h_∞ (∑_j∈ (g*ϕ̌_,m,j)^2)^1/2_p'≲_,p m^2/p'-1 f_p g_p' h_∞ ,
    where we made use again of the Fefferman-Stein's inequality (<cit.>) (for the second inequality) and standard Littlewood-Paley theory (for the third inequality).
    
    
    
    
    
     §.§ Bounds on the segment (CD): p=q=2r with 1<r<∞
    
    
    
    The main result of this subsection is
    
    	Let 1≤ r<∞. For m∈, the following holds
    	
        T_j(x),m(f,g)(x)_L^r(dx)≲_γ,r(1+m^1-1/r) 2^-m/20r f_2r g_2r.
    
    
    
    
    
    In order to prove our Theorem <ref> we will need the following
    
    
    	Let p, q, r be as in (<ref>) and (<ref>). Then for any j, m∈, we have
    	
        T_j,m(f,g)_r≲_γ,p,q(1+m^1-2/p) 2^-m/10(1-1/p*) f_p g_q.
    
    Using the notation from (<ref>) and <cit.>, we choose h∈ L^r' such that
    
        Λ_j,m(f,g,h)= ∫_ T_j,m(f,g)(x)h(x)dx.
    
    Therefore
    
        |Λ_j,m(f,g,h)|≤
        	∫_ |(f*ϕ̌_m+j)(x) |M(∑_p_0=2^m^2^m+1|(g*ϕ̌_j, p_0)(·) (h*ϕ̌_j, p_0)(·)|)(x) dx,
    
    and
    
        |Λ_j,m(f,g,h)|≤
        	∫_(∑_p_0=2^m^2^m+1|(g*ϕ̌_j, p_0)(x)(h*ϕ̌_j, p_0)(x)|^2)^1/2(∑_p_0=2^m^2^m+1|(f*ϕ̌_m+j*ψ̌_m, p_0, j)(x)|^2)^1/2dx.
    
    
    Hence, by following the second part of the proof of <cit.> and <cit.>, we deduce that the results in <cit.> hold for Λ_j,m. In particular, the analogue of <cit.> follows:
    
    
    Let 1<p<∞. Then the following estimates hold
    
        Edge (AC):        |Λ_j,m(f,g,h)|≲_γ,p (1+m^1-2/p)2^-m/10(1-1/p*)f_pg_∞h_p',
    
    and
    
        Edge (AB):        |Λ_j,m(f,g,h)|≲_γ,p (1+m^1-2/p)2^-m/10(1-1/p*)f_pg_p'h_∞.
    
    
    Thus, Proposition <ref> follows from the Proposition <ref> after applying real interpolation.
    
    
    We are now ready for the following:
    
    	
    	As we did for the proof of (<ref>), we write (<ref>) as
    
        T_j,m(f,g)_r≲_γ(1+m^1-1/r)2^-m/20rf*ϕ̌_m+j_2rg*ϕ̌_γ,m,j_2r,
    
    where here we used that p=q=2r and that (2r)^*=2r/2r-1 since r≥ 1.
    
    Since
    
        |T_j(x),m(f,g)(x)|≤(∑_j∈|T_j,m(f,g)|^r)^1/r,
    
    it follows from Hölder inequality and Proposition <ref> that
    
        T_j(x),m(f,g)(x)_L^r(dx)   ≤(∑_j∈T_j,m(f,g)_r^r)^1/r
           ≤ (1+m^1-1/r)2^-m/20r(∑_j∈f*ϕ̌_m+j_2r^rg*ϕ̌_γ,m,j_2r^r)^1/r.
    
    
    Notice that for r∈ [1,∞) and for any f, ϕ∈(), we have
    
        (∑_j∈f*ϕ_j_2r^2r)^1/2r   ≤(∫_∑_j∈|(f*ϕ_j)(x)|^2rdx)^1/2r
           ≤(∫_(∑_j∈|(f*ϕ_j)(x)|^2)^rdx)^1/2r = (∫_{(∑_j∈|(f*ϕ_j)(x)|^2)^1/2}^2rdx)^1/2r
           =(∑_j∈|f*ϕ_j(x)|^2)^1/2_2r≲f_2r,
    
    here for the last relation we used standard Littlewood-Paley theory.
    
    By Cauchy Schwartz, the hypothesis on ∈ and the result above, we obtain
    
        (∑_j∈f*ϕ̌_m+j_2r^rg*ϕ̌_γ,m,j_2r^r)^1/r   ≤(∑_j∈f*ϕ̌_m+j_2r^2r)^1/2r(∑_j∈g*ϕ̌_γ,m,j_2r^2r)^1/2r
           ≤_,rf_2rg_2r.
    
    
    Inserting (<ref>) in (<ref>) we get
    
        T_j(x),m(f,g)(x)_L^r(dx)≤_,r (1+m^1-1/r)2^-m/20rf_2rg_2r,
    
    as desired.
    
    In order to prove our main Theorem <ref> in the interior of ABC it is in fact not necessary to pursue the boundedness of our operator along the edge (CD). Indeed, one could apply interpolation between the bounds corresponding to the edges (AB) and (AC) instead, with the latter discussed in the next section. However, we wanted to offer a less expected but more interesting alternative approach to the trivial bounds one gets along the segment (AC). This is especially useful in situations in which one deals with similar type operators but for which one does not have a good control on the edges (BC) and (AC).
    
    
    § BOUNDEDNESS PROPERTIES OF T ON ∂( ABC)
    
    In this section we will prove positive boundedness results along the edges (AC], (BC] and negative results for the vertices {A} and {B} (see Figure <ref>).[We remind the reader that the boundedness along the edge (AB) was proved in Subsection <ref>.]
    
     §.§ Bounds on the edge (AC]: 1< p≤∞, r=p and q=∞
    
    
    From the definition (<ref>) of M_Γ it is trivial to notice that for all x∈,
    
        |M_Γ(f,g)(x)|   ≤g_∞ sup_>01/2∫_-^ |f(x-t)|dt
           ≤g_∞  Mf(x).
    
    Hence, applying the classical results on the Hardy-Littlewood maximal operator we get
    
    	For any 1<p≤∞, one has
    	
        M_Γ(f,g)_L^p≲_p f_p g_∞.
    
    
     §.§ Bounds on the edge (BC]: 1< q≤∞, r=q and p=∞
    
    
    Our goal in this section is to prove the following:
    
    	For any 1<q≤∞, one has
    	
        M_Γ(f,g)_L^q≲_,qf_∞ g_q.
    
    
    From (<ref>), recalling that γ∈ and our choice of j_0 in (<ref>) - (<ref>), we notice that for all x∈,
    
        |M_Γ(f,g)(x)|   ≤f_∞ M_γ  g(x).
    
    where here
    
        M_γ  g(x):=sup_j∈|j|>j_01/2^j+1∫_2^j≤|t|≤2^j+1  |g(x+γ(t))|dt .
    
    
    
    Thus, Theorem <ref> follows from the result below
    
    	For any 1<q≤∞, one has
    	
        M_γ(g)_L^q≲_γ,qg_q.
    
    From the properties of  and the choice of j_0 in (<ref>) - (<ref>), it is straightforward to check that for any k∈ with |k|>j_0  and t∈[2^k,2^k+1] one has
    
        |γ(2^k)|≈_γ 2^k|γ'(t)|≈_γ|γ(2^k+1)| .
    
    
    From (<ref>), we deduce that γ is strictly monotone and invertible over each of the intervals
    (-∞,-2^j_0), (-2^-j_0,0), (0,2^-j_0)  and (2^j_0, ∞). Wlog we consider from now on that our entire discussion takes place relative to J_0:=(2^j_0, ∞) and that both ' and ” are strictly positive on J_0. Thus, for k∈ with k>j_0, one has
    
        1/2^k+1∫_2^k^2^k+1|g(x+γ(t))|dt    =1/2^k+1∫_γ([2^k,2^k+1])|g(x+u)|du/|γ'(γ^-1(u))|
           ≤1/2^k+1|γ'(2^k)|∫_γ(2^k)^γ(2^k+1)|g(x+u)|du .
    
    Appealing now to (<ref>), we deduce
    
        1/2^k+1∫_2^k^2^k+1|g(x+γ(t))|dt ≤1/2^k+1|γ'(2^k)|∫_-|γ(2^k)|-|γ(2^k+1)|^|γ(2^k)|+|γ(2^k+1)||g(x-u)|du
        ≲_γ Mg(x).
    
    
    Analogously,
    
        1/2^k+1∫_-2^k+1^-2^k|g(x+γ(t))|dt   ≲_γ Mg(x).
    
    
    Therefore
    
        sup_j∈|j|>j_0 1/2^j+1∫_2^j≤|t|≤2^j+1 |g(x+γ(t))|dt≲_γ Mg(x).
    
    Using now the standard theory on Hardy Littlewood maximal operator we conclude our Theorem <ref>.
    
    
     §.§ Behavior on {A}∪{B}: triples p=r=1,  q=∞ and p=∞,  q=r=1
    
    
    In this section we show that our operator[Throughout this section we return to the original definition of our operator M_Γ(f,g) in (<ref>).] is unbounded at the points {A} and {B} (see Figure <ref>).
    
    Case 1.:
    
    
    We start by focusing on the vertex A, , p=r=1 and q=∞. In this case, we show that there are functions f∈ L^1(), g∈ L^∞(), and a map γ∈ such that M_Γ(f,g) is not bounded on L^1.
    
    Let f=χ_[-1,1] and g=χ_(-∞,1) be characteristic functions, and let γ(t)=t^2∈. Note that f∈ L^1(), g∈ L^∞() with  f_1=2, and g_∞=1. Then
    
        M_Γ(f,g)(x)=sup_>01/2∫_-^ |f(x-t)g(x-t^2)|dt≥1/2(|x|+1)∫_-(|x|+1)^|x|+1f(x-t)g(x-t^2)dt.
    
    Suppose x>2, then
    
        M_Γ(f,g)(x)≥1/2(x+1)∫_x-1^x+1f(x-t)g(x-t^2)dt=1/2(x+1)∫_x-1^x+1g(x-t^2)dt≳1/x+1.
    
    Therefore, M_Γ(f,g) is not absolutely integrable.
    
    
    Notice that if f∈ L^1 and g∈ L^∞ we have by (<ref>)
        {x∈:M_Γ(f,g)(x)>α}⊆{x∈:Mf(x)>α/g_∞},
    
    for any α>0.
    Thus, from the classical theory, we get that for any α>0 the following holds:
    
        |{x∈:M_Γ(f,g)(x)>α}|≲1/αf_1 g_∞ .
    Case 2.:
    
    
    In this situation we deal with the vertex B, given by p=∞,  q=r=1. We will show that there are functions f∈ L^∞() and g∈ L^1() such that M_Γ(f,g) is not bounded on L^1 for, say, γ(t)=t^2.
    
    Let f be the constant function 1, and let g=χ_[-1,1]∈ L^1(). Then
    
        M_Γ(f,g)(x)=sup_>01/2ϵ∫_-^|g(x-t^2)|dt ≃
        sup_>01/∫_^2 |g(x-t^2)|dt.
    
    
    Notice that for any >0, we have
    
        1/∫_^2|g(x-t^2)|dt=1/∫_^2^4^2|g(x-u)|/2√(u)du≥1/4^2∫_^2^4^2|g(x-u)|du.
    
    
    
    Then, for x>2 we have
    
        1/√((x+1)/4)∫_√((x+1)/4)^2 √((x+1)/4)|g(x-t^2)|dt≥1/x+1∫_(x+1)/4^4(x+1)/4  |g(x-u)|du ≳1/x+1.
    
    
    Therefore, M(f,g) ∉L^1().
    
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