The Leray transform: factorization, dual $CR$ structures and model hypersurfaces in $\mathbb{C}\mathbb{P}^2$
D. E. Barrett L. D. Edholm
math.CV 32A26, 32F17, 32A25, 32V05, 42A38

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Leray Transform]The Leray Transform: factorization, dual CR structures, and model hypersurfaces in ℙ^2



We compute the exact norms of the Leray transforms for a family _β of unbounded hypersurfaces in two complex dimensions.  The _β generalize the Heisenberg group, and provide  local projective approximations to any smooth, strongly -convex hypersurface  to two orders of tangency.    This work is then examined in the context of projective dual CR-structures and the corresponding pair of canonical dual Hardy spaces associated to , leading to a universal description of the Leray transform and a factorization of the  transform through orthogonal projection onto the conjugate dual Hardy space.


The first author was supported in part by NSF grant number DMS-1500142
2010 Mathematics Subject Classification: 32A26
Department of Mathematics
University of Michigan, Ann Arbor, MI, 48109-1043 USA
barrett@umich.edu
edholm@umich.edu

[
    David E. Barrett & Luke D. Edholm  
     
=======================================







§ INTRODUCTION


This paper is the first in a series aimed toward better understanding the Leray transform on smooth, strongly -convex hypersurfaces.    Here, the initial focus is on the following family of models.  For 0 ≤β <1, define the hypersurface 

    _β := { (ζ_1,ζ_2)∈^2Im(ζ_2) = |ζ_1|^2 + β(ζ_1^2)},

along with the (unbounded) domain lying on its -convex side

    Ω_β := { (z_1,z_2)∈^2Im(z_2) > |z_1|^2 + β Re(z_1^2)}.


Let ⊂ℙ^n be a strongly -convex hypersurface bounding a domain Ω on its -convex side.  The Leray transform L_ (see Remark <ref> regarding terminology) is a member of the Cauchy-Fantappiè class of integral operators re-capturing key properties of the familiar one-variable Cauchy transform.  Its applications include analysis of the Hardy space on the domain Ω:  if σ is a measure on  and L_ maps L^2(,σ) → L^2(,σ) boundedly, 
then the transform identity shows  how functions in the Hardy space H^2(,σ) are built from certain rational functions.

As is typical in Hardy space constructions, care must be taken to specify the measure σ, especially when  is unbounded.  For the _β defined above, the natural measure arising from the Leray transform is σ = dx_1 ∧ dy_1 ∧ dx_2, where (x_1,y_1,x_2,y_2) are the usual affine coordinates on ^4 ≅^2.  This choice of σ is a constant multiple of Fefferman hypersurface measure (see <cit.>) on _β, and consequently has many desirable invariance properties.  (Note that σ is not comparable to the Euclidean surface measure on _β.)

The main computation in the first half of the paper is the following:


Let L_β denote the Leray transform of _β, and σ =  dx_1 ∧ dy_1 ∧ dx_2.  Then L_β:L^2(_β,σ) → L^2(_β,σ) is a bounded operator with norm

    L_β= 1/√(1-β^2).

 

It is rare to compute exact norms of operators, so this result is interesting in its own right.  But this computation also has significant import since the _β serve as models in local geometric considerations.  Given a smooth, strongly -convex hypersurface ⊂ℙ^2, the _β can be used to locally approximate  to two orders of tangency.  Indeed for each fixed ζ^*∈, there is an automorphism of ℙ^2 – see equation (<ref>) – 
moving ζ^* to the origin such that the degree two Taylor expansion of  after this coordinate change is given by a unique _β(ζ^*), 0≤β(ζ^*)<1.  The precise formulation of this statement is given in Section <ref>.  

The mapping properties of the one-dimensional Cauchy transform are well-studied.  In <cit.>, Kerzman and Stein relate the Cauchy transform, C, on a smooth bounded domain Ω⊂ to the Szegő projection and show that the two operators coincide if and only if Ω is a   disc.  It follows that C_2≥1 with equality if and only if Ω is a disc.  However, it can also be extracted from this work that the essential L^2-norm C_ess = 1 for every smooth Ω⊂.  (Recall that the essential norm  – see <cit.> – measures the distance to the set of compact operators; it often occurs in localized analysis.)  See <cit.> for related results about C.

Much recent work has been done to understand L_ in dimensions two and higher.  In <cit.>, the first author and Lanzani study the essential spectrum of this operator on a class of Reinhardt domains in ^2, and their results are in the same vein as Theorem <ref>.  Lanzani and Stein have written a series of recent articles exploring many aspects of this operator.  (See <cit.>.)  In <cit.>, it is shown that the Leray transform of any bounded strongly -convex hypersurface  that is at least C^1,1 smooth maps L^p() to itself for all 1<p<∞.  Counterexamples to the L^2-boundedness of L_ exist if either the strong -convexity hypothesis or the C^1,1 smoothness hypothesis is dropped.  See <cit.> and <cit.> for more information.  We note that the _β are unbounded and fail to be even C^1 at infinity (see Remark <ref>).  

In <cit.>, the first author shows that the L^2-norm of L_ measures the effectiveness of pairing two natural Hardy spaces associated to a hypersurface .  This is analogous to the role played by the Cauchy transform in an L^2-pairing of functions holomorphic inside a Jordan curve with functions holomorphic outside the curve.  (See David's appendix in Meyer's monograph <cit.>.)  Recalling the Lanzani-Stein result in the previous paragraph, insight into the interaction of these two Hardy spaces is gained by investigating the norm of L_.  





Let ⊂ℙ^2 be a smooth bounded, strongly -convex hypersurface and L_ denote its Leray transform.  The essential L^2-norm is given by

    L__ess = sup_ζ∈1/√(1-β(ζ)^2).



We postpone further discussion of this conjecture to a subsequent work in this series, other than a quick run through of evidence for why we believe it to be true.  First, note the similarity between this statement and the situation for the Cauchy transform as described above.  A disc of varying radius will osculate any smooth, bounded domain Ω in the plane.  The essential norm of an operator frequently arises in local considerations, and the fact that C_D_2=1 on every disc D leads to the corresponding result for the essential L^2-norm of the Cauchy transform on Ω.  In two complex dimensions the osculating hypersurfaces giving local approximation are the _β, so Theorem <ref> suggests the form of this conjecture.  Additionally, it is shown in <cit.> that Conjecture <ref> holds for all smooth bounded, strongly convex Reinhardt domains.  

After proving Theorem <ref>, we continue to develop the general theory with the use of projective dual coordinates.  These coordinates depend on the choice of a matrix M, corresponding to a particular affinization of projective space.  The Leray transform is shown to be given by a universal formula involving projective dual coordinates in Proposition <ref>.  On every bounded -convex hypersurface enclosing the origin, this formula resembles the form of the Szegő projection of the unit ball – compare the formula (<ref>) with (<ref>).  On every unbounded -convex hypersurface which can be written as a graph of a smooth function over ×, this formula resembles the form of the Szegő projection of the Siegel upper half space – compare  (<ref>) with (<ref>).  This general procedure leads to the identification of a preferred measure on  in (<ref>).  It should be emphasized that while L_ only coincides with the Szegő projection in  special cases (affine images of the unit ball or Siegel upper half space <cit.>), the resemblance of the formulas suggests a relationship between the two operators that should be further explored.

The dual coordinates may be used to induce a secondary CR structure on , called the projective dual CR structure.  This is carried out in Section <ref>.  From the two CR structures and the preferred measure mentioned above, we obtain a pair of Hardy spaces on , denoted H^2() and H^2_().  The restriction Q_ of L_ to 
H^2_() provides an explicit invertible map H^2_()→ H^2(), and the full map L_ factors through Q_.  (Most of the objects mentioned just above depend on the choice of M – with transparent transformation laws – but the dual CR structure is independent of M.)



The paper is structured as follows.  Section <ref> collects notation, definitions and necessary background material.  It also motivates our problem by recalling results from one complex variable.  Section <ref> begins with the analysis of  L_β via one Fourier transform together with size estimates.  We then use certain projective automorphisms of _β to obtain a new parametrization of L_β in Section <ref>, leading to the use of a second Fourier transform in the proof of Theorem <ref> in Section <ref>.  The projective dual coordinates and related constructions are developed in Section <ref>.  In Sections <ref> and <ref> we explain the factorization results for L_, with special attention drawn to the case of L_β.   Appendix <ref> contains  further analysis  of automorphisms of _β leading to basic L^2-estimates related to L_β.

The authors would like to acknowledge the two anonymous referees for many useful suggestions to improve this paper.







§ BACKGROUND


One motivating factor for the study of the Leray transform is the desire for a higher dimensional analogue of the Cauchy transform in one complex variable.  For any smooth bounded domain Ω⊂, recall that the Cauchy transform of a function f defined on the boundary bΩ is given by  

    Cf(z):= 1/2π i∫_bΩf(ζ)/ζ-z  dζ.

Denoting the Cauchy kernel by C(z,ζ) = 1/2 π i (ζ-z)  dζ, we highlight three essential properties:  


  (i) Reproduction.  Given z∈Ω and f∈(Ω), we have Cf(z) = f(z). 


  (ii) Holomorphicity. For each fixed ζ∈ bΩ, the kernel C(z,ζ) is holomorphic in z, and thus can be used to construct holomorphic functions. 


  (iii) Domain Independence. The kernel C(z,ζ) is independent of Ω, in that no explicit reference to a defining function of Ω is made in the coefficient function of dζ.



Unfortunately, these three properties never simultaneously hold in this form for a kernel-operator pair in higher dimensions.  See <cit.> for an excellent survey detailing these matters.  We shall insist on keeping the reproduction property (i), and will look for higher dimensional successors of the Cauchy transform by dropping one of the other conditions. The Bochner-Martinelli formula  satisfies (i) and (iii), but fails to be holomorphic in the parameter z (see <cit.>).  Though useful in many respects, this formula cannot be used to create holomorphic functions from more general boundary data.


Restricting to strongly -convex hypersurfaces ⊂ℙ^n, we are able to define the Leray kernel-operator pair which satisfies properties (i) and (ii).  As above, property (ii) allows construction of holomorphic functions from rather general boundary data.  And while the Leray kernel is  domain dependent when  is viewed as a subset of ℙ^n, a universal description involving projective dual variables is presented in Section <ref> (see Remark <ref>).









 §.§ Strong -convexity

















Underlying the Leray transform is the geometric notion of -convexity.  There are several equivalent definitions (see <cit.>), but we focus here on a differential condition along .  We first note the following proposition and refer to Section 5.2 in <cit.> for its proof.


Let  be a smooth, strongly pseudoconvex real hypersurface in ℙ^n and let p∈.  Then there is an automorphism of ℙ^n moving p to 0 so that the transformed  takes the following form near 0.

    Im(ζ_n) = ∑_j,k=1^n-1α_j,kζ_jζ̅_k + Re(∑_j,k=1^n-1β_j,kζ_j ζ_k ) + c Re(ζ_n)^2 + O((ζ_1,…,ζ_n-1,Re(ζ_n))^3).

The real constant c may be set arbitrarily, but the sum ∑α_j,kζ_jζ̅_k + Re(∑β_j,kζ_j ζ_k ) is determined up to a scalar multiple and a -linear change of coordinates in (ζ_1, …, ζ_n-1).


We now use this normal form to set up our definition.


Let  be connected and strongly pseudoconvex in ℙ^n and suppose (after a projective automorphism) that  is given in the form of Proposition <ref> above.  We say that  is strongly -convex if and only if

    ∑_j,k=1^n-1α_j,kζ_jζ̅_k + Re(∑_j,k=1^n-1β_j,kζ_j ζ_k )

is positive definite on (ζ_1,…,ζ_n-1).  This is equivalent to saying that the complex tangent hyperplane ^n-1×{0} has minimal order of contact with .


Because the real constant c in Proposition <ref> may be chosen arbitrarily, set c=0 and diagonalize the form to 

    Im(ζ_n)=∑_j=1^n-1α_j|ζ_j|^2 + Re(∑_j=1^n-1β_j ζ_j^2 ) + O((ζ_1,…,ζ_n-1,Re(ζ_n))^3),

with each β_j≥0.  Strong -convexity at the origin is equivalent to saying that each β_j < α_j.  When n=2, we may set α_1=1 which leads to our hypersurfaces

    _β := { (ζ_1,ζ_2)∈^2: Im(ζ_2) = |ζ_1|^2 + β Re(ζ_1^2)}.


Every strongly -convex hypersurface is (by our definition) strongly pseudoconvex, and osculating biholomorphic images of the sphere  (or of _0) are a useful tool in the  study  of strongly pseudoconvex hypersurfaces.  The Leray transform has good transformation properties under automorphisms of ℙ^2 (but not under general biholomorphic maps); working with this smaller set of mappings we need a larger set of models, and the projective images of the _β suitably osculate any strongly -convex hypersurface. ◊













 §.§ The Leray transform

Let  be a compact strongly -convex C^2 hypersurface in ^n, and let Ω be the bounded domain  with boundary .  If f is a function on , the Leray transform maps f to a holomorphic function on Ω whenever the following integral makes sense.

    L_f(z) := ∫_ f(ζ)  ℒ_(z,ζ), 
       ℒ_(z,ζ) := 1/(2π i)^n∂ρ(ζ) ∧ (∂̅∂ρ(ζ))^n-1/< ∂ρ(ζ),(ζ - z) >^n.

Note that the Leray kernel ℒ_ is a form of bi-degree (n,n-1).  Here ρ is a defining function for , and <·,·> is the natural bilinear pairing between (1,0)-forms and vectors.  This definition is independent of the choice of defining function.  See chapter IV of <cit.> for more information. 

Formula (<ref>) can actually be defined for a more general class of hypersurfaces. If  is bounded and has the property that all complex tangent hyperplanes never intersect Ω, the integral is defined for all z ∈Ω.  Such hypersurfaces are simply called -convex.  (See Proposition 2.5.9 in <cit.>.  In the context of smooth connected hypersurfaces the notion of -convexity coincides with the notion of -linear convexity, but these notions diverge in more general situations – see Chapter 2 in <cit.> for a complete discussion of these matters.)

Now consider a compact strongly -convex C^2 hypersurface  in ℙ^n.  The complex tangent hyperplane  to  at some ζ∈ meets  only at ζ
(see Remark 2.5.11 in <cit.>).  Pushing the hyperplane away from  by a small multiple of the normal vector to  at ζ we obtain a hyperplane disjoint from  (lying on the concave side of ).  This hyperplane may be sent to the hyperplane at infinity by means of a projective transformation.  Thus  is projectively equivalent to a  compact strongly -convex C^2 hypersurface in ^n; Bolt's transformation law (previously mentioned in Remark <ref> above) can now be used to defined the Leray transform in terms of its euclidean counterpart.
 

Once it has been established that (<ref>) converges for z ∈Ω, it is of interest to think of L_ as a map from functions on  to functions on Ω, and to understand the boundary values of L_f(z).  For z ∈, (<ref>) is a singular integral and must be interpreted in a suitable way.

For unbounded -convex hypersurfaces, it is not a priori clear that (<ref>) converges even for z ∈Ω.  However when = _β, we show the following hold:



  (a) Given f ∈ L^2(_β, σ), L_βf is a holomorphic function on Ω_β.  (Appendix <ref>.)


  (b)  L_β is a bounded operator from L^2(_β,σ) → L^2(_β,σ). (Proposition <ref>.)


  (c) L_β_L^2(_β,σ) = 1/√(1-β^2). (Theorem <ref>, proved in Section <ref>.)


  (d) The Leray transform of _β is a projection, i.e., L_β∘L_β = L_β. (Corollary <ref>.)  It is not orthogonal, except when β = 0, in which case the Leray transform coincides with the Szegő projection.


  (e) The closure of S_β in complex projective space fails to be C^1.  (Remark <ref>.)



The measure used above is σ = dx_1 ∧ dy_1 ∧ dx_2.  We postpone the discussion of the surface measures considered on a more general  until Section <ref>.
 


Equation (<ref>) in dimension one reduces to (<ref>). ◊



L_ is often referred to in the literature as the Cauchy-Leray transform, the Leray-Aizenberg transform or the Cauchy transform for convex domains.  ◊





 §.§ The Fourier transform

The Fourier transform is a standard tool with varying normalizations.  We include this section to collect basic facts for the version used in this paper. 

Use  and ^-1 to denote the Fourier and inverse Fourier transforms on , respectively, with the following convention:  given h∈ L^1(),

    h(ξ):=  ∫_-∞^∞ h(x)e^-2π i xξ  dx,
          ^-1h(ξ):=  ∫_-∞^∞ h(x)e^2π i xξ  dx.

It is standard to extend  and ^-1 to operators on L^2() using the density of L^1 ∩ L^2 functions in L^2.  The of placement of the 2π in (<ref>) guarantees that both  and ^-1 are isometries of L^2, i.e., for g ∈ L^2(), we have Plancherel's identity:

    g_2 =  g_2 = ^-1 g_2.


Equations (<ref>) and (<ref>) below are written using the inverse Fourier transform ^-1 because this operator is used extensively in later sections.  When dealing with functions of more than one variable, we often subscript both the transform and the phase space variables.  Starting with a two variable function H(x,y),

    ^-1_x,yH(ξ_x,ξ_y) := ∫_-∞^∞∫_-∞^∞ H(x,y) e^2π i (xξ_x + yξ_y)  dx dy.

Also note that the formulas in (<ref>) transform convolutions in the following way:

    ^-1(F*G)    = ^-1(F)·^-1(G).

Finally, note the following integral which arises frequently in computations below

    ∫_-∞^∞ e^-π x^2 e^2π i xξ dx = e^-πξ^2.










§ THE LERAY TRANSFORM OF _Β


The Lanzani-Stein machinery in <cit.> cannot be directly applied to the Leray transform of _β because these models fail to be C^1,1 at ∞.  (In fact, they are not even C^1 at ∞; see Remark <ref>.)  In this section, we first use a single Fourier transform along with size estimates to establish the L^2-boundedness of L_β.  However, sharpness is lost in the computation of the exact norm when oscillatory cancellation is ignored.  This is remedied in Section <ref> when certain projective automorphisms  are used to re-parametrize _β, ultimately leading to the sharp result given in Theorem <ref>.  

The verification that integral (<ref>) converges and defines a holomorphic function on Ω_β for all f∈ L^2(_β,σ) is postponed to Appendix <ref>.





 §.§ L^2-boundedness via Fourier methods and size estimates


Starting from the definition of _β in (<ref>), choose defining function ρ(ζ) = |ζ_1|^2 + βζ_1^2-(ζ_2).  The pieces of the Leray kernel in (<ref>) are calculated to be

    1/(2π i)^2ρ∧ρ(ζ) = 1/8π^2 i dζ_2∧ dζ̅_1∧ dζ_1,
    < ∂ρ(ζ),(ζ - z) > = (ζ̅_1 +βζ_1)(ζ_1 - z_1)+  i2(ζ_2-z_2),

and the Leray transform (<ref>) takes the form

    L_βf(z) = 1/8π^2i∫__βf(ζ)  dζ_2 ∧ dζ̅_1 ∧ dζ_1/[(ζ̅_1 +βζ_1)(ζ_1 - z_1)+ i/2(ζ_2-z_2)]^2.


We parametrize _β by ^3, letting ζ_j = x_j+iy_j and z_j = u_j+iv_j.  The denominator of the integrand becomes ( C + i/2(x_2-u_2) )^2, where 

    C    = ((1+β)x_1(x_1-u_1) + (1-β)y_1(y_1-v_1) ) - 1/2(y_2 - v_2) 
        +i((1+β)x_1(y_1-v_1) - (1-β)y_1(x_1-u_1)).

When both ζ,z ∈_β, this simplifies to

    C    = 1+β/2(u_1-x_1)^2 + 1-β/2(v_1-y_1)^2 
        +i((1+β)x_1(y_1-v_1) - (1-β)y_1(x_1-u_1)) .

In particular, note that Re(C) ≥ 0, with equality if and only if both x_1=u_1 and y_1=v_1.  

It can be checked that dζ_2 ∧ dζ̅_1 ∧ dζ_1 = 2i   dx_1 ∧ dy_1 ∧ dx_2.  Using (<ref>), we see 

    (<ref>)   = L_βf(u_1,v_1,u_2) 
       = 1/4π^2∫_^3f(x_1,y_1,x_2)/(C + i/2(x_2-u_2))^2  dx_1 ∧ dy_1 ∧ dx_2 
       = -1/π^2∫_^2( ∫_-∞^∞f(x_1,y_1,x_2)/((u_2-x_2)+2iC)^2  dx_2 ) dx_1 ∧ dy_1 
       = -1/π^2∫_^2 F*G(u_2)   dx_1 ∧ dy_1 ,

where F(x_2):=f(x_1,y_1,x_2) and G(x_2) := 1/(x_2+2iC)^2.


We now calculate the inverse Fourier transform of the function G.


Let C be a complex number with Re(C)>0.  The inverse Fourier transform of G(x) = 1/(x+2iC)^2 is given by

    ^-1G(ξ) := ∫_-∞^∞ G(x)e^2π i x ξ  dx
    	= 
    		0    ξ≥ 0
    
    
    		4π^2ξ e^4πξ C   ξ < 0.



The condition Re(C)>0 means the function H(z) := e^2π i z ξ/(z+2iC)^2 has a single pole of order 2 in the lower half plane.  For any R>0, the integral

    ∫_-R^R e^2π i x ξ/(x+2iC)^2 dx

can be thought of as a piece of a contour integral around a semicircle with base on the x-axis.

When ξ>0, consider such a semicircle in the upper half plane traversed counterclockwise.  The radial portion of the integral tends to 0 as R→0.  On the other hand, this function is holomorphic inside the contour.  Thus Cauchy's theorem implies ^-1G(ξ) = 0 for ξ>0.

When ξ<0, consider a semicircle in the lower half plane traversed clockwise.  For sufficiently large R, this contour encloses the pole of H.  As above, the radial portion of this integral tends to 0 as R→0.  Thus, Cauchy's integral formula shows

    lim_R→∞∫_-R^R e^2π i x ξ/(x+2iC)^2 dx    = -2π i d/dx( e^2π i x ξ)|_x=-2iC = 4π^2 ξ e^4πξ C.




Equation (<ref>) and Proposition <ref> now show that

    ^-1_u_2L_βf(u_1,v_1,ξ_u_2) 
    	= 
    		0,    ξ_u_2≥ 0
    
    
    		-4 ξ_u_2∫_^2_u_2^-1f(x_1,y_1,ξ_u_2) e^4πξ_u_2 C dx_1 dy_1,    ξ_u_2 < 0.

For ξ_u_2<0, equation (<ref>) and the triangle inequality show

    | ^-1_u_2L_βf(x_1,y_1,ξ_u_2) |    ≤∫_^2|_u_2^-1f(x_1,y_1,ξ_u_2)| K(u_1-x_1,v_1-y_1,ξ_u_2) dx_1 dy_1  
       = |_u_2^-1f(·,·,ξ_u_2)| *_(u_1,v_1) K(·,·,ξ_u_2) ,

where

    K(x,y,ξ_u_2) := -4ξ_u_2Exp[ 2πξ_u_2(1+β)x^2 + 2πξ_u_2(1-β)y^2 ].


Equation (<ref>) is in convolution form because the triangle inequality lets us ignore the oscillatory piece of the exponential.  Now apply a two-dimensional inverse Fourier transform.  Making use of integral (<ref>), a computation shows

    ^-1_u_1,v_1K(ξ_u_1,ξ_v_1,ξ_u_2) = 2/√(1-β^2)Exp[π/2ξ_u_2(ξ^2_u_1/1+β + ξ^2_v_1/1-β) ],

and it follows that

    sup{^-1_u_1,v_1K(ξ_u_1,ξ_v_1,ξ_u_2) : ξ_u_1∈, ξ_v_1∈, ξ_u_2<0 } = 2/√(1-β^2) .

We now deduce that

    _u_1,v_1^-1[|_u_2^-1f(·,·,ξ_u_2)| *_(u_1,v_1) K(·,·,ξ_u_2) ]  
    
    = _u_1,v_1^-1(|_u_2^-1f|)(ξ_u_1,ξ_v_1,ξ_u_2)  ·2/√(1-β^2) Exp[π/2ξ_u_2(ξ^2_u_1/1+β + ξ^2_v_1/1-β) ].

Equations (<ref>) and (<ref>) give

    _u_1,v_1^-1[|_u_2^-1f(·,·,ξ_u_2)| *_(u_1,v_1) K(·,·,ξ_u_2) ]_L^2(^3, σ)
    ≤2/√(1-β^2)_u_1,v_1^-1(|_u_2^-1f|)(ξ_u_1,ξ_v_1,ξ_u_2)_L^2(^3, σ).


This is now summarized to confirm item (b) from Section <ref>:


The Leray transform L_β is bounded from L^2(_β,σ) → L^2(_β,σ).



First note that from repeated application of Plancherel's identity (<ref>), we have that

    f(u_1,v_1,u_2)_L^2(^3, σ)   = _u_2^-1 f(u_1,v_1,ξ_u_2)_L^2(^3, σ)
       = |_u_2^-1 f(u_1,v_1,ξ_u_2)|_L^2(^3, σ)
       = _u_1,v_1^-1(|_u_2^-1 f|)(ξ_u_1,ξ_v_1,ξ_u_2)_L^2(^3, σ).


Since _u_2^-1L_βf(u_1,v_1,ξ_u_2) = 0 for ξ_u_2≥ 0, we only need to consider ξ_u_2<0 in the norm computations below.  Now,

    L_βf(u_1,v_1,u_2)_L^2(^3, σ)   = _u_2^-1L_βf(u_1,v_1,ξ_u_2)_L^2(^3, σ)
       ≤|_u_2^-1f(·,·,ξ_u_2)| *_(u_1,v_1) K(·,·,ξ_u_2)_L^2(^3, σ)
       = _u_1,v_1^-1[|_u_2^-1f(·,·,ξ_u_2)| *_(u_1,v_1) K(·,·,ξ_u_2) ]_L^2(^3, σ)
       ≤2/√(1-β^2)_u_1,v_1^-1(|_u_2^-1f|)(ξ_u_1,ξ_v_1,ξ_u_2)_L^2(^3, σ)
       = 2/√(1-β^2)f(u_1,v_1,u_2)_L^2(^3, σ).

This is equivalent to saying

    L_β_L^2(_β, σ)≤2/√(1-β^2).

Inequality (<ref>) follows from (<ref>) and (<ref>) is just (<ref>). 


Note. A version of (<ref>) was shown previously to the first author by Jennifer Brooks.


The bound on L_β_2 given in equation (<ref>) is never sharp, given the validity of Theorem <ref>.  When β=0, we can sharpen the estimate L_0_2<2 simply by observing the Leray transform is identical to the Szegő projection on _0 – see equation (<ref>).  Consequently, L_0_2=1.  The operators do not coincide for  β 0.  Note that while _β is biholomorphically equivalent to _0 via the map (z_1,z_2)↦(z_1,z_2-iβ z_1^2), they are not projectively equivalent.  C.f. Remark <ref>.  ◊






While Proposition <ref> is weaker than Theorem <ref>, it is included for two reasons.  First, it is worthwhile to see that the L^2-boundedness of L_β follows straightforwardly from the application of a single Fourier transform (<ref>), without the use of the second transform facilitated by the re-parametrization of _β in Section <ref> below.  Additionally, it is of great interest to the authors to understand how much sharpness is lost when applying the triangle inequality in (<ref>) and similar situations.  For instance, is the sharp norm directly attainable from equation (<ref>)?  _β may be identified with the Heisenberg group (the boundary of the Siegel upper half space) when β = 0.  This allows for the use of the Weyl transform, which is tailored to handle the twisted convolution occurring in (<ref>).  See chapter 1 of <cit.> for more information on these topics.  This machinery gives a direct way to show that L_0_2=1 (as opposed to the indirect observation in the preceding remark), but (yet again) does not apply in the β 0 case. ◊






 §.§ Parametrizing _β with projective automorphisms


The work in this section is inspired by what is known in the case of the Heisenberg group, which corresponds to _0.   Immediately from its definition in equation (<ref>), we see that _β is invariant under translations of the form (ζ_1,ζ_2) ↦ (ζ_1, ζ_2 + s), where s is a real number.  We would like to find less trivial automorphisms of this hypersurface.  We seek maps ϕ:_β→_β of the form ϕ(ζ_1,ζ_2) = (ζ_1+c, ·), c ∈.  The second component is now determined.


Let c∈ and s∈.  A complex affine map preserving _β which translates the first coordinate ζ_1 ↦ζ_1 +c must be of the form

    ϕ_(c,s)(ζ_1,ζ_2) = (ζ_1+c, ζ_2 + 2i(c̅ + β c)ζ_1 + i(|c|^2+β Re(c^2))+ s ).



Write ϕ_(c,s)(ζ_1,ζ_2) = (ζ_1+c, ϕ_2(ζ_1,ζ_2)).  Since the image of this map lies in _β,

    Im(ϕ_2(ζ_1,ζ_2))    = |ζ_1+c  |^2 + β Re((ζ_1+c)^2) 
       = |ζ_1|^2 + |c|^2 + 2 Re(ζ_1c̅) + β Re(ζ_1^2) + 2β Re(ζ_1 c) + β Re(c^2)  
       = Im(ζ_2) + 2  Re((c̅+β c)ζ_1) + |c|^2 + β Re(c^2).

Since ϕ_2 is holomorphic, we must have

    ϕ_2(ζ_1,ζ_2) = ζ_2 + 2i(c̅ + β c)ζ_1 + i(|c|^2+β (c^2))+ s(ζ_1,ζ_2),

where s(ζ_1,ζ_2) is a real valued function.  But this implies s(ζ_1,ζ_2)=s is constant.


We now see how these maps compose.


Let ϕ_(c,s): _β→_β be defined as in equation (<ref>) above.  Then composition of these maps gives

    ϕ_(c_1,s_1)∘ϕ_(c_2,s_2) = ϕ_(c_1+c_2,  s_1+s_2-2 Im((c̅_1 + β c_1)c_2)).



From equation (<ref>),

    ϕ_(c_1,s_1)∘ϕ_(c_2,s_2)(ζ)    = ϕ_(c_1,s_1)(ζ_1+c_2   , ζ_2 + 2i(c̅_2 + β c_2)ζ_1 + i(|c_2|^2+β Re(c_2^2))+ s_2 ) 
       = (ζ_1+(c_1 +c_2)   , ζ_2 + 2i(c̅_2 + β c_2)ζ_1 + i(|c_2|^2+β Re(c_2^2))+ s_2 
                    + 2i(c̅_1+ β c_1)(ζ_1 +c_2) + i(|c_1|^2+β Re(c_1^2))+ s_1).

From here, the second component in equation (<ref>) can be written as

    ζ_2 + 2i(c_1+c_2 +β(c_1+c_2))ζ_1    + i(|c_1+c_2|^2 + β Re((c_1+c_2)^2) ) 
       + (s_1+s_2) -2 Im((c̅_1+β c_1)c_2).
 


Note that the automorphisms described above are affine maps preserving both volume and our distinguished boundary measure σ = dx_1 ∧ dy_1 ∧ dx_2.  Thus, it is unnecessary to rescale the Leray transform of _β when the hypersurface is re-parametrized using these maps.  Affine maps form a subgroup of the automorphisms of ℙ^n.  In <cit.>, Bolt proves a transformation law of the Leray kernel under projective automorphisms, when the kernel is expressed in terms of Fefferman hypersurface measure.  We return to this point in Appendix <ref>  ◊







It will be desirable to consider an abelian subgroup of the group of the automorphisms defined by equation (<ref>).  Notice that the term -2  Im((c̅_1 + β c_1)c_2) vanishes when both c_1,c_2 ∈.  This immediately implies the following corollary, where we've changed all instances of c to r to emphasize this parameter is now restricted to real values. 


The collection of automorphisms := {ϕ_(r,s) : r,s∈} is a closed abelian subgroup of Aut (_β).  In fact, given two maps in this subgroup,

    ϕ_(r_1,s_1)∘ϕ_(r_2,s_2) = ϕ_(r_1+r_2,s_1+s_2).



We now use the action of  on a one-dimensional curve γ lying in _β to re-parametrize this hypersurface.


Consider the curve γ: →_β given by γ(t) = (i t, i(1-β)t^2 ). The action of the group = {ϕ_(r,s) : r,s∈} on the image of γ gives a parametrization of _β, i.e., for each ζ∈_β, there is a unique (r,s,t) ∈^3 such that ζ = ϕ_(r,s)(γ(t)).


From equation (<ref>) we can check that 

    ϕ_(r,s)(γ(t)) = (r +it,   s - 2(1+β)r t + i[(1+β)r^2 +(1-β)t^2]).

The first coordinate can attain any complex number by specifying the parameters r and t.  Once these values are decided, we can appropriately choose s to adjust the real part of the second coordinate.







 §.§ Re-parametrizing the Leray kernel
  We now make use of the automorphisms in the previous subsection.  Recall that

    L_βf(z) = 1/8π^2i∫__βf(ζ)  dζ_2 ∧ dζ̅_1 ∧ dζ_1/[(ζ̅_1 +βζ_1)(ζ_1 - z_1)+ i/2(ζ_2-z_2)]^2.

In order to circumvent the loss of sharpness on the norm coming from application of the triangle inequality, re-write (<ref>) in terms of the parameterization described in Theorem <ref>.  For notational purposes, we use (r_z,s_z,t_z) and (r_ζ,s_ζ,t_ζ) to correspond to the respective z and ζ variables.  In other words,

    z    = (r_z +it_z,   s_z - 2(1+β)r_z t_z + i[(1+β)r_z^2 +(1-β)t_z^2]), 
    ζ   = (r_ζ +it_ζ,   s_ζ - 2(1+β)r_ζ t_ζ + i[(1+β)r_ζ^2 +(1-β)t_ζ^2]).


The wedge product of differentials appearing in the numerator of equation (<ref>) can be written as

    dζ_2 ∧ dζ̅_1 ∧ dζ_1 = 2i   ds_ζ∧ dr_ζ∧ dt_ζ.

The major advantage of using this parametrization is seen when considering the denominator of the integrand in (<ref>).  After the change of variables,

    (ζ̅_1 +βζ_1)(ζ_1 - z_1)    = ((1+β)r_ζ - i (1-β)t_ζ)((r_ζ-r_z) + i(t_ζ-t_z)) 
       = (1+β)r_ζ(r_ζ-r_z) + (1-β)t_ζ(t_ζ-t_z) 
                    + i[(1+β)r_ζ(t_ζ-t_z) - (1-β)t_ζ(r_ζ-r_z) ],

and

    i/2(ζ_2-z_2)    =  -1/2[(1+β) (r_ζ^2-r_z^2) + (1-β)(t_ζ^2-t_z^2)] 
                    +i/2[ (s_ζ - s_z) - 2(1+β)(r_ζt_ζ-r_z t_z) ].

Putting the pieces together, we obtain

    (<ref>) + (<ref>)   = 1+β/2(r_z-r_ζ)^2 + i(r_z-r_ζ)(t_z+t_ζ + β(t_z-t_ζ)) 
                    + 1-β/2(t_z-t_ζ)^2 - i/2(s_z-s_ζ) 
       := A - i/2 (s_z - s_ζ)  ,

where we've collected all terms not involving (s_z - s_ζ) into the temporary label

    A = 1+β/2(r_z-r_ζ)^2 + 1-β/2(t_z-t_ζ)^2 + i(r_z-r_ζ)(t_z+t_ζ + β(t_z-t_ζ)).


The fact that the term involving the s variables appears in convolution form suggests the use of an (inverse) Fourier transform in this variable.  Note that all terms involving the r variables also appear in convolution form.  But rather than performing a two-dimensional inverse Fourier transform, we make use of the computations in Section <ref>.  From (<ref>), (<ref>) and (<ref>),

    L_βf(r_z,s_z,t_z)    = 1/4π^2∫_^3f(r_ζ,s_ζ,t_ζ)/(A - i/2(s_z-s_ζ) )^2  ds_ζ∧ dr_ζ∧ dt_ζ
       = -1/π^2∫_^2( ∫_-∞^∞f(r_ζ,s_ζ,t_ζ)/((s_z-s_ζ)+2iA)^2  ds_ζ)  dr_ζ∧ dt_ζ
       = -1/π^2∫_^2 F*G(s_z)   dr_ζ∧ dt_ζ,

where F(s):= f(r_ζ,s,t_ζ) and G(s):= 1/(s+2iA)^2.


At this point we compare (<ref>) to (<ref>) and see that they are identical except that A = (<ref>) and C = (<ref>) are different.



A glance at equation (<ref>) shows Re(A) ≥ 0, with equality if and only if both r_z = r_ζ and t_z = t_ζ. When Re(A)=0, F*G is defined by a singular integral that should be interpreted in a principal-value sense.  This is reminiscent of the Cauchy transform and relates to topics like the Plemelj jump formula as described in <cit.>.  We can sidestep this issue, however, since Re(A) vanishes only on a set of three-dimensional measure zero. ◊


(A)>0 almost everywhere and for all such A, Proposition <ref> says the inverse Fourier transform of G(s) = 1/(s+2iA)^2 is given by

    _s^-1G(ξ_s) := ∫_-∞^∞ G(s)e^2π i s ξ_s  ds
    	= 
    		0    ξ_s ≥ 0
    
    
    		4π^2ξ_s e^4πξ_s A   ξ_s < 0.



As a consequence of equations (<ref>) and (<ref>), we are able to reduce the dimension of the integral in calculation of the L^2-norm of L_βf(r_z,s_z,t_z).  Starting from (<ref>), the inverse Fourier transform in the s variable yields a statement equivalent to (<ref>): 

    _s^-1L_βf(r_z,ξ_s,t_z) = 
    		0    ξ_s ≥ 0
    
    
    		-4 ξ_s ∫_^2_s^-1 f(r_ζ,ξ_s,t_ζ)  e^4πξ_s A  dr_ζ dt_ζ   ξ_s < 0.






 §.§ A second inverse Fourier transform and Hilbert-Schmidt operators


The payoff to the re-parametrization in the previous subsection comes from the fact that A = (<ref>) is in convolution form in the r variables.  Motivated by (<ref>), for ξ_s<0 we define

    H(r) := -4  ξ_s  e^2πξ_s[(1+β)r^2+(1-β)(t_z-t_ζ)^2+2ir(t_z+t_ζ+β(t_z-t_ζ))].

Now take the inverse Fourier transform.  Using the scaling r ↦√(-2ξ_s(1+β)) r and integral (<ref>), a computation shows that for ξ_s<0,

    _r^-1 H(ξ_r)    = -4 ξ_s ∫_-∞^∞  e^2π i r ξ_r e^2πξ_s[(1+β)r^2+(1-β)(t_z-t_ζ)^2+2ir(t_z+t_ζ+β(t_z-t_ζ))]  dr 
       = 2√(2)√(-ξ_s)/√(1+β) Exp[π/2(1+β)ξ_s(ξ_r^2 + 4(1+β)(ξ_sξ_rt_z +  2ξ_s^2 t_z^2) 
                                        + 4(1-β)(ξ_sξ_rt_ζ +  2ξ_s^2 t_ζ^2)) ] 
       = 2√(2)√(-ξ_s)/√(1+β) Exp[π/2(1+β)ξ_s{ξ_r^2 + 8(1+β)ξ_s^2(t_z+ξ_r/4ξ_s)^2 - (1+β)ξ_r^2/2
                                        + 8(1-β)ξ_s^2(t_ζ+ξ_r/4ξ_s)^2 - (1-β)ξ_r^2/2}] 
       =  2√(2)√(-ξ_s)/√(1+β) Exp[4πξ_s(t_z+ξ_r/4ξ_s)^2 ] ·Exp[4πξ_s(1-β)/1+β(t_ζ+ξ_r/4ξ_s)^2 ].



The key observation from (<ref>) is that the t_z and t_ζ variables are decoupled, i.e., _r^-1 H(ξ_r) 
breaks into a product of functions of these respective variables.  Now define 

    m_0,ξ_r,ξ_s(t_z) :=  m_0(t_z)
       = 2√(2)√(-ξ_s)/√(1+β) Exp[4πξ_s(t_z+ξ_r/4ξ_s)^2 ] ·1_{ξ_s <  0}
 
and

    m_1,ξ_r,ξ_s(t_ζ) := m_1(t_ζ)    = Exp[4π(1-β)ξ_s/1+β(t_ζ + ξ_r/4ξ_s)^2 ] ·1_{ξ_s <  0},

where 1_{ξ_s <   0} is the indicator function of the interval (-∞,0) in the ξ_s variable.  These definitions were set up so that for each ξ_s<0,

    _r^-1 H(ξ_r) = m_0(t_z)  m_1(t_ζ).

Now (<ref>), (<ref>) and (<ref>) give that


    _r^-1_s^-1L_βf(ξ_r,ξ_s,t_z) = 
    		0    ξ_s ≥ 0
    
    
    		m_0(t_z) ∫_-∞^∞ m_1(t_ζ) _r^-1_s^-1 f(ξ_r,ξ_s,t_ζ)  dt_ζ   ξ_s < 0.




We finish our analysis by studying the family of operators _ξ_r,ξ_s := defined by

    g(t_z) := m_0(t_z) ∫_-∞^∞ m_1(t_ζ) g(t_ζ) dt_ζ.



For each ξ_r∈, ξ_s<0, the operator _ξ_r,ξ_s = is rank-one Hilbert-Schmidt with norm 

    _HS = 1/√(1-β^2).



Observe 

    _HS^2    = ∫_^2 |m_0(t_z)  m_1(t_ζ)|^2 dt_zdt_ζ
       = -8 ξ_s/1+β (∫_-∞^∞Exp[8πξ_s(t_z+ξ_r/4ξ_s)^2]  dt_z) 
               ·(∫_-∞^∞Exp[8π(1-β)ξ_s/1+β(t_ζ + ξ_r/4ξ_s)^2 ]  dt_ζ)   
       = -8 ξ_s/1+β( 1/2√(2)√(-ξ_s)) ·(1/2√(2)√(-ξ_s)√(1+β/1-β)) 
       =1/√(1-β^2).

Taking square roots, we are done.


We are ready to prove Theorem <ref>, but first consider the action of _ξ_r,ξ_s = on a function g_ξ_r,ξ_s = g ∈ L^2() for a fixed pair ξ_r ∈, ξ_s <0. Recall that the Hilbert-Schmidt norm of an operator dominates its operator norm.  Indeed, 

    g_L^2()^2    = ∫_-∞^∞| m_0(t_z) ∫_-∞^∞ m_1(t_ζ) g(t_ζ) dt_ζ|^2   dt_z
       ≤(∫_-∞^∞|m_0(t_z)|^2 dt_z) ( ∫_-∞^∞|m_1(t_ζ)|^2 dt_ζ) ( ∫_-∞^∞|g(t_ζ)|^2  dt_ζ) 
       = 1/√(1-β^2)g_L^2()^2,
 
with equality in (<ref>) holding if and only if g is a multiple of m_1, i.e., 

    g_ξ_r,ξ_s(·) = φ(ξ_r,ξ_s)m_1,ξ_r,ξ_s(·).






 §.§ Proof of Theorem <ref>

The reparametrization in Section <ref> from (<ref>) through (<ref>) shows that L_βf_L^2(_β,σ) = L_βf(r_z,s_z,t_z)_L^2(^3,σ).  By repeated application of Plancherel's identity (<ref>), 

    L_βf(r_z,s_z,t_z)_L^2(^3,σ)   = _s^-1L_βf(r_z,ξ_s,t_z)_L^2(^3,σ)
       = _r^-1_s^-1L_βf(ξ_r,ξ_s,t_z)  _L^2(^3,σ)
       = _ξ_r,ξ_s( _r^-1_s^-1f(ξ_r,ξ_s,·))(t_z) _L^2(^3,σ)
       ≤1/√(1-β^2)_r^-1_s^-1f(ξ_r,ξ_s,t_z)_L^2(^3,σ)
       = 1/√(1-β^2)f(r_z,s_z,t_z)_L^2(^3,σ).

Noting that f(r_z,s_z,t_z)_L^2(^3,σ) = f_L^2(_β,σ) shows 1/√(1-β^2) is an upper bound.  

The norm of L_β is achieved when equality holds in (<ref>).  By (<ref>), this happens if and only if we choose f ∈ L^2(_β,σ) so that 

    _r^-1_s^-1f(ξ_r,ξ_s,·) = φ(ξ_r,ξ_s)m_1,ξ_r,ξ_s(·).

The square-integrablility of f ensures φ must satisfy

    ∫_-∞^∞∫_-∞^ 01/√(-ξ_s)|φ(ξ_r,ξ_s)|^2  dξ_s dξ_r < ∞.

This completes the proof and establishes item (c) on the list given in Section <ref>. 




 §.§ The Leray transform of _β is a projection operator


We establish item (d) from Section <ref> by showing L_β∘L_β = L_β. 


For each ξ_r∈, ξ_s<0, the operator _ξ_r,ξ_s = defined by equation (<ref>) is a projection.


Let g∈ L^2(), and m_0 and m_1 be given by equations (<ref>) and (<ref>), respectively.

    (∘)(g)(t_η)    = m_0(t_η)∫_-∞^∞ m_1(t_z)( m_0(t_z) ∫_-∞^∞ m_1(t_ζ) g(t_ζ) dt_ζ)   dt_z 
       = m_0(t_η) ∫_-∞^∞ m_1(t_ζ) g(t_ζ) dt_ζ∫_-∞^∞ m_1(t_z) m_0(t_z)    dt_z.

A computation shows that 

    ∫_-∞^∞ m_1(t_z) m_0(t_z)    dt_z    = 2√(2)√(-ξ_s)/√(1+β) ∫_-∞^∞Exp[ 8πξ_s/1+β(t_z+ξ_r/4ξ_s)^2 ] dt_z  
       = 1,

and thus (<ref>) =  g(t_η).



L_β is a projection operator from L^2(_β,σ) → L^2(_β,σ).


After Proposition <ref>, this amounts to symbol pushing


    L_β∘L_β   = _s ∘_r ∘^-1_r ∘^-1_s ∘L_β∘_s ∘_r ∘^-1_r ∘^-1_s ∘L_β
       = _s ∘_r ∘(∘^-1_r ∘^-1_s  ) ∘_s ∘_r ∘(∘^-1_r ∘^-1_s  ) 
       = _s ∘_r ∘∘∘^-1_r ∘^-1_s 
       = _s ∘_r ∘∘^-1_r ∘^-1_s   
       = _s ∘_r ∘^-1_r ∘^-1_s ∘L_β =  L_β.

A deeper analysis of these operators is provided in Section <ref>.


  To see that the closure of S_β in projective space fails to be a C^1 hypersurface when 0<β<1, apply the projective automorphism z_1=z_1/z_2, z_2=1/z_2; then the behavior of 

    S_β:= { (z_1,z_2)∈^2 -Im(z_2) = |z_1|^2 + β (z_2/z_2 z_1^ 2)}

 near the origin captures the behavior of S_β at infinity.

Setting z_j=x_j+iy_j, the cubic formula can be used to represent y_2 as a function of 
(x_1, y_1, x_2). Computing with the formula one can check that ∂y_2/∂x_2→ 0 along every line through the origin, whereas ∂y_2/∂x_2 is a non-zero constant along the parabola x_2=x_1^2, y_2=0; thus ∂y_2/∂x_2 is discontinuous at the origin.
◊





 §.§ Higher dimensional hypersurfaces

The natural generalization of _β to higher dimensions are hypersurfaces parametrized by vectors (β_1,…,β_n-1) as follows. Independently set each 0≤β_j <1 and define

    _(β_1,…,β_n-1) := { (ζ_1,…,ζ_n)∈^nIm(ζ_n) = ∑_j=1^n-1( |ζ_j|^2 + β_j (ζ_j^2))}.

This form is suggested by equation (<ref>), as such hypersurfaces give local projective approximations to any strongly -convex hypersurface to two orders of tangency.  A hypersurface of this kind can be converted to a bounded hypersurface using a projective automorphism.  However, the point at ∞ is always mapped to a point which is less than C^1-smooth – unless β_j = 0 for all j.  With the exception of this special case – in which (<ref>) is the Heisenberg surface –  the Lanzani-Stein results in <cit.> fail to imply the L^2-boundedness of the Leray transform because they require C^1,1 smoothness.

The L^2-boundedness of the Leray transform of (<ref>) may be seen, however, by mirroring the arguments in Section <ref>.  A sketch is now given.  As usual, define the Leray transform L_(β_1,…,β_n-1) = L by (<ref>).  L resembles (<ref>) and the measure ∂ρ(ζ) ∧∂̅∂ρ(ζ)^(n-1) is a constant multiple of

    σ = dx_1 ∧ dy_1 ∧…∧ dx_n-1∧ dy_n-1∧ dx_n.

Here the coordinates (ζ_1,…,ζ_n) have been identified with (x_1,y_1,…,x_n,y_n).  Parametrizing _(β_1,…,β_n-1) by ^2n+1 in this way gives the Leray transform as a one-variable convolution	as in (<ref>).  Apply the Fourier transform in this variable and use the triangle inequality to avoid the oscillatory pieces of the integral à la (<ref>).   This facilitates use of the Fourier transform in the remaining variables much like (<ref>).  Following this blueprint through to (<ref>) lets us deduce the following:  The Leray transform is a bounded operator on L^2(_(β_1,…,β_n-1),σ) with

    L≤2^n-1/√((1-β_1^2)(1-β_2^2)...(1-β_n-1^2)).



Computing the exact norm of the Leray transform of (<ref>) in higher dimensions presents more difficulties than when n=2.  One may start by studying affine automorphisms of _(β_1,…,β_n-1) analogous those in Section <ref>.  When n=2, the payoff is seen in Proposition <ref> when _β is re-parametrized by a (real) 2-dimensional abelian group of automorphisms acting on a (real) 1-dimensional curve in _β.  The higher dimensional version is a re-parametrization of _(β_1,…,β_n-1) by an n-dimensional abelian group of automorphisms acting on a (n-1)-dimensional surface in _(β_1,…,β_n-1).  It is unclear whether this approach will produce the exact norm of the Leray transform for n>2.
◊



The L^p-regularity (p ≠ 2) of the Leray transform on _β and the higher dimensional hypersurfaces in (<ref>) is an interesting open question.  These hypersurfaces are less than C^1-smooth at ∞, so the Lanzani-Stein machinery <cit.> is unable to imply any L^p-regularity.  In <cit.>, the same authors provide explicit examples of hypersurfaces which are C^m-smooth, 1<m<2, but for which to the L^p-regularity of the Leray transform fails for every 1<p<∞.  One might hope that a Calderón-Zygmund style approach to this problem will yield L^p results, but at this point the details are unclear and likely to involve non-trivial modifications.  Nonetheless, this problem is worthy of further pursuit.
◊














§ PROJECTIVE DUAL CR STRUCTURES






In this section we reinterpret the Leray transform L_ with the use of projective dual coordinates and  the projective dual CR structure on a general strongly -convex hypersurface .  The dual coordinates depend, in our presention, on the choice of a matrix M, but the dual CR structure will be independent of that choice.  (This follows from Lemma <ref> below.)

Let

    M=[  c_1  a_1  a_2;  b_1 m_11 m_12;  b_2 m_21 m_22 ]

be an invertible 3-by-3 complex matrix.  Use M to define a map Φ_M: ^2×^2→ by

    ((z_1,z_2),(w_1,w_2) )   ↦[   1 w_1 w_2 ] M [   1; z_1; z_2 ]
       =c_1+a_1z_1+a_2z_2+b_1w_1+b_2w_2 +[ w_1 w_2 ][ m_11 m_12; m_21 m_22 ][ z_1; z_2 ].


Given a smooth real hypersurface ⊂^2 with defining function ρ along with ζ∈, let 


    μ_1(ζ)   =ρ/ζ_2(ζ)
    μ_2(ζ)   =-ρ/ζ_1(ζ)


and note that

    L:= μ_1(ζ) /ζ_1 + μ_2(ζ) /ζ_2

 is a non-vanishing type-(1,0) vector field tangent to .  The (affine) complex tangent line for  at ζ may be described parametrically by a map from →^2 sending

    υ↦ζ+υ(μ_1(ζ),μ_2(ζ)),

or equivalently by

    {(z_1,z_2)ρ/ζ_1(ζ)(ζ_1-z_1)+ρ/ζ_2(ζ)(ζ_2-z_2)=0}.




We say the hypersurface   is M-admissible if for all ζ∈,

    [ m_11μ_1(ζ)+m_12μ_2(ζ)   m_11ζ_1+m_12ζ_2+b_1; m_21μ_1(ζ)+m_22μ_2(ζ)   m_21ζ_1+m_22ζ_2+b_2 ] 0
   


The motivation behind this definition comes from the following lemma.



If  is M-admissible then there are uniquely-determined functions w_1,M and w_2,M on  with the property that the  complex tangent line to  at ζ is given by 

    {(z_1,z_2)Φ_M( (z_1,z_2), (w_1,M(ζ),w_2,M(ζ)) )=0}.



 The set (<ref>) is either a complex line or the empty set (corresponding to the projective “line at infinity").
To prove the lemma, it suffices to check via (<ref>) that there are uniquely-determined w_1,M(ζ),w_2,M(ζ)  so that the set (<ref>) contains both ζ and ζ+(μ_1(ζ),μ_2(ζ)); this is equivalent to the system

    (m_11ζ_1+m_12ζ_2+b_1)w_1,M(ζ)+(m_21ζ_1+m_22ζ_2+b_2)w_2,M(ζ)   =-c_1-a_1ζ_1-a_2ζ_2
    
    (m_11μ_1(ζ)+m_12μ_2(ζ))w_1,M(ζ)+(m_21μ_1(ζ)+m_22μ_2(ζ))w_2,M(ζ)   =-a_1μ_1(ζ)-a_2μ_2(ζ).

Condition (<ref>) guarantees that this system is uniquely solvable for w_1,M(ζ),w_2,M(ζ).


We note that since ζ belongs to the tangent line (<ref>)
we have

    Φ_M( (ζ_1,ζ_2), (w_1,M(ζ),w_2,M(ζ)) )=0.




 §.§ Examples of M-admissibility



Setting 
    M_1=[ -1  0  0;  0  1  0;  0  0  1 ]
 we find that  is M_1-admissible if and only if ζ_1μ_2(ζ)ζ_2μ_1(ζ) on ; using (<ref>) we find that this is equivalent to the condition that no complex tangent line passes through the origin.

Every compact strongly -convex  enclosing the origin is M_1-admissible.  See <cit.> and Section 2.5 in <cit.> for further discussion.

Note also that in this case (<ref>) reads as 

    z_1w_1,M_1(ζ)+z_2w_2,M_1(ζ)=1

for z in the complex tangent line.  Similarly (<ref>) becomes

    ζ_1w_1,M_1(ζ)+ζ_2w_2,M_1(ζ)=1.

Comparing (<ref>) to (<ref>) we see that we may (and must) take

    w_1,M_1(ζ)   =ρ/ζ_1(ζ)/ζ_1ρ/ζ_1(ζ)+ζ_2ρ/ζ_2(ζ)
    
    w_2,M_1(ζ)   =ρ/ζ_2(ζ)/ζ_1ρ/ζ_1(ζ)+ζ_2ρ/ζ_2(ζ),

where the non-vanishing of the denominators follows from the M_1-admissibility. It follows now that

    < ∂ρ(ζ),(ζ-z) > = ( ζ_1ρ/ζ_1(ζ)+ζ_2ρ/ζ_2(ζ) ) (1-z_1w_1,M_1(ζ)-z_2w_2,M_1(ζ)),

where the left hand side appears in the denominator of equation (<ref>).  Thus, the Leray kernel 

    ℒ_(z,ζ)=1/(2π i)^2∂ρ(ζ) ∧∂̅∂ρ(ζ)/< ∂ρ(ζ),(ζ - z) >^2

may be written as

    1/(2π i)^2(ζ_1ρ/ζ_1(ζ)+ζ_2ρ/ζ_2(ζ))^-2 ∂ρ(ζ) ∧∂̅∂ρ(ζ)/(1-z_1w_1,M_1(ζ)-z_2w_2,M_1(ζ))^2.

To rewrite the numerator further we note that 

    (ζ_1ρ/ζ_1(ζ)+ζ_2ρ/ζ_2(ζ))^-1 ρ(ζ)=w_1,M_1 dζ_1+w_2,M_1 dζ_2,

from which we may deduce that

    (ζ_1ρ/ζ_1(ζ)+ζ_2ρ/ζ_2(ζ))^-2    ∂ρ(ζ) ∧∂̅∂ρ(ζ)
       = (w_1,M_1 dζ_1+w_2,M_1 dζ_2 )( w_1,M_1 dζ_1+ w_2,M_1 dζ_2 )
       = (w_2,M_1  w_1,M_1-w_1,M_1  w_2,M_1) dζ_1 dζ_2
       = (w_2,M_1 dw_1,M_1-w_1,M_1 dw_2,M_1) dζ_1 dζ_2,

so that 

    ℒ_(z,ζ)
    = 1/(2π i)^2(w_2,M_1 dw_1,M_1-w_1,M_1 dw_2,M_1) dζ_1 dζ_2/(1-z_1w_1,M_1-z_2w_2,M_1)^2.


In the special case when  is the unit sphere in ^2, we have

    w_1,M_1(ζ)   =ζ_1 
    
    w_2,M_1(ζ)   =ζ_2 
    ℒ_(z,ζ)
       = 1/(2π i)^2(ζ_2 dζ_1-ζ_1 dζ_2) dζ_1 dζ_2/(1-z_1ζ_1-z_2ζ_2)^2.


We note that this formula coincides with the Szegő kernel for the unit sphere.  






Setting 
    M_2=[  0  0  i;  0  2  0; -i  0  0 ]

we find that  is M_2-admissible if and only if μ_1(ζ)=ρ/ζ_2(ζ) 0 on ; equivalently,  has no “vertical" complex tangents.  Every  arising as a graph of a smooth function over × is M_2-admissible.

In this case (<ref>) reads as 

    2  z_1w_1,M_2(ζ)+iz_2-iw_2,M_2(ζ)=0

for z in the complex tangent line.  Similarly, (<ref>) becomes

    2  ζ_1w_1,M_2(ζ)+iζ_2-iw_2,M_2(ζ)=0.


Comparing (<ref>) to (<ref>) we find


    w_1,M_2(ζ)   =i/2ρ/ζ_1(ζ)/ρ/ζ_2(ζ), 
    
    w_2,M_2(ζ)   =ζ_1ρ/ζ_1(ζ)+ζ_2ρ/ζ_2(ζ)/ρ/ζ_2(ζ),


and

    < ∂ρ(ζ),(ζ-z) > 
    = iρ/ζ_2(ζ)·(  2  z_1w_1,M_2(ζ)+iz_2-iw_2,M_2(ζ) ).

Thus,

    ∂ρ(ζ) ∧∂̅∂ρ(ζ)/< ∂ρ(ζ),(ζ - z) >^2=  -(ρ/ζ_2(ζ))^-2 ∂ρ(ζ) ∧∂̅∂ρ(ζ)/(2  z_1w_1,M_2(ζ)+iz_2-iw_2,M_2(ζ))^2.

But note that

    (ρ/ζ_2(ζ))^-1 ∂ρ(ζ) 
    =2/iw_1,M_2 dζ_1+dζ_2,

from which we obtain

    (ρ/ζ_2(ζ))^-2 ∂ρ(ζ) ∧∂̅∂ρ(ζ)
       =( 2/iw_1,M_2 dζ_1+dζ_2)
    2/i  w_1,M_2 dζ_1
       = -2i  d w_1,M_2 dζ_1 dζ_2,

and so

    ℒ_(z,ζ)
    = 1/2π^2i d w_1,M_2 dζ_1 dζ_2 /(2  z_1w_1,M_2(ζ)+iz_2-iw_2,M_2(ζ))^2 .



In the special case when = _β we have

    w_1,M_2(ζ)   =ζ_1+βζ_1 
    
    w_2,M_2(ζ)   =2/iζ_1(ζ_1+βζ_1)+ζ_2 
    ℒ_β(z,ζ)
       = 1/2π^2i d ζ_1 dζ_1 dζ_2 /(2  z_1w_1,M_2(ζ)+iz_2-iw_2,M_2(ζ))^2 
       = 1/8π^2 idζ_2 ∧ dζ̅_1 ∧ dζ_1/((ζ̅_1 +βζ_1)(ζ_1 - z_1)+ i/2(ζ_2-z_2))^2,

which recovers the form of the Leray kernel given in (<ref>).  When β=0, this becomes

    w_1,M_2(ζ)   =ζ_1 
    
    w_2,M_2(ζ)   =2/iζ_1ζ_1+ζ_2=ζ_2 
    ℒ_0(z,ζ)
       = 1/2π^2i d ζ_1 dζ_1 dζ_2 /(2  z_1ζ_1+iz_2-iζ_2)^2 
       =1/π ^2 dx_1 dy_1 dx_2 /(2  z_1ζ_1+iz_2-iζ_2)^2 .

Since this is conjugate-CR with respect to ζ we find that ℒ_0 is the Szegő kernel of _0 with respect to the measure dx_1 dy_1 dx_2.  (See Chapter 10 in <cit.>.)  






The non-zero entries in M_2 are tailored specifically for use with _β. ◊




Setting 

    M_3=[  0  i  0; -i  0  0;  0  0  2 ],

(or any such matrix with non-zero entries in the same slots) we find that  is M_3-admissible if and only if  has no “horizontal" complex tangents.  In particular, each point in  has a neighborhood that is M_2-admissible or M_3-admissible. ◊



The formulas (<ref>) and (<ref>) reveal that the Leray transform does have some form of the domain independence property from Section <ref> if we allow the use of the dual variables.  Any two M-admissible hypersurfaces have identical Leray kernels in the variables (z,w) = (z_1,z_2,w_1,M, w_2,M).  The w variables, of course, are hypersurface dependent. ◊




 §.§ Universal dual coordinate description of the Leray transform


Recall that a projective automorphism is a (partially-defined) map from ^2 to ^2 extending to an automorphism of projective space.  These have the form 

    Υ^* (w_1,w_2)↦(  (1,w_1,w_2) Υ_1/ (1,w_1,w_2) Υ_0, (1,w_1,w_2) Υ_2/ (1,w_1,w_2) Υ_0),

where Υ_0,Υ_1,Υ_2 are the columns of an invertible 3-by-3 matrix Υ.


If  is both M-admissible and M'-admissible, then there is a projective automorphism  Υ^*  so that (w_1,M',w_2,M')=Υ^*(w_1,M,w_2,M).
 






The hypothesis guarantees that each complex tangent line for  may be written uniquely
as the zero set of 
( 1, w_1,M, w_2,M) M ( 1, z_1, z_2)^T
or of
( 1, w_1,M', w_2,M') M' ( 1, z_1, z_2)^T; 
 thus there is a relation of the form 
( 1, w_1,M', w_2,M') M' 
= κ(w_1,M,w_2,M) ·( 1, w_1,M, w_2,M) M 
or

    [ 1, w_1,M', w_2,M' ]  
    = κ(w_1,M,w_2,M) ·[ 1, w_1,M, w_2,M ] M (M')^-1.

  A computation shows that this works if we let Υ=M (M')^-1 and set
   κ(w_1,M,w_2,M)= 1/ (1,w_1,M,w_2,M) Υ_0.  The non-vanishing of the denominator follows from (<ref>).


We now obtain the universal dual coordinate description of the Leray transform.


If  is an M-admissible strongly -convex hypersurface in ^2 then the Leray integral from (<ref>) may be written as

    L_f(z) =  ∫_ζ∈ f(ζ)
    ν_M/((1, w_1,M, w_2,M) M (1, z_1, z_2)^T)^2,

where

    ν_M:= 1/(2π i)^2((  d w_1,M, d w_2,M) 
    [ m_11 m_12; m_21 m_22 ][  a_2; -a_1 ]
    
    +
     m_11    m_12
     m_21    m_22( w_2,M d w_1,M - w_1,M d w_2,M)) dζ_1 dζ_2.

Here a_j and m_jk are given by (<ref>).


Note that from the examples in Section <ref>, we already have this result for the special matrices M_1 and M_2.  In particular,

    ν_M_2=1/2π^2i d w_1,M_2 dζ_1 dζ_2.




The result is local.  Suppose that  is M_2-admissible at a particular point.  The transformation laws from the proof of Proposition <ref> (with M'=M_2)  yield 

    ν_M=1/2π^2i( (1,w_1,M,w_2,M) Υ_0 )^2   d(  (1,w_1,M,w_2,M) Υ_1/ (1,w_1,M,w_2,M) Υ_0) dζ_1 dζ_2,

with 
    Υ=[  -i a_2   a_1/2   i c_1; -i m_12  m_11/2   i b_1; -i m_22  m_21/2   i b_2 ].
  Routine computation then reduces ν_M to the form given in (<ref>).  If S fails to be M_2-admissible at some point then by Remark <ref> it will be M_3-admissible there and a similar computation will yield the result. 


 
Similarly, in the setting of Lemma <ref> we have the transformation law

    ν_M=( (1,w_1,M,w_2,M) Υ_0 )^2  ν_M'

with (1,w_1,M,w_2,M) Υ_0 non-vanishing.  ◊



If  is an M-admissible strongly -convex hypersurface in ^2 then the form ν_M from (<ref>) is nowhere-vanishing as a 3-form on .



In view of Remarks <ref> and <ref> it suffices to consider the case M=M_2 (or the similar case M=M_3).  The claim is equivalent to the non-vanishing of dρν_M_2 along .

Recalling (<ref>) and (<ref>) (and using subscripts to denote derivatives) we have

    dρν_M_2   = 1/4π^2(ρ_ζ̅_1 dζ̅_1 + ρ_ζ̅_2 dζ̅_2)
    d(  ρ_ζ_1/ρ_ζ_2)
    dζ_1 dζ_2 
       = 1/4π^2ρ_ζ_2^-2(ρ_ζ̅_1 dζ̅_1 + ρ_ζ̅_2 dζ̅_2)
           (ρ_ζ_2(ρ_ζ_1ζ̅_1 dζ̅_1+ ρ_ζ_1ζ̅_2 dζ̅_2 )
    - ρ_ζ_1(ρ_ζ_2ζ̅_1 dζ̅_1+ ρ_ζ_2ζ̅_2 dζ̅_2)     
    ) dζ_1 dζ_2 
       =1/4π^2ρ_ζ_2^-2[          0     ρ_ζ̅_1     ρ_ζ̅_2;      ρ_ζ_1 ρ_ζ_1 ζ̅_1 ρ_ζ_1 ζ̅_2;      ρ_ζ_2 ρ_ζ_2 ζ̅_1 ρ_ζ_2 ζ̅_2 ] dζ_1 dζ_2 dζ̅_1 dζ̅_2.

Since  is strongly -convex, it is also strongly pseudoconvex – see Remark <ref> – and strong pseudoconvexity is well-known to be equivalent to the negativity of the determinant above. This establishes the claim.



In view of Proposition <ref>, we write ν_M in the form ν_M·|ν_M| where ν_M is a unimodular scalar function on  (the phase function for ν_M) and |ν_M| is a positive 3-form on .  (We may view |ν_M|  as a measure on  that is a smooth positive multiple of the surface area measure.)

From Remark <ref> we have

    |ν_M|   =| (1,w_1,M,w_2,M) Υ_0 |^2  |ν_M'|,
    ν_M   = ( (1,w_1,M,w_2,M) Υ_0 )^2 /| (1,w_1,M,w_2,M) 
    Υ_0 |^2ν_M'.



For future reference we note that if  is M_2-admissible with defining function ρ that is independent of (ζ_2) then ρ_ζ_2 is purely imaginary and non-vanishing and thus  may be written locally in the form Im (ζ_2) = λ(ζ_1, ζ̅_1) (and thus  is (locally) rigid in the sense of Baouendi, Rothschild and Trèves – see <cit.>). From (<ref>) we see that in this situation the 4-form dρν_M_2 is positive along ; equivalently, ν_M_2 is positive as a 3-form on   and hence  |ν_M|=ν_M, ν_M=1. ◊





 §.§ Projective dual CR-structures

Recall the non-vanishing type-(1,0) tangent vector field

    L = μ_1(ζ) /ζ_1 + μ_2(ζ) /ζ_2.



L w_1,M and L w_2,M do not vanish simultaneously.




If  is M_2-admissible then from (<ref>) and Proposition <ref>
we have d w_1,M_2 dζ_1 dζ_2 0 as a 3-form on .  It follows that d w_1,M_2 fails to be -linear on the maximal complex subspace at any point of  , that is, 
L w_1,M_2 0.  A similar argument works if  is M_3-admissible.  The general case follows now from Remark <ref> and Lemma <ref>.




There is a uniquely (and locally) determined smooth function η on  so that if we set

    L_ dual   =L - ηL 
    L_ dual   = L - ηL,

we have L_ dual w_1,M = 0 = L_ dual w_2,M for all M for which  is M-admissible.





 At points where  is M_2-admissible,   Lemma <ref> allows us to assume that M=M_2. 
 
 From (<ref>) and (<ref>) we obtain w_1,M_2(ζ)=μ_2(ζ)/2iμ_1(ζ) and hence
 
    2i(Lζ_1)w_1,M_2= Lζ_2.

Applying L to (<ref>) and using (<ref>), we find that


    0    = 2(Lζ_1)w_1,M_2+2 ζ_1 (Lw_1,M_2)-i Lw_2,M_2+ i Lζ_2 
       =2 ζ_1 (Lw_1,M_2)-i Lw_2,M_2
    
     0    = 2 ζ_1 (Lw_1,M_2)-iLw_2,M_2.



By the proof of  Lemma <ref>  we have Lw_1,M_20, allowing us to define 

    η= L w_1,M_2/L w_1,M_2,

so that L_ dual w_1,M_2 = 0.  From (<ref>a), (<ref>b) we see that also L_ dual w_2,M_2 = 0 as required.


A similar argument holds in the M_3-admissible case, and the general case follows as before by application of Remark <ref>. 

We now define the (projective) dual CR-structure on :


We declare L_ dual to be dual-type (1,0); thus a function f on  is dual-CR if and only if L_ dual f=0.  Note that this definition is set up so that w_1,M and w_2,M are dual-CR for all choices of M for which  is M-admissible.
 

Referring now to Proposition <ref> (or to (<ref>) or (<ref>)) we see that L_f(z) is obtained by integrating
f against the dual-CR function  ((1, w_1,M, w_2,M) M (1, z_1, z_2)^T)^-2 with respect to the 3-form ν_M.  This may be compared with the corresponding Szegő projection, obtained by integrating against a conjugate-CR function.




§ FACTORIZATION OF L_S


For any M-admissible strongly -convex hypersurface , the three-form ν_M, the measure |ν_M| and the phase function ν_M are defined in (<ref>) and Remark <ref>.


In the special case of _β we have |ν_M_2|=ν_M_2=π^-2 dx_1 dy_1 dx_2
and ν_M_2=1 (with notation as in Section <ref>).  In fact, from Remark <ref> we have ν_M_2=1 for all rigid hypersurfaces, and the proof of Proposition <ref> (see equation (<ref>)) can be used to check that ν_M_2 is constant only for hypersurfaces that are ζ_2-rotations of rigid hypersurfaces.

For the remainder of this section, all L^2-norms are defined using the measure |ν_M| where  M is a matrix for which  is M-admissible.  The dependence of |ν_M| (and ν_M) on the choice of M was explained in Remark <ref>.







 §.§ Orthogonal and skew projections


Focus now on the case of smooth bounded strongly -convex  (which will be M_1-admissible after a translation).
Let Ω denote the domain bounded by .  Lanzani and Stein's main result in <cit.> guarantees that L_ defines a bounded projection operator from L^2(,|ν_M|) onto the Hardy space H^2(,|ν_M|) of L^2 boundary values of holomorphic functions on Ω.  We often omit the measure when writing these spaces. In particular, we have
that L_∘L_=L_ and 

    H^2()=(L_-I)=L_(L^2()).


Let P_ denote the orthogonal projection from L^2() onto (L_)^⊥. Define the space W_M()=W() by

    W()={ν_M h : h ∈(L_)^⊥},

and let R_ L^2()→() denote the surjective operator given by f↦ν_M·P_f.  Note that  R_=P_=1. 
For each z∈, define the dual-CR function

    g_z(ζ)    =Φ_M( (z_1,z_2), (w_1,M(ζ),w_2,M(ζ)) )^-2
       =((1, w_1,M(ζ), w_2,M(ζ)) M (1, z_1, z_2)^T)^-2.


 The conjugate  space W() is the closed span of {g_z z∈Ω}. 



From Proposition <ref>, L_f(z) = ∫_ f ν_M g_z = ∫_ f |ν_M| ν_M g_z. Hence, L_f=0 if and only if 

    f⊥_L^2(,|ν_M|){ν_M g_z z∈Ω}.

Consequently,
(L_)^⊥ is the closed span of {ν_M g_z z∈Ω} and thus W() is the closed span of {g_z z∈Ω}.  The claim follows.



We will show below in  Proposition <ref> that W() is in fact the Hardy space H^2_() corresponding to the projective dual CR structure.  ◊


For f∈ L^2(), h∈ (L_)^⊥ we have ν_M h∈W() and thus

    ∫_ f ν_M ν_M h = ∫_ f  |ν_M|  h   = ∫_(P_f) |ν_M|  h
       =∫_(R_f) |ν_M|  ν_M h.


Recalling Proposition <ref> and setting ν_M h=g_z above, we find that

    L_ f(z) = ∫_ f  ν_M   g_z =  ∫_(R_f)  |ν_M|   g_z    =  ∫_ν_M(R_f) ν_M   g_z
       = L_(  ν_M(R_f)) (z).











Define the operator Q_:()→ H^2() by 
f↦L_(ν_M· f).
Then
equation


  * L_ = Q_∘R_ 

  * L_=Q_

  * Q_ is invertible. 




The factorization N:Q-OpI:LQR follows from (<ref>).

To prove N:Q-OpI:LNormFromQNorm note that Q_≤L_ from the definition of Q_ and that L_≤Q_ from N:Q-OpI:LQR and R_=1.

To prove N:Q-OpI:QInv note first that surjectivity of Q_ follows from N:Q-OpI:LQR.  To verify injectivity, note that f=ν_Mh∈(), Q_ f=0 implies
h∈(L_)^⊥ and 
L_h=0 hence h=0=f.
  

Suppose now that  is the M-admissible strongly -convex boundary of an unbounded domain Ω.  As before, equip  with the measure |ν_M|.  Define L_, g_z etc. as above, and suppose that the following hold:
equation


  * g_z_ L^2() is a locally bounded function of  z∈Ω, hence L_ maps L^2() →(Ω)  ;

  * Taking boundary values, we obtain a projection operator L_ L^2()→ L^2().


If we then define the Hardy space H^2()⊂ L^2() to be the range of this operator we find that Theorem <ref> and the preceding discussion carries over immediately.  When = _β, compare (<ref>a) with Corollary (<ref>).





 §.§ Factorization of L_β


Restrict focus now to _β with the measure σ = dx_1 ∧ dy_1 ∧ dx_2, though we often omit σ below.  Recall that _β is M_2-admissible and  observe that σ = π^2|ν_M_2|.  From Section <ref> we have the Paley-Wiener-type result that  
    H^2(_β):=_r^-1_s^-1(H^2(_β))
 is the set of square-integrable functions
h(ξ_r,ξ_s,t) = φ(ξ_r,ξ_s)m_0,ξ_r,ξ_s(t). Recalling that m_0,ξ_r,ξ_s(t) vanishes for ξ_s ≥ 0,

    h_L^2(^3)^2    = ∫_-∞^∞∫_-∞^0∫_-∞^∞ |φ(ξ_r,ξ_s)m_0,ξ_r,ξ_s(t)|^2 dt dξ_s dξ_r 
       = 2√(2)/1+β∫_-∞^∞∫_-∞^0 √(-ξ_s)·|φ(ξ_r,ξ_s)|^2 dξ_s dξ_r<∞.





Recalling the general formula f(ξ)= f(-ξ) and noting that  m_1,ξ_r,ξ_s(t)=m_1,ξ_r,ξ_s(t) we similarly find that the spaces from Section <ref> correspond to 
 

    L̂^̂2̂(̂_̂β̂)̂   = L^2(^3),
    L_β   = {ψ∈ L^2(^3)∫_ψ(ξ_r,ξ_s,t)m_1,ξ_r,ξ_s(t) dt=0  a.e. (ξ_r,ξ_s)∈^2},
    (L_β)^⊥   = W(_β)
       = {φ(ξ_r,ξ_s)m_1,ξ_r,ξ_s(t)∫_-∞^∞∫_-∞^ 01/√(-ξ_s) |φ(ξ_r,ξ_s)|^2 dξ_s dξ_r<∞}, 
    W(_β)   = {φ(ξ_r,ξ_s)m_1,-ξ_r,-ξ_s(t)∫_-∞^∞∫_0^∞1/√(ξ_s) |φ(ξ_r,ξ_s)|^2 dξ_s dξ_r < ∞}.


Similarly, the operators correspond to
 

    L̂_̂β̂   = 
    R̂_̂β̂   =P̂_̂β̂ψ↦m_1,ξ_r,ξ_s(t)/m_1,ξ_r,ξ_s(·)_L^2()^2∫_ψ(ξ_r,ξ_s,u)m_1,ξ_r,ξ_s(u) du
    Q̂_̂β̂   = |_W(_β)


respectively.  The following diagram keeps track of these spaces and operators.





  (m) [matrix of math nodes,row sep=4em,column sep=4em,minimum width=2em]
  
     L^2(_β)          _r^-1_s^-11.0cm      L^2(^3)       

     W(_β)     H^2(_β)      _s_r1.0cm     W(_β)     H^2(_β)  
;
     


  [-stealth]
  
    (m-1-1) edge node [left] P_β=R_β (m-2-1)
            edge node [above] L_β
            (m-2-2)
            
    (m-2-1) edge node [below] Q_β
            (m-2-2)

    (m-1-4) edge node [left] R̂_̂β̂ (m-2-4) 
            edge node [above] L̂_̂β̂ = (m-2-5)
            
    (m-2-4) edge node [below] Q̂_̂β̂
            (m-2-5);
            
  
  
    
    







It is easy to use these formulas to provide a separate verification of the assertions of Theorem <ref> adapted to _β.  In fact, even more is true in this special case.


The operator √(1-β^2) Q_β(_β)→ H^2(_β) is an isometry.

This follows immediately from equation (<ref>).





§ DUAL HARDY SPACES
 


If ⊂^2 is an M-admissible compact strongly -convex hypersurface bounding a domain Ω, then the map w_M=(w_1,M,w_2,M) is a diffeomorphism from  onto a compact strongly -convex hypersurface _ bounding a domain Ω_⊂^2. 




Note that _ and Ω_ depend on M.  Whatever the choice of M, the CR functions on a relatively open subset of _ pull back with respect to w_M to functions on a relatively open subset of  that are dual-CR as defined in <ref>.


This is proved in Proposition 2.5.12 of <cit.>.



It is also true that _ is M^T-admissible and that the map w_M^T for _ is the inverse of the map w_M for  (see Section 6 in <cit.>). ◊




 §.§ Pullback operators and function spaces

Continuing with the assumptions of Theorem <ref>, a function on a relatively open subset of _ will be CR if and only if its pullback via w_M is CR with respect to the projective dual CR-structure described in Section <ref>. The pullback H^2_():=w_M^*( H^2(_)) is the projective dual Hardy space for .  The functions in H^2_() are L^2 boundary values of holomorphic functions on 
Ω_.


It should be emphasized that while the norms of functions in H_^2() depend on our choice of matrix M, the functions themselves are independent of this choice. ◊


The operator L__ pulls back to an operator L_^ L^2()→ L^2() satisfying 

    L_^∘L_^ =L_^

and

    H^2_()=(L_^-I)=L_^(L^2()).



For f,g∈ L^2() we have

    ∫_(L_f)ν_M  g = ∫_ f ν_M (L_^ g)




This is Theorem 25 in <cit.>.




The space W() defined by equation (<ref>) satisfies W()=H^2_().



We start by noting that

    f∈L_   ⇔∫_ (L_f)ν_M  g = 0  for all  g∈ L^2()
       ⇔∫_ f ν_M (L_^ g) = 0  for all  g∈ L^2()
          [by Prop. <ref>]
       ⇔ f ν_M⊥H^2_()
       ⇔ f ⊥ν_MH^2_().

Thus ( L_)^⊥ = ν_MH^2_().

Multiplying both sides by ν_M we have W()=H^2_(), as required.

Combining this with Lemma <ref> we find that span{g_z z∈Ω} is dense in H^2_().  Dualizing, we find it also true that the space span{h_w w∈Ω_} is dense in H^2(), where

    h_w(ζ)=Φ_M( (ζ_1,ζ_2), (w_1,w_2) )^-2.


The diagram below depicts the operators P_, R_, Q_ introduced in Section <ref>, when the Leray transform is decomposed into factors.  We now see that this factorization passes through the conjugate dual Hardy space.




  (m) [matrix of math nodes,row sep=5em,column sep=8em,minimum width=2em]
  
     L^2()           

     (L_)^⊥     H^2_()     H^2()
;
     
  [-stealth]
    (m-1-1) edge node [left] P_ (m-2-1)
            
            edge node [right] L_ (m-2-3)
            edge node [left] R_ (m-2-2)
            
    
        
    
            
            
    (m-2-1) edge node [below] (f ↦τ(ν_M)· f)
            (m-2-2)
            
    (m-2-2) edge node [below] Q_ (m-2-3);       







 §.§ The dual hypersurface of _β

Turning attention now to the _β, the assumptions of Theorem <ref> do not apply but we have from (<ref>) and (<ref>) that 

    w_M_2 (ζ_1,ζ_2)↦(ζ̅_̅1̅+βζ_1,2/iζ_1(ζ_1+βζ_1)+ζ_2)
    =(ζ̅_̅1̅+βζ_1, ζ_2-iβζ_1^2+iβζ_1^2   )

with inverse

    (w_1,w_2)↦(w_1-β w_1/1-β^2,w_2+2iw_1(w_1-β w_1)/1-β^2).

This leads to

    _β, := { (w_1,w_2)∈^2: -(1-β^2)Im(w_2) = |w_1|^2 - β Re(w_1^2)}.


Note that _β, is linearly equivalent to _β via the map (w_1,w_2)↦(iw_1/√(1-β^2),-w_2).  The space H^2_(_β) and operator L__β^ are induced from H^2(_β,) and L__β, as above.  The proof of Proposition <ref> carries over to show that we still have W(_β)=H^2_(_β).  

We previously discussed the inverse Fourier transforms of both this space and the original H^2(_β) in Section <ref>.  For reference, we include them below.  Recall that m_0,ξ_r,ξ_s(t) and m_1,ξ_r,ξ_s(t) are defined in (<ref>) and (<ref>).

    H^2(_β)   = {φ(ξ_r,ξ_s) m_0,ξ_r,ξ_s(t): ∫_-∞^∞∫_-∞^0 √(-ξ_s)·|φ(ξ_r,ξ_s)|^2  dξ_s dξ_r < ∞}, 
    H_^2(_β)   = {φ(ξ_r,ξ_s) m_1,-ξ_r,-ξ_s(t): ∫_-∞^∞∫_0^∞1/√(ξ_s)·|φ(ξ_r,ξ_s)|^2  dξ_s dξ_r < ∞}.


Note that the space H^2(_β) and the conjugate dual space H_^2(_β) coincide when β = 0.








§ L^2-NORMS OF THE KERNEL FUNCTION
 

Like the Cauchy transform in one complex variable, the Leray transform on a bounded -convex hypersurface  constructs holomorphic functions on the -convex domain it bounds.  But dealing with unbounded hypersurfaces requires more delicacy.  This appendix shows that L_β constructs holomorphic functions on Ω_β from boundary functions in L^2(_β,σ).



 §.§ Automorphisms of Ω_β


Partition Ω_β as an infinite union of translates of _β.  For all ϵ≥ 0, define the hypersurface 

    _β^ ϵ := {(z_1,z_2) : (z_1,z_2 - iϵ) ∈_β},

and note that if z ∈_β^ ϵ, then (z_2) = |z_1|^2 + β(z_1^2) + ϵ.  It is clear that _β^0 = _β and Ω_β = ∪_ϵ > 0 _β^ ϵ.  The maps ϕ_(c,s) defined by equation (<ref>) extend to automorphisms of Ω_β.  In fact, they preserve each shell:


The affine map 

    ϕ_(c,s)(z_1,z_2) = (z_1+c, z_2 + 2i(c̅ + β c)z_1 + i(|c|^2+β Re(c^2))+ s )

is an automorphism of each _β^ ϵ for all choices of c∈ and s ∈.


Choose any z ∈_β^ ϵ.  Then (z_2) = |z_1|^2 +β(z_1^2)+ϵ.  Writing the components of ϕ_(c,s)(z_1,z_2) as (ϕ_1,ϕ_2), we see

    |ϕ_1|^2 = |z_1|^2 + |c|^2 + 2(z̅_̅1̅c),     β(ϕ_1^2) = β(z_1^2) +2β(z_1c)+β(c^2),

and

    (ϕ_2) = (z_2) + β(z_1^2) + ϵ + 2(z̅_̅1̅ c) + 2β(z_1c) + |c|^2 + β(c^2).

This means ϕ_(c,s)(z_1,z_2) = (ϕ_1,ϕ_2)∈_β^ ϵ.



Fix z∈_β^ ϵ.  There is a unique ϕ_(c(z), s(z)) which sends z=(z_1,z_2) ↦ (0,iϵ).


It can be verified that the choice of

    c(z) = -z_1,      s(z) = -(z_2) - 2β(z_1^2)

sends z ↦ (0,iϵ).  Uniqueness follows from the form of (<ref>).
















It was mentioned in Remark <ref> that the maps ϕ_(c,s) preserve both volume and the boundary measure σ = dx_1∧ dy_1 ∧ dx_2.  The Leray kernel ℒ_ is defined in equation (<ref>) as an (n,n-1)-form, but when =_β we may think of as ℒ_β(z,ζ) = ℓ_β(z,ζ)σ(ζ), where ℓ_β is a function (i.e. a (0,0)-form) times the measure σ.  This coefficient function satisfies the following invariance property:


Let ℓ_β denote the coefficient function of ℒ_β written with respect to the measure σ.  Fix a point z ∈Ω_β, choose the unique ϵ such that z ∈_β^ ϵ, and let ϕ^* = ϕ_(c(z),s(z)) denote the map in Corollary <ref> sending z ↦ (0,iϵ).  Then

    ℓ_β(ϕ^*(z),ϕ^*(ζ)) = ℓ_β(z,ζ).



Following the parametrization of _β with the automorphisms ϕ_(r,s) in Section <ref>, write the points z ∈_β^ ϵ,  ζ∈_β as

    z    = (r_z +it_z,   s_z - 2(1+β)r_z t_z + i[(1+β)r_z^2 +(1-β)t_z^2 + ϵ]), 
    ζ   = (r_ζ +it_ζ,   s_ζ - 2(1+β)r_ζ t_ζ + i[(1+β)r_ζ^2 +(1-β)t_ζ^2]).

Starting from the definition of L_βf(z) in equation (<ref>), we first note that 

    dζ_2 ∧ dζ̅_1 ∧ dζ_1    = 2i   ds_ζ∧ dr_ζ∧ dt_ζ
       = 2i σ(ζ).


Repeating the steps from (<ref>) through (<ref>) – with the only difference arising from the fact that now z∈_β^ ϵ – we find that L_βf(z) = ∫__β f(ζ)ℓ_β(z,ζ)σ(ζ),
where 

    ℓ_β(z,ζ)    = π^-2[((1+β)(r_z-r_ζ)^2+ (1-β)(t_z-t_ζ)^2+ϵ) 
                       + i(2(r_z-r_ζ)(t_z+t_ζ+β(t_z-t_ζ)) - (s_z-s_ζ)) ]^-2.


Written with respect to this parametrization, the subscripts of the map ϕ^* =ϕ_(c(z),s(z)) defined in (<ref>) take the form

    c(z) = -r_z-it_z,      s(z) = -s_z+2(1-β)r_zt_z.

By construction, ϕ^*(z) = (0,iϵ).  This is equivalent to saying ϕ^* maps 

    r_z ↦ 0,      t_z ↦ 0,      s_z ↦ 0.

Calculating ϕ^*(ζ) is more involved, but starting from (<ref>) and (<ref>) it is seen that

    ϕ^*(ζ)    = ((r_ζ-r_z)+i(t_ζ-t_z)  ,  s_ζ-s_z + 2(r_z-r_ζ)((1-β)t_z + (1+β)t_ζ) 
                                +i((1+β)(r_z-r_ζ)^2 + (1-β)(t_z-t_ζ)^2) ).

This is equivalent to saying that ϕ^* maps

    r_ζ↦ r_ζ-r_z,      t_ζ↦ t_ζ-t_z,     s_ζ↦ s_ζ-s_z + 4t_z(r_z-r_ζ).


Substituting (<ref>) and (<ref>) into (<ref>) shows

    ℓ_β(ϕ^*(z),ϕ^*(ζ))    = π^-2[((1+β)(r_z-r_ζ)^2+ (1-β)(t_z-t_ζ)^2+ϵ) 
                + i(2(β-1)(r_z-r_ζ)(t_z-t_ζ) + s_ζ-s_z +4t_z(r_z-r_ζ)) ]^-2
       = π^-2[((1+β)(r_z-r_ζ)^2+ (1-β)(t_z-t_ζ)^2+ϵ) 
                       + i(2(r_z-r_ζ)(t_z+t_ζ+β(t_z-t_ζ)) - (s_z-s_ζ)) ]^-2.

This last line equals ℓ_β(z,ζ).


Theorem <ref> shows that for fixed ϵ > 0, the L^2(_β,σ) norm of ℓ(z,·) remains constant as z varies in _β^ ϵ.  Indeed, the map ϕ^* = ϕ_(c(z),s(z)) maps the point z to (0,iϵ).  Because the maps ϕ_(c,s) – and their inverses, which have the same form – preserve the measure σ,

    ∫__β |ℓ_β(z,ζ)|^2 σ(ζ)    = ∫__β|ℓ_β(ϕ^*(z),ϕ^*(ζ))|^2 σ(ζ) 
       = ∫__β|ℓ_β((0,iϵ),ζ)|^2 σ(ζ).



It can be shown that σ is a constant multiple of Fefferman hypersurface measure (see <cit.>) on _β.  Theorem <ref> can be deduced from the general transformation law given in <cit.> applying to ℒ_ written with respect to Fefferman measure.  ◊




For α>0, consider the non-isotropic dilation map

    δ_α(z_1,z_2) = (√(α) z_1,α z_2).

It can be checked that these maps are automorphisms of both _β and Ω_β.  More generally, δ_α is a bijection from _β^ϵ→_β^αϵ.  These maps no longer preserve volume or the surface measure σ, but the transformation law in <cit.> still applies to action of δ_α. ◊



























 §.§ Each f∈ L^2(_β,σ) generates a holomorphic function L_βf on Ω_β


Building on the identity (<ref>), we state the following proposition.


The integral

    ∫__β|ℓ_β((0,iϵ),ζ)|^2 σ(ζ) = 1/4π^2ϵ^2√(1-β^2).



The proof of Proposition <ref> is split into the following two computational lemmas.


For each ϵ>0,

    ∫__β|ℓ_β((0,iϵ),ζ)|^2 σ(ζ) = 1/ϵ^2∫__β|ℓ_β((0,i),ζ)|^2 σ(ζ).



From (<ref>), ℓ_β((0,iϵ),ζ) may be written by setting r_z=t_z=s_z=0.


    ∫__β|ℓ_β((0,iϵ),ζ)|^2 σ(ζ) 
       = 1/π^4∫_^3ds_ζ∧ dr_ζ∧ dt_ζ/|(1+β) r_ζ^2 + (1-β)t_ζ^2 + ϵ + i(s_ζ-2(1-β)r_ζt_ζ)|^4
       = 1/ϵ^2π^4∫_^3ds_ζ∧ dr_ζ∧ dt_ζ/|(1+β) r_ζ^2 + (1-β)t_ζ^2 + 1 + i(s_ζ-2(1-β)r_ζt_ζ)|^4
       = 1/ϵ^2∫__β|ℓ_β((0,i),ζ)|^2 σ(ζ).



The change of variable (r_ζ,s_ζ,t_ζ) ↦ (√(ϵ) r_ζ, ϵ s_ζ,√(ϵ) t_ζ ) used in (<ref>) is the non-isotropic dilation δ_ϵ in (<ref>).  It is also possible to use the transformation law alluded to in Remark <ref> to immediately deduce Lemma <ref>. ◊



The integral

    ∫__β|ℓ_β((0,i),ζ)|^2 σ(ζ) = 1/4π^2√(1-β^2).



Setting a(r_ζ,t_ζ)=(1+β)r_ζ^2 + (1-β)t_ζ^2 + 1, equation (<ref>) shows that

    π^4 ∫__β|ℓ_β((0,i),ζ)|^2 σ(ζ)    = ∫_^2[∫_(a(r_ζ,t_ζ)^2 + (s_ζ-2(1-β)r_ζt_ζ)^2 )^-2ds_ζ] dr_ζ∧ dt_ζ
       := ∫_^2 I(r_ζ,t_ζ)  dr_ζ∧ dt_ζ.
  
A computation shows that for each r_ζ,t_ζ∈, the quantity in brackets

    I(r_ζ,t_ζ)    =  π/2((1+β) r_ζ^2 + (1-β)t_ζ^2 + 1)^3.

Making the change of variable (r_ζ,t_ζ) ↦(r_ζ/√(1+β), t_ζ/√(1-β)), we see that 

    (<ref>)   = π/2√(1-β^2)∫_^2dr_ζ∧ dt_ζ/(r_ζ^2+s_ζ^2 +1)^3 = π^2/4√(1-β^2).

Dividing by π^4 gives the result.




The function Ω_β→ given by z↦∫__β |ℓ_β(z,ζ)|^2 σ(ζ) is uniformly bounded on compact subsets of Ω_β.



Every compact subset of Ω_β is contained in a union of shells ∪_ϵ > ϵ_0 _β^ ϵ with ϵ_0>0.  The desired conclusion then follows from (<ref>) and Proposition <ref>.




We are ready to prove the main result of Appendix <ref>.  This will verify item (a) from the list in Section <ref>.

 L_βf ∈(Ω_β) for each f ∈ L^2(_β,σ).



It will suffice to prove that L_βf is holomorphic on U for  each  relatively compact ball U⊂Ω_β.

In the special case of compactly-supported f this follows from a standard differentiate-the-integral argument.  

To handle general f we pick a sequence of compactly supported f_j∈ L^2(_β,σ) with 
f_j→ f in L^2.  Then with the use of Corollary <ref> we find that
L_βf_j→L_βf uniformly on U and thus that L_βf is indeed holomorphic on U.





























§ A CLOSER LOOK AT L_Β


In this section, we will take a closer look at families functions for which the norm is achieved as a supremum over the family.  In the previous section we took two Fourier transforms, but here we will pick up from our work after the first transform with equation (<ref>) and the definition of A in (<ref>):


    _s^-1L_β(f)(r_z,ξ_s,t_z) = 
    		0    ξ_s ≥ 0
    
    
    		-4 ξ_s ∫_^2_s^-1 f(r_ζ,ξ_s,t_ζ)  e^4πξ_s A  dr_ζ dt_ζ   ξ_s < 0,

where

    A    = 1+β/2(r_z-r_ζ)^2 + 1-β/2(t_z-t_ζ)^2 + i(r_z-r_ζ)(t_z+t_ζ + β(t_z-t_ζ)) 
       = 1+β/2(r_z-r_ζ)^2 + 1-β/2(t_z-t_ζ)^2 
                   + i((1+β)r_zt_z-(1+β)r_ζt_z+(1-β)r_zt_ζ-(1-β)r_ζt_ζ).




 §.§ Functions obtaining the norm


Consider the family of functions 

    _β := {ϕ(ξ_s) e^-(1+β)r_ζ^2-(1-β)t_ζ^2 +4π i ξ_s(1-β)r_ζt_ζ : ϕ∈ L^2() ∩supp(-∞,0) }
 
The oscillatory part in the exponential is included to exploit a convenient cancellation below.  It is easy to check that _β⊂ L^2(^3).  Indeed for h(r_ζ,ξ_s,t_ζ) ∈_β,

    ∫_^3|h(r_ζ,ξ_s,t_ζ)|^2 dV = π/2√(1-β^2)∫_-∞^0|ϕ(ξ)|^2 dξ< ∞.
  

Since the partial (one-variable) Fourier transform is an isomorphism of L^2(), we conclude that for each h∈_β, there is some f∈ L^2(^3) such that ^-1_sf = h.



    ^-1_sL_β(f)(r_z,ξ_s,t_z)    = -4 ξ_s ∫_^2^-1_3f(r_ζ,ξ_s,t_ζ)   e^4πξ_s A  dr_ζ dt_ζ
       = -4ξ_s e^4π i ξ_s (1+β)c_z t_z∫_^2 h(r_ζ,ξ_s,t_ζ) e^2πξ_s[(1+β)(r_z-r_ζ)^2 + (1-β)(t_z-t_ζ)^2]
                           · e^4π i ξ_s[-(1+β)r_ζt_z+(1-β)r_zt_ζ-(1-β)r_ζt_ζ] dr_ζ dt_ζ.

Substituting the general form for such an h given by (<ref>),


    (<ref>)   = -4 ξ_s  ϕ(ξ_s) e^4π i ξ_s (1+β)r_z t_z∫_^2 e^-(1+β)r_ζ^2-(1-β)t_ζ^2· e^2 πξ_s[(1+β)(r_z-r_ζ)^2 + (1-β)(t_z-t_ζ)^2]
                                           · e^4π i ξ_s[(1-β)r_zt_ζ -(1+β)r_ζt_z] dr_ζ dt_ζ
       =  -4πξ_sϕ(ξ_s) e^4π i ξ_s (1+β)r_z t_z/(1-2πξ_s)√(1-β^2)· e^2πξ_s[8i π r_z t_zβξ_s + r_z^2((1+β)-2πξ_s(1-β))+t_z^2((1-β)-2πξ_s(1+β))]/1-2πξ_s,

where the computation in the last line was carried out with the assistance of Mathematica.  Using Mathematica again, we calculate


    ^-1_sL(f)_2^2    = ∫_-∞^0 m_β(ξ_s)|ϕ(ξ_s)|^2   dξ_s,

where

    m_β(ξ_s) = -4π^2ξ_s/(1-β^2)(1-2πξ_s)√(((1+β)-2πξ_s(1-β))((1-β)-2πξ_s(1+β))).


It can be shown that m_β(ξ_s) achieves its maximum value over the interval (-∞,0) of π/2-2β^2 when ξ_s = -1/2π.  Thus for any ϵ > 0, choose ϕ(ξ_s) supported in a small enough neighborhood of ξ_s = -1/2π so that

    L(f)_2^2 = ^-1_s(L(f))_2^2    ≥(π/2-2β^2-ϵ)∫_-∞^0|ϕ(ξ_s)|^2 
       =1/√(1-β^2)h_2^2 - 2ϵ√(1-β^2)/πh_2^2 
       =1/√(1-β^2)f_2^2 - 2ϵ√(1-β^2)/πf_2^2,

where we made use of (<ref>) in the second line.  Now divide by f_2^2, take a square root and send ϵ→ 0 to establish the lower bound.



 §.§ Undoing the inverse Fourier Transform


We work with a concrete family of functions contained in _β.  Define

    h_ϵ(r_ζ, ξ_s, t_ζ) = χ_I_ϵ(ξ_s) e^-(1+β)r_ζ^2-(1-β)t_ζ^2 +4π i ξ_s(1-β)r_ζt_ζ,
 
where χ_I_ϵ(ξ_s) is the indicator function of the interval I_ϵ = [-1/2π-ϵ,-1/2π+ϵ].  Now we wish to compute the Fourier transform of h_ϵ.

    f(r_ζ, s_ζ, t_ζ)    = _s h_ϵ(r_ζ, s_ζ, t_ζ) 
       =e^-(1+β)r_ζ^2-(1-β)t_ζ^2∫_I_ϵ e^[4π i (1-β)r_ζt_ζ - 2π i s_ζ]ξ_s  dξ_s 
       = 2ϵ  e^-(1+β)r_ζ^2-(1-β)t_ζ^2· e^i[s_ζ - 2(1-β)r_ζt_ζ] ·Sinc[(4π(1-β)c_ζ t_ζ - 2π s_ζ)ϵ].





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