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THE MARTINGALE STRUCTURE OF THE BEURLING-AHLFORS TRANSFORM

˜uelos∗ Rodrigo Ban Arthur Lindeman II∗∗ Purdue University West Lafayette, Indiana Abstract. The Beurling-Ahlfors operator reveals a rich structure through its representation as a martingale transform. Using elementary linear algebra and martingale inequalities, we obtain new information on this operator. In particular, Ess´en-type inequalities are proved for the complex Beurling-Ahlfors operator and its generalization to higher dimensions. Moreover, a new estimate of their norms is obtained for dimensions n ≥ 3. Finally, we discuss a purely analytic approach to further investigate these norms that is suggested by the probabilistic method presented here.

§0. Introduction. The complex Beurling-Ahlfors transform and its generalization to higher dimensions have been used to study properties of quasiregular mappings for several years. It is of particular interest to identify the Lp -norms of these operators. These norms are directly connected to the regularity of quasiregular mappings, as well as conditions for a closed set to be removable under such maps. Identification of the norms would also have implications for the existence of minimizers of conformally invariant energy functionals and regularity of solutions to the generalized Beltrami system. See [IM1], [IM2], [IMNS] and [IL] for further discussion of these operators and their relationship to quasiregular mappings. The Beurling-Ahlfors operator in several dimensions was introduced by Donaldson and Sullivan as the “signature operator” in their paper Quasiconformal 4manifolds [DS]. The first systematic study of this operator as a Calder´ on-Zygmund singular integral in higher dimensions was made by Iwaniec and Martin in [IM1]. There it is observed that the operator possesses a rich invariance structure. The probabilistic study of the Beurling-Ahlfors transform was initiated in [BW] and [L]. Using some new sharp martingale inequalities inspired by the recent work of ∗,∗∗ Research

of both authors supported in part by the NSF under grant DMS9400854 Typeset by AMS-TEX 1

D. Burkholder [Bu1], [Bu2] on differential subordination of martingales, the Lp estimates of Iwaniec and Martin in [IM2] are substantially improved, giving the best known estimates. The results in [BW] and [L] reveal that the Beurling-Ahlfors transform also exhibits several interesting properties when viewed as a martingale transform, but this was not really explored there. The purpose of this paper is to present a deeper study of the martingale structure of the operators and from this improve the estimates obtained in [BW] and [L]. The paper is organized as follows. In §1, we recall the basic definitions and state the main results of the paper. In §2, we present the proofs. The martingale study of the Beurling-Ahlfors transform and the results obtained by these methods suggest a possible way to proceed analytically in the investigation of these Calder´ on-Zygmund singular integral operators. These questions and problems are given in §3. We find Question 1 particularly appealing given the absence, to the best of our knowledge, of any analytic techniques available to compute the norms of singular integrals of even kernel such as the Beurling-Ahlfors transform. The paper concludes with a conjecture. §1. Preliminaries and Statements of Results. First, we briefly recall some of the definitions and fix notation that will be used throughout the paper. The complex Beurling-Ahlfors transform acting on Lp (C), 1 < p < ∞, is the singular integral defined by

ZZ Bf (z) =

1 − 2πi

C

f (ζ) dζ ∧ dζ . (z − ζ)2

The Fourier multiplier of B is given by − ξξ , making it an isometry on L2 (C). The operator has the basic property that (1.1)

B◦

∂ ∂ = , ∂z ∂z

and if we denote the Laplacian by ∆ , then (1.2)

B=4

∂ 2 −1 ∆ , ∂z 2

where ∆ −1 is defined as a Green’s potential. 2

The Riesz transforms Rj , j = 1, 2, . . . , n acting on Lp (Rn ), 1 < p < ∞, are the singular integrals defined by Γ( n+1 2 ) Rj f (x) = (n+1)/2 π

Z Rn

xj − yj f (y) dy, |x − y|n+1

x = (x1 , . . . , xn ). The Fourier multiplier of Rj is given by

iξj |ξ|

which, along with

the usual identification of R2 and C, immediately shows B = R12 − R22 − 2iR1 R2 .

(1.3)

The Beurling-Ahlfors transform on Rn , n ≥ 3, acts on differential forms. For 0 ≤ k ≤ n, let I

n,k

n o {1,...,n} = I∈2 : I has k distinct elements

be the set of k-indices and p

n

k

L (R , Λ ) =

½ X

¾ αI (x) dx : αI (x) ∈ L (R ) , I

p

n

I∈I n,k

the class of complex-valued k-forms having Lp coefficients. Fix an order on I n,k ¡ ¢ and set ν = nk , its cardinality. We identify α ∈ Lp (Rn , Λk ) with the vector (αI1 , . . . , αIν )t and define the semi-norm ÃZ kαkLp (Rn ,Λk ) =

Rn

µX ν

!1/p

¶p/2 |αIi (x)|2

dx

.

i=1

The restriction of the Beurling-Ahlfors transform to k-forms is defined by (1.4)

Sk = (dδ − δd) ◦ ∆ −1 ,

where d is the exterior derivative taking k-forms to k + 1-forms and δ is its adjoint operator taking k + 1-forms to k-forms. As in (1.2), ∆−1 acts on the coefficients of the forms as a Green’s potential. Using the above identification of k-forms with vectors, the analogue of (1.3) is that Sk can be represented as the following ν × ν matrix of second-order Riesz transforms acting on the vector (αI1 , . . . , αIν )t : P P 2 2  / Rj , if I = J  i∈I Ri − j ∈I [Sk ]I,J = 2Ri Rj , if I \ J = {i} and J \ I = {j}   0, otherwise. 3

(See [L] or [IM1] for more details.) The full operator, S, is then defined as S = S0 ⊗ · · · ⊗ Sn and, of course, kSkLp (Rn ,Λ) = max kSk kLp (Rn ,Λk ) . 0≤k≤n

We now review the needed probabilistic tools. All martingales considered are with respect to the d-dimensional background radiation process (Wt )−∞