heating of the ahlfors-beurling operator: weakly ... - Project Euclid

i

i 2002/3/18

page 281 i

i

HEATING OF THE AHLFORS-BEURLING OPERATOR: WEAKLY QUASIREGULAR MAPS ON THE PLANE ARE QUASIREGULAR STEFANIE PETERMICHL and ALEXANDER VOLBERG

Abstract We establish borderline regularity for solutions of the Beltrami equation f z −µf z¯ = 0 on the plane, where µ is a bounded measurable function, kµk∞ = k < 1. What is the 1,q minimal requirement of the type f ∈ Wloc which guarantees that any solution of the Beltrami equation with any kµk∞ = k < 1 is a continuous function? A deep result of 1,1+k+ε K. Astala says that f ∈ Wloc suffices if ε > 0. On the other hand, O. Lehto and T. Iwaniec showed that q < 1 + k is not sufficient. In [2], the following question was asked: What happens for the borderline case q = 1 + k? We show that the solution is still always continuous and thus is a quasiregular map. Our method of proof is based on a sharp weighted estimate of the Ahlfors-Beurling operator. This estimate is based on a sharp weighted estimate of a certain dyadic singular integral operator and on using the heat extension of the Bellman function for the problem. The sharp weighted estimate of the dyadic operator is obtained by combining J. Garcia-Cuerva and J. Rubio de Francia’s extrapolation technique and two-weight estimates for the martingale transform from [26]. Contents Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 0. Introduction: Main objects and results . . . . . . . . . . . . . . . . . . 282 1. The sharp weighted estimate for the Ahlfors-Beurling operator in L 2 (w d A) 287 2. The sharp weighted estimate for the Ahlfors-Beurling operator in L p (w d A) for p > 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 3. The comparison of classical and heat A p -characteristics . . . . . . . . . 300 4. Injectivity at the critical exponent and regularity of solutions of the Beltrami equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 2, Received 8 December 2000. Revision received 2 May 2001. 2000 Mathematics Subject Classification. Primary 42B20, 42C15, 42A50, 47B35, 47B38. Volberg’s work partially supported by National Science Foundation grant number DMS-9900375 and by United States–Israel Binational Science Foundation grant number 00030. 281

i

i i

i

i

i 2002/3/18

page 282 i

i

282

PETERMICHL and VOLBERG

Notation We have the following notation: x = (x 1 , x2 ), |x| = (x 12 + x22 )1/2 ; B(x0 , r ) is the open ball, where B(x 0 , r ) := {x ∈ R2 : |x − x 0 | < r }; h f i B is the average of the function f over the ball B; x = (x 1 , x2 ), k(x, t) := (1/πt) exp(−(|x|2 /t)) is the heat kernel on the plane; RR RR −1 w(x − y)k(y, t) dy1 y2 · Q heat w (x − y)k(y, t) dy1 y2 is the w,2 = sup x∈R2 ,t>0 heat A2 characteristic of weight w (see Section 0); RR w(x − y)k(y, t) dy1 y2 = supx∈R2 ,t>0 Q heat w, RRp −(1/( p−1)) ·( w (x − y)k(y, t) dy1 y2 ) p−1 is the heat A p -characteristic of weight w (see Section 0); −1 Q class w,2 = sup B(x,r ) hwi B(x,r ) hw i B(x,r ) is the standard A 2 -characteristic of weight w (see Section 0); −(1/( p−1)) i p−1 is the standard A -characteristic Q class p B(x,r ) ) w, p = sup B(x,r ) hwi B(x,r ) (hw of weight w (see Section 0); dyadic

Q w,2 = sup I,I ∈D hwi I hw −1 i I is the dyadic A2 -characteristic of weight w (see Section 1); dyadic

Q w, p = sup I,I ∈D hwi I (hw −(1/( p−1)) i I ) p−1 is the dyadic A p -characteristic of weight w (see Section 1); D is a collection of dyadic intervals (see Section 1).

0. Introduction: Main objects and results Our goal is to present a sharp estimate of the weighted Ahlfors-Beurling operator. This estimate is sufficient to prove that any weakly quasiregular map is quasiregular. Thus, we are interested in the following Ahlfors-Beurling operator (d A denotes area Lebesgue measure on C): Z Z 1 ϕ(ζ ) d A(ζ ) T ϕ(z) := π (ζ − z)2 understood as a Calder´on-Zygmund operator. Our starting point is the following theorem of Astala, Iwaniec, and E. Saksman [2]. THEOREM

0.1

L ∞ , kµk

Let µ ∈ ∞ = k < 1. Then the operators I − µT and I − T µ are invertible in L p (C, d A) for p ∈ (1 + k, 1 + (1/k)).

i

i i

i

i

i 2002/3/18

page 283 i

i

HEATING OF THE AHLFORS-BEURLING OPERATOR

283

Remark. In [2], the authors suggest the following problem: Prove that these operators have dense range for p = 1+(1/k) and are injective for p = 1+k. Actually, this problem has a long history and has reappeared in many papers on regularity of quasiconformal homeomorphisms and quasiregular maps. The L p -theory of quasiregular mappings was essentially formulated by B. Bojarski [6] and [7]. Later this subject came under intensive investigation. In particular, the best integrability of K -quasiconformal mappings and the (in a sense dual; see [25]) problem of minimal regularity of the quasiregular mappings are discussed in [16], [13] – [15], [17], [20], [21], [22], [24], and [25]. The best integrability result was finally established in [1]. Notice that the p − 1 problem for the operator T is still open. We recall that this problem consists of proving that kT k L p →L p = p − 1, p > 2. A discussion of the fantastically beautiful connections of this problem with the calculus of variations and C. Morrey’s problem can be found in [3]. Until recently the best result for all p appeared in [5], where probabilistic methods were used to obtain the estimate kT k L p →L p ≤ 4( p − 1), p > 2. Discussion of related results can be found in [4] and [30]. Recently, in [27] this estimate was slightly improved to kT k L p →L p ≤ 2( p − 1), p > 2. The main result of this paper may serve as yet another indication that this norm is p − 1. We establish the dense range/injectivity property in the present paper. Our principal tool is a sharp weighted estimate of the Ahlfors-Beurling operator. The main consequence (see [2]) is the above-mentioned geometric fact: Every weakly quasiregular map is quasiregular. We first quote several results and notions that we use. The symbol f = f µ represents the homeomorphic solution of the Beltrami equation f z¯ − µf z = 0,

f (z) = z + o(1),

z → ∞.

Here we consider compactly supported µ, µ ∈ L ∞ (C), kµk∞ = k < 1. The function µ is called the Beltrami coefficient of the Beltrami equation written above. The constant K = (1 + k)/(1 − k) has the geometric meaning of the bound on the distortion of infinitely small discs by the map f . (The resulting infinitely small ellipses have the ratio of their axes at most K .) The map f is called K -quasiconformal. The image of a disc by a K -quasiconformal homeomorphism is called a K -quasidisc. We need the following results from Astala, Iwaniec, and Saksman [2]. THEOREM 0.2 Let f be a K -quasiconformal homeomorphism built by a certain µ as it is described above. Consider ω = | f z ◦ f −1 | p−2 for p ∈ (1 + k, 1 + (1/k)). Then k(I − µT )−1 k L p →L p , k(I − T µ)−1 k L p →L p are bounded by C(k)λω, p , where λω, p := kT k L p (ω d A)→L p (ω d A) .

i

i i

i

i

i 2002/3/18

page 284 i

i

284

PETERMICHL and VOLBERG

THEOREM 0.3 Let f be a K -quasiconformal homeomorphism. Let k = (K + 1)/(K − 1), and let p ∈ (0, 1 + (1/k)). Then for any disc or K -quasidisc B, one has Z  | f (B)|  p/2
p C(k) 1 | f z | + | f z¯ | d A ≤ . |B| B 1 + (1/k) − p |B|

R Remark. Obviously, we have the same type of estimate for (1/|B|) B | f z | p d A and R p/2 for (1/|B|) B J f d A, where J f := | f z |2 − | f z¯ |2 is the Jacobian of the mapping f . Next, a very elegant result from [2] reduces the critical exponent cases p = 1 + (1/k), p = 1 + k to a weighted estimate of the Ahlfors-Beurling operator. This result is quoted for the convenience of the reader. 0.4 Let k ∈ [0, 1), kµk∞ = k, f = f µ , ω = | f z ◦ f −1 | p−2 , p ∈ [2, 1 + (1/k)). If THEOREM

kT k L p (ω d A)→L p (ω d A) ≤

C , 1 + a(1/k) − p

(0.1)

then I − µT, I − T µ have dense ranges in L 1+1/k (C) and are injective in L 1+k (C). Our main result implies the weighted estimate (0.1). Theorem 0.4 served as motivation for our main result. In its turn, injectivity at the critical exponent 1 + k proves that weakly quasiregular maps are quasiregular (see [2]). Proof The operator T is invertible in all L q , q ∈ (1, ∞). We notice that I − T µ = T (I − µT )T −1 . Thus, it is enough to prove that I − µT has closed range in L 1+(1/k) (C). Let p0 := 1 + (1/k). Take any ϕ ∈ C 0∞ (C). By Theorem 0.1, ϕε := I − (1 − ε)µT By Theorem 0.2,


−1

ϕ ∈ L p0 (C).

kϕε k L p0 ≤ C(k)kT k L p0 (ωε d A)→L p0 (ωε d A) , where ωε = | f z ◦ f −1 | p−2 , and f denotes (as always) a homeomorphic solution of the Beltrami equation, but for Beltrami coefficient (1 − ε)µ instead of µ. Now assuming estimate (0.1), we conclude that kϕε k L p0 ≤

C C ≤ . 1 + (1/(1 − ε)k) − p0 ε

i

i i

i

i

i 2002/3/18

page 285 i

i

HEATING OF THE AHLFORS-BEURLING OPERATOR

285

Thus, kεϕε k L p0 ≤ C1 < ∞,

ε → 0.

(0.2)

On the other hand, the definition of ϕε implies that (I − µT )ϕε = ϕ − εµϕε

(0.3)

and kϕε k L 2 ≤ C(k) < ∞,

∀ε ∈ [0, 1].

(0.4)

Now (0.2) and (0.4) imply that εϕε converge weakly to zero in L p0 . Keeping this in mind, we see that (0.3) implies that the range of I − µT is weakly dense in L p0 . So the range of I − µT is automatically dense in L p0 in the norm topology. We are now ready to formulate one of our main results. THEOREM 0.5 Let k ∈ [0, 1), kµk∞ = k, f = f µ , ω = | f z ◦ f −1 | p−2 , p ∈ [2, 1 + (1/k)). Then

kT k L p (ω d A)→L p (ω d A) ≤

C . 1 + (1/k) − p

This result is obtained as a corollary of the following theorems, which may be of interest in their own right. Let ω be any weight on R2 , and denote its heat extension into R3+ by ω(x, t) = ω(x 1 , x2 , t): Z Z  kx − yk2  1 ω(x, t) = dy1 dy2 . ω(y) exp − πt t R2 We define Q heat ω, p :=

sup

ω(x, t) ω−(1/( p−1)) (x, t)

(x,t)∈R3+


p−1

.

The weights with finite Q heat ω, p are called A p -weights. There is an extensive theory of A p -weights (see, e.g., [29], [12]). The usual definition differs from the one above, but it describes the same class of weights. Actually, we say more about the relationship between the classical definition and ours. But first we state two more theorems, whose combined use gives Theorem 0.5. THEOREM 0.6 For any A p -weight w and any p ≥ 2, we have

kT k L p (w d A)→L p (w d A) ≤ C( p)Q heat ω, p .

i

i i

i

i

i 2002/3/18

page 286 i

i

286

PETERMICHL and VOLBERG

THEOREM 0.7 Let k ∈ [0, 1), kµk∞ = k, f = f µ , ω = | f z ◦ f −1 | p−2 , p ∈ [2, 1 + (1/k)). Then the weight ω belongs to A p and

Q heat ω, p ≤

C . 1 + (1/k) − p

Clearly, these last two results together imply Theorem 0.5. And Theorem 0.5 and Theorem 0.4 give positive answers to the Beltrami equation critical exponent questions discussed above and raised in [2]. At the end of our introduction, we discuss the connection between Q heat w, p and class denotes the following supremum over all discs in the plane: Q w, p . Here Q class w, p Z Z     p−1 1 1 Q class := sup ω d A · ω−(1/( p−1)) d A . w, p |B(x, R)| B(x,R) B(x,R) |B(x, R)| B(x,R) Obviously, there exists a positive absolute constant a such that for any function w, heat a Q class w, p ≤ Q w, p .

Remark. The opposite inequality is easy to prove, too. We are grateful to F. Nazarov for indicating this to us. Originally, we did not have the following two theorems. This was not an obstacle to Theorem 0.5 because we still could estimate (see Theorem 0.7) Q heat ω, p for special weights ω. THEOREM 0.8 There exists a finite absolute constant b such that class Q heat w, p ≤ bQ w, p .

In conjunction with our main Theorem 0.5, this gives us the following weighted estimate of the Ahlfors-Beurling operator, which answers another question of [2] positively. THEOREM 0.9 For any A p -weight w and any p ≥ 2, we have

kT k L p (w d A)→L p (w d A) ≤ C( p)Q class ω, p . In [2], it is proved that if k ∈ [0, 1), kµk∞ = k, f = f µ , ω = | f z ◦ f −1 | p−2 , and p ∈ [2, 1 + (1/k)), then the weight ω belongs to A p and Q class ω, p ≤

C . 1 + (1/k) − p

(0.5)

i

i i

i

i

i 2002/3/18

page 287 i

i

HEATING OF THE AHLFORS-BEURLING OPERATOR

287

So instead of using Theorem 0.7, one can use Theorem 0.9 in conjunction with (0.5). As a result, for the special weight function ω = | f z ◦ f −1 | p−2 , where p ∈ [2, 1 + (1/k)), f := f µ , kµk∞ = k, and k ∈ [0, 1), we always get Q heat ω, p ≤

C . 1 + (1/k) − p

(0.6)

Theorem 0.6 then proves our main Theorem 0.5. Plan of the paper In Section 1, we prove Theorem 0.6 for p = 2. In Section 2, we prove Theorem 0.6 for p > 2. In Section 3, we give an easy proof of Theorem 0.8. In Section 4, we repeat (for the convenience of the reader) the reasoning of [2] that deduces the quasiregularity of weakly quasiregular maps from the injectivity of I −µT at the critical exponent. 1. The sharp weighted estimate for the Ahlfors-Beurling operator in L 2 (w d A) In this section, w is an arbitrary positive function on the plane. We want the estimate kT k L 2 (w d A)→L 2 (w d A) ≤ C Q heat w,2 .

(1.1)

As we see in Section 3, estimate (1.1) readily implies the estimate kT k L 2 (w d A)→L 2 (w d A) ≤ C Q class w,2 .

(1.2)

However, at this moment we do not know how to get the latter estimate without using the first. So now our immediate goal is to obtain (1.1). The operator T is given in the Fourier domain (ξ1 , ξ2 ) by the multiplier ξ12 ξ22 ζ¯ 2 (ξ1 − iξ2 )2 ξ1 ξ2 ζ¯ = = = − − 2i 2 . 2 2 2 2 2 2 2 ζ |ζ | ξ1 + ξ 2 ξ1 + ξ 2 ξ1 + ξ 2 ξ1 + ξ22 Thus, T can be written as T = R12 − R22 −2i R1 R2 , where R1 , R2 are Riesz transforms on the plane (see [29] for their definition and properties). Another way of writing T is T = m 1 − im 2 , where m 1 , m 2 are Fourier multiplier operators. Notice that the multipliers themselves (as functions, not as multiplier operators) are connected by m 2 = m 1 ◦ ρ, where ρ is a π/4 rotation of the plane. So the multiplier operators are related by m 2 = Uρ m 1 Uρ−1 ,

i

i i

i

i

i 2002/3/18

page 288 i

i

288

PETERMICHL and VOLBERG

where Uρ is an operator of ρ-rotation in the (x 1 , x2 )-plane. But for any operator K , we have kUρ K Uρ−1 k L 2 (w d A)→L 2 (w d A) = kK k L 2 (w◦ρ −1 d A)→L 2 (w◦ρ −1 d A) . heat Combining this with the fact that Q heat w,2 = Q w◦ρ −1 ,2 for any rotation, we conclude

that we need the desired estimate (1.1) only for m 1 = R12 − R22 . Actually, we show that kRi2 k L 2 (w d A)→L 2 (w d A) ≤ C Q heat i = 1, 2. (1.3) w,2 , To prove (1.3) we fix, say, R12 and two test functions ϕ, ψ ∈ C 0∞ . We use heat extensions. For f on the plane, its heat extension is given by the formula f (y, t) :=

1 πt

ZZ

R2

 |x − y|2  f (x) exp − d x1 d x2 , t

(y, t) ∈ R3+ .

We usually use the same letter to denote a function and its heat extension. LEMMA 1.1 RRR Let ϕ, ψ ∈ C0∞ . Then the integral

ZZ

R12 ϕ

∂ϕ ∂ x1

· ψ d x 1 d x2 = −2

·

∂ψ ∂ x1

ZZZ

d x1 d x2 dt converges absolutely and ∂ϕ ∂ψ · d x1 d x2 dt. ∂ x1 ∂ x1

(1.4)

Proof The proof of this lemma is actually trivial. It is based on the well-known fact that a function is an integral of its derivative, and it also involves Parseval’s formula. Consider ϕ, ψ ∈ C 0∞ and now ZZ ψ R12 ϕ d x1 d x2 ZZ

ξ12

ˆ ϕ(ξ ˆ 1 , ξ2 )ψ(−ξ 1 , −ξ2 ) dξ1 dξ2 ξ12 + ξ22 ZZ Z ∞ 2 2 ˆ 1 , ξ2 ) dξ1 dξ2 dt =2 e−2t (ξ1 +ξ2 ) ξ12 ϕ(ξ ˆ 1 , ξ2 )ψ(ξ 0 Z ∞ ZZ 2 2 −t (ξ12 +ξ22 ) ˆ = −2 iξ1 ϕ(ξ ˆ 1 , ξ2 )e−t (ξ1 +ξ2 ) iξ1 ψ(−ξ dξ1 dξ2 dt 1 , −ξ2 )e 0 Z ∞ ZZ ∂ψ ∂ϕ = −2 (x1 , x2 , t) (x1 , x2 , t) d x 1 d x2 dt ∂ x1 ∂ x1 0 ZZZ ∂ϕ ∂ψ = −2 (x1 , x2 , t) (x1 , x2 , t) d x 1 d x2 dt. ∂ x1 R3+ ∂ x 1 =

i

i i

i

i

i 2002/3/18

page 289 i

i

HEATING OF THE AHLFORS-BEURLING OPERATOR

289

We used Parseval’s formula twice above, and we also used the absolute convergence of the integrals ZZZ 2 2 ˆ 1 , ξ2 ) dξ1 dξ2 dt, e−2t (ξ1 +ξ2 ) ξ12 ϕ(ξ ˆ 1 , ξ2 )ψ(ξ R3+

ZZZ

R3+

∂ψ ∂ϕ (x1 , x2 , t) (x1 , x2 , t) d x 1 d x2 dt. ∂ x1 ∂ x1

For the first integral this is obvious. The absolute convergence of the second integral can be easily proved. We leave this as an exercise for the reader. Our next goal is to estimate the right-hand side of (1.4) from above. THEOREM 1.2 For any ϕ, ψ ∈ C 0∞ , and any positive function w on the plane, we have

ZZZ

∂ϕ ∂ψ d x1 d x2 dt R3+ ∂ x 1 ∂ x 1 ≤

AQ heat w,2

 ZZ

2

|ϕ| w d x1 d x2 +

ZZ

|ψ|2

 1 d x1 d x2 , w

where A is an absolute constant. In the proof we use the following key result. (In what follows, d 2 f denotes the Hessian form that is the second differential form of f .) THEOREM 1.3 For any Q > 1, define the domain D Q := {0 < (X, Y, x, y, r, s) : x 2 < X s, y 2 < Y r, 1 < r s < Q}. Let K be any compact subset of D Q . Then there exists a function B = B Q,K (X, Y, x, y, r, s) infinitely differentiable in a small neighborhood of K such that (1) 0 ≤ B ≤ 5Q(X + Y ), (2) −d 2 B ≥ |d x||dy|.

We prove Theorem 1.3 later. Now we use it to obtain the proof of Theorem 1.2. Proof of Theorem 1.2 Given a nonconstant smooth w that is constant outside some large ball, we consider Q = Q heat w,2 . We treat only the case w ∈ A 2 , that is, Q < ∞, for otherwise there is nothing to prove. Consider two nonnegative functions ϕ, ψ ∈ C 0∞ . Now take B = B Q,K , where a compact K remains to be chosen.

i

i i

i

i

i 2002/3/18

page 290 i

i

290

PETERMICHL and VOLBERG

We are interested in
b(x, t) := B (ϕ 2 w)(x, t), (ψ 2 w −1 )(x, t), ϕ(x, t), ψ(x, t), w(x, t), w −1 (x, t) .

This is a well-defined function because the choice of Q ensures that the 6-vector v, defined by
v := (ϕ 2 w)(x, t), (ψ 2 w −1 )(x, t), ϕ(x, t), ψ(x, t), w(x, t), w −1 (x, t) ,

lies in D Q for any (x, t) ∈ R3+ . Also, we can fix any compact subset M of the open set R3+ and guarantee that for (x, t) ∈ M, the vector v lies in a compact K . In fact, notice that for our w and for compactly supported ϕ, ψ, the mapping (x, t) → v(x, t) maps compacts in R3+ to compacts in D Q . Now just take K large enough. We want to apply the Green formula to b(x, t). To do this, we introduce a Green function G(x, t), as in [11]:   ∂   + 1 G = −δ0,1/2 in C(1, 1) = B(0, 1) × (0, 1),   ∂t G=0 on ∂ 0 C(1, 1) = ∂ B(0, 1) × (0, 1),     G=0 when t = 1. Here δ0,1/2 is a δ-function at the point (0, 1/2). It is important to keep in mind

that
G(x, 0) ≥ a 1 − kxk ,

(1.5)

where a is a positive absolute constant. We also need a Green function in the cylinder C(R, R 2 ) = B(0, R) × (0, R 2 ):   ∂   + 1 G R = −δ0,R 2 /2 in C(R, R 2 ) = B(0, R) × (0, R 2 ),   ∂t GR = 0 on ∂ 0 C(R, R 2 ) = ∂ B(0, R) × (0, R 2 ),     R G =0 when t = R 2 . One can easily see that the following lemma holds.

LEMMA 1.4 Green functions in different-sized cylinders relate as follows:

G R (x, t) =

1 x t  G , . R R2 R2

We are ready to apply the Green formula to b(x, t).

i

i i

i

i

i 2002/3/18

page 291 i

i

HEATING OF THE AHLFORS-BEURLING OPERATOR

291

Let us first estimate b(0, R 2 /2) = B((ϕ 2 w)(0, R 2 /2), . . . , w −1 (0, R 2 /2)). Using the first property of Theorem 1.3, we get (x = (x 1 , x2 ), as always)   R2   R 2   R2  heat 2 2 −1 ≤ 5Q w,2 (ϕ w) 0, + (ψ w ) 0, . b 0, 2 2 2 Thus, ZZ  2|x|2  R2  2 2 5 heat b 0, ≤ Q w,2 d x1 d x2 (ϕ w)(x) exp − 2 π R2 R2 ZZ  2|x|2  2 5 2 −1 d x1 d x2 . + Q heat (ψ w )(x) exp − π w,2 R2 R2 

Now by the Green formula in C(R, R 2 ), ZZZ  ∂ R2  =− + 1 G R (x, t)b(x, t) d x 1 d x2 dt b 0, 2 C(R,R 2 )∩{t>ε} ∂t ZZZ ∂  = G R (x, t) − 1 b(x, t) d x 1 d x2 dt ∂t C(R,R 2 )∩{t>ε} ZZ b(x, ε)G R (x, ε) d x 1 d x2 + 

B(0,R)

 ∂b ∂G R  + GR − b ds dt ∂n outer ∂ 0 C(R,R 2 )∩{t>ε} ∂n outer ZZZ ∂  = G R (x, t) − 1 b(x, t) d x 1 d x2 dt ∂t C(R,R 2 )∩{t>ε} ZZ ZZ ∂G R b(x, ε)G R (x, ε) d x 1 d x2 + b ds dt + B(0,R) ∂ 0 C(R,R 2 )∩{t>ε} ∂n inner ZZZ ∂  ≥ G R (x, t) − 1 b(x, t) d x 1 d x2 dt. ∂t C(R,R 2 )∩{t>ε} ZZ

The last inequality is clear: the double integrals are both nonnegative because b is nonnegative and because G R is nonnegative and vanishes on the side boundary. Let us combine estimates of b(0, R 2 /2) into ZZZ ZZ ZZ  ∂  10  2 2 −1 G R (x, t) w + ψ w . − 1 b(x, t) ≤ ϕ ∂t π R2 C(R,R 2 )∩{t>ε} (1.6) Fix R and ε, and choose the compact set  M = (x, t) : x ∈ clos(B(0, R)), ε ≤ t ≤ R 2 . The next calculation is simple, but it is key to the proof.

i

i i

i

i

i 2002/3/18

page 292 i

i

292

PETERMICHL and VOLBERG

LEMMA

1.5

Let
v = (ϕ 2 w)(x, t), (ψ 2 w −1 )(x, t), ϕ(x, t), ψ(x, t), w(x, t), w −1 (x, t) ,

where all entries are heat extensions. If (x, t) ∈ M, then ∂    ∂v ∂v  ∂v ∂v  2 − 1 b(x, t) = (−d 2 B) , + (−d B) , . ∂t ∂ x 1 ∂ x 1 R6 ∂ x 2 ∂ x 2 R6

Proof We compute the derivative with respect to time and the Laplacian of b using the chain rule and obtain the following:  ∂ ∂v  , b = ∇ B, ∂t ∂t R6   ∂v ∂v  ∂v ∂v  2 1b = (d 2 B) , + (d B) , + (∇ B, 1v)R6 . ∂ x 1 ∂ x 1 R6 ∂ x 2 ∂ x 2 R6 Combining the two, we get ∂      ∂v ∂v ∂v  ∂v ∂v  2 − 1 b = ∇ B, − 1v 6 − (d 2 B) , − (d B) , . R ∂t ∂t ∂ x 1 ∂ x 1 R6 ∂ x 2 ∂ x 2 R6

However, the first term is zero because all entries of the vector v are solutions of the heat equation. By Theorem 1.3 in M, −d 2 B ≥ |d x||dy|. For (x, t) ∈ M, Lemma 1.5 gives ∂ϕ ∂ψ ∂ϕ ∂ψ ∂  − 1 b(x, t) ≥ + . ∂t ∂ x1 ∂ x1 ∂ x2 ∂ x2

(1.7)

Combining (1.6) and (1.7), we get

ZZ  ∂ϕ ∂ψ ∂ϕ ∂ψ  10Q heat  Z Z  w,2 2 2 −1 G R (x, t) w + ψ w . ϕ + ≤ ∂ x1 ∂ x1 ∂ x2 ∂ x2 π R2 M (1.8) Now it is time to use Lemma 1.4. So (1.8) implies ZZZ  x t  ∂ϕ ∂ψ ∂ϕ ∂ψ  G , + R R2 ∂ x1 ∂ x1 ∂ x2 ∂ x2 M ZZ  ZZ  10Q heat w,2 2 ≤ ψ 2 w −1 , (1.9) ϕ w+ π

ZZZ

i

i i

i

i

i 2002/3/18

page 293 i

i

HEATING OF THE AHLFORS-BEURLING OPERATOR

293

where M = {(x, t) : x ∈ clos(B(0, R)), ε ≤ t ≤ R 2 }. Let us fix any compact M0 in R3+ and choose R and ε in such a way that M0 ⊂ M. Restrict the integration in (1.9) to M0 , and let R → ∞. Taking into account that G(0, 0) = a > 0, we obtain Z Z Z  ZZ  10Q heat  Z Z  ∂ϕ ∂ψ ∂ϕ ∂ψ w,2 2 ψ 2 w −1 . ϕ w+ + ≤ ∂ x1 ∂ x1 ∂ x2 ∂ x2 πa M0

But M0 is an arbitrary compact set in the upper half-space. Therefore, ZZZ

ZZ  ∂ϕ ∂ψ ∂ϕ ∂ψ  10Q heat  Z Z  w,2 2 ψ 2 w −1 . (1.10) ϕ w+ + ≤ ∂ x1 ∂ x1 ∂ x2 ∂ x2 πa R3+

This immediately implies that ZZZ

 ∂ϕ ∂ψ ∂ϕ ∂ψ  20Q heat w,2 k f k L 2 (w) kψk L 2 (w−1 ) + ≤ 3 ∂ x1 ∂ x1 ∂ x2 ∂ x2 πa R+

(1.11)

by substituting ϕ/t, tψ into (1.10) and minimizing over t. This proves Theorem 1.2.

Now we couple inequality (1.11) with Lemma 1.1, getting the right weighted estimates for R12 , R22 , R1 R2 , T . In particular, we have proved Theorem 0.6 for the case p = 2 up to the proof of Theorem 1.3. Proof of Theorem 1.3 We start with a much simpler “model” operator, Tσ . The logic is the following. We want to get a sharp weighted estimate of kTσ k L 2 (w)→L 2 (w) via the A2 -characteristic of w. Strangely enough, we first consider a two-weight estimate of kTσ k L 2 (u)→L 2 (v) with different weights u, v. In the paper of Nazarov, S. Treil, A. Volberg [26], one can find that this norm is attained on some “simple” test functions—and that this holds for every pair u, v. Thus, the same is true for u = v = w. However, on the family T of test functions, one can compute N w,2 (Tσ ) := sup{kTσ tk L 2 (w) : t ∈ T , ktk L 2 (w) = 1}. It turns out that Nw,2 (Tσ ) ≈ Q class w,2 . J. Wittwer does this in [31] (see also [28]). Thus, we get kTσ k L 2 (w)→L 2 (w) = Nw,2 (Tσ ) ≈ Q class w,2 . This sharp one-weight estimate for the model operator (notice that its proof used a fact proved in two-weight theory) is used to construct the Bellman function for our operator. An application of the Green formula to this Bellman function is the content of the proof of Theorem 1.2.

i

i i

i

i

i 2002/3/18

page 294 i

i

294

PETERMICHL and VOLBERG

So let us show what the model operator is, what its sharp weighted estimate is, and how one obtains a special function (the Bellman function) from this estimate. Consider the family of dyadic singular operators Tσ : Tσ f = 6 I ∈D σ (I )( f, h I )h I . Here D is a dyadic lattice on R, h I is a Haar function associated with the dyadic interval I (h I is normalized in L 2 (R, d x)), and σ (I ) = ±1. We call the family Tσ the martingale transform. It is a dyadic analog of a Caldero´ n-Zygmund operator. The following are important questions about Tσ , the first one about two-weight estimates and the second one about one-weight estimates. (1) What are necessary and sufficient conditions for supσ kTσ k L 2 (u)→L 2 (v) < ∞? (2) What is the sharp bound on supσ kTσ k L 2 (w)→L 2 (w) in terms of w? How can one compute supσ kTσ k L 2 (w)→L 2 (w) ? These questions are dyadic analogs of notoriously difficult questions about classical Calder´on-Zygmund operators like the Hilbert transform, the Riesz transforms, and the Ahlfors-Beurling transform. The dyadic model is supposed to be easier than the continuous one. This turns out to be true. The answers to the questions above appeared in [26] and [31]. Moreover, these answers are key to answering questions about classical Calder´on-Zygmund operators. Strangely enough, the answer to the second question (which seems to be easier because it is about one-weight) seems to require the ideas from the two-weight case. Here is our explanation of this phenomenom. The necessary and sufficient conditions on (u, v) to answer the first question were given in [26]. They amount to the fact that supσ kTσ k L 2 (u)→L 2 (v) is almost attained on the family of simple test functions. This fact has beautiful consequences in the one-weight case. For then supσ kTσ k L 2 (w)→L 2 (w) is attainable (almost) on the family of simple test functions. One may try to compute supσ kTσ tk L 2 (w) for every element of this test family, thus getting a good estimate for the norm supσ kTσ k L 2 (w)→L 2 (w) . Test functions are rather simple, so this program can be carried out. This has been done in [31]. Here is the result. Recall that dyadic Q w,2 := sup hwi I hw −1 i I . I ∈D

THEOREM 1.6 For any A2 -weight w, we have the inequality dyadic

sup kTσ k L 2 (w)→L 2 (w) ≤ AQ w,2 . σ

Let us rewrite Theorem 1.6 as follows: dyadic sup 6 I ∈D σ (I )( f, h I )(g, h I ) ≤ AQ w,2 k f k L 2 (w) kgk L 2 (w−1 ) , σ (I )=±1

i

i i

i

i

i 2002/3/18

page 295 i

i

HEATING OF THE AHLFORS-BEURLING OPERATOR

295

or dyadic 6 I ∈D ( f, h I ) (g, h I ) ≤ AQ w,2 k f k L 2 (w) kgk L 2 (w−1 ) .

This inequality is scaleless, so we write it as J ∈ D,

1 6 I ∈D ,I ⊂J h f i I− − h f i I+ hgi I− − hgi I+ |I | |J | dyadic

1/2

1/2

≤ AQ w,2 h f 2 wi J hg 2 w −1 i J . (1.12)

Here I− , I+ are the left and the right halves of I , and h·il means averaging over l, as usual. Given a fixed J ∈ D and a number Q > 1, we wish to introduce the Bellman function of (1.12), B(X, Y, x, y, r, s) n 1 6 I ∈D ,I ⊂J h f i I− − h f i I+ hgi I− − hgi I+ |I | : = sup |J | h f i J = x, hgi J = y, hwi J = r, hw −1 i J = s, dyadic

h f 2 wi J = X, hg 2 w −1 i J = Y, w ∈ A2

dyadic

, Q w,2

o ≤Q .

Obviously, the function B does not depend on J , but it does depend on Q. Its domain of definition is the following:  R Q := 0 ≤ (X, Y, x, y, r, s), x 2 ≤ X s, y 2 ≤ Y r, 1 ≤ r s ≤ Q . By (1.12) it satisfies

0 ≤ B ≤ AQ X 1/2 Y 1/2 .

(1.13)

We prove that it also satisfies the following differential inequality. Denote v := (X, Y, x, y, r, s),

v− = (X − , Y− , x− , y− , r− , s− ),

v+ = (X + , Y+ , x+ , y+ , r+ , s+ ), and let v, v+ , v− lie in R Q such that v = 1/2(v− + v+ ). Then B(v) −


1 B(v+ ) + B(v− ) ≥ |x+ − x− ||y+ − y− |. 2

(1.14)

In fact, let f, g, w almost maximize B(v) (on the interval J ), let f + , g+ , w+ do this for B(v+ ), and let f − , g− , w− do this for B(v− ). The scalelessness of B allows us to put f + , g+ , w+ on J+ and f − , g− , w− on J− . Then we have “gargoyle” functions ( ( ( f + on J+ , g+ on J+ , w+ on J+ , F= G= W = f − on J− , g− on J− , w− on J− .

i

i i

i

i

i 2002/3/18

page 296 i

i

296

PETERMICHL and VOLBERG

Obviously, hFi J = 1/2(x + + x− ) = x, hGi J = y, hW i J = r, hW −1 i J = s, hF 2 W i J = X, hG 2 W =1 i J = Y . These numbers together form the vector v. In other words, F, G, W compete with f, g, w in definition (1.12) of Bellman function B(v). By this definition, B(v) ≥

1 6 I ∈D ,I ⊂J hFi I− − hFi I+ hGi I− − hGi I+ |I |. |J |

But the almost optimality of f + , g+ , w+ on J+ and f − , g− , w− on J− gives us B(v+ ) ≥ −ε + and B(v− ) ≥ −ε +

1 6 I ∈D ,I ⊂J+ hFi I− − hFi I+ hGi I− − hGi I+ |I |, |J+ | 1 6 I ∈D ,I ⊂J− hFi I− − hFi I+ hGi I− − hGi I+ |I | |J− |

(recall that F = f ± on J± , G = g± on J± ). Combining these, we get


1 B(v+ ) + B(v− ) ≥ −ε + hFi J− − hFi J+ hGi J− − hGi J+ 2 = −ε + h f − i J− − h f + i J+ hg− i J− − hg+ i J+ = −ε + x− − x+ y− − y+ .

B(v) −

We are done with (1.14) because ε is an arbitrary positive number. Therefore, our B is a very concave function. We modify B to have its Hessian satisfy the conclusion of Theorem 1.3. To do that we fix a compact K in the interior of R Q , and we choose ε such that 100ε < dist(K , ∂ R Q ). Consider the convolution of B with (1/ε 6 )ϕ(v/ε), v ∈ R6 , where ϕ is a bell-shaped infinitely differentiable function with support in the unit ball of R6 . It is now very easy to see that this convolution (we call it B K ,Q ) satisfies the following inequalities: 0 ≤ B K ,Q ≤ 6Q(X + Y ),

(1.15)

and for any vector ξ = (ξ1 , ξ2 , ξ3 , ξ4 , ξ5 , ξ6 ) ∈ R6 , −(d 2 B K ,Q ξ, ξ )R6 ≥ 2|ξ2 ||ξ3 |.

(1.16)

The factor 2 appears because B(v) − (1/2)(B(v+ ) + B(v− )) in (1.14) corresponds to −(1/2)d 2 B. Theorem 1.3 is completely proved.

i

i i

i

i

i 2002/3/18

page 297 i

i

HEATING OF THE AHLFORS-BEURLING OPERATOR

297

2. The sharp weighted estimate for the Ahlfors-Beurling operator in L p (w d A) for p > 2 Here again T is the Ahlfors-Beurling transform on the plane, and d A denotes Lebesgue measure on the plane. Now we prove that kT k L p (w d A)→L p (w d A) ≤ C( p)Q heat w, p .

(2.1)

In Section 3, it is shown how this implies kT k L p (w d A)→L p (w d A) ≤ C( p)Q class w, p .

(2.2)

Let us recall that for the weights on R, we can introduce dyadic

Q w, p

p−1

:= sup hwi I hw −(1/( p−1)) i I

.

I ∈D

The weight ρ := w −(1/( p−1)) is called the dual weight, and dyadic

dyadic

0

(Q ρ, p0 ) p = (Q w, p ) p .

(2.3)

We need the following result of S. Buckley. (We formulate its dyadic version.) Let us denote M d f (x) = supx∈I,I ∈D h| f |i I . This is the dyadic maximal function. THEOREM 2.1 For any weight on the line R, dyadic

kM d k L p (w d x)→L p (w d x) ≤ C( p)Q ρ, p0 ,

1 < p < ∞.

We use it now to extend Theorem 1.6 to p > 2. 2.2 For any weight on the line R, THEOREM

dyadic

sup kTσ k L p (w d x)→L p (w d x) ≤ C( p)Q w, p ,

2 < p < ∞.

σ

The proof that follows is word for word the extrapolation proof of Garcia-Cuerva and Rubio de Francia in [12]. We only need to be absolutely precise with exponents. Proof dyadic Let w ∈ A p . Let T stand for any of the Tσ ; let t = ( p − 2)/( p − 1). Then Z Z Z 2/ p = sup |T f |2 uw d x ≤ sup |T f |2 vw d x. |T f | p w d x u≥0,kuk

≤1 0 L p /t (w d x)

v∈F

(2.4)

i

i i

i

i

i 2002/3/18

page 298 i

i

298

PETERMICHL and VOLBERG

Here F is a family of functions v of the following type: v = v(u) := u +

Su S2u S3u + + + ··· , 2kSk 4kSk2 8kSk3 0

where u runs over the unit ball of L p /t (w d x), the nonlinear operator S is given by the formula
t S(u) := w−1 M d (u 1/t w) , and kSk denotes kSk L p0 /t (w d x)→L p0 /t (w d x) . Now let us prove the following: (1) u ≤ v, (2) kvk L p0 /t (w d x) ≤ 2kuk L p0 /t (w d x) , dyadic

dyadic

dyadic

, Q vw,2 ≤ Q w, p . (3) vw ∈ A2 The first inequality is clear. To prove the second inequality, we notice that if kSk is finite, then the second inequality follows immediately. So let us prove this finiteness. Now dyadic kSk ≤ C( p)(Q w, p )t . (2.5)

Z

In fact, using Theorem 2.1 we get Z
p0 0 p 0 /t (Su) w d x = M d (u 1/t w) w 1− p d x Z Z
p0 = M d (u 1/t w) w −(1/( p−1)) d x = Z 0 0 0 dyadic p 0 ≤ C( p)(Q w, p ) u p /t w p w 1− p d x Z 0 dyadic p 0 = C( p)(Q w, p ) u p /t w d x. Therefore, (2.5) is proved. Let us now check that


−1  dyadic huwi I S(u)w ≤ (Q w, p )1−t , I

M d (u 1/t w)


p0

ρ dx

∀I ∈ D , u ≥ 0.

(2.6)

In fact, (1 − t)(1 − p) = 1. Using this and the pointwise estimate on ≥ hu 1/t wi I , ∀x ∈ I , we get


−t −(1/( p−1)) huwi I M d (u 1/t w) w −(1/( p−1)) I ≤ hu 1/t witI hwi1−t · hu 1/t wi−t iI . I I hw

(M d (u 1/t w))(x)

Therefore,


−t

 −(1/( p−1)) (1−t)(1− p) iI huwi I M d (u 1/t w) w −(1/( p−1)) I ≤ hwi1−t I hw dyadic

≤ (Q w, p )1−t .

i

i i

i

i

i 2002/3/18

page 299 i

i

HEATING OF THE AHLFORS-BEURLING OPERATOR

299

We are done with (2.6). Now at last we have our third estimate as a combination of (2.5), (2.6), and the fact that by its definition, S(v) ≤ 2kSkv: dyadic

∀v ∈ F ,

dyadic

Q vw,2 ≤ C( p)Q w, p .

(2.7)

Now using (2.7) and inequality kvk L p0 /t (w d x) ≤ 2kuk L p0 /t (w d x) ≤ 2, we get Z

p

|T f | w d x

2/ p

Z

dyadic A(Q vw,2 )2

Z

|T f | vw d x ≤ | f |2 vw d x Z 2/ p dyadic 2 ≤ AC( p)(Q w, p ) | f |pw d x kvk L p0 /t (w d x)

≤ sup

v∈F

2

dyadic

≤ 2AC( p)(Q w, p )2 k f k2L p (w d x) .

In the second inequality we used Theorem 1.6. Thus, dyadic kTσ k L p (w d x)→L p (w d x) ≤ 2AC( p)Q w, p , and Theorem 2.2 is completely proved. Theorem 1.3 has the following ( p > 2)-version. THEOREM 2.3 p For any Q > 1, p > 2, define the domain D Q := {0 < (X, Y, x, y, r, s) : x p < 0

0

p

X s p−1 , y p < Y r p −1 , 1 < r s p−1 < Q}. Let K be any compact subset of D Q . Then ( p)

there exists a function B = B Q,K (X, Y, x, y, r, s) infinitely differentiable in the small neighborhood of K such that 0 (1) 0 ≤ B ≤ C( p)Q X 1/ p Y 1/ p , (2) −d 2 B ≥ |d x||dy|. Proof The proof repeats verbatim the proof of Theorem 1.3. But now we base the considerations on Theorem 2.2 rather than Theorem 1.6. Now we are ready to prove our main estimate (2.1). The proof repeats the one given in the previous section for p = 2, but instead of b(x, t) we consider b( p) (x, t) := B ( p) v(x, t), where v(x, t) is the 6-vector
0 ( f p w)(x, t), (g p w −(1/( p−1)) )(x, t), f (x, t), g(x, t), w(x, t), w −(1/( p−1)) (x, t) .

i

i i

i

i

i 2002/3/18

page 300 i

i

300

PETERMICHL and VOLBERG

The rest is identical, but the use of b ( p) explains why one needs to choose Q at −(1/( p−1)) (x, t)) p−1 . least as large as Q heat w, p = sup(x,t)∈R3+ w(x, t)(w Inequality (2.1) is completely proved. We have also proved our main Theorem 0.6. 3. The comparison of classical and heat A p -characteristics In this section we give an easy proof of the following theorem. THEOREM 3.1 There exists a finite absolute constant b such that class Q heat w, p ≤ bQ w, p .

Proof Constants are denoted by the letters c, C; they may vary from line to line and even within the same line. We introduce the following notation. Bk denotes B(0, 2k ), k = R 0, 1, 2, . . . , h f i B stands for the average (1/|B|) B f d A, and f (B) stands for R RR h 2 B f d A. If
B = B(0, r ), then h f i B stands for (1/(πr )) R2 f (x) exp − (kxk2 /(r 2 )) d x1 d x2 . LEMMA 3.2 Suppose that f and g, positive functions on the plane, are such that sup B h f i B hgi B = A. Then there exists a finite absolute constant c such that

h f i B hgihB ≤ c A for any disc B. Proof Scale invariance allows us to prove this for only one disc, B = B(0, 1). We start the estimate: A h f i B hgihB ≤ ch f i B 6k 22k exp(−22k−2 ) . h f i Bk On the other hand, h f i Bk > ch f i Bk−1 > · · · > ck h f i B . (Recall that B is the unit disc.) Plugging this into the inequality above, we get h f i B hgihB ≤ ch f i B 6k C k exp(−22k−2 )

A . h f iB

In other words, h f i B hgihB ≤ c A6k C k exp(−22k−2 ) = c A, and the lemma is proved.

i

i i

i

i

i 2002/3/18

page 301 i

i

HEATING OF THE AHLFORS-BEURLING OPERATOR

301

Now we want to prove Theorem 3.1. Fix B. Again, by scale invariance it is enough to consider B = B(0, 1). By the previous lemma, we know that h f i Bk hgihBk ≤ c A

(3.1)

for any k. Now h f ihB hgihB ≤ chgihB 622k exp(−22k−2 )h f i Bk ≤ chgihB 622k exp(−22k−2 )

cA hgihBk

.

The last inequality uses (3.1). On the other hand, hgihBk > chgihBk−1 > · · · > ck hgi B . (Recall that B is the unit disc.) Plugging this into the inequality above, we get h f ihB hgihB ≤ chgihB 6k C k exp(−22k−2 )

cA hgihB

.

In other words, h f ihB hgihB ≤ c2 A6k C k exp(−22k−2 ) = c2 A. Theorem 3.1 is completely proved. 4. Injectivity at the critical exponent and regularity of solutions of the Beltrami equation Theorem 0.4 was proved in [2]. It reduces the question of the injectivity of I − µT , I − T µ in L 1+k (C), k = kµk∞ < 1, to estimate (0.1): kT k L p (ω d A)→L p (ω d A) ≤

C , 1 + (1/k) − p

where ω = | f z ◦ f −1 | p−2 , f is a ((1 + k)/(1 − k))-quasiconformal homeomorphism, and p ∈ [2, 1 + (1/k)). This estimate has been proved twice in the present paper. We first proved the heat class estimate of the norm kT k via AQ heat w, p , then noticed that Q w, p ≈ Q w, p , and finally noticed that (see [2]) Q class w, p ≤ C/(1+(1/k)− p) for special ω’s that are certain powers of Jacobians of quasiconformal homeomorphisms. The second time, we started with heat the estimate of the norm kT k via AQ heat w, p . Then we directly computed Q w, p for ω’s that are certain powers of Jacobians of quasiconformal homeomorphism and again saw that Q heat w, p ≤ C/(1 + (1/k) − p) (see Theorem 0.7). Thus, we can be sure that injectivity holds. This fact is immediately applicable to so-called weakly quasiregular maps. Recall (see [2]) that the solution of Fz¯ − µFz = 0

in  ⊂ C

i

i i

i

i

i

“112i2˙03” 2002/3/18 page 302

i

i

302

PETERMICHL and VOLBERG 1,q

can be called q-weakly quasiregular if it belongs to Wloc . Here µ ∈ L ∞ (), kµk∞ = k < 1. It is known (see [2]) that if q > 1 + k, then a q-weakly quasiregu1,2+ε lar map is actually in Wloc for a certain positive . In particular, it is quasiregular. Therefore, it is continuous, open, discrete, and so on. However, the proof of this “selfimprovement” fact is very subtle. It is based on Astala’s sharp estimates of smoothness of quasiconformal homeomorphisms, which were obtained as a solution of a famous problem of F. Gehring and E. Reich. In [2], one can find an easy example that shows this is no longer true for q < 1+k. Thus, the remaining question is about the critical exponent q = 1 + k. THEOREM 4.1 Let kµk∞ = k < 1. Then (1 + k)-weakly quasiregular maps are also quasiregular.

We repeat the argument from [2]. This is done for the sake of the convenience of the reader. Proof Choose ϕ ∈ C∞ 0 (). Set G = Fϕ. Then G z¯ − µG z = (ϕz¯ − µϕz )F.

(4.1)

Obviously, G is a Cauchy transform of the compactly supported function ψ := G z¯ . Then (4.1) can be rewritten as (F0 := (ϕz¯ − µϕz )F): (I − µT )ψ = F0 ∈ L 2 (C).

(4.2)

1,1+k In fact, F ∈ Wloc , so by Sobolev’s theorem (ϕz¯ −µϕz )F ∈ L ((2+2k)/(1−k)) (C), and (ϕz¯ − µϕz )F has compact support. So it is in L 2+ε (C). Looking at (4.2), we can see that µ = 0 outside of supp ϕ. Our equation (4.2) has an obvious solution as a Neumann series. It converges in L 2+ε (C) (we use the fact that kT k in L p is close to 1 when p is close to 2) and has its support in supp ϕ. So it is in L 1+k (C) as well. Let us call it ψ0 . Now we have two solutions of (4.2) in L 1+k (C) − ψ and ψ0 . However, we have already proved the injectivity of I − µT in L 1+k (C). Therefore, ψ = ψ0 . Thus, G z¯ = ψ ∈ L 2+ε (C). Therefore, the compactly supported function G is in W 1,2+ε (C). This means that G is quasiregular and, in particular, continuous, open, discrete, and so on. Theorem 4.1 is completely proved.

Acknowledgments. We are deeply grateful to Kari Astala, Eero Saksman, Michael Frazier, Tadeusz Iwaniec, Fedor Nazarov, and Jacob Plotkin for discussions and criticism.

i

i i

i

i

i

“112i2˙03” 2002/3/18 page 303

i

i

HEATING OF THE AHLFORS-BEURLING OPERATOR

303

References [1]

K. ASTALA, Area distortion of quasiconformal mappings, Acta Math. 173 (1994),

[2] [3]

K. ASTALA, T. IWANIEC, and E. SAKSMAN, Beltrami operators, preprint, 1999.

37 – 60. MR 95m:30028b

[4] [5]

[6] [7]

[8]

[9]

[10] [11]

[12]

[13]

[14] [15]

[16]

A. BAERNSTEIN II and S. J. MONTGOMERY-SMITH, “Some conjectures about integral

means of ∂ f and ∂ f ” in Complex Analysis and Differential Equations (Uppsala, Sweden, 1999), ed. Ch. Kiselman, Acta. Univ. Upsaliensis Univ. C Organ. Hist. 64, Uppsala Univ. Press, Uppsala, Sweden, 1999, 92 – 109. MR 2001i:30002 ˜ R. BANUELOS and A. LINDEMAN, A martingale study of the Beurling-Ahlfors transform in Rn , J. Funct. Anal. 145 (1997), 224 – 265. MR 98a:30007 ˜ R. BANUELOS and G. WANG, Sharp inequalities for martingales with applications to the Beurling-Ahlfors and Riesz transforms, Duke Math. J. 80 (1995), 575 – 600. MR 96k:60108 B. V. BOJARSKI, Homeomorphic solutions of Beltrami systems (in Russian), Dokl. Akad. Nauk. SSSR (N.S.) 102 (1955), 661 – 664. MR 17:157a , Generalized solutions of a system of differential equations of first order and of elliptic type with discontinuous coefficients, Mat. Sb. (N.S.) 85, no. 43 (1957), 451 – 503. MR 21:5058 , “Quasiconformal mappings and general structure properties of systems of non linear equations elliptic in the sense of Lavrentiev” in Convegno sulle Transformazioni Quasiconformie Questioni Connesse (Rome, 1974), Sympos. Math. 18, Academic Press, London, 1976, 485 – 499. MR 58:22542 B. V. BOJARSKI and T. IWANIEC, Quasiconformal mappings and non-linear elliptic equations in two variables, I, II, Bull. Acad. Polon. Sci. S´er. Sci. Math. Astronom. Phys. 22 (1974), 473 – 484. MR 51:1110 S. BUCKLEY, Estimates for operator norms on weighted spaces and reverse Jensen inequalities, Trans. Amer. Math. Soc. 340 (1993), 253 – 272. MR 94a:42011 R. FEFFERMAN, C. KENIG, and J. PIPHER, The theory of weights and the Dirichlet problem for elliptic equations, Ann. of Math. (2) 134 (1991), 65 – 124. MR 93h:31010 J. GARCIA-CUERVA and J. RUBIO DE FRANCIA, Weighted Norm Inequalities And Related Topics, North-Holland Math. Stud. 116, North-Holland, Amsterdam, 1985. MR 87d:42023 F. W. GEHRING, “Open problems” in Proceedings of the Romanian-Finnish Seminar on Teichmuller Spaces and Quasiconformal Mappings (Brasov, Romania), Acad. Soc. Rep. Romania, Bucharest, 1969, 306. MR 45:8826 , The L p -integrability of the partial derivatives of a quasiconformal mapping, Acta Math. 130 (1973), 265 – 277. MR 53:5861 , “Topics in quasiconformal mappings” in Proceedings of the International Congress of Mathematicians (Berkeley, 1986), Vols. I, II, Amer. Math. Soc., Providence, 1987, 62 – 80. MR 89c:30051 F. W. GEHRING and E. REICH, Area distortion under quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser A I No. 388 (1966), 1 – 15. MR 34:1517

i

i i

i

i

i

“112i2˙03” 2002/3/18 page 304

i

i

304

[17] [18] [19] [20]

[21] [22] [23] [24]

[25] [26]

[27]

[28]

[29]

[30]

[31]

PETERMICHL and VOLBERG

T. IWANIEC, Extremal inequalities in Sobolev spaces and quasiconformal mappings,

Z. Anal. Anwendungen 1 (1982), 1 – 16. MR 85g:30027 , The best constant in a BMO-inequality for the Beurling-Ahlfors transform, Michigan Math. J. 33 (1986), 387 – 394. MR 88b:42024 , Hilbert transform in the complex plane and the area inequalities for certain quadratic differentials, Michigan Math. J. 34 (1987), 407 – 434. MR 89a:42025 , “L p -theory of quasiregular mappings” in Quasiconformal Space Mappings, ed. Matti Vuorinen, Lecture Notes in Math. 1508, Springer, Berlin, 1992. CMP 1 187 088 T. IWANIEC and G. MARTIN, Quasiregular mappings in even dimensions, Acta Math. 170 (1993), 29 – 81. MR 94m:30046 , Riesz transforms and related singular integrals, J. Reine Angew. Math. 473 (1996), 25 – 57. MR 97k:42033 O. LEHTO, Remarks on the integrability of the derivatives of quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A I No. 371 (1965), 3 – 8. MR 31:5975 , “Quasiconformal mappings and singular integrals” in Convegno sulle Transformazioni Quasiconformi e Questioni Connesse (Rome, 1974), Sympos. Math. 18, Academic Press, London, 1976, 429 – 453. MR 58:11387 O. LEHTO and K. I. VIRTANEN, Quasiconformal Mappings in the Plane, 2d ed., Grundlehren Math. Wiss. 126, Springer, New York, 1973. MR 49:9202 F. NAZAROV, S. TREIL, and A. VOLBERG, The Bellman functions and two-weight inequalities for Haar multipliers, J. Amer. Math. Soc. 12 (1999), 909 – 928. MR 2000k:42009 F. NAZAROV and A. VOLBERG, Heating of the Ahlfors-Beurling operator and the estimates of its norms, preprint, 2000, http://www.math.msu.edu/˜volberg/papers/heating1/piminusone.html S. PETERMICHL and J. WITTWER, A sharp estimate for the weighted Hilbert transform via Bellman functions, preprint, 2000, http://www.williams.edu/mathematics/jwittwer E. STEIN, Harmonic Analysis: Real-variable Methods, Orthogonality, And Oscillatory Integrals, Princeton Math. Ser. 43, Monogr. Harmon. Anal. 3, Princeton Univ. Press, Princeton, 1993. MR 95c:42002 G. WANG, Differential subordination and strong differential subordination for continuous-time martingales and related sharp inequalities, Ann. Probab. 23 (1995), 522 – 551. MR 96b:60120 J. WITTWER, A sharp estimate on the norm of the martingale transform, Math. Res. Lett. 7 (2000), 1 – 12. MR 2001e:42022

Petermichl Institute for Advanced Study, 1 Einstein Drive, Princeton, New Jersey 08540, USA; [email protected]

i

i i

i

i

i

“112i2˙03” 2002/3/18 page 305

i

i

HEATING OF THE AHLFORS-BEURLING OPERATOR

305

Volberg Department of Mathematics, Michigan State University, East Lansing, Michigan 48824, USA; [email protected]; current: Equipe d’Analyse, Universit´e Pierre et Marie Curie-Paris 6, 4 Place Jussieu, 75252 Paris CEDEX 05, France; [email protected]

i

i i

i