Heating of the Beurling operator: weakly quasiregular maps on the plane are quasiregular

St. Petermichl and A. Volberg

Abstract

We establish the borderline regularity for the solutions of Beltrami equation on the plane. Namely, consider f_{z} − μ fz̄ = 0, where μ is a bounded measurable function, ∥μ∥_{∞} = k < 1. What is the minimal requirement of the type f ∈ W_{loc}^{1,q} that guarantees that any solution of the Beltrami equation with any ∥μ∥_{∞} = k < 1 will be a continuous function? By a deep result of Astala, f ∈ W_{loc}^{1,1+k+ε} suffices if ε > 0. On the other hand, Lehto and Iwaniec gave the example that q < 1+k will not be sufficient. In [1] the 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). The method of the proof is based on sharp weighted estimate of the Beurling-Ahlfors operator. This estimate in its turn is based on sharp weighted estimate of certain dyadic singular integral operator and on using heat extension of the Bellman function of the problem. The sharp weighted estimate of the dyadic operator is obtained by combination of ideas from Rubio de Francia extrapolation technique and from two-weight estimates for the martingale transform.

Contents

Notation 2

0 Introduction: main objects and results 2

1 The sharp weighted estimate for the Beurling operator in L^{2}(wdA) 6 1.1 Proof of Theorem 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 The sharp weighted estimate for the Beurling operator in L^{p}(wdA), p > 2 15

3 The comparison of classical and heat A_{p} characteristics 18

4 The injectivity at the critical exponent and the regularity of solutions of Beltrami equation 20

Notation

:= equal by definition; μ bounded function on ℝ^{2}; B(x_{0}, r) the open ball, B(x_{0}, r) := {x ∈ ℝ^{2} : |x − x_{0}| < r}; ⟨f⟩_{B} average of function f over the ball B; x := (x_{1}, x_{2}) ; k(x, t) := \frac{1}{πt}e^{− \frac{∥x∥^{2}}{t}} heat kernel on the plane; Q_{w,2}^{heat} = sup_{x∈ℝ^{2},t>0} ∫∫ w(x) k(x, t)dx_{1}x_{2} · ∫∫ w^{−1}(x) k(x, t)dx_{1}x_{2} heat A_{2} characteristic of weight

w, see Section 0;

Q_{w,2}^{heat} = sup_{x∈ℝ^{2},t>0} ∫∫ w(x) k(x, t)dx_{1}x_{2} · (∫∫ w^{− \frac{1}{p−1}} (x) k(x, t)dx_{1}x_{2})^{p−1} heat A_{p} character-

istic of weight w, see Section 0;

Q_{w,2}^{classical} = sup_{B(x,r)}⟨w⟩_{B(x,r)}⟨w^{−1}⟩_{B(x,r)} standard A_{2} characteristic of weight w, see Section

0;

Q_{w,2}^{classical} = sup_{B(x,r)}⟨w⟩_{B(x,r)} (⟨w^{− \frac{1}{p−1}} ⟩_{B(x,r)})^{p−1} standard A_{p} characteristic of weight w, see

Section 0;

Q_{w,2}^{dyadic} = sup_{I, I∈D}⟨w⟩_{I}⟨w^{−1}⟩_{I} dyadic A_{2} characteristic of weight w, see Section 1; Q_{w,2}^{dyadic} = sup_{I, I∈D}⟨w⟩_{I} (⟨w^{− \frac{1}{p−1}} ⟩_{I})^{p−1} dyadic A_{p} characteristic of weight w, see Section 1; N_{w,2}(T) weighted norm of operator T on special family of test functions, see Section 1 D a collection of dyadic intervals, see Section 1;

0. Introduction: main objects and results

The goal of this paper is to present a quite sharp estimate of weighted Beurling operator. This estimate turns out to be sufficient to prove that any weakly quasiregular map is quasiregular.

We are interested, thus, in the Beurling operator (dA denotes area Lebesgue measure on ℂ):

T φ(z) := \frac{1}{π} ∫ ∫ \frac{φ(ζ)dA(ζ)}{(ζ − z)^{2}}

understood as a Calderón-Zygmund operator. The starting point for us is the following theorem of K. Astala, T. Iwaniec, E. Saksman [1].

Theorem 0.1. Let μ ∈ L^{∞}, ∥μ∥_{∞} = k < 1. Then operators I − μT, I − T μ are invertible in L^{p}(ℂ, dA) for p ∈ (1 + k, 1 + \frac{1}{k}).

Remark. In the same paper [1] the authors suggested the problem: to prove that these operators have dense range at p = 1 + \frac{1}{k} and injective at p = 1 + k. Actually this problem has a long history, and has been reappearing in many papers on regularity of quasiconformal homeomorphism and quasiregular maps. The L^{p}–theory of quasiregular mappings was essentially started with the works of B. Bojarski [6]-[7]. Later this subject came under intensive investigation. In particular, the best integrability of K-quasiconformal mappings and the (in a sense dual, see [27]) problem of the minimal regularity of the quasiregular mappings were discussed in many papers. We mention here [16], [13]-[15], [17], [20], [21], [22], [25], [27]. The best integrability result was finally established in [2]. Notice that that p − 1 problem for the operator T is still open. Remind that it consists of proving that ∥T ∥_{L^{p}→L^{p}} = p − 1, p > 2. The discussion of fantastically beautiful connections of this problem with the Calculus of Variations and Morrey’s problem can be found in [5]. Till recently the best result for all p was in [3]. This paper uses probabilistic method to obtain the estimate ∥T ∥_{L^{p}→L^{p}} ≤ 4(p − 1), p > 2. The discussion of related results can be found in [4] and [31]. Recently, in [28] the estimate was slightly improved to ∥T ∥_{L^{p}→L^{p}} ≤ 2(p − 1), p > 2. Main result of this paper may serve as yet another indication that this norm is p − 1.

We are establishing this dense range/injectivity property in the present paper. The main tool is the sharp weighted estimate of the Beurling operator. The main consequence (see [1]) is the abovementioned geometric fact: every weakly quasiregular map is quasiregular.

Let us first quote several results and notions we will be using. We will save the symbol f = f^{μ} for the homeomorphic solution of the Beltrami equation

fz̄ − μf_{z} = 0, f(z) = z + o(1), z → ∞

Here we consider compactly supported μ, μ ∈ L^{∞}(ℂ), ∥μ∥_{∞} = k < 1. Function μ is called Beltrami coefficient of Beltrami equation written above. The constant K = \frac{1+k}{1−k} has the geometric meaning of the bound on the distortion of infinitely small discs by the map f (resulting infinitely small ellipses have the ratio of their axis at most K). The map f is called K-quasiconformal. The image of a disc by a K-quasiconformal homeomorphism is called K-quasidisc. We will keep calling f^{−1} by letter g. We will need the following results from Astala-Iwaniec-Saksman paper [1].

Theorem 0.2. Let f be a K-quasiconformal homeomorphism built by a certain μ as it was described above. Consider ω = |f_{z} ∘ f^{−1}|^{p−2} for p ∈ (1 + k, 1 + \frac{1}{k}).

Then ∥(I − μT)^{−1}∥_{L^{p}→L^{p}}, ∥(I − T μ)^{−1}∥_{L^{p}→L^{p}} are bounded by C(k)λ_{ω,p}, where λ_{ω,p} := ∥T ∥_{L^{p}(ωdA)→L^{p}(ωdA)}.

In what follows | · | denotes the area.

Theorem 0.3. Let f be a K-quasiconformal homeomorphism. Let k = \frac{K+1}{K−1} and let p ∈ (0, 1 + \frac{1}{k}). Then for any disc or K-quasidisc B one has

\frac{1}{|B|} ∫_{B}(|f_{z}| + |fz̄|)^{p}dA ≤ \frac{C(k)}{1 + \frac{1}{k} − p} (\frac{|f(B)|}{|B|})^{\frac{p}{2}}

Remark. Obviously we have the same type of estimate for \frac{1}{|B|} ∫_{B} |f_{z}|^{p}dA and for

\frac{1}{|B|} ∫_{B} Jf^{\frac{p}{2}} dA, where J_{f} := |f_{z}|^{2} − |fz̄|^{2} is the Jacobian of the mapping f.

The next very elegant consideration from [1] reduces the critical exponent cases p = 1 + \frac{1}{k}, p = 1 + k to a weighted estimate of the Beurling operator. This consideration is incorporated for the convenience of the reader.

Theorem 0.4. Let k ∈ [0, 1), ∥μ∥_{∞} = k, f = f^{μ}, ω = |f_{z} ∘ f^{−1}|^{p−2}, p ∈ [2, 1 + \frac{1}{k}). If

∥T ∥_{L^{p}(ωdA)→L^{p}(ωdA)} ≤ \frac{C}{1 + \frac{1}{k} − p},  (0.1)

then I − μT, I − T μ have dense ranges in L^{1+ \frac{1}{k}} (ℂ) and are injective in L^{1+k}(ℂ).

Our main result will imply this weighted estimate (0.1). Theorem 0.4 serves as a motivation for our main result. In its turn, injectivity at the critical exponent 1 + k proves that weakly quasiregullar maps are quasiregular—see [1].

Proof. Operator T is invertible in all L^{q}, q ∈ (1, ∞). We can 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+ \frac{1}{k}} (ℂ).

We denote p_{0} := 1 + \frac{1}{k}. We take any φ ∈ C_{0}^{∞}(ℂ). By Theorem 0.1

φ_{ε} := (I − (1 − ε)μT)^{−1}φ ∈ L^{p_{0}}(ℂ),

and by Theorem 0.2

∥φ_{ε}∥_{L^{p_{0}}} ≤ C(k)∥T ∥_{L^{p_{0}}(ω^{ε}dA)→L^{p_{0}}(ω^{ε}dA)},

where ω^{ε} = |f_{z} ∘ f^{−1}|^{p−2}, and f denotes (as always) a homeomorphic solution of Beltrami eaquation, but for Beltrami coefficient (1 − ε)μ instead of μ.

Assuming now the estimate (0.1) we conclude that

∥φ_{ε}∥_{L^{p_{0}}} ≤ \frac{C}{1 + \frac{1}{(1−ε)k} − p_{0}} ≤ \frac{C}{ε}

Thus

∥εφ_{ε}∥_{L^{p_{0}}} ≤ C_{1} < ∞, ε → 0  (0.2)

On the other hand, the definition of φ_{ε} implies

(I − μT)φ_{ε} = φ − εμφ_{ε}  (0.3)

∥φ_{ε}∥_{L^{2}} ≤ C(k) < ∞, ∀ε ∈ [0, 1]  (0.4)

Now (0.2), (0.4) imply that εφ_{ε} converge weakly to 0 in L^{p_{0}}. Keeping this in mind we see that (0.3) implies that the range of I − μT is weakly dense in L^{p_{0}}. So the range of I − μT is automatically dense in L^{p_{0}} in norm topology.

Now we are ready to formulate one of the main results of this work.

Theorem 0.5. Let k ∈ [0, 1), ∥μ∥_{∞} = k, f = f^{μ}, ω = |f_{z} ∘ f^{−1}|^{p−2}, p ∈ [2, 1 + \frac{1}{k}). Then

∥T ∥_{L^{p}(ωdA)→L^{p}(ωdA)} ≤ \frac{C}{1 + \frac{1}{k} − p}

This result will be obtained as a corollary of the following results, which might be of interest in their own right.

Let ω be any weight on ℝ^{2}, denote by ω(x, t) = ω(x_{1}, x_{2}, t) its heat extension into ℝ_{+}^{3}:

ω(x, t) = \frac{1}{πt} ∫ ∫_{ℝ^{2}} ω(y) exp(−\frac{∥x − y∥^{2}}{t})dy_{1}dy_{2}

We introduce

Q_{ω,p}^{heat} := sup_{(x,t)∈ℝ_{+}^{3}} ω(x, t) (ω^{− \frac{1}{p−1}} (x, t))^{p−1}

The weights with finite Q_{ω,p}^{heat} are called A_{p} weights. There is an extensive theory of A_{p} weights, see for example [30],[12]. The usual definition is different from the one above. But they describe the same class of weight. Actually we are going to say more about the relationship between the classical definition and our definition. But first we formulate two more theorems. Their combination gives Theorem 0.5.

Theorem 0.6. For any A_{p} weight w and any p ≥ 2 we have

∥T ∥_{L^{p}(wdA)→L^{p}(wdA)} ≤ C(p)Q_{ω,p}^{heat}

Theorem 0.7. Let k ∈ [0, 1), ∥μ∥_{∞} = k, f = f^{μ}, ω = |f_{z} ∘ f^{−1}|^{p−2}, p ∈ [2, 1 + \frac{1}{k}). Then the weight ω belongs to A_{p} and

Q_{ω,p}^{heat} ≤ \frac{C}{1 + \frac{1}{k} − p}

Clearly the combination of the last two results implies Theorem 0.5. And Theorem 0.5 and Theorem 0.4 give positive answers to Beltrami equation critical exponent questions discussed above and raised in [1].

At the end of our introduction we wish to discuss the connection between Q_{w,p}^{heat} and Q_{w,p}^{classical}. Here Q_{w,p}^{classical} denotes the following supremum over all discs on the plane:

Q_{w,p}^{classical} := sup_{B(x,R)} (\frac{1}{|B(x, R)|} ∫_{B(x,R)} ωdA) · (\frac{1}{|B(x, R)|} ∫_{B(x,R)} ω^{− \frac{1}{p−1}} dA)^{p−1}

Obviously, there exists a positive absolute constant a such that for any function w

a Q_{w,p}^{classical} ≤ Q_{w,p}^{heat}

Remark. But 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 did not hinder out principal Theorem 0.5 because we still could estimate Q_{ω,p}^{heat} for special weights ω.

Theorem 0.8. There exists a finite absolute constant b such that

Q_{w,p}^{heat} ≤ b Q_{w,p}^{classical}

In conjunction with our main Theorem 0.5 this gives us the following very sharp weighted estimate of the Beurling operator (this result answers positively another question of [1]):

Theorem 0.9. For any A_{p} weight w and any p ≥ 2 we have

∥T ∥_{L^{p}(wdA)→L^{p}(wdA)} ≤ C(p)Q_{ω,p}^{classical}

In [1] it is proved that if k ∈ [0, 1), ∥μ∥_{∞} = k, f = f^{μ}, ω = |f_{z} ∘ f^{−1}|^{p−2}, p ∈ [2, 1 + \frac{1}{k}), then the weight ω belongs to A_{p} and

Q_{ω,p}^{classical} ≤ \frac{C}{1 + \frac{1}{k} − p}  (0.5)

So instead of using Theorem 0.7 one can use Theorem 0.9 in conjunction with (0.5). As a result we always get that for the special weight function ω = |f_{z} ∘f^{−1}|^{p−2}, p ∈ [2, 1+ \frac{1}{k}), f := f^{μ}, ∥μ∥_{∞} = k, k ∈ [0, 1) we always have

Q_{ω,p}^{heat} ≤ \frac{C}{1 + \frac{1}{k} − p}  (0.6)

And the use of Theorem 0.6 proves the 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 [1] which deduces from the injectivity of I −μT at the critical exponent the quasiregularity of weakly quasiregular maps.

1. The sharp weighted estimate for the Beurling operator in L^{2}(wdA)

In this section w is an arbitrary positive function on the plane. We want the estimate

∥T ∥_{L^{2}(wdA)→L^{2}(wdA)} ≤ C Q_{w,2}^{heat}  (1.1)

1. The sharp weighted estimate for the Beurling operator in L^{2}(wdA) 7

As one will see in Section 3 estimate (1.1) implies readily a much more classically looking estimate

∥T ∥_{L^{2}(wdA)→L^{2}(wdA)} ≤ C Q_{w,2}^{classical}  (1.2)

However, at this moment, it is not clear to us how to get the last estimate without the first one. So to get (1.1) is our main goal now.

The operator T is given in Fourier domain (ξ_{1}, ξ_{2}) by the multiplier \frac{ζ̄}{ζ} = \frac{ζ̄^{2}}{|ζ|^{2}} = \frac{(ξ_{1}−iξ_{2})^{2}}{ξ_{1}^{2}+ξ_{2}^{2}} =

\frac{ξ_{1}^{2}}{ξ_{1}^{2}+ξ_{2}^{2}} − \frac{ξ_{2}^{2}}{ξ_{1}^{2}+ξ_{2}^{2}} − 2i\frac{ξ_{1}ξ_{2}}{ξ_{1}^{2}+ξ_{2}^{2}} . Thus, operator T can be written as T = R_{1}^{2} − R_{2}^{2} − 2iR_{1}R_{2}, where R_{1}, R_{2} are Riesz transforms on the plane, see [30] 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 π/4 rotation of the plane. So the multiplier operators are related by

m_{2} = U_{ρ} m_{1} U_{ρ}^{−1}

where U_{ρ} is an operator of ρ-rotation in (x_{1}, x_{2}) plane. But for any operator K we have

∥U_{ρ} K U_{ρ}^{−1}∥_{L^{2}(wdA)→L^{2}(wdA)} = ∥K∥_{L^{2}(w∘ρ^{−1}dA)→L^{2}(w∘ρ^{−1}dA)}

Combining this with the fact that Q_{w,2}^{heat} = Q_{w∘ρ^{−1},2}^{heat} for any rotation, we come to conclusion that we would need the desired estimate (1.1) only for m_{1} = R_{1}^{2} − R_{2}^{2}. Actually, we will get

∥R_{i}^{2}∥_{L^{2}(wdA)→L^{2}(wdA)} ≤ C Q_{w,2}^{heat}, i = 1, 2  (1.3)

To prove (1.2) we fix, say, R_{1}^{2} and two test functions φ, ψ ∈ C_{0}^{∞}. We will be using heat extensions. Given f on the plane, its heat extension is given by the formula

f(y, t) := \frac{1}{π t} ∫∫ _{ℝ^{2}} f(x) exp(−\frac{∥x − y∥^{2}}{t}) dx_{1}dx_{2}, (y, t) ∈ ℝ_{+}^{3}

We will usually denote by the same letter the function and its heat extension.

Lemma 1.1. Let φ, ψ ∈ C_{0}^{∞} . Then integral ∫∫∫ \frac{∂φ}{∂x_{1}} · \frac{∂ψ}{∂x_{1}} dx_{1}dx_{2}dt converges absolutely and

∫∫ R_{1}^{2}φ · ψ dx_{1}dx_{2} = −2 ∫∫∫ \frac{∂φ}{∂x_{1}} · \frac{∂ψ}{∂x_{1}} dx_{1}dx_{2}dt  (1.4)

Proof. The proof of this lemma is actually trivial. It is based on a well-known fact that a function is an integral of its derivative. Well, also Parceval’s formula is involved. Here is this proof. We consider φ, ψ ∈ C_{0}^{∞} and now

∫∫ ψR_{1}^{2}φdx_{1}dx_{2} = ∫∫ \frac{ξ_{1}^{2}}{ξ_{1}^{2} + ξ_{2}^{2}} φ̂(ξ_{1}, ξ_{2})ψ̂(−ξ_{1}, −ξ_{2})dξ_{1}dξ_{2} =

2 ∫∫ ∫_{0}^{∞} e^{−2t(ξ_{1}^{2}+ξ_{2}^{2})}ξ_{1}^{2}φ̂(ξ_{1}, ξ_{2})ψ̂(ξ_{1}, ξ_{2})dξ_{1}dξ_{2}dt =

−2 ∫_{0}^{∞} ∫∫ iξ_{1}φ̂(ξ_{1}, ξ_{2})e^{−t(ξ_{1}^{2}+ξ_{2}^{2})} · iξ_{1}ψ̂(−ξ_{1}, −ξ_{2})e^{−t(ξ_{1}^{2}+ξ_{2}^{2})}dξ_{1}dξ_{2}dt =

−2 ∫_{0}^{∞} ∫∫ \frac{∂φ}{∂x_{1}} (x_{1}, x_{2}, t)\frac{∂ψ}{∂x_{1}} (x_{1}, x_{2}, t)dx_{1}dx_{2}dt =

−2 ∫∫∫ _{ℝ_{+}^{3}} \frac{∂φ}{∂x_{1}} (x_{1}, x_{2}, t)\frac{∂ψ}{∂x_{1}} (x_{1}, x_{2}, t)dx_{1}dx_{2}dt

Here we used Parceval’s formula twice, we also used absolute convergence of the integrals ∫∫∫_{ℝ_{+}^{3}} e^{−2t(ξ_{1}^{2}+ξ_{2}^{2})}ξ_{1}^{2}φ̂(ξ_{1}, ξ_{2})ψ̂(ξ_{1}, ξ_{2})dξ_{1}dξ_{2}dt, ∫∫∫_{ℝ_{+}^{3}} \frac{∂φ}{∂x_{1}} (x_{1}, x_{2}, t)\frac{∂ψ}{∂x_{1}} (x_{1}, x_{2}, t)dx_{1}dx_{2}dt. For the first integral this is obvious. The absolute convergence of the second integral can be quite easily proved. We leave this to the reader as an exercise.

Our next goal will be to estimate the right part of (1.4) from above.

Theorem 1.2. For any φ, ψ ∈ C_{0}^{∞}, and any positive function w on the plane we have

∫∫∫ _{ℝ_{+}^{3}} |\frac{∂φ}{∂x_{1}} | |\frac{∂ψ}{∂x_{1}} | dx_{1}dx_{2}dt ≤ A Q_{w,2}^{heat} (∫∫ |φ|^{2}wdx_{1}dx_{2} + ∫∫ |ψ|^{2} \frac{1}{w}dx_{1}dx_{2})

where A is an absolute constant.

In the proof we will be using the following key result. In what follows d^{2}f denotes the Hessian form (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} < Xs, y^{2} < Y r, 1 < rs < 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 the small neighborhood of K such that

1) 0 ≤ B ≤ 5Q(X + Y)
2) − d^{2}B ≥ |dx||dy|

We will prove Theorem (1.3) a bit later. Let us now use it to obtain the proof of Theorem (1.2).

Proof. Given a non constant smooth w which is constant outside some big ball let us consider Q = Q_{w,2}^{heat}. We consider only the case w ∈ A_{2}, that is Q < ∞. Otherwise there is nothing to prove. Consider two nonnegative functions φ, ψ ∈ C_{0}^{∞}. Now we take B = B_{Q,K}, where compact K will be chosen later.

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 6-vector v,

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) ∈ ℝ_{+}^{3}. Also we can fix any compact subset M of the open set ℝ_{+}^{3} and guarantee that for (x, t) ∈ M vector v lies in compact K. Just take K large enough after noticing that for our w and for compactly supported φ, ψ, mapping (x, t) → v(x, t) maps compacts in ℝ_{+}^{3} to compacts in D_{Q}.

We want to apply Green’s formula to b(x, t). To do this we introduce Green’s function G(x, t) as in [11].

(\frac{∂}{∂t} + Δ) G = −δ_{0,1/2} in C(1, 1) = B(0, 1) × (0, 1)
G = 0 on ∂′C(1, 1) = ∂B(0, 1) × (0, 1)
G = 0 when t = 1

Here δ_{0,1/2} is a δ function at the point (0, 1/2). It is important to keep in mind that

G(x, 0) ≥ a(1 − ∥x∥)  (1.5)

where a is a positive absolute constant.

We also need Green’s function in the cylinder C(R, R^{2}) = B(0, R) × (0, R^{2}):

(\frac{∂}{∂t} + Δ) G^{R} = −δ_{0,R^{2}/2} in C(R, R^{2}) = B(0, R) × (0, R^{2})
G^{R} = 0 on ∂′C(R, R^{2}) = ∂B(0, R) × (0, R^{2})
G^{R} = 0 when t = R^{2}

One can easily see that the following connection holds:

Lemma 1.4.

G^{R}(x, t) = \frac{1}{R^{2}} G(\frac{x}{R}, \frac{t}{R^{2}} )

We are ready to apply Green’s formula to b(x, t). 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}, x_{2}) as always):

b(0, R^{2}/2) ≤ 5 Q_{w,2}^{heat} ((φ^{2}w)(0, R^{2}/2) + (ψ^{2}w^{−1})(0, R^{2}/2))

Thus,

b(0, R^{2}/2) ≤ \frac{5}{π} Q_{w,2}^{heat} (∫∫ \frac{2}{R^{2}} (φ^{2}w)(x)e^{− \frac{2∥x∥^{2}}{R^{2}}} dx_{1}dx_{2} + ∫∫ \frac{2}{R^{2}} (ψ^{2}w^{−1})(x)e^{− \frac{2∥x∥^{2}}{R^{2}}} dx_{1}dx_{2})

Now Green’s formula in C(R, R^{2}):

b(0, R^{2}/2) = − ∫∫∫ _{C(R,R^{2})∩\{t>ε\}} (\frac{∂}{∂t} + Δ) G^{R}(x, t) b(x, t) dx_{1}dx_{2}dt

= ∫∫∫ _{C(R,R^{2})∩\{t>ε\}} G^{R}(x, t) (\frac{∂}{∂t} − Δ) b(x, t) dx_{1}dx_{2}dt

+ ∫∫ _{B(0,R)} b(x, ε)G^{R}(x, ε)dx_{1}dx_{2}

+ ∫∫ _{∂′C(R,R^{2})∩\{t>ε\}} (\frac{∂b}{∂n_{outer}} G^{R} − \frac{∂G^{R}}{∂n_{outer}} b) dsdt

= ∫∫∫ _{C(R,R^{2})∩\{t>ε\}} G^{R}(x, t) (\frac{∂}{∂t} − Δ) b(x, t) dx_{1}dx_{2}dt

+ ∫∫ _{B(0,R)} b(x, ε)G^{R}(x, ε)dx_{1}dx_{2} + ∫∫ _{∂′C(R,R^{2})∩\{t>ε\}} \frac{∂G^{R}}{∂n_{inner}} b dsdt

≥ ∫∫∫ _{C(R,R^{2})∩\{t>ε\}} G^{R}(x, t) (\frac{∂}{∂t} − Δ) b(x, t) dx_{1}dx_{2}dt

The last inequality is clear–the double integrals are both nonnegative, because b is nonnegative and because G^{R} is nonnegative and vanishing on the side boundary.

Let us combine estimates on b(0, R^{2}/2) into

∫∫∫ _{C(R,R^{2})∩\{t>ε\}} G^{R}(x, t) (\frac{∂}{∂t} − Δ) b(x, t) ≤ \frac{10}{πR^{2}} (∫∫ φ^{2}w + ∫∫ ψ^{2}w^{−1})  (1.6)

Fix R and ε and choose compact M = {(x, t) : x ∈ clos(B(0, R)), ε ≤ t ≤ R^{2}}. The next calculation is simple but it is a key point of the proof. It uses the notations

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)).

Lemma 1.5. If (x, t) ∈ M then

(\frac{∂}{∂t} − Δ) b(x, t) = ((−d^{2}B) \frac{∂v}{∂x_{1}} , \frac{∂v}{∂x_{1}} )_{ℝ^{6}} + ((−d^{2}B) \frac{∂v}{∂x_{2}} , \frac{∂v}{∂x_{2}} )_{ℝ^{6}}

Proof.

\frac{∂}{∂t}b = (∇B, \frac{∂v}{∂t})_{ℝ^{6}}

Δb = ((d^{2}B)\frac{∂v}{∂x_{1}} , \frac{∂v}{∂x_{1}} )_{ℝ^{6}} + ((d^{2}B)\frac{∂v}{∂x_{2}} , \frac{∂v}{∂x_{2}} )_{ℝ^{6}} + (∇B, Δv)_{ℝ^{6}}

We just used chain rule. Now

(\frac{∂}{∂t} − Δ) b = (∇B, (\frac{∂v}{∂t} − Δv)_{ℝ^{6}} − ((d^{2}B)\frac{∂v}{∂x_{1}} , \frac{∂v}{∂x_{1}} )_{ℝ^{6}} − ((d^{2}B)\frac{∂v}{∂x_{2}} , \frac{∂v}{∂x_{2}} )_{ℝ^{6}}

But the first term is zero because all entries of vector v are solutions of the heat equation.

But in M by Theorem 1.3

−d^{2}B ≥ |dx||dy|

So Lemma 1.5 gives for (x, t) ∈ M:

(\frac{∂}{∂t} − Δ) b(x, t) ≥ |\frac{∂φ}{∂x_{1}} | |\frac{∂ψ}{∂x_{1}} | + |\frac{∂φ}{∂x_{2}} | |\frac{∂ψ}{∂x_{2}} |  (1.7)

Combining (1.6) (1.7) we get

∫∫∫ _{M} G^{R}(x, t) (|\frac{∂φ}{∂x_{1}} | |\frac{∂ψ}{∂x_{1}} | + |\frac{∂φ}{∂x_{2}} | |\frac{∂ψ}{∂x_{2}} |) ≤ \frac{10Q_{w,2}^{heat}}{πR^{2}} (∫∫ φ^{2}w + ∫∫ ψ^{2}w^{−1}) (1.8)

Now it is time to use Lemma 1.4. So (1.8) implies

∫∫∫ _{M} G(\frac{x}{R}, \frac{t}{R^{2}} ) (|\frac{∂φ}{∂x_{1}} | |\frac{∂ψ}{∂x_{1}} | + |\frac{∂φ}{∂x_{2}} | |\frac{∂ψ}{∂x_{2}} |) ≤ \frac{10Q_{w,2}^{heat}}{π} (∫∫ φ^{2}w + ∫∫ \frac{ψ^{2}}{w} ) (1.9)

But M = {(x, t) : x ∈ clos(B(0, R)), ε ≤ t ≤ R^{2}}. Let us fix any compact M_{0} in ℝ_{+}^{3} and choose R and ε in such a way that M_{0} ⊂ M. All large R and small ε will suit us. Restrict the integration in (1.9) to M_{0} and make R → ∞. Taking into account that

G(0, 0) = a > 0

We obtain

∫∫∫ _{M_{0}} (|\frac{∂φ}{∂x_{1}} | |\frac{∂ψ}{∂x_{1}} | + |\frac{∂φ}{∂x_{2}} | |\frac{∂ψ}{∂x_{2}} |) ≤ \frac{10Q_{w,2}^{heat}}{πa} (∫∫ φ^{2}w + ∫∫ ψ^{2}w^{−1})

But M_{0} is an arbitrary compact set in the upper half space. Therefore,

∫∫∫ _{ℝ_{+}^{3}} (|\frac{∂φ}{∂x_{1}} | |\frac{∂ψ}{∂x_{1}} | + |\frac{∂φ}{∂x_{2}} | |\frac{∂ψ}{∂x_{2}} |) ≤ \frac{10Q_{w,2}^{heat}}{πa} (∫∫ φ^{2}w + ∫∫ ψ^{2}w^{−1})  (1.10)

This implies immediately

∫∫∫ _{ℝ_{+}^{3}} (|\frac{∂φ}{∂x_{1}} | |\frac{∂ψ}{∂x_{1}} | + |\frac{∂φ}{∂x_{2}} | |\frac{∂ψ}{∂x_{2}} |) ≤ \frac{20Q_{w,2}^{heat}}{πa} ∥f∥_{L^{2}(w)}∥ψ∥_{L^{2}(w^{−1})}  (1.11)

To get (1.11) we just substitute into (1.10) φ/t, tψ and minimized over t. Theorem 1.2 is proved.

Now we couple inequality (1.11) with Lemma 1.1. Then we get the right weighted estimates for R_{1}^{2}, R_{2}^{2}, R_{1}R_{2}, T. In particular we proved Theorem 0.6 for the case p = 2 up to proving Theorem 1.3.

1.1. Proof of Theorem 1.3

We start with a much simpler “model” operator—T_{σ}. The logic will be the following. We want to get a sharp weighted estimate of ∥T_{σ}∥_{L^{2}(w)→L^{2}(w)} via A_{2} characteristic of w. Strangely enough, we first consider a two weighted estimate of ∥T_{σ}∥_{L^{2}(u)→L^{2}(v)} with different weights u, v. In the paper of Nazarov, Treil, Volberg one can find, see [29], that this norm is attained on some ”simple” test functions—and this holds for every pair u, v. So for u = v = w also. But on the family T of test functions one can compute the N_{w,2}(T_{σ}) := sup{∥T_{σ}t∥_{L^{2}(w)} : t ∈ T , ∥t∥_{L^{2}(w)} = 1}. It turns out that N_{w,2}(T_{σ}) ≈ Q_{w,2}^{classical}. This is done in J. Wittwer’s work [24]. See also [23]. Thus, we get ∥T_{σ}∥_{L^{2}(w)→L^{2}(w)} = N_{w,2}(T_{σ}) ≈ Q_{w,2}^{classical}.

This fact about sharp one weight estimate of the model operator (notice that its proof used the fact from two weight theory!) will be used to construct the Bellman function for our operator. Application of Green’s formula to this Bellman function was the content of the proof of Theorem 1.2.

So let us show what is the model operator, what is its sharp weighted estimate, and how one obtains a special function (Bellman function) from this estimate.

Consider the family of dyadic singular operators T_{σ}:

T_{σ}f = Σ_{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, dx)), and σ(I) = ±1. We call the family T_{σ} the martingale transform. It is a dyadic analog of Calderón-Zygmund operators. The following questions concerning T_{σ} were asked.

1) What is the necessary and sufficient conditions for sup_{σ} ∥T_{σ}∥_{L^{2}(u)→L^{2}(v)} < ∞? This type of questions will be referred to as “two-weight” ones.

2) What is the sharp bound on sup_{σ} ∥T_{σ}∥_{L^{2}(w)→L^{2}(w)} in terms of w? How to compute sup_{σ} ∥T_{σ}∥_{L^{2}(w)→L^{2}(w)}? These are questions about “one weight” estimates.

Notice that the logic behind asking such questions was that these are dyadic analogs of notoriously difficult questions about “classical” Calderón-Zygmund operators like the Hilbert transform, The Riesz transforms, the Beurling transform. Dyadic model is supposed to be easier than the continuous one. This turned out to be true. The answers to the question above were found, see [29], [24]. Moreover these answers give the key to answering questions about “classical” Calderón-Zygmund operators.

Strangely enough, the answer to the second question (which seems to be easier, because it is of a “one weight” type) so far required the ideas from “two-weight” case. Here is our explanation to this phenomenom. The necessary and sufficient condition on (u, v) to answer the first question were given in [29]. They amount to the fact that sup_{σ} ∥T_{σ}∥_{L^{2}(u)→L^{2}(v)} is almost attained on the family of simple test functions. This fact has a beautiful consequences in one weight case. In fact, then sup_{σ} ∥T_{σ}∥_{L^{2}(w)→L^{2}(w)} is attainable (almost) on the family of simple test functions. But one may try to compute sup_{σ} ∥T_{σ} t∥_{L^{2}(w)} for every element of this test family, thus getting a good estimate for the norm sup_{σ} ∥T_{σ}∥_{L^{2}(w)→L^{2}(w)}. Test functions are rather simple, so this program can be carried over. This has been done in [24]. Here is the result. Remind that

Q_{w,2}^{dyadic} := sup_{I∈D}⟨w⟩_{I}⟨w^{−1}⟩_{I}

Theorem 1.6.

sup_{σ} ∥T_{σ}∥_{L^{2}(w)→L^{2}(w)} ≤ A Q_{w,2}^{dyadic}

Let us write down Theorem 1.6 as follows

sup_{σ(I)=±1} |Σ_{I∈D}σ(I) (f, h_{I}) (g, h_{I})| ≤ AQ_{w,2}^{dyadic}∥f∥_{L^{2}(w)}∥g∥_{L^{2}(w^{−1})}

Or,

Σ_{I∈D}|(f, h_{I})| |(g, h_{I})| ≤ AQ_{w,2}^{dyadic}∥f∥_{L^{2}(w)}∥g∥_{L^{2}(w^{−1})}

This inequality is scaleless, so we will write it as

∀J ∈ D, \frac{1}{|J|}Σ_{I∈D, I⊂J}|⟨f⟩_{I_{−}} − ⟨f⟩_{I_{+}}| |⟨g⟩_{I_{−}} − ⟨g⟩_{I_{+}}||I| ≤ AQ_{w,2}^{dyadic}⟨f^{2}w⟩_{J}^{1/2}⟨g^{2}/w⟩_{J}^{1/2} (1.12)

Here I_{−}, I_{+} are the left and the right halves of I, and ⟨·⟩_{l} 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) = sup\{\frac{1}{|J|}Σ_{I∈D, I⊂J}|⟨f⟩_{I_{−}} − ⟨f⟩_{I_{+}}| |⟨g⟩_{I_{−}} − ⟨g⟩_{I_{+}}||I| :
⟨f⟩_{J} = x, ⟨g⟩_{J} = y, ⟨w⟩_{J} = r, ⟨w^{−1}⟩_{J} = s,
⟨f^{2}w⟩_{J} = X, ⟨g^{2}/w⟩_{J} = Y, w ∈ A_{2}^{dyadic}, Q_{w,2}^{dyadic} ≤ Q}

Obviously, 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} ≤ Xs, y^{2} ≤ Y r, 1 ≤ rs ≤ Q}

By (1.12) it satisfies

0 ≤ B ≤ AQX^{1/2}Y^{1/2}  (1.13)

We are going to prove that it also satisfies the following “differential” inequality. Denote v := (X, Y, x, y, r, s). Let v, v_{−} = (X_{−}, Y_{−}, x_{−}, y_{−}, r_{−}, s_{−}), v_{+} = (X_{+}, Y_{+}, x_{+}, y_{+}, r_{+}, s_{+}) ∈ R_{Q} and let v = \frac{1}{2}(v_{−} + v_{+}). Then

B(v) − \frac{1}{2} (B(v_{+}) + B(v_{−})) ≥ |x_{+} − x_{−}||y_{+} − y_{−}|  (1.14)

In fact, let f, g, w be almost maximizing B(v) (on the interval J), let f_{+}, g_{+}, w_{+} do this for B(v_{+}), f_{−}, g_{−}, w_{−} do this for B(v_{−}). But using scalelessness of B, lets us put f_{+}, g_{+}, w_{+} on J_{+} and f_{−}, g_{−}, w_{−} on J_{−}. Then we have a “gargoyle” functions

F = \begin{cases}f_{+} & onJ_{+} \\ f_{−} & onJ_{−} \end{cases}  G = \begin{cases}g_{+} & onJ_{+} \\ g_{−} & onJ_{−} \end{cases}  W = \begin{cases}w_{+} & onJ_{+} \\ w_{−} & onJ_{−} \end{cases}

Obviously, ⟨F ⟩_{J} = \frac{1}{2}(x_{+} + x_{−}) = x, ⟨G⟩_{J} = y, ⟨W ⟩_{J} = r, ⟨W^{−1}⟩_{J} = s, ⟨F^{2}W ⟩_{J} = X, ⟨G^{2}W^{=1}⟩_{J} = Y. These numbers together form the vector v. In other words F, G, W compete with f, g, w in the definition (1.12) of Bellman function B(v). By this definition,

B(v) ≥ \frac{1}{|J|}Σ_{I∈D, I⊂J}|⟨F ⟩_{I_{−}} − ⟨F ⟩_{I_{+}}| |⟨G⟩_{I_{−}} − ⟨G⟩_{I_{+}}||I|

But almost optimality of f_{+}, g_{+}, w_{+} on J_{+} and f_{−}, g_{−}, w_{−} on J_{−} gives us (remind that F = f_{±} on J_{±}, G = g_{±} on J_{±}):

B(v_{+}) ≥ −ε + \frac{1}{|J_{+}|}Σ_{I∈D, I⊂J_{+}}|⟨F ⟩_{I_{−}} − ⟨F ⟩_{I_{+}}| |⟨G⟩_{I_{−}} − ⟨G⟩_{I_{+}}||I|

2. The sharp weighted estimate for the Beurling operator in L^{p}(wdA), p > 2 15

B(v_{−}) ≥ −ε + \frac{1}{|J_{−}|}Σ_{I∈D, I⊂J_{−}}|⟨F ⟩_{I_{−}} − ⟨F ⟩_{I_{+}}| |⟨G⟩_{I_{−}} − ⟨G⟩_{I_{+}}||I|

Combining all together, we get

\begin{matrix}B(v) & − \frac{1}{2}(B(v_{+}) + B(v_{−})) ≥ −ε + |⟨F ⟩_{J_{−}} − ⟨F ⟩_{J_{+}}| |⟨G⟩_{J_{−}} − ⟨G⟩_{J_{+}} \\ = −ε+ & |⟨f_{−}⟩_{J_{−}} − ⟨f_{+}⟩_{J_{+}}| |⟨g_{−}⟩_{J_{−}} − ⟨g_{+}⟩_{J_{+}} = ε|x_{−} − x_{+}||y_{−} − y_{+}| \end{matrix}

We are done with (1.14) because ε is an arbitrary positive nimbler. Therefore, our B is a very concave function. We are going to mollify B to have its Hessian to 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 \frac{1}{ε^{6}} φ(\frac{v}{ε}), v ∈ ℝ^{6}, where φ is a bell shape infinitely differentiable function with support in the unit ball of ℝ^{6}. 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)

For any vector ξ = (ξ_{1}, ξ_{2}, ξ_{3}, ξ_{4}, ξ_{5}, ξ_{6}) ∈ ℝ^{6}

−(d^{2}B_{K,Q}ξ, ξ)_{ℝ^{6}} ≥ 2|ξ_{2}||ξ_{3}|  (1.16)

The factor 2 appears because B(v) − \frac{1}{2}(B(v_{+}) + B(v_{−})) in (1.14) corresponds to −\frac{1}{2}d^{2}B.

Theorem 1.3 is completely proved.

2. The sharp weighted estimate for the Beurling operator in L^{p}(wdA), p > 2

Here again T is a Beurling transform on the plane, dA denotes Lebesgue measure on the plane. Now we are going to prove

∥T ∥_{L^{p}(wdA)→L^{p}(wdA)} ≤ C(p) Q_{w,p}^{heat}  (2.1)

In Section 3 below it is shown how this implies

∥T ∥_{L^{p}(wdA)→L^{p}(wdA)} ≤ C(p) Q_{w,p}^{classical}  (2.2)

Let us remind that for the weights on ℝ we can introduce

Q_{w,p}^{dyadic} := sup_{I∈D}⟨w⟩_{I}⟨w^{− \frac{1}{p−1}} ⟩_{I}^{p−1}

Weight ρ := w^{− \frac{1}{p−1}} is called the dual weight, and

(Q_{ρ,p′}^{dyadic})^{p} = (Q_{w,p}^{dyadic})^{p′}  (2.3)

We will need the following result of St Buckley (we formulate its dyadic version). Let us denote by M^{d}f(x) = sup_{x∈I, I∈D}⟨|f|⟩_{I}. This is the dyadic maximal function.

Theorem 2.1. For any weight on the line ℝ

∥M^{d}∥_{L^{p}(wdx)→L^{p}(wdx)} ≤ C(p) Q_{ρ,p′}^{dyadic}, 1 < p < ∞

We will use it now to extend Theorem 1.6 to p > 2.

Theorem 2.2. For any weight on the line ℝ

sup_{σ} ∥T_{σ}∥_{L^{p}(wdx)→L^{p}(wdx)} ≤ C(p) Q_{w,p}^{dyadic}, 2 < p < ∞

Now we will present the proof which follows wordly the extrapolation proof of Rubio de Francia. We want only to be absolutely precise with constants.

Proof. Let w ∈ A_{p}^{dyadic}. Let T temporarily means any of T_{σ}, and let t = \frac{p−2}{p−1}.

(∫ |T f|^{p}wdx)^{\frac{2}{p}} = sup_{u≥0, ∥u∥_{L \frac{p′}{t} (wdx)}≤1} ∫ |T f|^{2}uwdx ≤ sup_{v∈F} ∫ |T f|^{2}vwdx  (2.4)

Here F is a family of functions v of the following type

v = v(u) := u + \frac{Su}{2∥S∥} + \frac{S^{2}u}{4∥S∥^{2}} + \frac{S^{3}u}{8∥S∥^{3}} + ⋯,

where u runs over the unit ball of L^{\frac{p′}{t}} (wdx), and where non-linear operator S is given by the formula

S(u) := (w^{−1}M^{d}(u^{\frac{1}{t}} w))^{t} ,

and ∥S∥ denotes ∥S∥_{L^{\frac{p′}{t}} (wdx)→L^{\frac{p′}{t}} (wdx)}.

Now let us prove

1) u ≤ v,

2) ∥v∥_{L^{\frac{p′}{t}} (wdx)} ≤ 2∥u∥_{L^{\frac{p′}{t}} (wdx)}, 3) vw ∈ A_{2}^{dyadic}, Q_{vw,2}^{dyadic} ≤ Q_{w,p}^{dyadic}. The first is clear. To prove the second inequality we notice that ∥S∥ is finite. Then the

second inequality follows immediately. So let us prove this finiteness.

∥S∥ ≤ C(p)(Q_{w,p}^{dyadic})^{t}  (2.5)

In fact, using Theorem 2.1 we get

∫ (Su)^{p′/t}wdx = ∫ (M^{d}(u^{1/t}w))^{p′} w^{1−p′}dx =
∫ (M^{d}(u^{1/t}w))^{p′} w^{− \frac{1}{p−1}} dx = ∫ (M^{d}(u^{1/t}w))^{p′} ρdx
≤C(p)(Q_{w,p}^{dyadic})^{p′} ∫ u^{p′/t}w^{p′}w^{1−p′}dx = C(p)(Q_{w,p}^{dyadic})^{p′} ∫ u^{p′/t}wdx

Therefore (2.5) is proved. Let us now check that

⟨uw⟩_{I}⟨(S(u)w)^{−1}⟩_{I} ≤ (Q_{w,p}^{dyadic})^{1−t}, ∀I ∈ D, u ≥ 0  (2.6)

In fact, (1−t)(1−p) = 1, and using this and the pointwise estimate on (M^{d}(u^{1/t}w)) (x) ≥ ⟨u^{1/t}w⟩_{I}, ∀x ∈ I, we get

⟨uw⟩_{I}⟨(M^{d}(u^{1/t}w))^{−t} w^{− \frac{1}{p−1}} ⟩_{I} ≤ ⟨u^{1/t}w⟩_{I}^{t}⟨w⟩_{I}^{1−t} · ⟨u^{1/t}w⟩_{I}^{−t}⟨w^{− \frac{1}{p−1}} ⟩_{I}

Therefore

⟨uw⟩_{I}⟨(M^{d}(u^{1/t}w))^{−t} w^{− \frac{1}{p−1}} ⟩_{I} ≤ ⟨w⟩_{I}^{1−t}⟨w^{− \frac{1}{p−1}} ⟩_{I}^{(1−t)(1−p)} ≤ (Q_{w,p}^{dyadic})^{1−t}

We are done with (2.6).

Now we have at last our third estimate:

∀v ∈ F Q_{vw,2}^{dyadic} ≤ C(p)Q_{w,p}^{dyadic}  (2.7)

In fact, by its definition, S(v) ≤ 2∥S∥v. So using (2.5) and (2.6), we get

(∫ |T f|^{p}wdx)^{\frac{2}{p}} ≤ sup_{v∈F} ∫ |T f|^{2}vwdx ≤ A(Q_{vw,2}^{dyadic})^{2} ∫ |f|^{2}vwdx

≤ AC(p)(Q_{w,p}^{dyadic})^{2} (∫ |f|^{p}wdx)^{2/p} ∥v∥_{L^{p′/t}(wdx)} ≤ 2AC(p)(Q_{w,p}^{dyadic})^{2}∥f∥_{L^{p}(wdx)}^{2} In the second inequality we used Theorem 1.6.

Thus

∥T_{σ}∥_{L^{p}(wdx)→L^{p}(wdx)} ≤ 2A C(p) Q_{w,p}^{dyadic}

Theorem 2.2 is completely proved.

Now Theorem 1.3 has the following p > 2 version.

Theorem 2.3. For any Q > 1, p > 2 define the domain D_{Q}^{p} := {0 < (X, Y, x, y, r, s) : x^{p} < Xs^{p−1}, y^{p′} < Y r^{p′−1}, 1 < rs^{p−1} < Q}. Let K be any compact subset of D_{Q}^{p}. Then there exists a function B = B_{Q,K}^{(p)}(X, Y, x, y, r, s) infinitely differentiable in the small neighborhood of K such that

1) 0 ≤ B ≤ C(p) Q X^{1/p}Y^{1/p′}

2) − d^{2}B ≥ |dx||dy|

Proof. The proof repeats wordly the proof of Theorem 1.3. Only now we base the considerations on Theorem 2.2 rather than Theorem 1.6.

Now we are ready to prove our main estimate (2.2). The proof repeats the one given in the previous Section for p = 2. Only instead of b(x, t) we consider

b^{(p)}(x, t) := B^{(p)}((f^{p}w)(x, t), (g^{p′}w^{− \frac{1}{p−1}} )(x, t), f(x, t), g(x, t), w(x, t), w^{− \frac{1}{p−1}} (x, t))

The rest stays identically the same, but the use of b^{(p)} explains why one would need to

choose Q at least as large as Q_{w,p}^{heat} = sup_{(x,t)∈ℝ_{+}^{3}} w(x, t) (w^{− \frac{1}{p−1}} (x, t))^{p−1}.

Inequality (2.2) is completely proved. We also proved our main Theorem 0.6.

3. The comparison of classical and heat A_{p} characteristics

In this section we are going to give an easy proof of

Theorem 3.1. There exists a finite absolute constant b such that

Q_{w,p}^{heat} ≤ b Q_{w,p}^{classical}

3. The comparison of classical and heat A_{p} characteristics 19

Proof. The constants will be called by letters c, C, they may vary from line to line and even within the same line. We introduce the following notations. B_{k} denotes B(0, 2^{k}), k = 0, 1, 2, ..., ⟨f⟩_{B} stands for the average \frac{1}{|B|} ∫_{B} fdA, f(B) stand for ∫_{B} fdA. If B = B(0, r),

then ⟨f⟩_{B}^{h} stands for \frac{1}{π r^{2}} ∫∫_{ℝ^{2}} f(x) exp(−\frac{∥x∥^{2}}{r^{2}} )dx_{1}dx_{2}.

Lemma 3.2. Suppose f, g are positive functions on the plane are such that sup_{B} ⟨f⟩_{B}⟨g⟩_{B} = A then there exists a finite absolute constant c such that

⟨f⟩_{B}⟨g⟩_{B}^{h} ≤ cA

for any disc B.

Proof. The scale invariance allows to prove this only for one disc B = B(0, 1). We start the estimate:

⟨f⟩_{B}⟨g⟩_{B}^{h} ≤ c ⟨f⟩_{B}Σ_{k}2^{2k} exp(−2^{2k−2})\frac{A}{⟨f⟩_{B_{k}}}

On the other hand ⟨f⟩_{B_{k}} > c⟨f⟩_{B_{k−1}} > … c^{k}⟨f⟩_{B} (remind that B means our unit disc). Plugging this in the inequality above we get

⟨f⟩_{B}⟨g⟩_{B}^{h} ≤ c⟨f⟩_{B}Σ_{k}C^{k} exp(−2^{2k−2})\frac{A}{⟨f⟩_{B}}

In other words

⟨f⟩_{B}⟨g⟩_{B}^{h} ≤ cAΣ_{k}C^{k} exp(−2^{2k−2}) = cA,

and the lemma is proved.

Now we want to prove Theorem 3.1. Fix B. Again by scale invarince it is enough to think that B = B(0, 1). By the previous lemma we know that

⟨f⟩_{B_{k}}⟨g⟩_{B_{k}}^{h} ≤ cA  (3.1)

for any k.

Now

⟨f⟩_{B}^{h}⟨g⟩_{B}^{h} ≤ c⟨g⟩_{B}^{h}Σ2^{2k} exp(−2^{2k−2})⟨f⟩_{B_{k}} ≤ c⟨g⟩_{B}^{h}Σ2^{2k} exp(−2^{2k−2})\frac{cA}{⟨g⟩_{B_{k}}^{h}}

The last inequality used (3.1).

On the other hand ⟨g⟩_{B_{k}}^{h} > c⟨g⟩_{B_{k−1}}^{h} > … c^{k}⟨g⟩_{B} (remind that B means our unit disc). Plugging this in the inequality above we get

⟨f⟩_{B}^{h}⟨g⟩_{B}^{h} ≤ c⟨g⟩_{B}^{h}Σ_{k}C^{k} exp(−2^{2k−2})\frac{cA}{⟨g⟩_{B}^{h}}

In other words

⟨f⟩_{B}^{h}⟨g⟩_{B}^{h} ≤ c^{2}AΣ_{k}C^{k} exp(−2^{2k−2}) = c^{2}A

And Theorem 3.1 is completely proved.

4. The injectivity at the critical exponent and the regularity of solutions of Beltrami equation

In [1] Theorem 0.4 has been proved. It was a conditional theorem reducing the question of injectivity of I − μT, I − T μ in L^{1+k}(ℂ) (k = ∥μ∥_{∞} < 1) to the estimate (0.1):

∥T ∥_{L^{p}(ωdA)→L^{p}(ωdA)} ≤ \frac{C}{1 + \frac{1}{k} − p}

where ω = |f_{z} ∘ f^{−1}|^{p−2}, f is a \frac{1+k}{1−k}-quasiconformal homeomorphism, p ∈ [2, 1 + \frac{1}{k}).

This estimate has been proved in the present paper. It has been considered even twice. First time when we proved the estimate of the norm ∥T ∥ via AQ_{w,p}^{heat}, then noticed that Q_{w,p}^{heat} ≈ Q_{w,p}^{classical}, and finally noticed that (see [1]) Q_{w,p}^{classical} ≤ C/(1 + \frac{1}{k} − p) for special ω’s which are certain powers of Jacobians of quasiconformal homeomorphisms. Second way of proving this estimate is again to start with the estimate of the norm ∥T ∥ via AQ_{w,p}^{heat}. Then we compute directly Q_{w,p}^{heat} for ω’s which are certain powers of Jacobians of quasiconformal homeomorphism, and again see that Q_{w,p}^{heat} ≤ C/(1 + \frac{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. Remind (see [1]) that the solution of

Fz̄ − μF_{z} = 0 in Ω ⊂ ℂ

can be called q-weakly quasiregular if it belongs to W_{loc}^{1,q}. Here μ ∈ L^{∞}(Ω), ∥μ∥_{∞} = k < 1. It is known (see [1]) that if q > 1 + k then q-weakly quasiregular map is actually in W_{loc}^{1,2+ε} for a certain positive ϵ. In particular, it is quasiregular. Therefore, it is continuous, open, discrete,... . By the way, the proof of this “self-improvement” 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 Gehring and Reich.

In [1] one can find an easy example which shows that this is no longer true for q < 1 + k. So the remaining question is about the critical exponent q = 1 + k.

Theorem 4.1. Let ∥μ∥_{∞} = k < 1. Then (1+k)-weakly quasiregular maps are also quasiregular.

We repeat the consideration from [1]. This is done for the sake of convenience of the reader.

Proof. Choose φ ∈ ℂ_{0}^{∞}(Ω). Set G = F φ.Then

Gz̄ − μG_{z} = (φz̄ − μφ_{z})F  (4.1)

Obviously G is a Cauchy transform of the compactly supported function ψ := Gz̄. Then (4.1) can be rewritten as (F_{0} := (φz̄ − μφ_{z})F):

(I − μT)ψ = F_{0} ∈ L^{2}(ℂ)  (4.2)

In fact, F ∈ W_{loc}^{1,1+k}, so by Sobolev’s theorem (φz̄ − μφ_{z})F ∈ L^{\frac{2+2k}{1−k}} (ℂ), and (φz̄ − μφ_{z})F has a 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 will converge in L^{2+ε}(ℂ) (we use the fact that ∥T ∥ in L^{p} is close to 1 when p is close to 2), and will have its support in supp φ. So it will be in L^{1+k}(ℂ) as well. Let us call it ψ_{0}. Now we have two solutions of (4.2) in L^{1+k}(ℂ)–ψ and ψ_{0}. But we already proved the injectivity of I − μT in L^{1+k}(ℂ). Therefore, ψ = ψ_{0}. Thus, Gz̄ = ψ ∈ L^{2+ε}(ℂ). Therefore, compactly supported function G is in W^{1,2+ε}(ℂ). This means that G is quasiregular, and, in particular, continuous, open, discrete,....

Theorem 4.1 is completely proved.

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