A gradient estimate for harmonic functions sharing the same zeros

Let u, v be two harmonic functions in the disk of radius two which have exactly the same set Z of zeros. We observe that the gradient of \log |u/v| is bounded in the unit disk by a constant which depends on Z only. In case Z is empty this goes back to Li-Yau's gradient estimate for positive harmonic functions. The general boundary Harnack principle gives H\"older estimates on \log |u/v|.


Background and statement
Consider a positive harmonic function u in the disk of radius two B 2 ⊂ C. The Harnack inequality gives a bound on | log u(z 1 ) − log u(z 2 )| where z 1 , z 2 run in the unit disk. This bound is independent of u. If we let v = 1 be the constant function then this is the same as saying that is bounded by a constant independent on u when z 1 , z 2 ∈ B 1 . More generally, it is natural to ask what can be said about the quotient u/v in case u, v are harmonic functions in B 2 which do change sign in B 2 and have exactly the same set of zeros. The aim of this short note is to give an answer to this question in two dimensions and to pose two related natural questions.
Let us assume for clarity of the introduction that u = f ·v for some smooth function f > 0 (in fact, below we show that this is always the case). The boundary Harnack principle (BHP) ([Anc78, Wu78, CFMS81, JK82, BV96,PV98]) applied to our situation shows that if Ω is a connected component of {u = 0} then log u/v is a C α -function near ∂Ω ∩ B 1 for some 0 < α < 1. In this note we make the observation, maybe known to experts, that in fact we have a C 1 -bound. Namely, we show that |∇ log u v | is bounded in B 1 . In the case v is the constant function and u is a positive harmonic function this goes back to Li-Yau's gradient estimate ( [LY86]).
Another point of view of our observation is to say that if in the BHP the harmonic functions can be extended across the boundary of the domain, then one has a C 1 -bound on log u/v. In fact, an example due to Carlos Kenig ( §8.2) shows that one cannot in general obtain a C 1 -bound even when the boundary of the domain is composed of straight lines. The precise result we show is Then for all u, v ∈ F (Z) u/v extends to a smooth nowhere vanishing function in B 2 and there exists a constant C Z > 0 such that

Two questions
We pose two natural questions which arise naturally from Theorem 1.2.
Deformations of the zero set. A related result close in spirit to Theorem 1.2 is proved in [Nad99]. In this paper Nadirashvili considers a variablesign harmonic function in B 2 , and proves the existence of a bound on |u| in B 1 in terms of the first 3k derivatives of u at 0, where k is the number of connected components of B 2 \ Z. Nadirashvili's result hints that it may be true that the constant C Z in Theorem 1.2 depends only on the number of components of B 2 \ Z. Equivalently, we ask whether it is true that the constant C Z depends only on the number of intersection points of Z with the circle {|z| = 3/2}.
Higher dimensions. We ask whether Theorem 1.2 is a coincidence of dimension two or stays true in dimension three. It seems that even the case where Z is the zero set of a quadratic harmonic polynomial in three variables is not known.
We hope to investigate both questions in the near future.

Idea of proof
We give two proofs of Theorem 1.2. Both proofs are built upon the cases where Z is in normal form (say, Z = {ℑz k = 0}). This is the reason why our method is restricted to two dimensions.
The first proof we give is based on the maximum principle in a similar spirit to [LY86]. In a normal case we find a certain positive definite quadratic form which leads to a Bochner type estimate. In the case where Z is empty (the Li-Yau case) the positivity of the quadratic form we define is evident. Then, we use the Bochner type estimate to get a gradient estimate.
The second (shorter) proof, due to Misha Sodin, is based on the Poisson formula for sectors of the plane.
We decided to keep both proofs in this note since the first proof makes almost no use of explicit formulas, and we hope it may be useful to extend Theorem 1.2 to situations where no explicit formulas exist.

Organization of the paper
The proof of the normal form case of Theorem 1.2 is given in Section 6. The reduction to the normal form case is done in Section 7. In Section 9 we give an alternative proof using Poisson formulas due to Misha Sodin. In sections 2-5 we develop different ingredients of the proof: In Section 2 we calculate the singular elliptic equation satisfied by the quotient of two harmonic functions. We also find a Bochner type formula. In Section 3 we define and prove the positivity of a certain quadratic form needed in the proof of a Bochner type Inequality proved in Section 4. In Section 5 we treat several expressions which involve logarithmic singularities. Finally, in Section 8 we give several examples which illustrate Theorem 1.2 and clarify it.

Acknowledgements
I am happy to thank Jozef Dodziuk for preliminary discussions on Li-Yau's gradient estimates. I am grateful to David Kazhdan and Leonid Polterovich for asking me questions which led me to the present work. I thank Benji Weiss for an interesting example ( §8). I thank Gady Kozma and Fedja Nazarov for their interest in this work and for referring me to the BHP. I especially thank Misha Sodin for his encouragement and for finding a second simple proof ( §9).
I am grateful to Carlos Kenig for his interest and for an illuminating example ( §8.2) calrifying the relation of this note to the BHP. I am grateful to Charlie Fefferman for his advice and interest. I thank S.-T. Yau for his support. This research was supported by ISF grant 225/10 and by BSF grant 2010214.

A non-negative quadratic form
Let v k (z) = ℑz k , where k is a non-negative integer. In this section we define a quadratic form related to v k and to equation (2.4) and show it is non-negative. This will play a key role in the proof of the Bochner type inequality in Lemma 4.1 Definition 3.1. Let X be a smooth vector field on B 2 . We define Proposition 3.2. For all vector fields X on B 2 the function Q k (X) is nonnegative. Moreover, we have More precisely, we show Proof. Let X, Y be two vector fields on B 2 , and let We compute in polar coordinates: The Levi-Civita connection is given by We see thatB We conclude that

An inequalityà la Bochner
In this section we prove an inequality of Bochner's type. This will be crucial to prove Theorem 1.2.

Study of ∇F, ∇v k /v k
We would like to show that 1 v k ∇F, ∇v k extends to a smooth function in B 2 . We will apply the following standard division lemma: Lemma 5.1. Let l(x, y) = ax + by. Let f ∈ C ∞ (B 2 ) be such that f (x, y) = 0 whenever l(x, y) = 0. Then there exists q ∈ C ∞ (B 2 ) such that f (x, y) = q(x, y)l(x, y).
Proof. We can assume without loss of generality that b = 0 and a = 0. Define Proof. v k can be expressed as a product of k linear factors. In fact v k (x, y) = a k k−1 l=0 y cos lπ k − x sin lπ k for some a k ∈ R. Hence, the lemma follows by induction from Lemma 5.1.
Lemma 5.3. There exists a smooth function G k such that Proof. By Lemma 5.2 it is enough to show that ∇F, ∇v k vanishes whenever v k does. This will follow from equation (2.2). Indeed, let p ∈ B 2 , p = 0 be such that v k (p) = 0. Then We will show that F ,θ (p) = 0. So, Since v k,θ (p) = 0 we conclude that h ,θ (p) = 0. Since, h ,θ (p) = 0 on the line passing through 0 and p, we also see that h ,rθ (p) = 0.
The next lemma shows that the expression ∇F, ∇v k /v k has the role of a second derivative of F on v k = 0. This will be important in the proof of Theorem 1.2.
Let p ∈ B 2 be a local maximum point of f . Then g(p) ≤ 0.
since p is a maximum point. If p = 0, let θ 0 = π 2k , then since 0 is a maximum point.
Existence of a positive quotient. We first show that |u/v k | defines a positive smooth function in B 2 . By Lemma 5.
If f (p) = 0 then u(p) = 0 and (∇u)(p) = 0. Hence, u has a zero of order d ≥ 2 at p, but since u is harmonic in a small ball B ε (p) centered at p this implies that Z ∩ B ε (p) is homeomorphic to the zero set of ℑz d . This is a contradiction. If p = 0 and f (p) = 0 then u(p) has a zero of order d ≥ k + 1 at 0 with Z k as a zero set which is impossible since u is harmonic.
A gradient estimate. We now proceed to proving the gradient estimate on log f . Let h = log f . h ∈ C ∞ (B 2 ) and u = e h v k . We let F = |∇h| 2 . At points q ∈ B 2 where v k (q) = 0 and ϕ(q) = 0 we have by Lemma 4.1 ∆(ϕF ) = (∆ϕ)F + 2 ∇ϕ, ∇F + ϕ∆F The last inequality follows from the Cauchy-Schwartz inequality.
Let p ∈ B 2 be a point where ϕF attains its maximum. Observe that by Lemma 5.3 and (6.1) It follows from (6.3), (6.4) and Lemma 5.4 that at the point p the following inequality is satisfied Dividing by F (p) and using (6.2) we get the following quadratic inequality in (ϕF ) 1/2 : from which we conclude that (ϕF ) 1/2 (p) ≤ 4(k+1) √ A. Since p is a maximum point, the same inequality is true for all q ∈ B 2 . In particular, in B 1 we get |∇h| ≤ 4(k + 1) √ A .

Proof of Theorem 1.2 -the general case
In this case we reduce the general case to the Z = Z k proved in Section 6.
Proof. Existence of a positive quotient. Fix v ∈ F (Z). Let u ∈ F (Z) be arbitrary. We will first show that |u/v| extends to a positive smooth function in B 2 . Let p ∈ Z. There exist k(p) ∈ N, a neighborhood N p ∋ p, an injective conformal map α p : N p → B 1 such that α p (p) = 0 and v•α −1 . v • α −1 p and u • α −1 p are harmonic functions both vanish exactly on Z k(p) ∩ W . By Section 6 we know that |(u • α −1 p )/(v • α −1 p )| defines a positive smooth function f 0 on W . Let f (z) := f 0 (α p (z)) be defined in N p . Then |f | > 0 and u = f v in N p . This shows that |u/v| extends to a smooth positive function in N p . Since p is arbitrary we conclude that |u/v| extends to a smooth positive function in B 2 .
A bound on ∇ log |u/v|. Let p ∈ Z ∩ B 1 . Let α p , k(p), N p , N ′ p be defined as above. Let us write u = e h v where h ∈ C ∞ (B 2 ). By Section 6 we know that By the chain rule it follows that |∇h| ≤ C p |α ′ p | in N ′ p . Since we can cover Z ∩ B 1 by a finite number of open sets of the form N ′ p we get that Since u, v do not vanish in a neighborhood of B 1 \ N Z , by [LY86] (or by the case Z = ∅ in section 6) we know that From inequalities (7.1) and (7.2) we get that |∇h| ≤ C Z in B 1 .

Harmonic functions sharing the same zeros
We give a few examples of harmonic functions with common zeros.
The zero set of u α in B 2 is the x-axis.
(ii) Let F : B 2 → C be holomorphic. Suppose |F | < π. The zero set of ℑe F is the same as the zero set of ℑF .
(iii) Let α ∈ R be such that 0 < |α| ≤ 1. Let f (z) be the branch of z α in C \ (−∞, 0] which admits positive values on the positive real axis. Define u(z) = ℑf (z + 2). The zero set of u in B 2 is the x-axis.
(iv) Let F : B 2 → B 2 be holomorphic. Let u = ℑ aF cF +d where a, c, d ∈ R are such that F = −d/c in B 2 . Then, the zero set of u in B 2 coincides with the zero set of ℑF .
(v) Let k ∈ N and let S k = {0 < arg z < π/k} ∩ B 2 . Let u be a positive harmonic function in S k , which is continuous on ∂S k and vanishes on ∂S k ∩ B 2 . One can extend u by reflections to a harmonic function in B 2 . The zero set of u coincides with the zero set of ℑz k .
(vi) (Due to Charlie Fefferman) Let u(x, y) = xy. Let v(x, y) = x 3 y − xy 3 . Then u and u + εv have the same zero set in B 2 if ε > 0 is sufficiently small.

An example clarifying the relation to the BHP
(Due to Carlos Kenig) Let t > 1 and let S = {0 < arg z < 2π/t} ⊂ C. Let v = ℑz t/2 . Observe that v is positive in S and v| ∂S = 0. Let p ∈ S be such that |p| > 2. Let G p be the Green function of S with singularity at p. We consider G p /v in S ∩ B 2 . Unless t is an integer, G p /v cannot be extended as a C 1 -function in a neighborhood of 0. Moreover, we note that |∇ log(G p /v)| is bounded in S ∩ B 1 if and only if t ≥ 2. A little simpler, we let u = ℑz t/2 + εℑz t for small ε > 0. Then log u/v has no bounded gradient in S ∩ B 1 unless t ≥ 2.
These families of examples (for 1 < t < 2) show that the BHP alone is not enough to obtain gradient estimates even if the boundary of the domain is nice (straight lines).

A second proof of the normal form case using Poisson's formula
This section is due to Misha Sodin. Let S k = {z ∈ C| |z| < 1, 0 < arg z < π/k} .
So we get |∇ log g| ≤ C 2 /C 1 in S k ∩ B 1/2 , and then from (9.1) |∇ log u r k sin kθ | ≤ Ck in S k ∩ B 1/2 . Finally we use reflection to get the stated result in the unit ball.