Bergman kernel and oscillation theory of plurisubharmonic functions

Based on Harnack’s inequality and convex analysis we show that each plurisubharmonic function is locally BUO (bounded upper oscillation) with respect to polydiscs of finite type but not for arbitrary polydiscs. We also show that each function in the Lelong class is globally BUO with respect to all polydiscs. A dimension-free BUO estimate is obtained for the logarithm of the modulus of a complex polynomial. As an application we obtain an approximation formula for the Bergman kernel that preserves all directional Lelong numbers. For smooth plurisubharmonic functions we derive a new asymptotic identity for the Bergman kernel from Berndtsson’s complex Brunn–Minkowski theory, which also yields a slightly better version of the sharp Ohsawa–Takegoshi extension theorem in some special cases.

whenever S is a ball or a polydisc, with center z. Here |S| denotes the Lebesgue measure of S and S means the Lebesgue integral. The above inequality implies φ ∈ L 1 loc ( ) and suggests to estimate the difference |φ − φ S |. The concept of BMO functions then enters naturally. Let S = S( ) be a family of relatively compact open subsets in . We say that φ ∈ L 1 loc ( ) has bounded mean oscillation (BMO) with respect to S if Let B M O( , S) denote the set of functions which are BMO with respect to S. When S is the set of balls in , this is the original definition of BMO functions due to John-Nirenberg [13]. A classical example of BMO functions is log |z|. It is also convenient to introduce local BMO functions as follows. For an open set 0 ⊂⊂ we define S| 0 to be the sets of all S ∈ S which are relatively compact in 0 . Let B M O loc ( , S) be the set of functions on which belong to B M O( 0 , S| 0 ) for every open set 0 ⊂⊂ . By using pluripotential theory, Brudnyi [6] was able to show that each psh function is locally BMO with respect to balls (see also [7] for stronger results concerning subharmonic functions in the plane). Recently, the first author found another approach to local BMO properties of psh functions by using the Riesz decomposition theorem and some basic facts of psh functions (cf. [9]). Benelkourchi et al. [1] showed that every function in the Lelong class L is globally BMO with respect to balls. Recall that L = u ∈ P S H(C n ) : lim sup |z|→∞ (u(z) − log |z|) < ∞ .
In this paper we propose a new and simpler approach based on the following basic observation: It is easier to look at the upper oscillation instead of the mean oscillation for psh functions.
To define the upper oscillation one simply uses sup S φ instead of φ S : Note that −U O S (−φ) is exactly the lower oscillation introduced by Coiffman-Rochberg (cf. [10], see also [16] for further properties). Since we see that bounded upper oscillation (BUO) implies BMO. One may define BU O( , S) and BU O loc ( , S) analogously as the case of BMO. Let P = P( ) denote the set of relatively compact polydiscs in and P N the set of polydiscs P ⊂⊂ of finite type N , i.e., where N > 0 and {r j } 1≤ j≤n is the polyradius of P.
Based on Harnack's inequality and convex analysis, we are able to show the following (2) P S H(D n ) B M O loc (D n , P) for n ≥ 2, where D n is the unit polydisc; (3) L ⊂ BU O(C n , P); more precisely, for every φ ∈ P S H(C n ) with where c is a constant, we have U O P (φ) < 3 n for all polydiscs P in C n .
For (deg p) −1 log | p| ∈ L where p is a complex polynomial, we even obtain a dimensionfree BUO estimate with respect to all compact convex sets.

Theorem 1.2 For every non-empty compact convex set A in
Remark (i) The above estimate is sharp, in fact, there exists a line segment A in C such that (ii) In particular, if A is a compact convex set in R n ⊂ C n and all coefficients of p are real, then we have which is closely related the classical Remez inequality for real polynomials. Theorem 1.2 also suggests to study the Remez inequality for complex polynomials (see [1] and [8] for related results). (iii) Notice that 1.278 < γ < 1.279. By (1.2) we have Such dimension-free estimate (with a slightly better constant 2 + log 2 ≈ 2.301) was first obtained by Nazarov et al. [15]. Our proof of Theorem 1.2 is elementary, however.
For φ ∈ P S H( ) we define the (weighted) Bergman kernel by For a vector a = (a 1 , . . . , a n ) with all a j > 0 we set P r a := {z ∈ C n : |z j | ≤ r a j , 1 ≤ j ≤ n}.
Here we will present an analogous but independent result, as an application of Theorem 1.1. For φ ∈ P S H(D n ) and t ∈ D n we define φ t (z) := φ(tz), tz := (t 1 z 1 , . . . , t n z n ).
A fundamental result of Berndtsson [2] implies that is psh on D n × D n . Theorem 1.3 For each a = (a 1 , . . . , a n ) with all a j > 0, there exists a number ε 0 = ε 0 (a, φ, ) such that holds for all ε ≤ ε 0 .
Although Theorem 1.3 makes sense only when φ is singular at the origin, it is of independent interest to study the relation between F(φ) and φ for smooth φ.

An enlightening example
To explain why BUO is easier than BMO, we will show that the upper oscillation of log |z| with respect to discs is computable. Recall that for every disc B in C. Then we have Proof If c ≤ |ẑ| then log |z| is harmonic in the disc {z : |z −ẑ| < c}, so that I (c) = log |ẑ|, in view of the mean-value equality. For c > |ẑ| we may write As log |z| is harmonic in {z : |z − c| < |ẑ|}, we get I (c) = log c.
Moreover, the bound is sharp.
It follows that

and the equality holds if and only if
This finishes the proof.
3 Proof of Theorem 1.1

One dimensional case
Let be a domain in C and φ a subharmonic function on . Recall that The idea is to use Harnack's inequality and a convexity lemma. Let us write where Applying Harnack's inequality to the nonpositive subharmonic function ψ : i.e., Here the constant 1/3 comes from the Poisson kernel of the unit disc since The following fact explains why we need such an estimate.
φ is continuous inẑ and r respectively; moreover, it is increasing with respect to r .
Proof Since sup B φ is a convex function of log r (see [11,Corollary 5.14]), it follows that J 1 is a continuous increasing function of r . The continuity of J 1 inẑ is obvious.
Let 0 be a relatively compact open subset in . Let δ 0 denote the distance between 0 and ∂ . By the above fact we see that if the radius r of B ⊂ 0 is less than δ 0 /2 then and if r ≥ δ 0 /2 then To estimate I 2 , we need the following convexity lemma which was communicated to the second author by Bo Berndtsson: Lemma 3.0.2 Let dμ be a probability measure on a Borel measurable subset S in R n with barycentert ∈ R n . Let f be a convex function on R n . Then where the first equality follows from the definition of barycenter.
With f (t) := φ {z:|z−ẑ|=e t r } we have Since f (t) is convex and d(e 2t ) is a probability measure on (−∞, 0) with barycenter at which implies Since f is convex, we get an analogous conclusion as Fact 1: Fact 2 J 2 is continuous inẑ and r respectively; moreover, it is increasing with respect to r . By a similar argument as above, we may verify that sup B⊂ 0 I 2 < ∞.

High dimensional case
The following result plays the role of Fact 1, 2.
We have where g j := ∂ g ∂t j . Notice that and ds is an increasing function of s ∈ (−∞, 0) by convexity of g. Thus we have Thus Since g is convex and increasing, we have which finishes the proof.
Let P := z ∈ C n : |z j −ẑ j | < r j , 1 ≤ j ≤ n ⊂ be a polydisc of type N , i.e., Similar as above, we write where and ∂ P := {z ∈ C n : |z j −ẑ j | = r j , 1 ≤ j ≤ n} is the Shilov boundary of P. Applying Harnack's inequality (see [14, p. 186]) n-times, we get the following Since both sup P φ and φ ∂ P are continuous inẑ j and convex increasing with respect to log r j for all j, it follows from Lemma 3.0.3 (through a similar argument as the onedimensional case) that for every open set 0 ⊂⊂ , which finishes the proof of the first part of Theorem 1.1.

A counterexample
For the second part of Theorem 1.1, we need to construct a counterexample. For the sake of simplicity, we only consider the case n = 2. It suffices to verify the following where D 2 r := (z, w) ∈ C 2 : |z| < r 1 , |w| < r 2 .
The following lemma shows that Fact 1, 2 is no more true for general bidiscs.
Proof The first conclusion follows by a straightforward calculation. For (3.2) it suffices to note that The proof is complete.
Let us first verify that φ / ∈ BU O loc (D 2 , P).
Integrate by parts with respect to t and s successively, we may write where and Obviously, By a similar argument as Lemma 3.1.2, we conclude the proof of Theorem 3.1.

Lelong class
In this section we shall prove the third part of Theorem 1.1. The key ingredient is the following counterpart of Lemma 3.0.3.

Lemma 3.1.3 Let g(t)
= g(t 1 , . . . , t n ) be a convex function on R n which is increasing in each variable. Assume that Then for every M > 0 we have By the assumption, we have for every s ≥ 0, so that The proof is complete.

Proof of the third part of Theorem 1.1 Again for any polydisc
we may write where By Lemma 3.0.4 we have Put P t := z ∈ C n : |z j −ẑ j | < e t j r j , 1 ≤ j ≤ n and f 1 (t) := sup P t φ. Since φ ∈ L, we know that for some constant c 1 1 the function f 1 − c 1 satisfies the assumption in Lemma 3.1.3, so that sup P φ − sup 1 2 P φ = f 1 (0) − f 1 (− log 2) ≤ log 2, which in turn implies Moreover, we infer from Lemma 3.0.5 that Applying Lemma 3.1.3 in a similar way as above, we have which finishes the proof.

Proof of Theorem 1.2
The starting point is the following Definition 4.0.1 (γ -constant) We shall define the constant γ as the BUO norm of log |z| on C with respect to all line segments. More precisely, where [a, b] denotes the line segment connecting a and b, and the upper oscillation is defined by The key step is to show the following Since log |z| is S 1 -invariant, by a rotation of z, we may assume that Thus It suffices to verify that γ satisfies (4.1). To see this, put and write Since it follows that f (t) = 0 if and only if i.e., 1 − a + log(−a) = 0.
Thus we have where a 0 is determined by which gives It is clear that (4.2) is equivalent to (4.1).
Since a translation of a line segment is still a line segment, we know that log |z − z 0 | and log |z| have the same line segment BUO norm. This fact can be used to estimate the line segment BUO norm of log | p| for general polynomials p. In fact, if we write p = a 0 (z − a 1 ) n 1 · · · (z − a k ) n k , then sup [a,b] log | p| ≤ log |a 0 | + k j=1 n j sup [a,b] log |z − a j | and (log | p|) [a,b] = log |a 0 | + k j=1 n j (log |z − a j |) [a,b] .
for all polynomials p and all a, b ∈ C. Now we may conclude the proof of Theorem 1.2 as follows. Since A is compact, we may For every ray (half line), say L, starting from z 0 , we see that A ∩ L is a line segment in view of convexity of A. Let L C be the complex line containing L. Apply (4.3) to p| L C , we have where dμ is a certain measure on the unit sphere S 2n−1 and we identify the set of rays L starting from z 0 with S 2n−1 . Notice that the above inequality gives from which the assertion immediately follows.
Although the argument is fairly standard, we will provide a proof in Appendix, because the result cannot be found in literature explicitly.
In the case of Proposition 5.0.1, we define ρ(z, w) = max k |z k − w k | 1/a k , z, w ∈ C n .
It is easy to verify that ρ is a quasi-distance on C n and B(ẑ, r ) = P r a (ẑ),ẑ ∈ C n , r > 0. Besides (7.1), the following properties also hold for B(ẑ, r ): |B(ẑ, c 1 r )| ≤ c For each open set U and each r > 0, the functionẑ → |B(ẑ, r ) ∩ U | is continuous.