Weighted Bergman kernel, directional Lelong number and John-Nirenberg exponent
aa r X i v : . [ m a t h . C V ] S e p WEIGHTED BERGMAN KERNEL, DIRECTIONAL LELONG NUMBER ANDJOHN-NIRENBERG EXPONENT
BO-YONG CHENA bstract . Let ψ be a plurisubharmonic function on the closed unit ball and K t ψ ( z ) the Bergmankernel on the unit ball with respect to the weight t ψ . We show that the boundary behavior of K t ψ ( z )is determined by certain directional Lelong number of ψ for all t smaller than the John-Nirenbergexponent of ψ associated to certain family of nonisotropic balls, which is always positive. K eywords : Weighted Bergman kernel, plurisubharmonic function, directional Lelong number,John-Nirenberg exponent. 1. I ntroduction Let B be the unit ball in C n and PS H ( B ) the set of plurisubharmonic (psh) functions on B (we always assume that psh functions are not identically −∞ ). For each ψ ∈ PS H ( B ) we define K t ψ ( z , w ) to be the weighted Bergman kernel of the Hilbert space A t ψ = ( f ∈ O ( B ) : Z B | f | e − t ψ < ∞ ) , t ≥ . Set K t ψ ( z ) = K t ψ ( z , z ). A cerebrated theorem of Demailly [6] states that ψ t : = t log K t ψ ( z ) → ψ ( z ) ( t → + ∞ )and ν ( ψ, z ) − n / t ≤ ν ( ψ t , z ) ≤ ν ( ψ, z )where ν ( ϕ, z ) denotes the Lelong number for a psh function ϕ at z .In this paper we consider the case when t is fixed and z approaches the boundary ∂ B . Wediscover that for all su ffi ciently small t the asymptotic behavior of K t ψ ( z ) at a boundary point ζ isdetermined by certain directional Lelong number of ψ at ζ . To state the results precisely, we needto introduce some notions. Let˜ B r = n ( z , z ′ ) ∈ C × C n − : | z | < r , | z ′ | < √ r o . For a bounded domain Ω ⊂ C n we define ˜ B ( Ω ) to be the set of all F ( ˜ B r ) ⊂ Ω where F is acomplex a ffi ne mapping composed by a translation and a unitary transformation. We define theJohn-Nirenberg exponent of ψ associated to the family ˜ B ( Ω ) by˜ ε Ω ( ψ ) : = sup ε : sup D ∈ ˜ B ( Ω ) ? D e ε | ψ − ψ D | < ∞ where ψ D = > D ψ is the mean value of ψ over D . For each ζ ∈ ∂ B we denote by T ζ the holomorphictangent space at ζ and N ζ the orthogonal complement of T ζ in C n . Let F ζ be the complex a ffi ne Supported by NSF grant 11771089 and Gaofeng grant support from School of Mathematical Sciences, FudanUniversity. mapping which is composed by a translation and a unitary transformation, and maps the z axis to N ζ and z = T ζ respectively. Theorem 1.1.
Let ψ be a psh function in a neighborhood of the closed ball B R : = {| z | ≤ R } whereR > . For each ≤ t < ˜ ε B R ( ψ ) and each ζ ∈ ∂ B we have (1.1) lim r → log K t ψ ((1 − r ) ζ )log 1 / r = n + − t ˜ ν ( ψ ζ ) where ψ ζ = ψ ◦ F ζ and ˜ ν ( ψ ζ ) = lim r → r sup θ , ··· ,θ n ψ ζ ( re i θ , √ re i θ , · · · , √ re i θ n ) . Note that the quantity 2 − n ˜ ν ( ψ ζ ) is essentially the directional Lelong number with coe ffi cients(1 , , · · · ,
2) of ψ ζ at 0 (see [7], p. 166).Of course, Theorem 1.1 is meaningless unless one has verified the following Theorem 1.2.
Let Ω be a bounded domain in C n . If ψ is psh in a neighborhood of Ω then ˜ ε Ω ( ψ ) > . The proof of Theorem 1.2 relies on the following local Bernstein type inequality(1.2) sup B ψ ≤ sup E ψ + C n ,α (cid:0) + | ψ B R | (cid:1) α (cid:2) + log ( | B | / | E | ) (cid:3) for each ball B ⊂ B R , measurable set E ⊂ B and negative psh function ψ on B R , where α > C n ,α depends only on n , α . Inequalities like (1.2) were obtained earlier by Brudnyi [3].Analogous global inequalities were obtained by Benelkourchi et al. [1] for the Lelong class of pshfunctions. The analysis in these papers relies heavily on (nonlinear) pluripotential theory. Here weshall present an entirely new approach, using only linear analysis: the Riesz decomposition theoremand some basic facts from the theory of weights (see [14]).2. P roof of T heorem Theorem 2.1.
Let ψ be a psh function in a neighborhood of B R where R > . For each ≤ t < ˜ ε B R ( ψ ) there exists a constant C > such that for all ζ ∈ ∂ B and < r ≪ , (2.1) C − r − n − e t ψ ˜ Br ( ζ ) ≤ K t ψ ((1 − r ) ζ ) ≤ Cr − n − e t ψ ˜ Br ( ζ ) where ˜ B r ( ζ ) = F ζ ( ˜ B r ) and F ζ is as in § We start with a few elementary lemmas. For each ζ ∈ B we denote by T ζ the holomorphicautomorphism of B which maps ζ onto the origin. Lemma 2.2.
Let < r < and ζ r : = (1 − r , , · · · , . Then we have (2.2) D r : = n z : | z − (1 − r ) | < r / , | z ′ | < p r / o ⊂ B (2.3) n z : | T ζ r ( z ) | < / √ o ⊂ D ′ r : = n z : | z − | < r , | z ′ | < √ r o . Proof. If z ∈ D r , then | z | < − r /
2, so that | z | < (1 − r / + r / = − r / + r / < , i.e. (2.2) is verified. EIGHTED BERGMAN KERNEL, DIRECTIONAL LELONG NUMBER AND JOHN-NIRENBERG EXPONENT 3
Now suppose | T ζ r ( z ) | < /
2. By the standard formula(2.4) 1 − | T ζ r ( z ) | = (1 − | z | )(1 − | ζ r | ) | − h z , ζ r i| (see [15], p. 5), we conclude that2 | − z | ≥ − | z | ) ≥ − | z | ≥ − | z | ≥ · | − (1 − r ) z | r − r ≥ ( | − z | − r | z | ) r − r . With c : = r | z | and c : = r − r we have | − z | ≤ c + c + q c c + c < r . On the other hand, we have | z ′ | < − | z | ≤ | − z | < r . Thus (2.3) is verified. (cid:3)
Lemma 2.3.
Let V be a measurable set in R n and ψ ∈ L ( V ) . For each measurable set W ⊂ V wehave (2.5) | ψ W − ψ V | ≤ | V || W | ? V | ψ − ψ V | (2.6) ? W | ψ − ψ W | ≤ | V || W | ? V | ψ − ψ V | . Proof.
First of all, we have | ψ W − ψ V | ≤ | W | Z W | ψ − ψ V |≤ | W | Z V | ψ − ψ V |≤ | V || W | ? V | ψ − ψ V | . Next we have ? W | ψ − ψ W | ≤ ? W (cid:2) | ψ − ψ V | + | ψ W − ψ V | (cid:3) ≤ | V || W | ? V | ψ − ψ V | . (cid:3) Lemma 2.4.
Let ψ be a psh function in a neighborhood of B R where R > . For each ≤ t < ˜ ε B R ( ψ ) there exists a constant C > such that (2.7) ? D e t ψ ≤ Ce t ψ D and ? D e − t ψ ≤ Ce − t ψ D for all D ∈ ˜ B ( B R ) . BO-YONG CHEN
Proof.
From the inequality ? D e t | ψ − ψ D | ≤ C , we immediately get (2.7). (cid:3) Remark.
By Jensen’s inequality (2.8) e t ψ D ≤ ? D e t ψ and e − t ψ D ≤ ? D e − t ψ , one may call (2.7) the reverse Jensen inequality.Proof of Theorem 2.1. Without loss of generality, we assume that ζ = (1 , , · · · , ζ r = (1 − r ) ζ .For each f ∈ A t ψ we have | f ( ζ r ) | ≤ ? D r | f | ≤ " ? D r | f | e − t ψ / " ? D r e t ψ / ≤ "Z B | f | e − t ψ / " | D r | ? D r e t ψ / ≤ "Z B | f | e − t ψ / h Cr − n − e t ψ Dr i / where the last inequality follows from Lemma 2.4. As (cid:12)(cid:12)(cid:12) ψ D r − ψ ˜ B r ( ζ ) (cid:12)(cid:12)(cid:12) ≤ C , r ≪ K t ψ ( ζ r ) ≤ sup f ∈ A t ψ | f ( ζ r ) | k f k r ψ ≤ Cr − n − e t ψ ˜ Br ( ζ ) . For the lower bound of K t ψ we shall use L − estimates of the ¯ ∂ − equation in a standard way(compare [5]). Let g ( z , w ) be the pluricomplex Green function of B with pole at w , i.e. g ( z , w ) = log | T w ( z ) | where T w is the holomorphic automorphism of B which maps w onto the origin. Set g r ( z ) = g ( z , ζ r ).Choose a smooth cut-o ff function χ : R → [0 ,
1] such that χ | ( −∞ , − = χ | [0 , ∞ ) =
0. By theDonnelly-Fe ff erman estimate (see e.g. [2]), we may find a solution of¯ ∂ u = ¯ ∂ h χ (cid:16) − log( − g r ) + log log √ (cid:17)i = : v which satisfies Z B | u | e − t ψ − ng r ≤ C Z B | v | − i ∂ ¯ ∂ log( − g r ) e − t ψ − ng r ≤ C n Z {| T ζ r | < / √ } e − t ψ ≤ C n Z D ′ r e − t ψ in view of Lemma 2.2, where C is a universal constant and C n depends only on n . Set f : = χ (cid:16) − log( − g r ) + log log √ (cid:17) − u . EIGHTED BERGMAN KERNEL, DIRECTIONAL LELONG NUMBER AND JOHN-NIRENBERG EXPONENT 5
It follows that f ∈ O ( B ), f ( ζ r ) =
1, and Z B | f | e − t ψ ≤ C n Z D ′ r e − t ψ . Thus(2.9) K t ψ ( ζ r ) ≥ | f ( ζ r ) | k f k t ψ ≥ C − n "Z D ′ r e − t ψ − ≥ C − r − n − e t ψ D ′ r in view of Lemma 2.4. As (cid:12)(cid:12)(cid:12) ψ D ′ r − ψ ˜ B r ( ζ ) (cid:12)(cid:12)(cid:12) ≤ C in view of Lemma 2.3, we establish the desired lower bound. (cid:3) Proof of Theorem 1.1.
Set ˜ P r = { z : | z | < r , max j ≥ | z j | < √ r } . Then we havelim inf r → ψ ˜ B r ( ζ ) / log r = lim inf r → ( ψ ζ ) ˜ B r / log r = lim inf r → ( ψ ζ ) ˜ P r / log r ≥ lim inf r → r sup θ , ··· ,θ n ψ ζ ( re i θ , r / e i θ , · · · , r / e i θ n ) = ˜ ν ( ψ ζ )where the second equality follows from Lemma 2.3 and the inequality follows from the maximumprinciple for psh functions. On the other hand, for each r we choose a r = ( re i θ ( r ) , r / e i θ ( r ) , · · · , r / e i θ n ( r ) )such that ψ ζ ( a r ) = sup θ , ··· ,θ n ψ ζ ( re i θ , r / e i θ , · · · , r / e i θ n ) . Set ˜ P r + a = { z + a : z ∈ ˜ P r } . By Lemma 2.3 we havelim sup r → ψ ˜ B r ( ζ ) / log r = lim sup r → ( ψ ζ ) ˜ B r / log r = lim sup r → ( ψ ζ ) ˜ P r + a r / log r ≤ lim sup r → ψ ζ ( a r ) / log r = ˜ ν ( ψ ζ )where the inequality follows from the mean value inequality. Thus by (2.1) we establish (1.1). (cid:3)
3. A local
BMO estimate of psh functions
A function ψ ∈ L ( Ω ) is of BMO (bounded mean oscillation) if(3.1) k ψ k BMO( Ω ) : = sup B ? B | ψ − ψ B | < ∞ , where the supremum is taken over all balls B ⊂⊂ Ω . BMO was first introduced by John-Nirenberg[12] in connection with PDE, who also proved a crucial inequality:(3.2) sup B ⊂ Ω ? B e c n | ψ − ψ B | / k ψ k BMO( Ω ) ≤ C n BO-YONG CHEN where c n , C n > n . The BMO space became well-known after Fe ff erman provedthat it is the dual of the real-variable Hardy space H (cf. [9]). A famous unbounded example ofBMO( R n ) is log | x | . We refer to Stein [14] for further examples and properties.For a domain Ω ⊂ C n we define PS H − ( Ω ) to be the set of negative psh functions on Ω . Thepurpose of this section is to show the following BMO estimate for psh functions. Theorem 3.1.
Let α > . If ψ ∈ PS H − ( B R ) , then (3.3) k ψ k BMO( B R ) ≤ C n ,α (cid:0) + | ψ B R | (cid:1) α where C n ,α > depends only on n , α . Theorem 3.1 will be deduced from a number of lemmas.
Lemma 3.2.
Let ψ, φ be two real C function on a domain Ω ⊂ R n . Let η : R → (0 , ∞ ) be a C function with η ′ > . If either φ or ψ has compact support in Ω , then (3.4) Z Ω φ " ∆ ψη ( − ψ ) + η ′ ( − ψ ) η ( − ψ ) |∇ ψ | ≤ Z Ω |∇ φ | η ′ ( − ψ ) . Proof.
Integration by parts gives Z Ω φ η ( − ψ ) ∆ ψ = − Z Ω ∇ ψ · ∇ " φ η ( − ψ ) = − Z Ω φ ∇ ψη ( − ψ ) · ∇ φ − Z Ω φ η ′ ( − ψ ) η ( − ψ ) |∇ ψ | , so that Z Ω φ η ( − ψ ) ∆ ψ + Z Ω φ η ′ ( − ψ ) η ( − ψ ) |∇ ψ | = − Z Ω φ ∇ ψη ( − ψ ) · ∇ φ ≤ Z Ω φ η ′ ( − ψ ) η ( − ψ ) |∇ ψ | + Z Ω |∇ φ | η ′ ( − ψ ) , from which (3.4) immediately follows. (cid:3) Lemma 3.3.
Let α > . If ψ is a negative subharmonic function on a domain Ω ⊂ R n , then Z Ω φ ∆ ψ ≤ α − Z Ω (1 + | ψ | ) α |∇ φ | , φ ∈ C ∞ ( Ω ) . Proof.
We take a decreasing sequence of smooth subharmonic functions ψ j < φ such that ψ j ↓ ψ . Applying (3.4) with η ( t ) = − (1 + t ) − α , we have Z Ω φ η ( − ψ j ) ∆ ψ j ≤ α − Z Ω (1 + | ψ j | ) α |∇ φ | ≤ α − Z Ω (1 + | ψ | ) α |∇ φ | . As η <
2, we have Z Ω φ ∆ ψ = lim j →∞ Z Ω φ ∆ ψ j ≤ α − Z Ω (1 + | ψ | ) α |∇ φ | . (cid:3) EIGHTED BERGMAN KERNEL, DIRECTIONAL LELONG NUMBER AND JOHN-NIRENBERG EXPONENT 7
Lemma 3.4. If ψ ∈ S H − (2 B ) , then (3.5) Z B | ψ | ≤ n Z B | ψ | . Proof.
Let σ n be the volume of the unit sphere in R n . Write B = B ( a , r ). Since ψ is a subharmonicfunction, it follows that the mean value M ψ ( a , t ) = Z | y | = ψ ( a + ty ) d σ ( y ) /σ n is an increasing function of t ∈ (0 , r ) (see [11], Theorem 3.2.3), i.e. M | ψ | ( a , t ) = Z | y | = | ψ | ( a + ty ) d σ ( y ) /σ n is a decreasing function of t . Then we have Z B \ B | ψ | = Z rr M | ψ | ( a , t ) t n − σ n dt ≤ (2 n − σ n n r n M | ψ | ( a , r ) ≤ (2 n − Z r M | ψ | ( a , t ) t n − σ n dt = (2 n − Z B | ψ | , from which (3.5) immediately follows. (cid:3) For the proof of Theorem 3.1 we consider at first the one-dimensional case.
Lemma 3.5.
Let α > . If n = and ψ ∈ S H − ( B R ) , then (3.6) ? B | ψ − ψ B | ≤ C α ? B R / (1 + | ψ | ) α for all balls B ⊂ B R . Here C α > depends only on α .Proof. Applying Lemma 3.3 with φ ∈ C ∞ ( B R / ) such that φ | B R / = |∇ φ | ≤ / R , we concludethat(3.7) Z B R / ∆ ψ ≤ C α ? B R / (1 + | ψ | ) α . Let R ′ = R /
3. Recall that the (negative) Green function g R ′ of B R ′ is given by g R ′ ( z , w ) = log | z − w | + log R ′ | R ′ − z ¯ w | . The Riesz decomposition theorem (cf. [11], Theorem 3.3.6) gives ψ ( z ) = π Z ζ ∈ B R ′ g R ′ ( z , ζ ) ∆ ψ ( ζ ) + h ( z ) = π Z ζ ∈ B R ′ log | z − ζ | ∆ ψ ( ζ ) + π Z ζ ∈ B R ′ log R ′ | R ′ − z ¯ ζ | ∆ ψ ( ζ ) + h ( z ) = : u ( z ) + v ( z ) + h ( z )where h is the smallest harmonic majorant of ψ , which naturally satisfies ψ ≤ h ≤ . BO-YONG CHEN
Since h ≤ B R ′ , it follows from the mean value property that for each z ∈ B R − h ( z ) = ? B ( z , R / ( − h ) ≤ π ( R / Z B R ′ ( − ψ ) ≤ C α ? B R / (1 + | ψ | ) α . (3.8)For each ball B ⊂ B R and z ∈ B , we have2 π [ u ( z ) − u B ] = Z ζ ∈ B R ′ log | z − ζ | ∆ ψ ( ζ ) − | B | Z w ∈ B Z ζ ∈ B R ′ log | w − ζ | ∆ ψ ( ζ ) = Z ζ ∈ B R ′ [log | z − ζ | − (log | · − ζ | ) B ] ∆ ψ ( ζ ) . As the BMO norm on C n is invariant under translations, it follows from Fubini’s theorem that ? B | u − u B | ≤ π k log | z | k BMO( C n ) Z B R ′ ∆ ψ ≤ C α ? B R / (1 + | ψ | ) α . (3.9)Analogously, as log R ′ | R ′ − z ¯ ζ | = log R ′ | ζ | − log (cid:12)(cid:12)(cid:12) z − R ′ / ¯ ζ (cid:12)(cid:12)(cid:12) for ζ ,
0, it follows that the BMO norms (in z ) of log R ′ | R ′ − z ¯ ζ | and − log | z | coincide, while for ζ = R ′ | R ′ − z ¯ ζ | ≡ log 1 / R ′ , so that its BMO norm is zero. Thus(3.10) ? B | v − v B | ≤ C α ? B R / (1 + | ψ | ) α . Clearly, (3.8)-(3.10) imply (3.6). (cid:3)
For each a ∈ C n and each r = ( r , · · · , r n ) where r j >
0, we define the polydisc P ( a , r) = n z ∈ C n : | z j − a j | < r j , ≤ j ≤ n o . Set P ( a , r ) = P ( a , ( r , · · · , r )). Then we have Lemma 3.6.
Let α ≥ . If ψ ∈ PS H − ( P (0 , , then ? P (0 , r) | ψ | α ≤ C n ,α ? | z k | < r k | ψ (0 , · · · , , z k , , · · · , | α for all ≤ k ≤ n, where C n ,α > depends only on n , α .Proof. It su ffi ces to consider the case k =
1. The Riesz decomposition theorem implies that if u < C then(3.11) Z | z | < / | u | α ≤ C α | u (0) | α EIGHTED BERGMAN KERNEL, DIRECTIONAL LELONG NUMBER AND JOHN-NIRENBERG EXPONENT 9 where C α depends only on α (see [11], p. 230). In the case of n complex variables we consider anegative psh function u in a neighborhood of the unit closed polydisc in C n . Then we have Z P (0 , / | u | α ≤ C α Z | z | < / · · · Z | z n − | < / | u ( z , · · · , z n − , | α ≤ · · · ≤ C n − α Z | z | < / | u ( z , ′ ) | α , so that ? P (0 , / | u | α ≤ C n ,α ? | z | < / | u ( z , ′ ) | α . It su ffi ces to apply the above inequality with u ( z ) = ψ (4 r z / , · · · , r n z n / (cid:3) Lemma 3.7.
Let α > . If ψ ∈ PS H − ( P (0 , , then (3.12) ? P | ψ − ψ P | ≤ C n ,α n X k = ? | z k | < R k / (1 + | ψ (0 , · · · , , z k , , · · · , | ) α for any polydisc P = P (0 , r) with r ≤ R , i.e. r k ≤ R k for all k.Proof. We write P = Q nj = B j where B j = { z j : | z j | < r j } . For each z ∈ P we have ψ ( z ) − ψ P = ψ ( z , z , · · · , z n ) − ψ ( · , z , · · · , z n ) B + · · · + ψ ( · · · , z k , · · · , z n ) B ··· B k − − ψ ( · · · , z k + , · · · , z n ) B ··· B k + · · · + ψ ( · · · , z n ) B ··· B n − − ψ B ··· B n where ψ ( · · · , z k , · · · , z n ) B ··· B k − = ? ζ ∈ B · · · ? ζ k − ∈ B k − ψ ( ζ , · · · , ζ k − , z k · · · , z n ) . Since (cid:12)(cid:12)(cid:12) ψ ( · · · , z k , · · · , z n ) B ··· B k − − ψ ( · · · , z k + , · · · , z n ) B ··· B k (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) ψ ( · · · , z k , · · · , z n ) − ψ ( · · · , z k + , · · · , z n ) B k (cid:12)(cid:12)(cid:12) B ··· B k − , it follows that ? z k ∈ B k (cid:12)(cid:12)(cid:12) ψ ( · · · , z k , · · · , z n ) B ··· B k − − ψ ( · · · , z k + , · · · , z n ) B ··· B k (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ? z k ∈ B k (cid:12)(cid:12)(cid:12) ψ ( · · · , z k , · · · , z n ) − ψ ( · · · , z k + , · · · , z n ) B k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) B ··· B k − ≤ C α " ? | z k | < R k / (1 + | ψ ( · · · , z k , · · · , z n ) | ) α B ··· B k − in view of Lemma 3.5. Thus ? P | ψ − ψ P | ≤ C α n X k = " ? | z k | < R k / (1 − ψ ( · · · , z k , · · · )) α B ··· B k − B k + ··· B n ≤ C n ,α n X k = ? | z k | < R k / (1 − ψ (0 , · · · , , z k , , · · · , α in view of Lemma 3.6, for ψ − (cid:3) Lemma 3.8. If ψ ∈ PS H − ( B R ) , then there exists C n > depending only on n, such that there arecomplex lines L , · · · , L n through the origin, which are orthogonal each other and satisfy L j ∩ S , ∅ for all j, where S = n z ∈ B R / n : ψ ( z ) > C n ψ B R / o . Proof.
Set S m = n z ∈ B R / : ψ ( z ) > m ψ B R / o , S cm = B R / − S m . By Chebychev’s inequality Z B R / | ψ | ≥ − m ψ B R / | S cm | , we establish | S m | ≥ | B R / | − | B R / | / m > | B R / | / m > . We choose a universal constant 0 < c < / (cid:12)(cid:12)(cid:12) { z ∈ B R / : | z | < c R } (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) { z ∈ B R / : | z | < c R } (cid:12)(cid:12)(cid:12) < | B R / | / . Set S ′ m = S m ∩ { z : min {| z | , | z |} > c R } . Then we have | S ′ m | ≥ | S m | − | B R / | / > | S m | / ff eomorphism F on S ′ m as follows: w j = z j for j > w = − ( | z | + · · · + | z n | ) / ¯ z . Clearly, the vector F ( z ) is orthogonal to z in C n and satisfies | F ( z ) | ≤ | z | / | z | < (9 c ) − R , i.e. (3 c ) · F ( z ) ∈ B R / . Since the real Jacobian J R ( F ) of F satisfies J R ( F )( z ) = − ( | z | + · · · + | z n | ) / | z | for z ∈ S ′ m , it follows that | J R ( F )( z ) | ≥ (3 c ) and | (3 c ) · F ( S ′ m ) | ≥ (3 c ) n + | S ′ m | >
12 (3 c ) n + | S m | . Thus if we choose m > + (3 c ) n + (3 c ) n + so that | S m | > " +
12 (3 c ) n + − | B R / | in view of (3.13), then S m ∩ (cid:2) (3 c ) · F ( S ′ m ) (cid:3) , ∅ . In other words, there exists a complex line L such that both L and its orthogonal complement L ⊥ in C n intersect S m . Suppose a ∈ S m ∩ L ⊥ . The mean value inequality for the psh function ψ implies ? B ( a , R / ∩ L ⊥ | ψ | ≤ | ψ ( a ) | . EIGHTED BERGMAN KERNEL, DIRECTIONAL LELONG NUMBER AND JOHN-NIRENBERG EXPONENT 11
Since B R / ⊂ B ( a , R / ? B R / ∩ L ⊥ | ψ | ≤ C n m ? B R / | ψ | where C n > n . By repeating the previous argument, we obtain the remainingcomplex lines L , · · · , L n . (cid:3) Proof of Theorem 3.1.
Given a ∈ B R , we have B ( a , R ) ⊂ B R and B ( a , r ) ⊂ P ( a , r ) ⊂ B ( a , R / , r ≤ R / (3 n / ) . We assume a = L j , 1 ≤ j ≤ n , be chosen as Lemma 3.8. By aunitary transformation, we may assume that L j is the z j − axis for each j . By Lemma 3.7 and Lemma2.3, we see that ? B (0 , r ) (cid:12)(cid:12)(cid:12) ψ − ψ B (0 , r ) (cid:12)(cid:12)(cid:12) ≤ C n ,α n X k = ? | z k | < R / (1 + | ψ (0 , · · · , , z k , , · · · , | ) α . Let b ( k ) = (0 , · · · , , b k , , · · · , ∈ L k ∩ S . It follows from (3.11) and Lemma 3.8 that ? | z k − b k | < R / + R / n | ψ (0 , · · · , , z k , , · · · , | α ≤ C n ,α | ψ ( b ( k ) ) | α ≤ C n ,α | ψ B (0 , R / | α . As { z k : | z k | < R / } ⊂ { z k : | z k − b k | < R / + R / n } , we have ? | z k | < R / | ψ (0 , · · · , , z k , , · · · , | α ≤ C n ,α | ψ B (0 , R / | α ≤ C n ,α | ψ B R | α ≤ C n ,α | ψ B R | α in view of Lemma 2.3 and Lemma 3.4. Thus ? B (0 , r ) (cid:12)(cid:12)(cid:12) ψ − ψ B (0 , r ) (cid:12)(cid:12)(cid:12) ≤ C n ,α (cid:0) + | ψ B R | (cid:1) α . On the other hand, for each ball B ( a , r ) ⊂ B R with r > R / (3 n / ), we naturally have ? B ( a , r ) (cid:12)(cid:12)(cid:12) ψ − ψ B ( a , r ) (cid:12)(cid:12)(cid:12) ≤ | ψ B ( a , r ) | ≤ (cid:0) + | ψ B ( a , r ) | (cid:1) α ≤ C n ,α (cid:0) + | ψ B R | (cid:1) α ≤ C n ,α (cid:0) + | ψ B R | (cid:1) α in view of Lemma 2.3. Thus we have (3.3). (cid:3)
4. P roof of T heorem ω ≥ Ω in R n is said to satisfy the A p condition if(4.1) " ? B ω · " ? B ω − / ( p − p − ≤ A < ∞ for all balls B ⊂⊂ Ω . The smallest constant A for which (4.1) holds is called the A p constant of ω ,which is denoted by A p ( ω ). It is known that ω ∈ A p if and only if(4.2) ( f B ) p ≤ C "Z B f p ω · "Z B ω −
12 BO-YONG CHEN for all nonnegative f ∈ L ( Ω ) and all balls B ⊂⊂ Ω ; moreover the smallest C for which (4.2)is valid equals A p ( ω ) (see [14], p. 195). Let E be a measurable set in B and χ E the characteristicfunction of E . Applying (4.2) with f = χ E we establish(4.3) Z B ω ≤ A p ( ω ) ( | B | / | E | ) p Z E ω. In particular, ω satisfies a doubling property(4.4) Z B ω ≤ np A p ( ω ) Z B ω. Let ψ ∈ BMO( Ω ) and u : = c n ψ/ k ψ k BMO( Ω ) , where c n is the constant in (3.2). Then we have ? B e u − u B ≤ C n , ? B e u B − u ≤ C n , so that(4.5) " ? B e u · " ? B e − u = " ? B e u − u B · " ? B e u B − u ≤ C n , i.e. e u , e − u ∈ A . Applying (4.3) with ω = e u we establish(4.6) Z B e u ≤ C n ( | B | / | E | ) Z E e u . Now we can prove the following inequality mentioned in § Theorem 4.1.
Let B R = { z ∈ C n : | z | < R } and α > . If ψ ∈ PS H − ( B R ) , then for each ball B ⊂ B R and each measurable set E ⊂ B one has (4.7) sup B ψ ≤ sup E ψ + C n ,α (cid:0) + | ψ B R | (cid:1) α (cid:2) + log ( | B | / | E | ) (cid:3) where C n ,α > depends only on n , α .Proof. Let B be a ball in B R and E a measurable set in B . Then we have(4.8) Z E e u ≤ | E | e sup E u where u = c n ψ/ k ψ k BMO( B R ) . On the other hand, we choose a point a ∈ B such that u ( a ) = sup B u .Let r be the radius of B . The doubling property (4.4) implies(4.9) Z B e u ≥ C − n Z B e u ≥ C − n Z B ( a , r ) e u ≥ C − n | B ( a , r ) | e u ( a ) ≥ C − n | B | e sup B u where the third inequality follows from the mean value inequality for the psh function e u . Combin-ing (4.6), (4.8) and (4.9) yields(4.10) sup B u ≤ sup E u + log ( | B | / | E | ) + C n . This inequality combined with Theorem 3.1 gives (4.7). (cid:3)
Theorem 4.1 implies a new interpretation of the Lelong number.
Corollary 4.2.
Let ψ be a psh function on a domain Ω ⊂ C n . Let ν ( ψ, z ) denote the Lelong numberof ψ at z ∈ Ω . Let E r , < r ≪ , be a family of measurable sets satisfying E r ⊂ B ( z , r ) and log ( | B ( z , r ) | / | E r | ) = o(log 1 / r ) , r → . EIGHTED BERGMAN KERNEL, DIRECTIONAL LELONG NUMBER AND JOHN-NIRENBERG EXPONENT 13
Then we have (4.11) ν ( ψ, z ) = lim r → (sup E r ψ ) / log r . Proof.
After subtracting a constant to ψ we may assume ψ < B ( z , r ) for some r < d ( z , ∂ Ω ).As ν ( ψ, z ) = lim r → (sup B ( z , r ) ψ ) / log r , we have ν ( ψ, z ) ≤ lim inf r → (sup E r ψ ) / log r . On the other hand, (4.7) implies ν ( ψ, z ) ≥ lim sup r → (sup E r ψ ) / log r . (cid:3) Theorem 4.3.
Let α > and γ > . If ψ ∈ PS H − ( ˜ B R ) , then there exist positive constants c n ,α,γ depends only on n , α, γ and C n depending only on n such that (4.12) ? ˜ B r + a e ε | ψ − ψ ˜ Br + a | ≤ C n for all ˜ B r + a : = { z + a : z ∈ ˜ B r } ⊂ ˜ B R , where ε = c n ,α,γ " + ? ˜ B R / | ψ | α − γ/α . Let us first observe that Theorem 1.2 follows from Theorem 4.3. Let U be a neighborhood of Ω such that ψ is psh on U . Let D ∈ ˜ B ( Ω ). By a change of complex coordinates we may assume that D is of form ˜ B r for some r >
0. As ˜ B R ⊂ U for R : = d ( D , ∂ U ) / d ( D , ∂ U ) ≤ ψ replaced by ˜ ψ : = ψ − sup U ψ to get ? D e ε | ψ − ψ D | ≤ C n provided ε = c n ,α,γ " + ? ˜ B R / | ˜ ψ | α − γ/α ≥ c n ,α,γ " + d ( D , ∂ U ) − n + Z U | ˜ ψ | α − γ/α . This completes the proof of Theorem 1.2.Theorem 4.3 will be deduced from the following inequalities.
Lemma 4.4.
Let α > and γ > . If ψ ∈ PS H − ( ˜ B R ) , then there exists a number < λ ≤ C n ,α,γ " + ? ˜ B R / | ψ | α γ/α such that for each r ≤ R / and a with ˜ B r + a ⊂ ˜ B R , (4.13) " ? ˜ B r + a e ψ/λ / ≤ C n ? ˜ B r + a e ψ/λ (4.14) Z ˜ B r + a e ψ/λ ≤ C n Z ˜ B r / + a e ψ/λ . Proof.
For each r and a with ˜ B r + a ⊂ ˜ B R we define R a , r to be the supremum of all t ≥ r such that˜ B t + a ⊂ ˜ B R / . Clearly we have c n R ≤ R a , r ≤ R / ? ˜ B Ra , r + a | ψ | α ≤ C n ? ˜ B R / | ψ | α . It su ffi ces to verify (4.13) and (4.14) with λ ≤ C n ,α,γ + ? ˜ B Ra , r + a | ψ | α γ/α . For the sake of simplicity we assume a = R a , r as R . Set ϕ ( ζ ) = ψ ( ζ , ζ ′ ) and B ∗ r = (cid:8) ζ ∈ C n : | ζ | < r , | ζ ′ | < r (cid:9) . Let r ′ = √ r and γ >
1. By Theorem 4.1 we conclude that if r ′ ≤ R ′ / √ B ∗ r ′ ϕ ≤ sup B (0 , √ r ′ ) ϕ ≤ sup B ∗ r ′ / ϕ + C n ,γ (cid:0) + | ϕ | B (0 , R ′ ) (cid:1) γ + log | B (0 , √ r ′ ) || B ∗ r ′ / | ≤ sup B ∗ r ′ / ϕ + C n ,γ (cid:16) + | ϕ | B ∗ R ′ (cid:17) γ = : sup B ∗ r ′ / ϕ + λ. It follows that sup B ∗ r ′ e ϕ/λ ≤ e sup B ∗ r ′ / e ϕ/λ , i.e. sup ˜ B r e ψ/λ ≤ e sup ˜ B r / e ψ/λ . Then we have " ? ˜ B r e ψ/λ / ≤ sup ˜ B r e ψ/λ ≤ e sup ˜ B r / e ψ/λ ≤ C n ? ˜ B r e ψ/λ ? ˜ B r e ψ/λ ≤ sup ˜ B r e ψ/λ ≤ e sup ˜ B r / e ψ/λ ≤ C n ? ˜ B r / e ψ/λ in view of the mean value inequality for the psh function e ψ/λ .Let α ′ be the dual exponent of α . As ? B ∗ R ′ | ϕ | ≤ | B ∗ R ′ | Z B ∗ R ′ | ζ | − α ′ /α /α ′ Z B ∗ R ′ | ϕ | α | ζ | /α = | B ∗ R ′ | Z B ∗ R ′ | ζ | − α ′ /α /α ′ " Z ˜ B R | ψ | α /α ≤ C n ,α R − ( n + /α "Z ˜ B R | ψ | α /α ≤ C n ,α " ? ˜ B R | ψ | α /α , EIGHTED BERGMAN KERNEL, DIRECTIONAL LELONG NUMBER AND JOHN-NIRENBERG EXPONENT 15 we have λ ≤ C n ,α,γ " + ? ˜ B R | ψ | α γ/α . (cid:3) Proof of Theorem 4.3.
It is a standard fact that the reverse H ¨older inequality like (4.13) and thedoubling property like (4.14) for Euclidean balls would imply the A p property for some p >
1. Weshall show that the same is true for nonisotropic balls ˜ B r + a by using Calder´on’s work [4]. Wedefine ̺ ( z , w ) = max n | z − w | , | z ′ − w ′ | o , z , w ∈ C n . It is easy to verify that ̺ satisfies the following properties(1) ̺ ( z , z ) = ̺ ( z , w ) = ̺ ( w , z ) > z , w ;(3) ̺ ( z , w ) ≤ ̺ ( z , ζ ) + ̺ ( ζ, w )) for all z , w , ζ ∈ C n .Note also that ˜ B r + a = { z : ̺ ( z , a ) < r } . Set ω = e ψ/λ and d µ = ω dV where dV is the Lebesguemeasure in C n . Let | · | µ be the volume associated to d µ . Then we may rewrite (4.14) as(4.15) (cid:12)(cid:12)(cid:12) ˜ B r + a (cid:12)(cid:12)(cid:12) µ ≤ C n (cid:12)(cid:12)(cid:12) ˜ B r / + a (cid:12)(cid:12)(cid:12) µ . Let E be a measurable set in ˜ B r + a . By (4.13) we have | E | µ = Z E ω ≤ "Z E ω / | E | / ≤ "Z ˜ B r + a ω / | E | / ≤ C n (cid:12)(cid:12)(cid:12) ˜ B r + a (cid:12)(cid:12)(cid:12) − / "Z ˜ B r + a ω | E | / , i.e. | E | (cid:12)(cid:12)(cid:12) ˜ B r + a (cid:12)(cid:12)(cid:12) ≥ C − n | E | µ (cid:12)(cid:12)(cid:12) ˜ B r + a (cid:12)(cid:12)(cid:12) µ . According to Calder´on (see [4], the proof of Theorem 1), the above inequality implies a reverseH ¨older inequality w.r.t. the measure d µ (noting that dV = ω − d µ ) (cid:12)(cid:12)(cid:12) ˜ B r + a (cid:12)(cid:12)(cid:12) µ Z ˜ B r + a ω − p n d µ / p n ≤ C n (cid:12)(cid:12)(cid:12) ˜ B r + a (cid:12)(cid:12)(cid:12) µ Z ˜ B r + a ω − d µ, for some p n >
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