A characterization of the degenerate complex Hessian equations for functions with bounded (p,m) -energy
aa r X i v : . [ m a t h . C V ] M a r A CHARACTERIZATION OF THE DEGENERATE COMPLEXHESSIAN EQUATIONS FOR FUNCTIONS WITH BOUNDED ( p, m ) -ENERGY PER ÅHAG AND RAFAŁ CZYŻ
Abstract.
By proving an estimate of the sublevel sets for ( ω, m ) -subharmonicfunctions we obtain a Sobolev type inequality that is then used to characterizethe degenerate complex Hessian equations for such functions with bounded ( p, m ) -energy. March 16, 2020 Introduction
Ever since the 1930s when the interest in Kähler geometry gained momentumwith the publication of Erich Kähler’s article [13], the attention has been immenseboth from mathematicians and physicists. Take, for example, the works of Aubin [2]and Yau [20], as well as the highly regarded Seiberg-Witten theory [18, 19] ofphysics. In the mentioned work of Aubin and Yau they showed how geometricinformation of a Kähler manifold can be retrieved by solving certain partial differ-ential equation of Monge-Ampère type. This is part of the explanation of why theseequations and associated methods have been of great interest in recent decades. Ourmotivation is instead from a pluripotential theoretical background and the highlyinfluential work of Bedford and Taylor [3, 4], and Cegrell [7, 8].Combing the ideas of Cegrell’s energy classes with globally defined plurisubhar-monic functions known as ω -plurisubharmonic functions Guedj and Zeriahi intro-duced and studied weighted energy classes of ω -plurisubharmonic functions ([14]).In particular, they proved the existence of solutions to the Dirichlet problem for thecomplex Monge-Ampère operator, and later Dinew proved the uniqueness ([10]).Here we shall also use the idea of energy classes, but for the interpolation spacesof m -subharmonic functions. These spaces interpolate between subharmonic andplurisubharmonic functions, and the differential operator is the complex Hessianoperator. The idea of these interpolation spaces goes back to Caffarelli et al. [6],and pluripotential methods were introduced by Błocki in [5].The general setting of this paper is that n ≥ , p > , and ≤ m ≤ n . Further-more, we shall use ( X, ω ) to denote a connected and compact Kähler manifold ofcomplex dimension n , where ω is a Kähler form on X such that R X ω n = 1 . Theenergy classes of ( ω, m ) -subharmonic functions with bounded ( p, m ) -energy that is Mathematics Subject Classification.
Primary 32U05, 31C45, 46E35; Secondary 53C55,35J60.
Key words and phrases. compact Kähler manifolds, complex Hessian equation, ( ω, m ) -subharmonic functions, Sobolev type inequalities. central for this paper is defined by E pm ( X, ω ) := { u ∈ E m ( X, ω ) : u ≤ , e p,m ( u ) < ∞} , where e p,m ( u ) = Z X ( − u ) p H m ( u ) , and H m denote the complex Hessian operator (see Section 2 for details). For ahistorical account and references see e.g. [1, 17].By proving in Lemma 4.1 an estimate of the sublevel sets for ( ω, m ) -subharmonicfunctions we obtain the following Sobolev type inequality. Theorem 5.1.
Let n ≥ , p > , and let ≤ m ≤ n . Assume that ( X, ω ) isa connected and compact Kähler manifold of complex dimension n , where ω is aKähler form on X such that R X ω n = 1 . Furthermore, assume that µ is a Borelmeasure defined on X . Fix a constant β such that > β > max (cid:16) pn − npn − n + m , pp +1 (cid:17) ,for p > , and β = pp +1 for p ≤ . The following conditions are then equivalent: (1) E pm ( X, ω ) ⊂ L q ( X, µ ) ;(2) there exists a constant C > such that for all u ∈ E m ( X, ω ) ∩ L ∞ ( X ) with sup X u = − it holds Z X ( − u ) q dµ ≤ Ce p,m ( u ) qβp ; (3) there exists a constant C > such that for all u ∈ E pm ( X, ω ) with sup X u = − it holds Z X ( − u ) q dµ ≤ Ce p,m ( u ) qβp . Theorem 5.1 is then used in Theorem 5.2 to characterize the degenerate complexHessian equation for ( ω, m ) -subharmonic functions with bounded ( p, m ) -energy.This equation was first considered for smooth solutions, and later for continuousfunctions (see e.g. [11, 16, 17] and references therein). In [16], Lu and Nguyenrecently solved the Dirichlet problem for the complex Hessian equation in E m ( X, ω ) .In their paper, they used the variational method. By instead using our Sobolev typeinequality we can in Theorem 5.2 generalize Lu and Nguyen results to p > . Theorem 5.2.
Let n ≥ , p > , and let ≤ m ≤ n . Assume that ( X, ω ) isa connected and compact Kähler manifold of complex dimension n , where ω is aKähler form on X such that R X ω n = 1 . Furthermore, assume that µ is a Borelprobability measure defined on X . The following conditions are then equivalent: (1) E pm ( X, ω ) ⊂ L p ( X, µ ) ;(2) there exists unique ( ω, m ) -subharmonic function u in E pm ( X, ω ) such that sup X u = − and H m ( u ) = µ . Preliminaries
Let Ω ⊂ C n , n ≥ , be a bounded domain, ≤ m ≤ n , and define C (1 , to bethe set of (1 , -forms with constant coefficients. We then define Γ m = (cid:8) α ∈ C (1 , : α ∧ β n − ≥ , . . . , α m ∧ β n − m ≥ (cid:9) , where β = dd c | z | is the canonical Kähler form in C n . CHARACTERIZATION OF THE DEGENERATE COMPLEX HESSIAN EQUATIONS 3
Definition 2.1.
Let n ≥ , and ≤ m ≤ n . Assume that Ω ⊂ C n is a boundeddomain, and let u be a subharmonic function defined on Ω . Then we say that u is m -subharmonic if the following inequality holds dd c u ∧ α ∧ · · · ∧ α m − ∧ β n − m ≥ , in the sense of currents for all α , . . . , α m − ∈ Γ m . With SH m (Ω) we denote theset of all m -subharmonic functions defined on Ω .Let σ k be k -elementary symmetric polynomial of n -variable, i.e., σ k ( x , . . . , x n ) = X ≤ j < ··· Let n ≥ , and let ≤ m ≤ n . Assume that ( X, ω ) is a connectedand compact Kähler manifold of complex dimension n , where ω is a Kähler form on X such that R X ω n = 1 . A function u : X → R ∪{−∞} is called ( ω, m ) -subharmonic if in any local chart Ω of X , the function f + u is m -subharmonic, where f is a localpotential of ω . We shall denote by SH m ( X, ω ) the set of all ( ω, m ) -subharmonicfunctions on X .The following notation is convenient: for any u ∈ SH m ( X, ω ) let ω u = dd c u + ω. With this notation we have that a smooth function u is ( ω, m ) -subharmonic if, andonly if, ω ku ∧ ω n − k ≥ , for all k = 1 , . . . , m. In the following proposition we list useful properties of ( ω, m ) -subharmonic func-tions. For proofs see e.g. [16] and the references therein. Proposition 2.3. Let ( X, ω ) be a compact Kähler manifold. Then (1) if u, v ∈ SH m ( X, ω ) , t ∈ [0 , , then tu + (1 − t ) v ∈ SH m ( X, ω ) ; (2) if u ∈ SH m ( X, ω ) , t ∈ [0 , , then tu ∈ SH m ( X, ω ) ; (3) if u, v ∈ SH m ( X, ω ) , then max( u, v ) ∈ SH m ( X, ω ) ; (4) if u j ∈ SH m ( X, ω ) , j ∈ N then (sup j u j ) ∗ ∈ SH m ( X, ω ) . Here ( ) ∗ denotesthe upper semicontinuous regularization; (5) if u ∈ SH m ( X, ω ) , then there exists a decreasing sequence u j ∈ SH m ( X, ω ) ∩C ∞ ( X ) such that u j → u , j → ∞ . Following the idea from [14] one can define the complex Hessian operator for ( ω, m ) -subharmonic through the following construction. First assume that u ∈SH m ( X, ω ) ∩ L ∞ ( X ) , then H m ( u ) := ω mu ∧ ω n − m , PER ÅHAG AND RAFAŁ CZYŻ which is a non-negative (regular) Borel measure on X . For an arbitrary (not neces-sarily bounded) ( ω, m ) -subharmonic function u let u j = max( u, − j ) be the canon-ical approximation of u . Then define H m ( u ) := lim j →∞ χ { u> − j } H m ( u j ) . The complex Hessian operator is then used to construct the following function class. Definition 2.4. Let E m ( X, ω ) be the class of all ( ω, m ) -subharmonic functionsdefined as E m ( X, ω ) = (cid:26) u ∈ SH m ( X, ω ) : Z X H m ( u ) = 1 (cid:27) . Remark. Note that u ∈ E m ( X, ω ) if, and only if, H m ( u j )( { u j ≤ − j } ) → , as j → ∞ . Here, u j = max( u, − j ) .Let us collect some properties of the class E m ( X, ω ) . Proofs can be found in [12]. Theorem 2.5. Let ( X, ω ) be a compact Kähler manifold. (1) If u, v ∈ E m ( X, ω ) , t ∈ [0 , , then tu +(1 − t ) v ∈ E m ( X, ω ) , and max( u, v ) ∈E m ( X, ω ) . In particular, if u, v ∈ SH m ( X, ω ) , u ∈ E m ( X, ω ) and u ≤ v ,then v ∈ E m ( X, ω ) . (2) If u j ∈ E m ( X, ω ) is decreasing sequence converging to u ∈ E m ( X, ω ) , then H m ( u j ) converges weakly to H m ( u ) . (3) If u, v ∈ E m ( X, ω ) , then χ { u For any Borel set A ⊂ X define the m -capacity of A as cap m ( A ) := sup (cid:26)Z A H m ( u ) : u ∈ SH m ( X, ω ) , − ≤ u ≤ (cid:27) . We say that a Borel set A ⊂ X is m -polar if cap m ( A ) = 0 . CHARACTERIZATION OF THE DEGENERATE COMPLEX HESSIAN EQUATIONS 5 Proposition 2.7. If u ∈ E m ( X, ω ) , u ≤ , then for any t < it holds cap m ( { u < − t } ) ≤ Ct , where the constant C does not depend on u . A central part of the proof of Lemma 4.2 is the following estimate due to Dinewand Kołodziej [11]. Lemma 2.8. For any < α < nn − m there exits a constant C ( α ) > such that forany Borel set A ⊂ X it holds V ( A ) ≤ C ( α ) cap m ( A ) α , where V ( A ) = R A ω n . Functions with bounded ( p, m ) -energy In this section we focus on ( ω, m ) -subharmonic functions with bounded ( p, m ) -energy, and prove some necessary properties that is needed for the rest of thispaper. Definition 3.1. Let n ≥ , p > , and let ≤ m ≤ n . Assume that ( X, ω ) isa connected and compact Kähler manifold of complex dimension n , where ω is aKähler form on X such that R X ω n = 1 . We define the class of ( ω, m ) -subharmonicfunctions with bounded ( p, m ) -energy as E pm ( X, ω ) := { u ∈ E m ( X, ω ) : u ≤ , e p,m ( u ) < ∞} , where e p,m ( u ) = Z X ( − u ) p H m ( u ) . Remark. It was proved in [14, 16] (see also [7, 9]) that u ∈ E pm ( X, ω ) if, and onlyif, sup j e p,m ( u j ) < ∞ , where u j = max( u, − j ) is the canonical approximation of u .Furthermore, e p,m ( u j ) → e p,m ( u ) as j → ∞ . Lemma 3.2. Let ( X, ω ) be a compact Kähler manifold, and p > . Furthermore,let ≤ j ≤ m , and let T be a positive current of the type T = ω ψ ∧ · · · ∧ ω ψ m − j ∧ ω n − m , where ψ , . . . , ψ m − j ∈ E m ( X, ω ) . Then for any u, v ∈ E m ( X, ω ) ∩ L ∞ ( X ) , u, v ≤ ,it holds Z X ( − u ) p ω jv ∧ T ≤ p Z X ( − u ) p ω ju ∧ T + 2 p Z X ( − v ) p ω jv ∧ T. Proof. Note that we have the following { u < − s } ⊂ { u < v − s } ∪ { v < − s } , andtherefore by Theorem 2.5 (4) Z { u Let ( X, ω ) be a compact Kähler manifold, and p > . Let T be apositive current of the type T = ω ψ ∧ · · · ∧ ω ψ m − ∧ ω n − m , where ψ , . . . , ψ m − ∈ E m ( X, ω ) . Then for any u, v ∈ E m ( X, ω ) ∩ L ∞ ( X ) such that u ≤ v ≤ it holds Z X ( − u ) p ω v ∧ T ≤ ( p + 1) Z X ( − u ) p ω u ∧ T. (3.1) In particular, if u, v ∈ E pm ( X, ω ) are such that u ≤ v ≤ , then e p,m ( v ) ≤ ( p + 1) m e p,m ( u ) . Proof. From Proposition 2.3 it follows that we can assume that u is smooth and u ≤ v < . Case 1: ( p ≥ ). We have Z X ( − u ) p ω v ∧ T = Z X ( − u ) p ω ∧ T + Z X vdd c ( − u ) p ∧ T = I + I . (3.2)Note that I = Z X ( − u ) p ω ∧ T ≤ Z X ( − u ) p ω ∧ T + p Z X ( − u ) p − du ∧ d c u ∧ T = Z X ( − u ) p ω u ∧ T, (3.3)and dd c ( − u ) p = p ( p − − u ) p − du ∧ d c u − p ( − u ) p − dd c u ≥ − p ( − u ) p − dd c u. The integral I can be estimated as follows I = Z X vdd c ( − u ) p ∧ T ≤ p Z X ( − v )( − u ) p − dd c u ∧ T ≤ p Z X ( − v )( − u ) p − ω u ∧ T ≤ p Z X ( − u ) p ω u ∧ T. (3.4)Combining the inequalities (3.2), (3.3) and (3.4) we get (3.1). CHARACTERIZATION OF THE DEGENERATE COMPLEX HESSIAN EQUATIONS 7 Case 2: ( < p < ). We have by (3.3) Z X ( − u ) p ω v ∧ T = Z X ( − u ) p ω ∧ T + Z X ( − v ) dd c ( − ( − u ) p ) ∧ T ≤ Z X ( − u ) p ω ∧ T + Z X ( − v ) (cid:2) p ( − u ) p − ω + dd c ( − ( − u ) p ) (cid:3) ∧ T ≤ Z X ( − u ) p ω ∧ T + Z X ( − u ) (cid:2) p ( − u ) p − ω + dd c ( − ( − u ) p ) (cid:3) ∧ T = p Z X ( − u ) p ω ∧ T + Z X ( − u ) p ω u ∧ T ≤ ( p + 1) Z X ( − u ) p ω u ∧ T. The last statement of this lemma follows from the canonical approximation, andinequality (3.1) applied m -times e p,m ( v ) = Z X ( − v ) p ω mv ∧ ω n − m ≤ Z X ( − u ) p ω mv ∧ ω n − m ≤ ( p + 1) Z X ( − u ) p ω u ∧ ω m − v ∧ ω n − m ≤ · · · ≤ ( p + 1) m Z X ( − u ) p ω mu ∧ ω n − m = ( p + 1) m e p,m ( u ) . (cid:3) Corollary 3.4. Let ( X, ω ) be a compact Kähler manifold, and p > . The followingconditions are then equivalent (1) u ∈ E pm ( X, ω ) ; (2) for any decreasing sequence u j ∈ E pm ( X, ω ) , u j ց u we have sup j e p,m ( u j ) < ∞ ; (3) there exists a decreasing sequence u j ∈ E pm ( X, ω ) , u j ց u such that sup j e p,m ( u j ) < ∞ . Proof. The equivalence (1) ⇔ (3) follows from the remark after Definition 3.1, andimplication (2) ⇒ (3) is immediate. Finally, we prove (3) ⇒ (2). Assume that thereexists a decreasing sequence u j ∈ E pm ( X, ω ) , u j ց u such that sup j e p,m ( u j ) < ∞ , and let v j be any sequence decreasing to u . Then for any j there exists k j suchthat v j ≥ u k j . Therefore by Lemma 3.3 the sequence e p,m ( v j ) is also bounded. (cid:3) Lemma 3.5. Let ( X, ω ) be a compact Kähler manifold, and p > . Then thereexists a constant C > such that for any u , u , . . . , u m ∈ E pm ( X, ω ) it holds Z X ( − u ) p ω u ∧ · · · ∧ ω u m ∧ ω n − m ≤ C max j =1 ,...,m e p,m ( u j ) . (3.5) Proof. By using the canonical approximation we can assume without lost of gener-ality that all functions u , . . . u m are bounded. For T = ω u ∧ · · · ∧ ω u m ∧ ω n − m ,Lemma 3.2 yields Z X ( − u ) p ω u ∧ T ≤ p Z X ( − u ) p ω u ∧ T + 2 p Z X ( − u ) p ω u ∧ T. PER ÅHAG AND RAFAŁ CZYŻ Therefore we can assume that u = u . Set u = ǫ P mj =1 u j , where ǫ is a smallpositive constant that will be specified later. It is sufficient to estimate integrals ofthe type R X ( − u ) p ω mu ∧ ω n − m , since ω mu ∧ ω n − m ≥ ǫ m ω u ∧ · · · ∧ ω u m ∧ ω n − m . (3.6)Again by using Lemma 3.2 Z X ( − u ) p ω mu ∧ ω n − m ≤ p e p,m ( u ) + 2 p e p,m ( u ) , and e p,m ( u ) = Z X − ǫ m X j =1 u j p ω mu ∧ ω n − m ≤ max( ǫ p , ǫ ) m X j =1 Z X ( − u j ) p ω mu ∧ ω n − m ≤ max( ǫ p , ǫ )2 p m (cid:18) max j =1 ,...,m e p,m ( u j ) + e p,m ( u ) (cid:19) . (3.7)Now take ǫ such that − p m max( ǫ, ǫ p ) > , then by (3.6) and (3.7) we get Z X ( − u ) p ω u ∧ · · · ∧ ω u m ∧ ω n − m ≤ ǫ − m Z X ( − u ) p ω mu ∧ ω n − m ≤ p mǫ m (1 − p m max( ǫ, ǫ p )) max j =1 ,...,m e p,m ( u j ) + 2 p ǫ − m e p,m ( u j ) ≤ p +1 ǫ − m max j =1 ,...,m e p,m ( u j ) . (cid:3) Remark. Assume that the functions u j ∈ E pm ( X, ω ) are such that sup X u j = − ,and sup j ∈ N e p,m ( u j ) < ∞ . Then u = ∞ X j =1 j u j ∈ E pm ( X, ω ) . (3.8)By using Corollary 3.4 it is sufficient to construct a decreasing sequence of functions v j ∈ E pm ( X, ω ) , v j ց u , j → ∞ , such that sup j ∈ N e p,m ( v j ) < ∞ . Let us next define v j = j X k =1 a k u k , where a k = 2 j k (2 j − . Then v j ∈ SH m ( X, ω ) , v j ց u , and by Lemma 3.5 we get e p,m ( v j ) = Z X − j X k =1 a k u k ! p ω mv j ∧ ω n − m ≤ j X k =1 max ( a k , a pk ) X k + ··· + k = m (cid:18) mk . . . k j (cid:19) a k · · · a k j j Z X ( − u k ) p ω k u ∧· · ·∧ ω k j u j ∧ ω n − m ≤ j X k =1 max ( a k , a pk ) X k + ··· + k = m (cid:18) mk . . . k j (cid:19) a k · · · a k j j C max k =1 ,...,j e p,m ( u k ) ≤ C sup k ∈ N e p,m ( u k ) , CHARACTERIZATION OF THE DEGENERATE COMPLEX HESSIAN EQUATIONS 9 which means that v j ∈ E pm ( X, ω ) , and sup j ∈ N e p,m ( v j ) < ∞ . Thus, (3.8) holds.4. A Sobolev type inequality The aim of this section is to prove the Sobolev type inequality in Theorem 4.5.We shall first need to prove the estimates of the sublevel sets for ( ω, m ) -subharmonicfunctions with bounded ( p, m ) -energy in Lemma 4.4. Lemma 4.1. If u ∈ E m ( X, ω ) , t ∈ [0 , , s > then t m cap m ( { u < − s − t } ) ≤ Z { u< − s } H m ( u ) ≤ s m cap m ( { u < − s } ) . (4.1) Furthermore, if u ∈ E pm ( X, ω ) , p > and s > , then cap m ( { u < − s } ) ≤ ( s − − p e p,m ( u ) . (4.2) Proof. Let v ∈ E m ( X, ω ) be such that − ≤ v ≤ . Then for t ∈ [0 , we get that tv ∈ E m ( X, ω ) . Note that we have { u < − s − t } ⊂ { u < − s + tv } ⊂ { u < − s } , and therefore by Theorem 2.5 (4) Z { u< − s − t } H m ( v ) ≤ Z { u< − s + tv } H m ( v ) ≤ t − m Z { u< − s − t } H m ( s + tv ) ≤ t − m Z { u< − s + tv } H m ( u ) ≤ t − m Z { u< − s } H m ( u ) . This proves the left inequality in (4.1).To prove the right inequality in (4.1) we assume for a moment that u is continuousand let ≤ s < s . Then Z { u< − s } H m (max( u, − s )) = − Z { u ≥− s } H m (max( u, − s ))+ Z X H m (max( u, − s ))= − Z { u ≥− s } H m ( u ) + Z X H m ( u ) = Z { u< − s } H m ( u ) . Note that s max( u, − s ) ∈ E m ( X, ω ) , and − ≤ s max( u, − s ) ≤ , and therefore cap m ( { u < − s } ) ≥ Z { u< − s } H m (cid:18) s max( u, − s ) (cid:19) ≥ s − m Z { u< − s } H m (max( u, − s )) = s − m Z { u< − s } H m ( u ) . If s ց s , then we get cap m ( { u < − s } ) ≥ s − m Z { u< − s } H m ( u ) . For the general situation take a smooth decreasing sequence u j ց u and observethat Z { u< − s } H m ( u ) ≤ lim inf j →∞ Z { u j < − s } H m ( u j ) ≤ s m lim inf j →∞ cap m ( { u j < − s } ) ≤ s m cap m ( { u < − s } ) . To prove inequality (4.2) assume that u ∈ E pm ( X, ω ) , and s > . We shall use(4.1) to obtain t m cap m ( { u < − s − t } ) ≤ Z { u< − s } H m ( u ) ≤ s − p Z { u< − s } ( − u ) p H m ( u ) ≤ s − p e p,m ( u ) . For t = 1 we get the desired conclusion. (cid:3) Lemma 4.2. If u ∈ E pm ( X, ω ) , then u ∈ L q ( X ) for < q < max( p, nn − m .Proof. Let u ∈ E pm ( X, ω ) . Assume first that p ≥ . From Lemma 2.8 and Lemma 4.1we have for α < nn − m Z X ( − u ) q ω n ≤ q + q Z ∞ t q − V ( { u < − t } ) dt ≤ q + qC ( α ) Z ∞ t q − (cid:0) ( t − − p e p,m ( u ) (cid:1) α dt = 2 q + qC ( α ) e p,m ( u ) α Z ∞ t q − ( t − − pα dt. The right hand side is finite if, and only if, q < pα < pnn − m .For p < we use Proposition 2.7 to obtain, in a similar way as above, that Z X ( − u ) q ω n ≤ q + q Z ∞ t q − ( t − − α dt. The right hand side is finite if, and only if, q < α < nn − m . (cid:3) We shall need the following elementary fact. Proposition 4.3. Let α > and F : [0 , ∞ ) → [0 , ∞ ) a decreasing function suchthat Z ∞ t α F ( t ) dt < ∞ . Then there exists a constant C > such that for all t > we have F ( t ) ≤ Ct − α − .Proof. Using integration by parts we get C = Z ∞ t α F ( t ) dt = Z s t α F ( t ) dt + Z ∞ s t α F ( t ) dt = s α +1 F ( s ) α + 1 − α + 1 Z s t α +1 F ′ ( t ) dt + Z ∞ s t α F ( t ) dt ≥ s α +1 F ( s ) α + 1 . (cid:3) Remark. From Lemma 4.2 and Proposition 4.3 it follows that for all u ∈ E pm ( X, ω ) there exists a constant C ( u, q ) depending only on u and q such that V ( { u < − t } ) ≤ C ( u, q ) t q , for < q < max( p, nn − m . (4.3)In Theorem 4.4 we prove estimates of the sublevel sets of ( ω, m ) -subharmonicfunctions with bounded ( p, m ) -energy. For the case p = 1 , Theorem 4.4 givessharper estimates than those proved in [16]. Theorem 4.4. Let n ≥ , and let ≤ m ≤ n . Assume that ( X, ω ) is a connectedand compact Kähler manifold of complex dimension n , where ω is a Kähler formon X such that R X ω n = 1 . If u ∈ E pm ( X, ω ) , then CHARACTERIZATION OF THE DEGENERATE COMPLEX HESSIAN EQUATIONS 11 (1) there exists a constant C ( u ) depending only on u such that for all t > m ( { u < − t } ) ≤ C ( u ) t p +1 ; (2) there exists a constant C ( u, q ) depending only on u and q such that for all t > , and < q < ( p +1) nn − m , V ( { u < − t } ) ≤ C ( u, q ) t q ; (3) for all < q < ( p +1) nn − m , we have that u ∈ L q ( X ) .Proof. By Lemma 4.2 we know that if u ∈ E pm ( X, ω ) , then u ∈ L q ( X ) for < q < max( p, nn − m . Fix u ∈ E pm ( X, ω ) , u ≤ − , and v ∈ E m ( X, ω ) , − ≤ v ≤ , and let t ≥ .Then ut ∈ E m ( X, ω ) , and { u < − t } ⊂ n ut < v − o ⊂ { u < − t } . By Theorem 2.5 (4) we obtain Z { u< − t } ω mv ∧ ω n − m ≤ Z { ut Let X be a connected and compact Kähler manifold of complexdimension n , where ω is a Kähler form on X such that R X ω n = 1 . Also let ≤ m ≤ n and p ≥ . Then for any < q < pnn − m , and any ǫ > , there existsconstant C ( ǫ ) such that for any u ∈ E pm ( X, ω ) , sup X u = − , we have that Z X ( − u ) q ω n ≤ C ( ǫ ) e p,m ( u ) n ( q − np − n + m + ǫ . Proof. Take u ∈ E pm ( X, ω ) , sup X u = − , and fix q < pnn − m . Also, let q < Q < pα ,where < α < nn − m . Then we have by Lemma 4.1 Z X ( − u ) q ω n = q Z ∞ t q − V ( { u < − t } ) dt ≤ q (cid:18)Z ∞ t Q − V ( { u < − t } ) dt (cid:19) q − Q − (cid:18)Z ∞ V ( { u < − t } ) dt (cid:19) Q − qQ − ≤ q (cid:18) Q Q + Z ∞ t Q − e p,m ( u ) α ( t − − pα dt (cid:19) q − Q − (cid:18)Z X ( − u ) ω n (cid:19) Q − qQ − ≤ Ce p,m ( u ) α q − Q − (cid:18)Z X ( − u ) ω n (cid:19) Q − qQ − . It follows from [15] that if u is ( ω, m ) -subharmonic function such that sup X u = − , then there exists a constant C ′ that does not depending on u such that sup j R X ( − u ) ω n ≤ C ′ . Note that inf α,Q (cid:26)(cid:18) α q − Q − (cid:19) : q < Q < pα and < α < nn − m (cid:27) = n ( q − np − n + m . Therefore, for any ǫ > there exists constant C ( ǫ ) that does not depending on u such that Z X ( − u ) q ω n ≤ C ( ǫ ) e p,m ( u ) n ( q − np − n + m + ǫ . (cid:3) At the end of this section we can prove the following partial characterization ofnegative ( ω, m ) -subharmonic functions with bounded ( p, m ) -energy. CHARACTERIZATION OF THE DEGENERATE COMPLEX HESSIAN EQUATIONS 13 Proposition 4.6. Let X be a connected and compact Kähler manifold of complexdimension n , where ω is a Kähler form on X such that R X ω n = 1 . Also let ≤ m ≤ n and p > . Then (cid:26) u ∈ SH − m ( X, ω ) : Z ∞ t m + p − cap m ( { u < − t } ) dt < ∞ (cid:27) ⊂ E pm ( X, ω ) . In particular, if for u ∈ SH − m ( X, ω ) there exist constants C ( u ) > and ǫ > suchthat cap m ( { u < − t } ) ≤ C ( u ) t p + m + ǫ , then u ∈ E pm ( X, ω ) .Proof. Let u ∈ SH − m ( X, ω ) , and assume that Z ∞ t m + p − cap m ( { u < − t } ) dt < ∞ . Then without lost of generality we can assume that u ≤ − . Let us define u t =max( u, − t ) , t ≥ , then v = u t t ∈ SH m ( X, ω ) , − ≤ v ≤ , and ω mv ∧ ω n − m ≥ t − m ω mu t ∧ ω n − m . We then have by Proposition 4.3 ( ω mu t ∧ ω n − m )( { u < − t } ) ≤ t m ( ω mv ∧ ω n − m )( { u < − t } ) ≤ t m cap m ( { u < − t } ) → , as t → ∞ . Thus, u ∈ E m ( X, ω ) . Furthermore, ( ω mu ∧ ω n − m )( { u ≤ − t } ) = Z X ω n − ( ω mu ∧ ω n − m )( { u > − t } )= Z X ω n − ( ω mu t ∧ ω n − m )( { u > − t } ) = ( ω mu t ∧ ω n − m )( { u ≤ − t } ) . Finally, Z X ( − u ) p H m ( u ) = p Z ∞ t p − ( ω mu ∧ ω n − m )( { u < − t } ) dt ≤ p Z ∞ t p − ( ω mu t ∧ ω n − m )( { u ≤ − t } ) dt ≤ p Z ∞ t m + p − cap m ( { u ≤ − t } ) dt < ∞ . (cid:3) The complex Hessian equations In this section we consider complex Hessian equations for E pm ( X, ω ) . We needthe following generalization of Theorem 4.5. Theorem 5.1. Let n ≥ , p > , and let ≤ m ≤ n . Assume that ( X, ω ) isa connected and compact Kähler manifold of complex dimension n , where ω is aKähler form on X such that R X ω n = 1 . Furthermore, assume that µ is a Borelmeasure defined on X . Fix a constant β such that > β > max (cid:16) pn − npn − n + m , pp +1 (cid:17) ,for p > , and β = pp +1 for p ≤ . The following conditions are then equivalent: (1) E pm ( X, ω ) ⊂ L q ( X, µ ) ; (2) there exists a constant C > such that for all u ∈ E m ( X, ω ) ∩ L ∞ ( X ) with sup X u = − it holds Z X ( − u ) q dµ ≤ Ce p,m ( u ) qβp ; (3) there exists a constant C > such that for all u ∈ E pm ( X, ω ) with sup X u = − it holds Z X ( − u ) q dµ ≤ Ce p,m ( u ) qβp . Proof. The implication (2) ⇒ (1) is obvious. The equivalence (2) ⇔ (3) follows byapproximation. We shall prove (1) ⇒ (2).Assume first that p > . To prove this implication assume that condition (2) isnot true, i.e., there exists a sequence u j ∈ E m ( X, ω ) ∩ L ∞ ( X ) , sup X u j = − , suchthat Z X ( − u j ) q dµ ≥ jq e p,m ( u j ) qβp . (5.1) Case 1. If the sequence { e p,m ( u j ) } is bounded (or it contains a bounded subse-quence), then let us define u = ∞ X j =1 j u j . Then u belongs to SH m ( X, ω ) , and by Lemma 3.5 it follows Z X ( − u ) p H m ( u ) ≤ C ( p, m ) sup j ∈ N e p,m ( u j ) < ∞ . Hence, u ∈ E pm ( X, ω ) . On the other hand by (5.1) Z X ( − u ) q dµ ≥ jq Z X ( − u j ) q dµ ≥ jq jq e p,m ( u j ) qβp ≥ jq → ∞ , as j → ∞ . Thus, u / ∈ L q ( X, µ ) . Case 2. Now assume that e p,m ( u j ) → ∞ . Let us define v j = t j u j , where t j = e p,m ( u j ) − βp . (5.2)Then we have by Theorem 4.5 and Lemma 3.5 e p,m ( v j ) = t pj Z X ( − u j ) p ω n + t pj m X k =1 (cid:18) mk (cid:19) t kj (1 − t j ) m − k Z X ( − u j ) p ω ku j ∧ ω n − k ≤ t pj Z X ( − u j ) p ω n + C m t p +1 j e p,m ( u j ) ≤ C ′ t pj e p,m ( u j ) β + C m t p +1 j e p,m ( u j ) < + ∞ . Therefore, we can repeat the argument from the first case to show that function v = ∞ X j =1 j v j CHARACTERIZATION OF THE DEGENERATE COMPLEX HESSIAN EQUATIONS 15 belongs to SH pm ( X, ω ) , but v / ∈ L q ( X, µ ) , since Z X ( − v ) q dµ ≥ jq t qj Z X ( − u j ) q dµ ≥ jq jq t qj e p,m ( u j ) qβp = 2 jq → ∞ , as j → ∞ . Next, assume that p ≤ and β = pp +1 . By [15] it follows that if u is ( ω, m ) -subharmonic function such that sup X u = − , then there exists a constant C ′ whichdoes not depending on u such that Z X ( − u ) p ω n ≤ (cid:18)Z X ( − u ) ω n (cid:19) p ≤ ( C ′ ) p , and then we repeat the above proof for the case when p > . (cid:3) By making the best use of Theorem 5.1 we prove the following theorem. Theo-rem 5.2 was in the case p = 1 proved in [16]. Theorem 5.2. Let n ≥ , p > , and let ≤ m ≤ n . Assume that ( X, ω ) isa connected and compact Kähler manifold of complex dimension n , where ω is aKähler form on X such that R X ω n = 1 . Furthermore, assume that µ is a Borelprobability measure defined on X . The following conditions are then equivalent: (1) E pm ( X, ω ) ⊂ L p ( X, µ ) ; (2) there exists unique ( ω, m ) -subharmonic function u in E pm ( X, ω ) such that sup X u = − and H m ( u ) = µ .Proof. Implication (2) ⇒ (1) follows from Lemma 3.5. Next, we shall prove impli-cation (1) ⇒ (2). To do so let us define the following collection of Borel probabilitymeasures M = { µ : µ ( X ) = 1 , µ ( K ) ≤ cap m ( K ) , K ⊂ X } . It was proved in [16] that M is convex and compact. Furthermore, for any Borelprobability µ measure we have the following decomposition µ = f ν + σ, where ν ∈ M , σ ⊥M , f ∈ L ( ν ) . If we assume that µ vanishes on m -polar sets, then σ = 0 . By assumption µ is aBorel probability measure defined on X such that E pm ( X, ω ) ⊂ L p ( X, µ ) . Thus, µ vanishes on m -polar sets, so there exist ν ∈ M and f ∈ L ( ν ) such that µ = f ν .Set µ j = c j min( f, j ) ν, where c j > is such that µ j ( X ) = 1 . It follows from [16] that there exists u j ∈E m ( X, ω ) such that H m ( u j ) = µ j and sup X u j = − . Without loss of generality wecan assume that u j → u in L ( X ) . Next, we define u j,k = max( u j , − k ) ∈ L ∞ ( X ) . This construction implies that u j,k ∈ E pm ( X, ω ) , and u j,k ց u j , as k → ∞ . Hence, e p,m ( u j,k ) → e p,m ( u j ) , k → ∞ . By Theorem 5.1 it follows for some β < that Z X ( − u j,k ) p H m ( u j ) ≤ Z X ( − u j,k ) p dµ j ≤ C (cid:18)Z X ( − u j,k ) p H m ( u j,k ) (cid:19) β , and since we have R X ( − u j,k ) p H m ( u j ) → e p,m ( u j ) and e p,m ( u j,k ) → e p,m ( u j ) , as k → ∞ , we get sup k e p,m ( u j,k ) < ∞ . Thus, u j ∈ E pm ( X, ω ) . Theorem 5.1 yields, again for some β < , that Z X ( − u j ) p H m ( u j ) ≤ c j Z X ( − u j ) p dµ ≤ c j C (cid:18)Z X ( − u j ) p H m ( u j ) (cid:19) β . Thus, sup j e p,m ( u j ) < ∞ . Let us define v j = (cid:0) sup k ≥ j u k (cid:1) ∗ . Here ( ) ∗ denotes theupper semicontinuous regularization. Then v j is a decreasing sequence of functionfrom E pm ( X, ω ) , v j ց u , j → ∞ . Furthermore, since sup j e p,m ( v j ) < ∞ , then we canconclude that u ∈ E pm ( X, ω ) . Then by [16] we conclude that H m ( v j ) ≥ min( f, j ) ν ,after passing to the limit with j we get H m ( u ) ≥ µ , but since both measure H m ( u ) and µ have the same total mass we conclude that H m ( u ) = µ . (cid:3) At the end of this section we shall prove the following proposition. Proposition 5.3. Assume the same conditions as in Theorem 5.2. a ) If there exist constants α > pp +1 and C > such that for all Borel sets E it holds µ ( E ) ≤ C cap m ( E ) α , then E pm ( X, ω ) ⊂ L p ( X, µ ) . b ) If E pm ( X, ω ) ⊂ L p ( X, µ ) , then for fixed β > max (cid:16) pn − npn − n + m , pp +1 (cid:17) , if p > ,and β = pp +1 if p ≤ , there exists a constant C > such that for all Borelsets E it holds µ ( E ) ≤ C cap m ( E ) β . Proof. a) Let u ∈ E pm ( X, ω ) with sup X u = − . From Theorem 4.4 it follows that Z X ( − u ) p dµ = p Z ∞ t p − µ ( { u < − t } ) dt ≤ pC Z ∞ t p − cap m ( { u < − t } ) dt ≤ pC Z ∞ t p − (cid:18) C ′ ( u ) t p +1 (cid:19) α dt = pC ( C ′ ( u )) α Z ∞ t p − − αp − α dt < ∞ . b) Assume that E pm ( X, ω ) ⊂ L p ( X, µ ) . From Theorem 5.1 it follows that thereexists a constant C > such that for all v ∈ E pm ( X, ω ) ∩ L ∞ ( X ) , with sup X v = − ,it holds Z X ( − v ) p dµ ≤ Ce p,m ( v ) β . (5.3)Let E be a Borel set, and let h m,E be the m -extremal function for the set E , i.e. h m,E = (cid:0) sup { u ∈ SH m ( X, ω ) : u ≤ − on E, u ≤ on X } (cid:1) ∗ , where ( ) ∗ denotes the upper semicontinuous regularization (see e.g. [16, Section 4]for further information). Using h m,E in (5.3) we arrive at µ ( E ) ≤ Z X ( − h m,E ) p dµ ≤ Ce p,m ( h m,E ) β = C cap m ( E ) β . 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Department of Mathematics and Mathematical Statistics, Umeå University, SE-901 87 Umeå, Sweden E-mail address : [email protected] Institute of Mathematics, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków,Poland E-mail address ::