Pluripotential Theory and Convex Bodies: A Siciak-Zaharjuta theorem
aa r X i v : . [ m a t h . C V ] N ov PLURIPOTENTIAL THEORY AND CONVEX BODIES:A SICIAK-ZAHARJUTA THEOREM
T. BAYRAKTAR,* S. HUSSUNG, N. LEVENBERG** AND M. PERERA
Abstract.
We work in the setting of weighted pluripotential the-ory arising from polynomials associated to a convex body P in( R + ) d . We define the logarithmic indicator function on C d : H P ( z ) := sup J ∈ P log | z J | := sup J ∈ P log[ | z | j · · · | z d | j d ]and an associated class of plurisubharmonic (psh) functions: L P := { u ∈ P SH ( C d ) : u ( z ) − H P ( z ) = 0(1) , | z | → ∞} . We first show that L P is not closed under standard smoothingoperations. However, utilizing a continuous regularization due toFerrier which preserves L P , we prove a general Siciak-Zaharjutatype-result in our P − setting: the weighted P − extremal function V P,K,Q ( z ) := sup { u ( z ) : u ∈ L P , u ≤ Q on K } associated to a compact set K and an admissible weight Q on K can be obtained using the subclass of L P arising from functions ofthe form deg P ( p ) log | p | (appropriately normalized). Introduction
A fundamental result in pluripotential theory is that the extremalplurisubharmonic function V K ( z ) := sup { u ( z ) : u ∈ L ( C d ) , u ≤ K } associated to a compact set K ⊂ C d , where L ( C d ) is the usual Lelongclass of all plurisubharmonic (psh) functions u on C d with the propertythat u ( z ) − log | z | = 0(1) as | z | → ∞ , may be obtained from the subclassof L ( C d ) arising from polynomials: V K ( z ) = max[0 , sup { deg ( p ) log | p ( z ) | : p polynomial , || p || K ≤ } ] . *Supported by The Science Academy BAGEP, **Supported by Simons Founda-tion grant No. 354549. More generally, given an admissible weight function Q on K ( Q islowersemicontinuous and { z ∈ K : Q ( z ) < ∞} is not pluripolar), V K,Q ( z ) := sup { u ( z ) : u ∈ L ( C d ) , u ≤ Q on K } = max[0 , sup { deg ( p ) log | p ( z ) | : p polynomial , || pe − deg ( p ) · Q || K ≤ } ] . We refer to this as a
Siciak-Zaharjuta type result. Standard proofs oftenreduce to a sufficiently regular case by regularization; i.e., convolvingwith a smooth bump function.In recent papers, a (weighted) pluripotential theory associated to aconvex body P in ( R + ) d has been developed. Let R + = [0 , ∞ ) andfix a convex body P ⊂ ( R + ) d ( P is compact, convex and P o = ∅ ).An important example is when P is a non-degenerate convex polytope,i.e., the convex hull of a finite subset of ( R + ) d with nonempty interior.Associated with P we consider the finite-dimensional polynomial spaces P oly ( nP ) := { p ( z ) = X J ∈ nP ∩ ( Z + ) d c J z J : c J ∈ C } for n = 1 , , ... where z J = z j · · · z j d d for J = ( j , ..., j d ). For P = Σwhere Σ := { ( x , ..., x d ) ∈ R d : 0 ≤ x i ≤ , d X i =1 x i ≤ } , we have P oly ( n Σ) is the usual space of holomorphic polynomials ofdegree at most n in C d . For a nonconstant polynomial p we define(1.1) deg P ( p ) = min { n ∈ N : p ∈ P oly ( nP ) } . We define the logarithmic indicator function of P on C d H P ( z ) := sup J ∈ P log | z J | := sup J ∈ P log[ | z | j · · · | z d | j d ] . Note that H P ( z , ..., z d ) = H P ( | z | , ..., | z d | ). As in [3], [4], [7], we makethe assumption on P that(1.2) Σ ⊂ kP for some k ∈ Z + . In particular, 0 ∈ P . Under this hypothesis, we have(1.3) H P ( z ) ≥ k max j =1 ,...,d log + | z j | ICIAK-ZAHARJUTA 3 where log + | z j | = max[0 , log | z j | ]. We use H P to define generalizationsof the Lelong classes L ( C d ) and L + ( C d ) = { u ∈ L ( C d ) : u ( z ) ≥ max j =1 ,...,d log + | z j | + C u } where C u is a constant depending on u . Define L P = L P ( C d ) := { u ∈ P SH ( C d ) : u ( z ) − H P ( z ) = 0(1) , | z | → ∞} , and L P, + = L P, + ( C d ) = { u ∈ L P ( C d ) : u ( z ) ≥ H P ( z ) + C u } . For p ∈ P oly ( nP ) , n ≥ n log | p | ∈ L P ; also each u ∈ L P, + islocally bounded in C d . Note L Σ = L ( C d ) and L Σ , + = L + ( C d ).Given E ⊂ C d , the P − extremal function of E is given by V ∗ P,E ( z ) :=lim sup ζ → z V P,E ( ζ ) where V P,E ( z ) := sup { u ( z ) : u ∈ L P ( C d ) , u ≤ E } . Introducing weights, let K ⊂ C d be closed and let w : K → R + be anonnegative, uppersemicontinuous function with { z ∈ K : w ( z ) > } nonpluripolar. Letting Q := − log w , if K is unbounded, we addition-ally require that(1.4) lim inf | z |→∞ , z ∈ K [ Q ( z ) − H P ( z )] = + ∞ . Define the weighted P − extremal function V ∗ P,K,Q ( z ) := lim sup ζ → z V P,K,Q ( ζ )where V P,K,Q ( z ) := sup { u ( z ) : u ∈ L P ( C d ) , u ≤ Q on K } . If Q = 0 we simply write V P,K,Q = V P,K as above. For P = Σ, V Σ ,K,Q ( z ) = V K,Q ( z ) := sup { u ( z ) : u ∈ L ( C d ) , u ≤ Q on K } is the usual weighed extremal function.A version of a Siciak-Zaharjuta type result has been given in [1] inthe case where it is assumed that V P,K,Q is continuous. Here we give acomplete proof of the general version:
Theorem 1.1.
Let P ⊂ ( R + ) d be a convex body, K ⊂ C d closed, and w = e − Q an admissible weight on K . Then V P,K,Q = lim n →∞ n log Φ n = lim n →∞ n log Φ n,P,K,Q T. BAYRAKTAR,* S. HUSSUNG, N. LEVENBERG** AND M. PERERA pointwise on C d where (1.5) Φ n ( z ) := sup {| p n ( z ) | : p n ∈ P oly ( nP ) , max ζ ∈ K | p n ( ζ ) e − nQ ( ζ ) | ≤ } . If V P,K,Q is continuous, we have local uniform convergence on C d . In the next section, we show that standard convolution does not necessarily preserve the L P classes. Thus the transition from the Siciak-Zaharjuta type result for V P,K,Q continuous to general V P,K,Q is notimmediate. In section 3, we recall the Ferrier regularization procedurefrom [10] and show that it does preserve the L P classes. Then in sections4 and 5 we present a complete proof of Theorem 1.1 together withremarks on regularity of P − extremal functions.2. Approximation by convolution
We fix a standard smoothing kernel(2.1) χ ( z ) = χ ( z , ..., z d ) = χ ( | z | , ..., | z d | )with 0 ≤ χ ≤ R χdV = 1where dV is the standard volume form on C d . Let χ /j ( z ) = j d χ ( jz ).For which P does u ∈ L P imply u j := u ∗ χ /j ∈ L P for j sufficientlylarge? To determine this, it clearly suffices to consider u = H P . Thuswe write u j ( z ) := ( H P ∗ χ /j )( z ).For general P we know that u j ≥ H P ; u j ↓ H P pointwise on C d anduniformly on compact sets. Thus if u j ∈ L P then, in fact, u j ∈ L P, + .Fix δ > j ≥ j ( δ ) we have u j ( z ) ≤ H P ( z ) + δ if | z | , ..., | z d | ≤ δ and(2) u j ( z ) ≤ H P ( z ) + C ( δ ) if | z | , ..., | z d | ≥ δ for all j where C ( δ )depends only on δ .Property (1) follows from the local uniform convergence. For (2),(2.2) u j ( z ) ≤ max D ( z, /j ) H P ≤ H P ( | z | + 1 /j, ..., | z d | + 1 /j )where D ( z, /j ) is the polydisk of polyradius (1 /j, ..., /j ) centered at z . Then for | z k | > δ ,log( | z k | + 1 /j ) = log | z k | + log(1 + 1 j | z k | ) ≤ log | z k | + log(1 + 1 jδ ) ≤ log | z k | + log(1 + 1 δ ) ICIAK-ZAHARJUTA 5 and since H P ( z ) = sup ( j ,...,j d ) ∈ P log[ | z | j · · · | z d | j d ] = sup ( j ,...,j d ) ∈ P d X k =1 j k log | z k | , for | z | , ..., | z d | ≥ δ we havemax D ( z, /j ) H P ≤ sup ( j ,...,j d ) ∈ P d X k =1 j k log( | z k | + 1 /j ) ≤ sup ( j ,...,j d ) ∈ P d X k =1 j k log | z k | + log(1 + 1 δ ) · sup ( j ,...,j d ) ∈ P d X k =1 j k which gives (2).For simplicity we work in C with variables ( z , z ). From the abovecalculations, we see that, given δ >
0, fixing j ≥ j ( δ ), to show u j ∈ L P it suffices to show there is a constant A ( δ ) depending only on δ such thatfor ( z , z ) with | z | < δ and | z | > /δ and for ( z , z ) with | z | > /δ and | z | < δ , we have(2.3) u j ( z , z ) ≤ H P ( z , z ) + A ( δ ) . Proposition 2.1.
If there exists δ > so that H P ( z , z ) ≥ H P ( δ, z ) for | z | < δ and | z | > /δ as well as H P ( z , z ) ≥ H P ( z , δ ) for | z | < δ and | z | > /δ then u j = H P ∗ χ /j ∈ L P for j sufficiently large.Proof. We need to prove (2.3); to do this it suffices to show u j ( z , z ) ≤ H P ( δ, z ) + A ( δ ) , | z | < δ and | z | > /δ ( A )and u j ( z , z ) ≤ H P ( z , δ ) + A ( δ ) , | z | > /δ and | z | < δ. ( B )We verify (A); (B) is the same. To verify (A), for such ( z , z ), from(2.2), we need the appropriate upper bound onsup ( j ,j ) ∈ P [ j log( | z | + 1 /j ) + j log( | z | + 1 /j )] . Now j log( | z | + 1 /j ) + j log( | z | + 1 /j ) ≤ j log( δ + 1 /j ) + j log( | z | + 1 /j )= j [log δ + log(1 + 1 δj )] + j [log | z | + log(1 + 1 | z | j )] ≤ j [log δ + log(1 + 1 δj )] + j [log | z | + log(1 + δj )] T. BAYRAKTAR,* S. HUSSUNG, N. LEVENBERG** AND M. PERERA ≤ j [log δ + log(1 + 1 δ )] + j [log | z | + log(1 + δ )] . Thus u j ( z , z ) ≤ sup ( j ,j ) ∈ P [ j log( | z | + 1 /j ) + j log( | z | + 1 /j )] ≤ sup ( j ,j ) ∈ P (cid:0) j [log δ + log(1 + 1 δ )] + j [log | z | + log(1 + δ )] (cid:1) ≤ sup ( j ,j ) ∈ P [ j log δ + j log | z | ] + A ( δ )where A ( δ ) = sup ( j ,j ) ∈ P [ j log(1 + 1 δ ) + j log(1 + δ )] . (cid:3) We call a convex body P ⊂ ( R + ) d a lower set if for each n = 1 , , ... ,whenever ( j , ..., j d ) ∈ nP ∩ ( Z + ) d we have ( k , ..., k d ) ∈ nP ∩ ( Z + ) d for all k l ≤ j l , l = 1 , ..., d . Clearly H P for such P ⊂ ( R + ) satisfy thehypotheses of Proposition 2.1. Corollary 2.2. If P ⊂ ( R + ) is a lower set, then H P ∗ χ /j ∈ L P for j sufficiently large. Indeed, it appears this condition is necessary for H P ∗ χ /j ∈ L P asthe following explicit example indicates. Example 2.3.
Let P be the quadrilateral with vertices (0 , , (1 , , (0 , , P is not a lower set. We show that for ǫ > H P ∗ χ ǫ L P . Here, H P ( z , z ) = max[0 , log | z | , log | z | , log | z | + 2 log | z | ] . Consider the regions A := { ( z , z ) : | z z | < , | z | > } and B := { ( z , z ) : | z z | > , | z | > } . In A , H P ( z , z ) = log | z | while in B we have H P ( z , z ) = log | z | +2 log | z | = log | z | + log | z z | . Fixing ǫ >
0, we take any large C . Weclaim we can find a point ( x C , y C ) at which H P ∗ χ ǫ ( x C , y C ) − H P ( x C , y C ) > C. ICIAK-ZAHARJUTA 7
If 0 < | x | < / | y | < x, y ) ∈ A . For such ( x, y ), let D ǫ ( x, y ) := { ( z , z ) : | z − x | , | z − y | < ǫ } and S ǫ ( x, y ) := { ( z , z ) ∈ D ǫ ( x, y ) : | z | > ǫ/ } . We first choose y C with | y C | sufficiently large so that for any choice of x C with | x C | < / | y C | – so that Z := ( x C , y C ) ∈ A – we have | y C | > max[ 4 ǫ , ǫ e C/A ǫ ] + ǫ where A ǫ := R ˜ S ǫ χ ǫ dV and˜ S ǫ := { ( z , z ) : ǫ/ ≤ | z | ≤ ǫ, | z | < ǫ } . Note then that | x C | < ǫ/ S ǫ ( x C , y C ) contains the set of points { ( z , z ) : ǫ/ ≤ | z | ≤ ǫ, | z − y C | < ǫ } which is a translation of ˜ S ǫ , centered at (0 , , y C ). The choice of y C insures that S ǫ ( x C , y C ) ⊂ B andfor ( ζ , ζ ) ∈ S ǫ ( x C , y C ) we have | ζ ζ | > | ζ | ≥ ǫ e C/A ǫ . Writing ζ := ( ζ , ζ ), we have H P ∗ χ ǫ ( Z ) − H P ( Z ) = Z D ǫ ( x C ,y C ) H P ( ζ ) χ ǫ ( Z − ζ ) dV ( ζ ) − H P ( Z )= Z D ǫ ( x C ,y C ) [log | ζ | + log + | ζ ζ | ] χ ǫ ( Z − ζ ) dV ( ζ ) − log | y C | = Z D ǫ ( x C ,y C ) log + | ζ ζ | χ ǫ ( Z − ζ ) dV ( ζ )since ζ → log | ζ | is harmonic on D ǫ ( x C , y C ) and R D ǫ χ ǫ ( Z − ζ ) dV ( ζ ) =1. But then Z D ǫ ( x C ,y C ) log + | ζ ζ | χ ǫ ( Z − ζ ) dV ( ζ ) ≥ Z S ǫ ( x C ,y C ) log | ζ ζ | χ ǫ ( Z − ζ ) dV ( ζ ) ≥ Z S ǫ ( x C ,y C ) log[ ǫ ǫ e C/A ǫ ] χ ǫ ( Z − ζ ) dV ( ζ )= log e C/A ǫ Z S ǫ ( x C ,y C ) χ ǫ ( Z − ζ ) dV ( ζ ) T. BAYRAKTAR,* S. HUSSUNG, N. LEVENBERG** AND M. PERERA ≥ CA ǫ Z ˜ S ǫ χ ǫ ( ζ ) dV ( ζ ) = 2 C. We will use this standard regularization procedure in the proof ofTheorem 1.1 but in our application we only utilize the monotonicityproperty u j ↓ u . In the next section, we discuss an alternate regular-ization procedure which always preserves L P classes and which will beneeded to complete the proof of Theorem 1.1.3. Ferrier approximation
We can do a global approximation of u ∈ L P from above by contin-uous u t ∈ L P following the proof of Proposition 1.3 in [12] which itselfis an adaptation of [10]. Proposition 3.1.
Let u ∈ L P . For t > , define (3.1) u t ( x ) := − log (cid:2) inf y ∈ C d { e − u ( y ) + 1 t | y − x |} (cid:3) . Then for t > sufficiently small, u t ∈ L P ∩ C ( C d ) and u t ↓ u on C d .Proof. The continuity of u t follows from continuity of δ t ( x ) := e − u t ( x ) which follows from the estimate δ t ( x ) − δ t ( y ) = e − u t ( x ) − e − u t ( y ) ≤ t | x − y | . Note that δ t ↑ so u t ↓ . Since inf y ∈ C d { e − u ( y ) + t | y − x |} ≤ e − u ( x ) , wehave u t ( x ) ≥ u ( x ). To show that u t ↓ u on C d , fix x ∈ C d . By addinga constant we may assume u ( x ) = 0. Given δ >
0, we want to showthere exists t ( δ ) > u t ( x ) < δ for t < t ( δ ). Thus we want(3.2) inf y ∈ C d { e − u ( y ) + 1 t | y − x |} > e − δ for t < t ( δ ) . Since e − u is lowersemicontinuous and e − u ( x ) = 1 > e − δ , we can find ǫ > e − u ( y ) > e − δ for | y − x | < ǫ. For such y , we have e − u ( y ) + t | y − x | > e − δ for any t >
0. Choosing t ( δ ) > t ( δ ) < ǫe δ achieves (3.2).The proof that u t is psh follows [10]; for the reader’s convenience weinclude this in an appendix. Given this, we are left to show u t ∈ L P ICIAK-ZAHARJUTA 9 for t > u = H P .Thus, let u t ( x ) := − log[inf y { e − H P ( y ) + 1 t | y − x |} ] . For t >
R >> < C <
1, both depending only on t and P , so that for each x ∈ C d with e H P ( x ) > R we have e − u t ( x ) ≥ Ce − H P ( x ) . Unwinding this last inequality, we require e − H P ( y ) + 1 t | y − x | ≥ Ce − H P ( x ) for all y ∈ C d . This is the same as(3.3) Ce H P ( y ) − e H P ( x ) e H P ( x ) e H P ( y ) ≤ t | y − x | for all y ∈ C d . Fix x and fix C with 0 < C <
1. For any y with e H P ( y ) ≤ C e H P ( x ) ,(3.3) is clearly satisfied. If e H P ( y ) ≥ C e H P ( x ) , since Ce H P ( y ) − e H P ( x ) e H P ( x ) e H P ( y ) ≤ Ce H P ( y ) e H P ( x ) e H P ( y ) = Ce H P ( x ) , we would like to have(3.4) Ce H P ( x ) ≤ t | y − x | . To estimate | y − x | , note that x lies on the set L x := { z : e H P ( z ) = e H P ( x ) } while y lies outside the larger level set L C,x := { z : e H P ( z ) = 1 C e H P ( x ) } . Thus | y − x | ≥ dist ( L x , L C,x )and it suffices, for (3.4), to have(3.5) Ce H P ( x ) ≤ t dist ( L x , L C,x ) . Note that L x depends only on x (and P ) while L C,x depends only on C and x (and P ). But for any fixed C with 0 < C < dist ( L x , L C,x )is bounded below by a positive constant as H P ( x ) → ∞ for a convex body P ⊂ ( R + ) d satisfying (1.2). Thus taking t > x with H P ( x ) sufficiently large. (cid:3) Remark 3.2. If u ∈ L P, + , there exists c with u ( y ) ≥ c + H P ( y ) on C d .Henceinf y ∈ C d { e − u ( y ) + 1 t | y − x |} ≤ inf y ∈ C d { e − [ c + H P ( y )] + 1 t | y − x |} ≤ e − [ c + H P ( x )] which gives u t ( x ) ≥ c + H P ( x ) . Thus u t ∈ L P, + We use Proposition 3.1 in the next sections in proving Theorem 1.1.4.
Proof of Main Result
Let P be a convex body in ( R + ) d satisfying (1.3). As in the case P = Σ, for K unbounded and Q satisfying (1.4), V P,K,Q = V P,K ∩ B R ,Q for B R := { z : | z | ≤ R } with R sufficiently large (cf., [3]). Thus inproving Theorem 1.1 we may assume K is compact. Theorem 2.10in [1] states a Siciak-Zaharjuta theorem for K, Q such that V P,K,Q iscontinuous (without assuming Q is continuous): Theorem 4.1.
Let K be compact and Q be an admissible weight func-tion on K such that V P,K,Q is continuous. Then V P,K,Q = ˜ V P,K,Q where ˜ V P,K,Q ( z ) := lim n →∞ [sup { N log | p ( z ) | : p ∈ P oly ( N P ) , || pe − NQ || K ≤ } ] with local uniform convergence in C d . Remark 4.2.
The fact that the limit ˜ V P,K,Q ( z ) := lim n →∞ Φ n ( z ) existspointwise follows from the observation that Φ n · Φ m ≤ Φ n + m (here weare using the notation from (1.5)). Convexity of P is crucial as thisproperty implies that P oly ( nP ) · P oly ( mP ) ⊂ P oly (cid:0) ( n + m ) P (cid:1) . Note we can also write˜ V P,K,Q ( z ) = sup { deg P ( p ) log | p ( z ) | : p polynomial , || pe − deg P ( p ) Q || K ≤ } . where deg P ( p ) is defined in (1.1) and clearly V P,K,Q ≥ ˜ V P,K,Q . ICIAK-ZAHARJUTA 11
The proof of Theorem 4.1 is given in [1], Theorem 2.10 but the proofis omitted from the final version [2]. Here we provide complete detailsincluding a proof of the following version of Proposition 2.9 from [2],[1] which is stated but not proved in these references. Below dV is thestandard volume form on C d . Proposition 4.3.
Let P ⊂ ( R + ) d be a convex polytope and let f ∈O ( C d ) such that (4.1) Z C d | f ( z ) | e − NH P ( z ) (1 + | z | ) − ǫ dV ( z ) < ∞ for some ǫ ≥ sufficiently small. Then f ∈ P oly ( N P ) . Proof.
Since P is a convex polytope it is given by P = { x ∈ R d : ℓ j ( x ) ≤ , j = 1 , . . . , k } where ℓ j ( x ) := h x, r j i − α j . Here r j = ( r j , ..., r dj ) ∈ R d is the primitive outward normal to the j − thcodimension one face of P ; α j ∈ R ; and < · , · > is the standard innerproduct on R d . Recall that the support function h P : R d → R of P isgiven by h P ( x ) = sup p ∈ P h x, p i . Fix an index J ∈ Z d + \ N P . Replacing P with N P above, this meansthat h NP ( r j ) ≤ N α j < h J, r j i for some j . Define Log : ( C ∗ ) d → R d via Log ( z ) := (log | z | , . . . , log | z d | ) . The pre-image of r j ∈ R d under Log is the complex d -torus S r j := {| z | = e r j } × · · · × {| z d | = e r dj } . We conclude that(4.2)
N H P ( z ) = h NP ( r j ) < h J, Log ( z ) i for every z ∈ S r j . Clearly, the above inequality is true for every positivemultiple of r j and hence on the set of tori S tr j for t > Write f ( z ) = P L ∈ ( Z + ) d a L z L . By the Cauchy integral formula a J = 1(2 πi ) d Z S trj f ( ζ ) ζ ( J + I ) dζ where z J = z j . . . z j d d and I = (1 , . . . , a J = 0for J ∈ Z d + \ N P . We write d | ζ | = Q i e tr ij dθ i . Then by the Cauchy-Schwarz inequality and (4.2) we have | a J | ≤ π ) d ( Z S trj | f ( ζ ) | e − NH P ( ζ ) (1 + | ζ | ) ǫ d | ζ | ) (cid:0) Z S trj e NH P ( ζ ) | ζ J + I ) | (1 + | ζ | ) ǫ d | ζ | (cid:1) ≤ π ) d ( Z S trj | f ( ζ ) | e − NH P ( ζ ) (1 + | ζ | ) ǫ d | ζ | ) (cid:0) Z S trj (1 + | ζ | ) ǫ | ζ I | d | ζ | (cid:1) . Thus, Q di =1 exp( tr ij )(1 + P di =1 exp(2 tr ij )) ǫ | a J | ≤ (2 π ) − d Z S trj | f ( ζ ) | e − NH P ( ζ ) (1+ | ζ | ) − ǫ d | ζ | . Note that some components r ij of r j could be negative and some couldbe nonnegative; e.g., for Σ we have r j = (0 , ... , − , , ...,
0) = − e j for j = 1 , ..., d and r d +1 = (1 /d, ..., /d ). Writing ζ i = ρ i e iθ where ρ i := e tr ij ,the above inequality becomes(4.3) Q di =1 ρ i (1 + P di =1 ρ i ) ǫ | a J | ≤ (2 π ) − d Z | ζ | = ρ i · · · Z | ζ d | = ρ d | f ( ζ ) | e − NH P ( ζ ) (1 + | ζ | ) ǫ Y i ρ i dθ i . From (1.2), P contains a neighborhood of the origin and hence for some i ∈ { , . . . , d } we have r ij ≥
0; i.e., we cannot have all r ij < Case 1 : There is some i ∈ { , ..., d } for which r ij >
0. Then for each i = 1 , . . . , d , we integrate both sides of (4.3) over ( ≤ ρ i ≤ T if r ij ≥ /T ≤ ρ i ≤ r ij < T → ∞ we see that a J = 0. Case 2: r ij ≤ i = 1 , . . . , d. Note that since r j = ~ i such that r ij <
0. Since P is a convex polytope this implies that Z C d | z J | e − NH P ( z ) (1 + | z | ) − ǫ dV ( z ) = ∞ . ICIAK-ZAHARJUTA 13
On the other hand, since H P ( z , ..., z d ) = H P ( | z | , ..., | z d | ), the mono-mials a L z L occurring in f are orthogonal with respect to the weighted L − norm in (4.1). Hence for each such L we have | a L | Z C d | z L | e − NH P ( z ) (1 + | z | ) − ǫ dV ( z ) ≤ Z C d | f ( z ) | e − NH P ( z ) (1 + | z | ) − ǫ dV ( z ) < ∞ from which we conclude that a J = 0. (cid:3) Remark 4.4.
Clearly if P is a convex body in ( R + ) d and f ∈ O ( C d )satisfies (4.1) then for any convex polytope P ′ containing P , f satisfies(4.1) with P ′ so that f ∈ P oly ( N P ′ ).We will use the following version of H¨ormander’s L -estimate ([8,Theorem 6.9] on page 379) for a solution of the ∂ equation: Theorem 4.5.
Let Ω ⊂ C d be a pseudoconvex open subset and let ϕ be a psh function on Ω . For every r ∈ (0 , and every (0 , form g = P dj =1 g j dz j with g j ∈ L p,q (Ω , loc ) , j = 1 , ..., d such that ∂g = 0 and Z Ω | g | e − ϕ (1 + | z | ) dV ( z ) < ∞ where | g | := P dj =1 | g j | there exists f ∈ L (Ω , loc ) such that ∂f = g and Z Ω | f | e − ϕ (1 + | z | ) − r dV ( z ) ≤ r Z Ω | g | e − ϕ (1 + | z | ) dV ( z ) < ∞ . Moreover, we can take f to be smooth if g and ϕ are smooth. Finally we will use the following result, which is Lemma 2.2 in [2].
Lemma 4.6.
Let P be a convex body in ( R + ) d and ψ ∈ L P, + ( C d ) . Thenfor every p ∈ P ◦ there exists κ, C ψ > such that ψ ( z ) ≥ κ max j =1 ,...,d log | z j | + log | z p | − C ψ for every z ∈ ( C ∗ ) d . Proof of Theorem 4.1.
From Remark 4.2, given z ∈ C d and ǫ > N large and p N ∈ P oly ( N P ) with1 N log | p N ( z ) | ≤ Q ( z ) , z ∈ K and 1 N log | p N ( z ) | > V P,K,Q ( z ) − ǫ. STEP 1 : We write V := V P,K,Q . Since V is continuous, we can fix δ > V ( z ) > V ( z ) − ǫ/ z ∈ B ( z , δ ) . If V is not smooth on C d then we approximate V by smooth psh func-tions V t := χ t ∗ V ≥ V on C d with χ as in (2.1). Since V is continuous, V t converges to V locally uniformly as t → . Let η be a test function with compact support in B ( z , δ ) = { z : | z − z | < δ } such that η ≡ B ( z , δ ) . For a fixed point p ∈ P ◦ , wedefine ψ N,t ( z ) := ( N − dκ )( V t ( z ) − ǫ dκ log | z p | + d max j =1 ,...,d log | z j − z ,j | where κ > dκ ≪ N. Note that ψ N,t is psh on C d , and smooth away from z . Applying Theorem 4.5 with the weightfunction ψ N,t , for every r ∈ (0 ,
1] there exists a smooth function u N,t on C d such that ∂u N,t = ∂η and(4.4) Z C d | u N,t | e − ψ N,t (1 + | z | ) − r dV ( z ) ≤ r Z C d | ∂η | e − ψ N,t (1 + | z | ) dV ( z ) . Note that the (0 ,
1) form ∂η is supported in B ( z , δ ) \ B ( z , δ ); thereforeboth integrals are finite. Since ψ N,t ( z ) = d max j =1 ,...,d log | z j − z ,j | +0(1)as z → z we conclude that u N,t ( z ) = 0 . Moreover, since V t ≥ V byLemma 4.6 and (4.4) we obtain Z C d | u N,t | e − N ( V t − ǫ ) (1 + | z | ) − r dV ( z ) ≤ C e − N ( V ( z ) − ǫ ) where C > N or t .Next, we let f N,t := η − u N,t . Then f N,t is a holomorphic function on C d such that f N,t ( z ) = 1 . Furthermore, Z C d | f N,t | e − N ( V t − ǫ ) (1 + | z | ) − r dV ( z ) ≤ C e − N ( V ( z ) − ǫ ) and these bounds are uniform as C > N and t . Weextract a convergent subsequence f N,t k → f N as t k → f N is a ICIAK-ZAHARJUTA 15 holomorphic function on C d satisfying f N ( z ) = 1 and(4.5) Z C d | f N | e − N ( V − ǫ ) (1 + | z | ) − r dV ( z ) ≤ C e − N ( V ( z ) − ǫ ) . Finally, using V ∈ L P, + ( C d ) we see that Z C d | f N | e − NH P (1 + | z | ) − r dV ( z ) < ∞ . Taking r > f N ∈ P oly ( N P ). STEP 2 : We want to modify f N ∈ P oly ( N P ) satisfying (4.5) and f N ( z ) = 1 to get p N . Note V ( z ) ≤ Q ( z ) on all of K . Fix ρ > r > C r := min z ∈ K ρ (1 + | z | ) − r where K ρ = { z : dist ( z, K ) ≤ ρ } . There exists β = β ( ρ ) > | V ( z ) − V ( y ) | < ǫ if y, z ∈ K ρ with | y − z | < β . Without loss of generality we may assume β ≤ ρ (or elsereplace β by min[ β, ρ ]).For z ∈ K , applying subaveraging to | f N | on B ( z, β ) ⊂ K ρ we have | f N ( z ) | ≤ C β Z B ( z,β ) | f N ( y ) | dV ( y ) . Thus, for every z ∈ KC r | f N ( z ) | e − NQ ( z ) ≤ C r | f N ( z ) | e − NV ( z ) ≤ C β Z B ( z,β ) | f N ( y ) | e − NV ( z ) (1 + | y | ) − r dV ( y ) ≤ C β Z B ( z,β ) | f N ( y ) | e − N ( V ( y ) − ǫ ) (1 + | y | ) − r dV ( y ) ≤ C β C e − N ( V ( z ) − ǫ ) from (4.5). Thus taking p N := q C r C β C e N ( V ( z ) − ǫ ) f N we have p N ∈ P oly ( N P ) and max z ∈ K | p N ( z ) e − NQ ( z ) | ≤ . Finally, 1 N log | p N ( z ) | = V ( z ) − ǫ + 12 N log C r C β C . Since none of C r , C β , C depend on N , we have1 N log | p N ( z ) | > V ( z ) − ǫ for N sufficiently large . This completes the proof of the pointwise convergence of1 N log Φ N ( z ) := [sup { N log | p ( z ) | : p ∈ P oly ( N P ) , || pe − NQ || K ≤ } ]to V P,K,Q ( z ). The local uniform convergence follows as in the proof ofLemma 3.2 of [6]; this utilizes the observation that Φ N · Φ M ≤ Φ N + M . (cid:3) Remark 4.7.
Let u ∈ L P, + ∩ C ( C d ) with u ≤ Q on K . The sameargument as in Steps 1 and 2 applies to u to show: given z ∈ C d and ǫ > N large and p N ∈ P oly ( N P ) with1 N log | p N ( z ) | ≤ Q ( z ) , z ∈ K and 1 N log | p N ( z ) | > u ( z ) − ǫ. Note we have not assumed continuity of Q in Theorem 4.1. Weproceed to do the general case (Theorem 1.1) using Theorem 4.1; i.e.,having proved if V P,K,Q is continuous, then V P,K,Q = ˜ V P,K,Q , we verifythe equality without this assumption. We begin with an elementaryobservation.
Lemma 4.8.
For any K compact and Q admissible, V P,K,Q ( z ) = sup { u ( z ) : u ∈ L P, + : u ≤ Q on K } . Proof.
We have Q is bounded below on K ; say Q ≥ m on K . Now K is bounded and so K ⊂ D R = { z ∈ C d : | z j | ≤ R, j = 1 , ..., d } for all R sufficiently large. Then for u ∈ L P with u ≤ Q on K we have˜ u ( z ) := max[ u ( z ) , m + H P ( z/R )] ∈ L P, + with ˜ u ≤ Q on K . (cid:3) Proof of Theorem 1.1.
First we show: if Q is continuous, then V P,K,Q =˜ V P,K,Q . From Lemma 4.8 and Remark 4.7, it suffices to show that if u ∈ L P, + with u ≤ Q on K , given ǫ >
0, for t > u t defined in (3.1) satisfies u t ∈ L P, + with u t ≤ Q + ǫ on K . That u t ∈ L P, + follows from Proposition 3.1 and Remark 3.2. Since u t ↓ u ICIAK-ZAHARJUTA 17 on K , u t | K ∈ C ( K ), u is usc on K , and K is compact, by Dini’stheorem, given ǫ >
0, there exists t such that for all t < t we have u t ≤ Q + ǫ on K , as desired.Finally, to show V P,K,Q = ˜ V P,K,Q in the general case, i.e., where Q isonly lsc and admissible on K , we utilize the argument in [5], Lemma7.3 (mutatis mutandis) to obtain the following. Proposition 4.9.
Let K ⊂ C d be compact and let w j = e − Q j be ad-missible weights on K with Q j ↑ Q . Then lim j →∞ V P,K,Q j ( z ) = V P,K,Q ( z ) for all z ∈ C d . Taking Q j ∈ C ( K ) with Q j ↑ Q , since V P,K,Q j = ˜ V P,K,Q j ≤ ˜ V P,K,Q for all j , we conclude from the proposition that V P,K,Q = ˜ V P,K,Q . Thisconcludes the proof of Theorem 1.1. (cid:3)
We finish this section with some remarks on regularity of P − extremalfunctions. Recall that a compact set K is L − regular if V K = V Σ ,K iscontinuous on K (and hence on C d ) and K is locally L − regular if it islocally L − regular at each point a ∈ K ; i.e., if for each r > V K ∩ B ( a,r ) is continuous at a where B ( a, r ) = { z : | z − a | ≤ r } . For aconvex body P ⊂ ( R + ) d we define the analogous notions of P L − regular and locally P L − regular by replacing V K by V P,K . For any such P thereexists A > P ⊂ A Σ; hence V P,K ( z ) ≤ A · V K ( z ) and V P,K ∩ B ( a,r ) ( z ) ≤ A · V K ∩ B ( a,r ) ( z )so if K is L − regular (resp., locally L − regular) then K is P L − regular(resp., locally P L − regular). Note for P satisfying (1.2) there exist0 < a < b < ∞ with a Σ ⊂ P ⊂ b Σ so that K is locally P L − regular ifand only if K is locally L − regular. Corollary 4.10.
For K compact and locally L − regular and Q contin-uous on K , V P,K,Q is continuous.Proof.
From Theorem 1.1, V P,K,Q is lowersemicontinuous. We show V ∗ P,K,Q ≤ Q on K from which it follows that V ∗ P,K,Q ≤ V P,K,Q and henceequality holds and V P,K,Q is continuous.Since K is locally L − regular, it is locally P L − regular. Given a ∈ K and ǫ >
0, choose r > Q ( z ) ≤ Q ( a ) + ǫ for z ∈ K ∩ B ( a, r ). Then V P,K,Q ( z ) ≤ V P,K ∩ B ( a,r ) ,Q ( a )+ ǫ ( z ) = Q ( a ) + ǫ + V P,K ∩ B ( a,r ) ( z ) for all z ∈ C d . Thus, at a , V P,K,Q ( a ) ≤ Q ( a )+ ǫ . Moreover, by continuityof V P,K ∩ B ( a,r ) at a , we have V P,K ∩ B ( a,r ) ( z ) ≤ ǫ for z ∈ B ( a, δ ), δ > V ∗ P,K,Q ( a ) ≤ Q ( a ) + 2 ǫ which holds for all ǫ > (cid:3) Remark 4.11.
The converse-type result that for a compact set K ⊂ C d , if V P,K,Q is continuous for every Q continuous on K then K is locally P L − regular, follows exactly as in [9] Proposition 6.1.5. Appendix
We provide a version of the lemma from Ferrier [10] appropriate forour purposes to show u t in Proposition 3.1 is psh. For λ >
0, we use thedistance function d λ : C d × C → [0 , ∞ ) defined as d λ ( z, w ) = λ | z | + | w | (in our application, t = 1 /λ ). Lemma 5.1.
Let δ : C d → [0 , ∞ ) be nonnegative. For λ > , define b δ λ ( s ) := inf s ′ ∈ C d [ δ ( s ′ ) + λ | s ′ − s | ] . Let Ω := { ( s, t ) ∈ C d × C : | t | < δ ( s ) } . Then (5.1) b δ λ ( s ) = d λ (cid:0) ( s, , ( C d × C ) \ Ω (cid:1) . Furthermore, if δ is lsc, then Ω is open. Moreover, Ω = { ( s, t ) : − log δ ( s ) + log | t | < } so that if, in addition, − log δ is psh in C d , then Ω is pseudoconvex in C d × C .Proof. This is straightforward; first observe d λ (cid:0) ( s, , ( C d × C ) \ Ω (cid:1) = inf { λ | s − s ′ | + | t | : ( s ′ , t ) ∈ ( C d × C ) \ Ω } = inf { λ | s − s ′ | + | t | : | t | ≥ δ ( s ′ ) } = inf { λ | s − s ′ | + δ ( s ′ ) : s ′ ∈ C d } = b δ λ ( s ) . Next, Ω := { ( s, t ) ∈ C d × C : | t | < δ ( s ) } = { ( s, t ) ∈ C d × C : − log δ ( s ) + log | t | < } . (cid:3) ICIAK-ZAHARJUTA 19
Corollary 5.2.
Under the hypotheses of the lemma, if − log δ is psh in C d then − log b δ λ is psh.Proof. Since Ω is pseudoconvex in C d × C and d λ : C d × C → [0 , ∞ )is a distance function, we have U ( s, t ) := − log d (cid:0) ( s, t ) , ( C d × C ) \ Ω (cid:1) is psh. Thus U ( s,
0) = − log b δ λ ( s ) is psh. (cid:3) References [1] T. Bayraktar, Zero distribution of random sparse polynomials,https://arxiv.org/abs/1503.00630 (version 4).[2] T. Bayraktar, Zero distribution of random sparse polynomials,
Michigan Math.J. , (2017), no. 2, 389-419.[3] T. Bayraktar, T. Bloom, N. Levenberg, Pluripotential theory and convex bod-ies, Mat. Sbornik , (2018), no. 3, 67-101.[4] T. Bayraktar, T. Bloom, N. Levenberg and C. H. Lu, Pluripotential Theoryand Convex Bodies: Large Deviation Principle, Arkiv for Mat. , , (2019),247-283.[5] T. Bloom and N. Levenberg, Weighted pluripotential theory in C n , Amer. J.Math. , , (2003), 57-103.[6] T. Bloom and B. Shiffman, Zeros of random polynomials on C n , Math. Res.Lett. , , (2007), no. 3, 469-479.[7] L. Bos and N. Levenberg, Bernstein-Walsh theory associated to convex bod-ies and applications to multivariate approximation theory, Comput. MethodsFunct. Theory , (2018), 361-388.[8] J.-P. Demailly, Complex analytic and differential geometry. ∼ demailly/manuscripts/agbook.pdf.[9] N. Q. Dieu, Regularity of certain sets in C n , Annales Polonici , , vol. 3,(2003), 219-232.[10] J. P. Ferrier, Spectral theory and complex analysis , North-Holland MathematicsStudies, No. 4, Notas de Matem´atica (49). North-Holland Publishing Co., NewYork, 1973.[11] E. Saff and V. Totik,
Logarithmic potentials with external fields , Springer-Verlag, Berlin, 1997.[12] J. Siciak, Extremal plurisubharmonic functions,
Annales Polonici ,39