Weighted polynomials and weighted pluripotential theory

We study the relation between weighted pluripotential on a compact set E in C^N and the pluripotential theory of an associated circed set Z in C^(N+1)


Introduction
An admissible weight on a compact set E ⊂ C N is a function w ≥ 0 which is strictly positive on a non-pluripolar subset of E. Associated to (E, w) is a weighted pluripotential theory involving weighted polynomials, i.e, functions of the form w d p where p is a polynomial of degrees ≤ d, a weighted pluricomplex Green function V E,Q and a weighted equilibrium measure dµ eq (E, w). The definitions of these concepts are given in section 1.
In the one-dimesional case (N = 1) the book of Saff and Totik [SaTo] has many basic results. In the one-dimensional case, weighted polynomials arise in diverse problems -approximation theory, orthognal polynomials, random matrices, statistical physics. For an example of recent developments see [Dei]. In the higher dimensional case, weighted pluripotential theory was used in [BL2] to obtain results on directional Tchebyshev constants of compact sets -the main procedure being an inductive step from circular compact sets to a weighted problem in one less variable.
In this paper we further develop the relation between weighted pluripotential theory on a compact set E ⊂ C N with admissible (see (1.10)) weight w and the potential theory of a canonically associated circular set Z ⊂ C N+1 (defined in (2.1)).
We show that V Z , the pluricomplex Green function of Z, and dµ eq (Z) the equilibrium measure of Z, are related to the weighted pluricomplex Green function and the weighted equilibrium measure of E with weight w.
The main results are: Theorem 2.1.
Here t is the first coordinate of C N+1 and L : C N+1 − {(t, z)|t = 0} → C N is given by (2.4). L * is the push-forward of measures under L.
Special cases of the above results may be found in the paper of DeMarco [DeM].
In particular theorem 2.1 generalizes examples of section 4 of [DeM] and theorem 2.2 generalizes lemma 2.3 of [DeM].
The advantage of considering weighted pluripotential theory is that (up to a limiting procedure described in section 5 of this paper) the potential theory of a general compact circular set in (N + 1) variables may be reduced to the weighted case in N variables.
In section 3 we consider the Bernstein-Markov (B-M) inequality (for the definition see (3.1)). This inequality may be used to relate asymptotics of orthonormal polynomials with respect to a measure µ on E to potential theoretic invariants of E. We introduce a weighted version of the B-M inequality (see(3.2)). We show (theorem 3.1) that the weighted B-M inequality holds on E with weight w and measure µ if and only if the B-M inequality holds for an associated measure on Z.
Then we give the following cases where the B-M inequality holds.
Corollary 3.1. dµ eq (E, w) for E regular and weight w.
In section 4 we obtain asymptotics for the leading coefficients of orthonormal polynomials with respect to certain exponentially decreasing measures on R n . As in the known procedure in the one-variable case, we first scale the problem to obtain a problem on the asymptotics of weighted polynomials. Using the weighted B-M inequality (a special case of theorem 3.2) gives the asymptotics (see example 4.1 and equation (4.24)).

Preliminaries
Let E be a bounded subset of C N . The pluricomplex Green function of E is defined by is the Lelong class of PSH functions of logarithmic growth. (We use the notation A set E ⊂ C N is said to be pluripolar if for all points a ∈ E, there is a neighborhood U of a and a function v which is PSH on U such that E ∩ U ⊂ {z ∈ U |v(z) = −∞}. A property of a set E is said to hold quasi-everywhere (q.e.) if there is a pluripolar set P ⊂ E and the property holds at all points of E \ P .
For G an open subset of C N and f a real-valued function on G, we let f * denote its uppersemicontinuous (u.s.c.) regularization, defined by V * E ∈ L if and only if E is non-pluripolar [K]. For E non-pluripolar, the equilibrium measure of E is defined by where (dd c ) N is the complex Monge-Ampère operator. dµ eq is a positive Borel measure of total mass (2π) N and with supp(dµ eq ) ⊂ E. [K].
In the case that E is a compact set, it is a result of Siciak and Zaharyuta ( [K], It follows that V E (z) = VÊ(z) whereÊ denotes the polynomially convex hull of E.
We will also use the class H of logarithmically homogeneous PSH functions on For E a bounded set in C n , we define (1.7) H E (z) := sup{u(z)|u ∈ H, u ≤ 0 onE}.
For E compact (see [Si2]) we have |p is a homogeneous holomorphic polynomial, deg p ≥ 1 and p E ≤ 1}.
For E a compact set in C N , an admissible weight function is a real-valued function In particular, if E admits on admissible weight function then E itself is nonpluripolar.
There is a "weighted" version of the pluricomplex Green function (see [S.1], [SaTo, appendix B]) defined as follows: let Then Q is lowersemicontinuous (l.s.c.) on E. The weighted pluricomplex Green function of E with weight w is defined by The weighted equilibrium measure of E is defined by It is a positive Borel measure with supp (dµ eq (E, w)) ⊂ E. and total mass (2π) N .
A weighted polynomial on E is defined to be a function of the form w d p where d is an integer ≥ 0 and p is a holomorphic polynomial of degree ≤ d. Note that if w d p E ≤ 1 then 1 d log |p(z)| ≤ Q(z) on E and since 1 d log |p(z)| ∈ L we have, It is known (see [Si1] or [SaTo], appendix B) that (1.14) Si1]) to be locally regular at a point a ∈ E if for each r > 0, V E∩B(a,r) is continuous at a. Here B(a, r) := {z ∈ C N |z − a| ≤ r} denotes the ball center a, radius r. It is sufficient, for E to be locally regular at a, that V E∩B(a,r) be continuous at a for all r > 0 sufficiently small.
E is said to be locally regular if it is locally regular at each point of E.
For E compact and locally regular and w a continuous admissible weight function on E then V E,Q is continuous [Si1].
For u ∈ L we define its Robin function ρ u by Then ρ(z) ∈ H.

Equilibrium measures
Let E be a compact set in C N and w an admissible weight function on E. We associate the set Z = Z(E, w) ⊂ C N+1 defined as follows: (2.1) We will relate the weighted potential theory on E with weight w to potential theory on Z. We will use the notation (t, z) for a point in C N+1 where t ∈ C and z ∈ C N .
We denote by C λ the complex line in C N+1 given by (the closure of Z) is compact and circular.
Z is non-pluripolar since E is non-pluripolar ( [BL2], lemma 6.1). Note that for u ≤ 0 on Z then u ≤ 0 onZ. But since w is u.s.c. we have, for λ ∈ E: Applying the maximum principle to the subharmonic function t → u(t, λ 1 t, · · · , λ N t) we have u ≤ 0 onZ.
The second statement follows similarly.
Proof: Since H * Z ∈ H and H * Z (0) = −∞ then H * Z < 0 in a neighborhood of the origin (estimates on the size of that neighborhood, know as the Sibony-Wong inequality, can be found in ( [A], [Si2]).
Thus, the origin is an interior point ofẐ. But dµ eq (Z) places no mass on the interior ofẐ so the result follows.
Let L denote the mapping L : If we consider P N (complex projective N -space) as the space of lines through the origin in C N+1 , then L gives one of the standard coordinate charts for P N . Note We recall the "H-principle" of Siciak [Si3]. There is a natural 1 − 1 correspondence between H(C N+1 ) and L(C N ) as follows: Then u ∈ L(C N ). Conversely, given u ∈ L(C N ) we let Furthermore, let P d (t, z) be a homogeneous polynomial of degree d on C × C N .
is a homogeneous polynomial on C N+1 of degree d in (t, z). We use the notation: Given a weighted polynomial w d G d (λ) on E we can relate its norm on E with the norm of the associated polynomial P d (t, z) on Z (or equivalently, Z). Specifically, we have The result follows.
Theorem 2.1 below gives the relation between the pluricomplex Green function on E and the homogeneous pluricomplex Green function of Z.
Hence u ≤ V E,Q and so, using (2.6) Taking the pointwise sup in (t, z) over all suchũ we have It remains to prove the reverse inequality.
Taking the pointwise sup over all such u gives the reverse inequality to (2.12).
Then, by theorem 2.1, . This proves the result for t = 0 but since both sides (in the statement of corollary 2.1) are PSH functions on C N+1 and agree for t = 0 they must agree on C N+1 .
Note that the result in ( [K], prop. 2.9.16) is similar but not immediately applicable.
Corollaries 2.2, 2.3 and 2.4 deal with the converge of sequences of pluricomplex Green functions for sequences of weights converging in various manners (see also lemma 7.3 [BL2]).
Corollary 2.2. Let E ⊂ C N be compact and {w j } j=1,2,··· a sequence of admissible weights on E and let w also be an admissible weight on E. Suppose that w j ↓ w.
Proof: Let Z j := Z j (E, w j ) and Z := Z(E, w) be the associated circular sets in z) |t| ≤ w} differ by a pluripolar set so the result follows from ( [K], cor. 5.2.5).
Corollary 2.4. Let E, {w j }, w be as in corollary 2.2 except that w j ↑ w q.e. Then Proof: For some pluripolar set F we have Z j ∪ F ↑ Z ∪ F . Hence by ( [K], cor.

and 5.2.6)
Hence, using homogeneity, H * Z ↓ H Z and by corollary 2.1, By proposition 2.3, we may consider L * (dµ eq (Z))-the push forward of the equilibrium measure dµ eq (Z) under L. Since supp(dµ eq (Z)) ⊂ Z ⊂ λ∈E C λ we have supp(L * (dµ eq (Z)) ⊂ E. There is however a more precise relation. Assume that Z is regular. The equilibrium measure on Z and the weighted equilibrium measure on E are related by: Theorem 2.2. L * 1 2π dµ eq (Z) = dµ eq (E, w).
Proof: The proof is based on lemma 3.3 in [DeM] which itself is based on work of Briend. (Note that we use the convention of Klimek's book [K] for d c := i(∂ − ∂) not that of [DeM]. This results in the factor 1 2π in the statement of theorem 2.2). H Z is continuous by proposition 2.2 so, as a consequence of Theorem 2.1 V E,Q is continuous. Then Let φ be a smooth compactly supported function on C N+1 − {t = 0}. then (2.16) Max(H Z (z), ǫ) > ǫ then V Z = H Z . So, in the expression on the right of (2.17) we may replace V Z by H Z to obtain. (2.18) For λ ∈ E, we let dm λ be the Lebesgue measure on the circle |t| = w(λ) in C λ normalized to have total mass 1. Then 1 2π dd c V Z/C λ = dm λ . The right side of (2.19) is thus equal to (2.20) 1 2π which proves theorem 2.2.
The next corollary shows that the assumption ofZ being regular may be dropped from the hypothesis of theorem 2.2.
Proof: We need only find a sequence of locally regular compact sets E j , admissible, continuous, weights w j on E j such that E j ↓ E and w j ↓ w on E. Then Z(E j , w j ) ↓ Z(E, w). Applying theorem 2.2 to each Z(E j , w j ) and E j and taking limits gives the result.
To construct such a sequence of E j and w j we may follow the procedure of ([BL2], section 7).

The Bernstein-Markov inequality
Given a compact set E ⊂ C N and a finite positive Borel measure µ on E, we say that (E, µ) satisfies the Bernstein-Markov (B-M) inequality if, for every ǫ > 0, there exists a constant C = C(ǫ) > 0 such that, for all holomorphic polynomials p we have This inequality may be used to relate L 2 properties of polynomials with potential theoretic invariants of E (see [B1] and [BL2] for conditions under which the inequality holds).
We will introduce a "weighted" version of the B-M inequality.
Given a compact set E ⊂ C N , an admissible weight w on E and a finite positive Borel measure µ on E, we say that (E, w, µ) satisfies the weighted B-M inequality if for all ǫ > 0, that exists a constant C = C(ǫ) > 0 such that, for all weighted polynomials w d p we have Of course, for w ≡ 1, (3.2) reduces to (3.1).
We will relate the weighted B-M inequality for (E, w, µ) to a B-M inequality onZ with respect to a certain associated measure ν. The measure ν is defined as follows: Proof: First (using the notation of lemma 2.1) we prove Lemma 3.1.
Now, suppose (Z, ν) satisfies the B-M inequality. Applying that inequality to homogeneous polynomials, using lemmas 2.1 and 3.1 we obtain the weighted B-M inequality for (E, w, µ).
For the converse, suppose (E, w, µ) satisfies the weighted B-M inequality.
We first note that if two monomials are of different degrees, they are orthogonal in L 2 (ν) since their restrictions to any C λ are orthogonal in L 2 (dm λ ). Hence for a polynomial p on C N+1 , written as a sum of homogeneous polynomials Hence, for any ǫ > 0 there is a C > 0 such that where the second inequality comes from lemmas 2.1, 3.1 and the weighted B-M inequality for (E, w, µ). The third inequality in (3.7) comes from (3.6). The B-M inequality for (Z, ν) follows from (3.7).
We will give another general situation in which the weighted B-M inequality holds (see also [StTo], theorem 3.2.3. (vi)). Proof: log w is continuous on E and so may be, by the Weierstrass theorem, approximated by (real) polynomials. That is, given ǫ > 0 there exists g ǫ = g ǫ (x 1 , ·, x n ), a real polynomial, such that log w − g ǫ E ≤ ǫ Taking exponentials, we have We consider g ǫ as a holomorphic polynomial Taking sufficent many terms in the power series for exp(g ǫ ) we get a holomorphic polynomial H such that, for ǫ sufficiently small Now, consider a weighted polynomial w d G and let Now, by the B-M inequality for (E, σ) we have, given ǫ 1 > 0 a constant C 1 > 0 such that (3.14) where h := deg H. Hence the left inequality in (3.13).
Example 3.1 Let B R = {x ∈ R N |x| ≤ R} be the (real) ball of radius R (center the origin). Then (B R , dx) satisfies the BM inequality (see [B]) where dx denotes Lebesgue measure.
Let w(x) be any continuous positive function on B R . Then by theorem 3.2 (B R , w, dx) satisfies the weighted B-M inequality.

L 2 theory of weighted polynomials
Let E be a compact non-pluripolar subset of C N , w an admissible weight on E, and µ a finite positive Borel measure with supp(µ) = E. For d a positive integer, the monomials are linearly independent in L 2 (w 2d µ) ([Bl1], prop. 3.5 adapts to this situation). Ordering via a lexicographic ordering on their multi-index exponents and applying the Gram-Schmidt procedure we obtain orthonomal polynomials We can write (4.2) p d α (z, µ) = a d α z α + (monomials of lower lexicographic order) where a d α > 0 We will only consider these polynomials where |α| = d.
Proof: First, we recall the definition of weighted directional Tchebyshev constant (see [BL2]). For α a multiindex we let P (α) = {q|q = z α + β<α c β z β } where c β ∈ C and the notation β < α is used to denote the fact that the multiindex β preceeds α in the lexicographic ordering on the multi-indices.
For α a multiindex with |α| = d we let t d α denote a (Tchebyshev) polynomial which minimizes { w d q E q ∈ P(α)}. That is, t d α ∈ P(α) and (4.5) Then (see [BL2]) it is known that for a sequence of multi-indices {α(j)} j=1,2,··· satisfying (4.4) the limit exist and is called the weighted Tchebyshev constant in the direction θ ∈ Σ 0 . Now, it follows from general Hilbert space theory that where q d α is the unique polynomial in P(α) satisfying.
(4.8) w d q d α L 2 (µ) = inf{ w d q L 2 (µ) q ∈ P(α)} Now, for ǫ > 0, there is a C > 0 such that (4.9) by the weighted B-M inequality by (4.8) since the sup norm estimates the L 2 norm for a finite measure with compact support.
Hence, for every ǫ > 0 there is a constant C 1 > 0 such that (4.10) Now, given a sequence of multi-indices {α(j)} satisfying (4.4), taking the 1/d powers of the expressions in (4.10), letting j → ∞, using (4.6), (4.7) and the fact that ǫ > 0 is arbitrary, the result follows. i) H(x) is homogeneous of degree γ > 0. That is ii) H(x) > 0 for all x = 0 We let {p α (x)} α∈N N denote the orthonormal polynomials obtained, by applying the Gram-Schmidt procedure to the (real) monomials ordered via a lexicographic ordering of their exponents. Then (4.12) for any two multi-indices α, β.
We write (4.13) p α (x) = a α x α + (sum of monomials of lower lexicographic order). a α > 0 We will obtain asymptotic estimates (see (4.24)) for |a α | 1 |α| for a sequence of multi-indices satisfying (4.4). In the case N = 1, these estimates are Theorem VII, 1.2 of [SaTo]. In that case explicit knowledge of the set S w (defined below) yields an explicit form to the right hand side of (4.24). It would be of interest to find S w explicitly in the case N > 1.
In the one-dimensional case (N = 1) this gives a version of so-called weak asymptotics and in this case considerably more detailed asymptotic results are known (see [SaTo] or [Dei]).
For |α| = d we scale by x = d 1 γ y. We get Consider the weight w(y) = e − H(y) 2 on R N ⊂ C N . This weight is admissible in the sense of ( [SaTo], appendix B) although, since R N is not compact, not in the sence of 1.10. We let Q(y) = H(y) 2 . The following is known ( [SaTo], appendix B). S w := (dd c V R N ,Q ) N has compact support. For any weighted polynomial w d p we have (4.16) In particular (4.17) sup Now V R N ,Q ∈ L so, fixing R > 0 large, using (4.16), there is a constant A > 0 such that.
We may assume S w ⊂ B R . Then (4.20) We get However for ǫ > 0 sufficiently small, the expression on the right of (4.22) is bounded in d. It follows that for a sequence of multi-indices {α(j)} satisfying (4.4) But, by (the proof of) theorem 4.1 the limit on the right side of (4.23) exists and it may be identified with τ w (S w , θ) using (4.1). Hence we obtain

General Circular Sets
The circular sets which arise in the formẐ(E, w) (i.e. the polynomially convex hull of a set of the form Z(E, w)) are i) polynomially convex ii) circular iii) compact iv) non-pluripolar However, they are not the most general sets with the properties i), ii), iii), iv).
We will show, however, that the most general set with those properties is, in an appropriate sense, a limit of sets of the formẐ(E, w) Let Z ⊂ C N+1 be a set with properties i), ii), iii), iv) above. Then the origin is an interior point of Z. We associate to Z the a function on C N defined by (5.1) w(λ) := sup{|t| (t, z) ∈ Z ∩ C λ } Then w(λ) > 0 for all λ and w is bounded above. We let Q(λ) := − log w(λ).
Proposition 5.1. w is u.s.c. on C N .
Hence V * Z R ↓ V * Z as R → ∞. The result follows from ( [K], cor 5.2.5 and 5.2.6) and the continuity of the Monge-Ampère operator under decreasing limits [K].