ALEXANDER S CAPACITY FOR INTERSECTIONS OF ELLIPSOIDS IN C N. s N := dσ. = inf

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1 UNIVERITATI IAGELLONICAE ACTA MATHEMATICA, FACICULU XL 00 ALEXANDER CAPACITY FOR INTERECTION OF ELLIPOID IN C N by Mieczysław Jędrzejowski Abstract. Alexander s projective capacity for some intersections of ellipsoids in C N is computed.. Introduction. Let be the unit sphere in C N. Let σ denote the Lebesgue surface area measure on. Let s N := dσ. Let H n = H n (C N ) denote the set of all homogeneous polynomials of degree n (with complex coefficients) in N complex variables. Let K be a compact subset of C N. Let f K := sup{ f(z) : z K}, where f : K C is a continuous function. Definition.. (see [], [5] ). Alexander s projective capacity γ(k) is γ(k) := lim n (γ n(k)) n = inf n (γ n(k)) n, where γ n (K) := inf { Q K }, the infimum being taken over all homogeneous polynomials Q H n, normalized so that ( ) log Q(z) n dσ(z) = κ N, s N 000 Mathematics ubject Classification. Primary 3C5. Key words and phrases. Ellipsoid, projective capacity, extremal function. Research partially supported by KBN grant No. PO3 A 0454.

2 40 where It is known that κ N := log z N dσ(z). s N κ N = N j= j. Let L j > P j > 0 (j =,..., N). Let d ij > 0 (i, j =,..., N). For i =,..., N let N g i (z) := d ij z j, j= where z = (z,..., z N ) C N. Let E i := {z C N : g i (z) }. Let g(z) := max{g i (z) : i =,..., N}. Let E := {z C N : g(z) }. Obviously, (.) E = E E N. In this paper we take d ii := L i P i for i =,..., N and d ij := L j for i, j =,..., N (i j). Then we compute Alexander s projective capacity of the set E.. Preliminaries. Let K be a compact subset of C N. Definition.. (see [4], [5] ). The iciak homogeneous extremal function Ψ K is defined as follows: Ψ(z) = Ψ K (z) := lim (Ψ n(z)) n, z C N, n where Ψ n (z) := sup{ Q(z) }, the supremum being taken over all Q H n, normalized so that Q K =. The function g(z) is a norm in C N and therefore we have: Proposition.. (see [4]). If the set E is given by (.) then Ψ E (z) = g(z), z C N. Definition.3. (see [5], p. 53). The constant τ(k) is given by the formula: ( τ(k) := exp ) log Ψ K (z) dσ(z). s N Theorem.4. (see []). If K is a compact subset of C N then γ(k) = exp(κ N )τ(k).

3 4 Let N D = D N := (r,..., r N ) R N : r 0,..., r N 0, rj =, N Σ = Σ N := (θ,..., θ N ) R N : θ 0,..., θ N 0, θ j =. Lemma.5. (see [3], Lemma 3.3). If f : D R is a continuous function then f ( z,..., z N ) dσ(z) = ( f θ,..., ) θ N dω(θ), s N vol(σ) Σ where ω is the Lebesgue surface area measure on the hyperplane N (θ,..., θ N ) R N : θ j = and vol(σ) := Σ dω(θ). Lemma.6. (see [3], Lemma 3.6). Let a j > 0 for j =,..., N. Let F (a,..., a N ) := z N Log z dz πi (z a )... (z a N ), C where C is any contour in the right half-plane enclosing all the points a,..., a N, and Log z is the principal branch of the logarithm. Then N N vol (Σ N log a j θ j dω(θ) = ) Σ N j + F (a,..., a N ). j= 3. Main result. For j =,..., N let L j > P j > 0 and let b j := /P j, c N := N i= b i, t j := b j /c N, w := + N i= L ib i, T := w/c N, j= j= R j := F (L,..., L j, T, L j+,..., L N ). Theorem 3.. If the set E is given by (.) then log γ(e) = N t j R j. j= j= j=

4 4 Proof. For θ R N let u(θ) := N j= L jθ j and v i (θ) := u(θ) P i θ i. We first observe that (3.) log γ(e) (3.) = N j= j vol(σ) Σ max(log v (θ),..., log v N (θ)) dω(θ). Indeed, combining Theorem.4 with Proposition. and Lemma.5 gives log γ(e) = κ N + log τ(e) = κ N log Ψ E (z) dσ(z) s N = κ N log g(z) dσ(z) = s N N j= For i =,..., N define Obviously, j vol(σ) Σ max(log v (θ),..., log v N (θ)) dω(θ). Λ i := { θ Σ N : P i θ i P j θ j, j =,..., N }, m i := max(log v (θ),..., log v N (θ)) dω(θ) vol(σ) Λ i = log v i (θ) dω(θ). vol(σ) Λ i max(log v (θ),..., log v N (θ)) dω(θ) = vol(σ) Σ We next show that (3.3) m i = t i N j= j + R i. N m i. Without loss of generality we can assume that i =. Clearly, (3.4) m = log v (θ) dω(θ), vol(σ) where Λ := Λ. Λ i=

5 43 An analysis similar to that in the proof of Theorem 3. ([3], p. 57) shows that log v (θ) dω(θ) vol(σ) Λ (3.5) N = t log T η + L j η j dω(η). vol(σ) Indeed, we change the variables: Σ θ = t η, j= θ j = t j η + η j for j =,..., N. Let us observe that the simplex Λ has the vertices: (t, t,..., t N ), (0,, 0,..., 0), (0, 0,,..., 0),..., (0, 0, 0,..., ). It is easy to see that η,..., η N are the barycentric coordinates on Λ. Applying (3.4), (3.5) and Lemma.6 we get m = t = t N j= N j= j + F (T, L, L 3,..., L N ) j + R. Now (3.3) is proved for i =. In the same manner we can show that (3.3) is true for i =,..., N. Combining (3.) with (3.) and (3.3) we get log γ(e) = which completes the proof. = = N j= N j= N t i R i, i= N j i= N j + j= m i N j t i R i i=

6 44 Remark 3.. Now suppose that P j = 0 for at least one j. Then the set E is an ellipsoid: E = { (z,..., z N ) C N : L z + + L N z N }. Therefore in this case (see [3], Theorem 3.): log γ(e) = F (L,..., L N ). References. Alexander H., Projective capacity, Conference on everal Complex Variables, Ann. of Math. tud., 00, Princeton Univ. Press, 98, Cegrell U., Kołodziej., An identity between two capacities, Univ. Iagel. Acta Math., 30 (993), Jędrzejowski M., Alexander s projective capacity for polydisks and ellipsoids in C N, Ann. Polon. Math., 6 (995), iciak J., Extremal plurisubharmonic functions in C n, Ann. Polon. Math., 39 (98), , Extremal Plurisubharmonic Functions and Capacities in C n, ophia Kokyuroku in Math., 4, ophia University, Tokyo, 98. Received December 0, 999 Jagiellonian University Institute of Mathematics Reymonta Kraków, Poland jedrzejo@im.uj.edu.pl