The paper is dedicated to approximation of plurisubharmonic functions in the Lelong class L(C n) by the functions of the form 1 dj log |pj |, where pj are polynomials in C n with degree deg pj ≤ dj . We must say that this result is inspired by Theorem 15.1.6 in 7. Some applications to pluripolar sets are also given
A THEOREM ON THE APPROXIMATION OF PLURISUBHARMONIC FUNCTIONS IN LELONG CLASSES KIEU PHUONG CHI Abstract. The paper is dedicated to approximation of plurisubharmonic functions in the Lelong class L(Cn ) by the functions of the form d1j log |pj |, where pj are polynomials in Cn with degree deg pj ≤ dj . We must say that this result is inspired by Theorem 15.1.6 in [7]. Some applications to pluripolar sets are also given. 1. Introduction The set of plurisubharmonic functions with logarithmic growth is defined by 1 (1) L(Cn ) = {u plurisubharmonic on Cn : u(z) ≤ log(1+|z|2 )+Cu }, 2 where Cu is a constant depending only on u. We also define a more restricted class 1 L+ (Cn ) = {u plurisubharmonic on Cn : u(z) − log(1 + |z|2 ) ≤ Cu }. 2 n + n L(C ), L (C ) are sometimes referred to as the Lelong classes in Cn . These classes have been studied by many authors, including Leja, Lelong, Sadullaev, Siciak, Zaharjuta, Bedford and Taylor,... in connection with problems concerning polynomials in n complex variables (see [2, 9, 15] and the references given therein). Of particular interest for the Lelong classes is the Robin function defined by (2) ρu (z) = lim |λ|→∞,λ∈C sup u(λz) − log+ |λz| . The Robin function plays an important role in approximation problems of holomorphic functions by polynomials (see [1],[17] and the reference therein). One of important results concerning approximation of plurisubharmonic functions is the following theorem due to H¨ormander in [7] (Theorem 15.1.6). 2010 Mathematics Subject Classification. 32U05, 32E99, 32U15. Key words and phrases. plurisubharmonic, H¨ormander’s theorem, Lelong classes,L2 -estimates, pluripolar. 1 Theorem 1.1. (H¨ormander’s theorem) Let PA be the set of all func1 tions of the form log |f (z)|, where N is a positive integer and f = 0 N is an entire function. Then the closure of PA in L1loc (Cn ) consists of all plurisubharmonic functions. Some refinements of the above result were given for the functions in L(Cn ). In fact, if u ∈ L(Cn ) then we are able to obtain approximation results in precise form. One of them is due to Siciak [16, 17]. Theorem 1.2. (Siciak’s theorem) Let u ∈ L(Cn ). One can find for every positive integer ν a function uν ∈ L(Cn ) with the following properties: a) (3) uν := sup 1≤j≤sν 1 log |pj |, nj where pj are holomorphic polynomials in Cn , nj are positive integers satisfying deg pj ≤ nj . b) uν+1 ≤ uν and limν→∞ uν = u. The main goal of the present work is to explore a variation of H¨ormander’s result on approximation in the Lelong classes. More precisely, in Theorem 2.1 we show that every u ∈ L(Cn ) can be log |p| approximated in the L1loc (Cn ) topology by quotients , where d p is a polynomial and d is some integer larger than deg p. The proof relies heavily on the solution to the ∂−problem and H¨ormander L2 −estimates. As a consequence of the theorem, we give in Corollary 2.3 a characterization of closed complete pluripolar sets in Cn . In the same vein, a complete description of closed complete pluripolar subsets of a pseudoconvex domain in Cn is also given in Proposition 2.5. Acknowledgment. The author would like to thank Professor Nguyen Quang Dieu for his valuable suggestions. This paper was revised during a stay of the author at Vietnam Institute for Advance Study in Mathematics. He wishes to express his gratitude to the institute for the support. ¨ rmander’s theorem 2. A variation of Ho The main result of this paper is the following theorem. Theorem 2.1. Let u ∈ L(Cn ) and K a subset of Cn which is at most countable. Then there exist a sequence {ϕm }m≥1 of polynomials in Cn , 2 a sequence {dm }m≥1 of positive integers with deg ϕm ≤ dm such that, 1 for all m ≥ 1 and ϕm = log |ϕm |, the following hold dm (a) ϕm → u in L1loc (Cn ), (b) ϕm → u pointwise on K, (c) lim sup ϕm ≤ u on Cn , m→∞ (d) lim sup ρϕm ≤ ρu on Cn , m→∞ (e) For every r > 0 and every z ∈ K \ {0} ρu (z) ≤ lim inf m→∞ sup ϕm (λz) − log+ |λz| . |λ|>r We need the following elementary fact, which may not be original. Lemma 2.2. Let f be a holomorphic function on Cn . Assume that there exists a > 0 such that |f (z)|2 dλn (z) := C < ∞. (1 + |z|2 )a (4) Cn where λn is the Lebesgue measure in Cn . Then f is a polynomial of degree not greater than a − n. Proof. By subharmonicity of |f |2 on Cn , we have for any z ∈ Cn with |z| ≥ 1 and r > 0, |f (z)|2 ≤ ≤ Cn r2n Cn r2n |f (w)|2 dλn (w) (Cn is independent of z) |w−z| 0, and Aj,m is a quadratic holomorphic polynomial satisfying Aj,m (zj ) = um (zj ). Let χ be a test function in Cn with compact support in |z| < 1 and equal to 1 if |z| < 1/2. Fix ν ≥ 1, 1 choose δ > 0 such that δ < minj=1,2,...,ν rj,m and δ < |zj − zj | when 2 1 ≤ j < j ≤ ν. Set αν,m = min1≤j≤ν εj,m and ν χ((z − zj )/δ)ekAj,m (z) . fk,m,ν (z) = j=1 Clearly ν ∂χ((z − zj )/δ)ekAj,m (z) := gk,m,ν (z). ∂fk,m,ν (z) = j=1 We are going to solve the equation ∂uk,m,ν = gk,m,ν with respect to the plurisubharmonic weight 2kum . Observe that, by the choice of χ and 4 (6), we can find C1 > 0, C2 > 0 independent of k such that |gk,m,ν |2 e−2kum dλn ≤ C1 δ −2 e−αν,m kδ (7) 2 /2 Cn and |fk,m,ν |2 e−2kum dλn ≤ C2 (2kαν,m )−n . (8) Cn By (7) and Theorem 15.1.2 in [7], we can find a solution uk,m,ν such that (9) Cn |uk,m,ν (z)|2 e−2kum (z) 2 dλn (z) ≤ C1 δ −2 e−αν,m kδ /2 . 2 2 (1 + |z| ) Since uk,m,ν is holomorphic on the ball |z − zj | < δ/2, if j ≤ ν, by applying the submean value inequality to the subharmonic function |uk,m,ν |2 and the ball B(zj , t), where t is chosen such that 0 < t < δ/2 δ2 , from (9) we and the oscillation of um on the ball is smaller than αν,m 8 get for all 1 ≤ j ≤ ν C3 |uk,m,ν |2 dλn 2n t B(zj ,t) αν,m δ 2 C4 2 ≤ 2n 2 e−αν,m kδ /2 e2k(um (zj )+ 8 ) , t δ |uk,m,ν (zj )|2 ≤ where C3 , C4 are positive constants independent of k and t. Since fk,m,ν (zj ) = ekum (zj ) for all 1 ≤ j ≤ ν, we infer from the above inequalities that for all k large enough (10) |uk,m,ν (zj )| < ekum (zj ) ekum (zj ) , hence |pk,m,ν (zj )| > , ∀1 ≤ j ≤ ν, 2 2 where pk,m,ν := fk,m,ν − uk,m,ν . Combining the elementary inequality |pk,m,ν |2 ≤ 2(|fk,mν |2 + |uk,m,ν |2 ) with (8) and (9) we infer for all k large enough (11) Cn |pk,m,ν (z)|2 e−2kum (z) dλn (z) ≤ C5 , (1 + |z|2 )2 where C5 > 0 is independent of k. Since um ≤ (1/2 + 1/m) log(1 + |z|2 ) + Cu on Cn , we deduce from (11) that (12) Cn |pk,m,ν (z)|2 dλn (z) ≤ C6 . (1 + |z|2 )2+k+2k/m 5 Here C6 > 0 is independent of k. By Lemma 2.2, pk,m,ν is a polynomial of degree not exceeding b(k, m), where b(k, m) is the largest integer not 2k greater than k + 2 + − n. m Next, we let z be an arbitrary point in Cn . Using (11) and the subharmonicity of the function |pk,m,ν |2 , for every t > 0 we obtain |pk,m,ν (z)|2 ≤ C6 (1 + |z|2 + t2 )2 e2k supB(z,t) um t2n This implies that lim sup k→∞ log |pk,m,ν (z)| ≤ sup um . k B(z,t) By letting t tend to 0 we obtain (13) lim sup k→∞ log |pk,mν (z)| ≤ um (z), ∀z ∈ Cn . k In view of (10) we can choose k = k(m, ν) ≥ m + ν so large that for 1 ≤ j ≤ ν the following inequalities hold um (zj ) − log |pk(m,ν),m,ν (zj )| log 2 ≤ . k(m, ν) k(m, ν) For simplicity of notation, we put qm,ν = pk(m,ν),m,ν . From the last inequality and (13) we obtain log |qm,ν (zj )| = um (zj ), ∀j ≥ 1. ν→∞ k(m, ν) lim Since um is continuous and plurisubharmonic on Cn and K is dense log |qm,ν | converges to um in in Cn , by Lemma 15.1.7 in [7] we have k(m, ν) L1loc (Cn ) when ν tends to ∞. So we can choose ν(m) ≥ m so large that Bm log |qm,ν(m) | 1 − um dλn < , k(m, ν(m)) m where Bm is the ball of radius m centered at 0. Set ϕm = qm,ν(m) and lm = k(m, ν(m)). Since um ↓ u, in particular um → u in L1loc (Cn ), log |ϕm | we infer that converges to u in L1loc (Cn ). Because u is, in lm particular, subharmonic on Cn , by Theorem 3.2.13 in [8] we get (14) lim sup k→∞ log |ϕm (z)| ≤ u(z), ∀z ∈ Cn . k 6 Putting (10) and (14) together we get log |ϕm (zj )| (15) lim = u(zj ), ∀j ≥ 1. m→∞ lm Since deg ϕm ≤ dm := b(lm , m) (where b(k, m) is the largest integer 2k not greater than k + 2 + − n) and limm→∞ dm /lm = 1, from (14) m and (15) we conclude that the sequences {ϕm }m≥1 and {dm }m≥1 satisfy (a), (b) and (c) of the theorem. For (d), we are going to repeat a reasoning due to Bloom and Zeriahi (see Theorem 2.1 in [1]) for the readers convenience. First, we claim that if ϕ ∈ L(C) then lim sup(ϕ(λ) − log |λ|) = inf (max ϕ(λ) − log r) r≥1 |λ|=r |λ|→∞ To see this, for λ = 0 we set ψ(λ) = ϕ(1/λ) + log |λ|. It is easy to see that ψ is a subharmonic function on C\{0}, which is also bounded from above near 0. Thus ψ extends through 0 to a subharmonic function on C. We still denote this extension by ψ. On one hand we have ψ(0) = lim sup ψ(λ) = lim sup(ϕ(λ) − log |λ|). |λ|→∞ λ→0 On the other hand, by the maximum principle we obtain ψ(0) = inf max{ψ(z) : |z| = r} = inf (max ϕ(λ) − log r). 0 1, applying the claim 1 just proven to the function λ → ϕm (λz), where ϕm = log |ϕm |, we dm obtain ρϕm (z) + log |z| ≤ max(ϕm (λz) − log r). |λ|=r Taking lim sup of both sides when m → ∞ and using Hartogs’ lemma ([8, 9]) and (c) of the theorem on the righthand side we obtain log |z| + lim sup ρϕm ≤ max(u(λz) − log r). |λ|=r m→∞ Letting r → ∞, we get (d) of the theorem. Finally, fix z ∗ ∈ K \ {0}. Then by (5) we can choose a sequence {λz∗ ,j }j≥1 ↑ ∞ such that ρu (z ∗ ) = lim (u(λz∗ ,j z ∗ ) − log+ |λz∗ ,j z ∗ |). j→∞ This implies (e). The proof of the theorem is complete. As a consequence of the theorem, we have a characterization of complete pluripolar sets in Cn . 7 Corollary 2.3. Let E be a closed set in Cn . Then the following assertions are equivalent (a) E is complete pluripolar in Cn . (b) For every closed ball U in Cn , every point z0 ∈ U \ E, and every > 0, there exists a constant δ > 0 such that for all m ≥ 1 we can find a polynomial pm satisfying (i) |pm (z0 )| > δ dm , where dm = deg pm . (ii) ||pm ||U ∩E ≤ (1/m)dm , ||pm ||U < 1, λn {z ∈ U : |pm (z)| < δ dm } < ε. Recall that a subset E of Cn is called pluripolar if for every a ∈ E and every neighbourhood U of a we can find a plurisubharmonic function u on U such that u ≡ −∞ on every connected component of U and u|E∩U ≡ −∞. A subset E of an open set D ⊂ Cn is said complete pluripolar in D if there exists a plurisubharmonic function u in D such that u ≡ −∞ and E = {z ∈ D : u(z) = −∞}. We need the following lemma due to Zeriahi (Proposition 2.1 in [18], see also Proposition 3.1 in [12]). Lemma 2.4. Let D be a pseudoconvex domain in Cn and E be a subset ∗ , the pluripolar hull of D which is of Fσ and Gδ type. Assume that ED of E relative to D, coincides with E. Then E is complete pluripolar in D. Recall that if E is a pluripolar subset of D then ∗ ED := {z ∈ D : u(z) = −∞ if u is plurisubharmonic on D, u ≡ −∞ on E}. Proof of Corollary 2.3. (a) ⇒ (b). According to a theorem of Siciak in [15] (see also Theorem 7.1 in [2]), we can find u ∈ L(Cn ) such that {u = −∞} = E. By subtracting a large constant, we may assume that u < 0 on a neighbourhood of U . Since λn (E) = 0, we may choose c < u(z0 ) such that λn {z ∈ U : u(z) < c} < ε/2. Given m ≥ 1, by Hartogs’ lemma and Theorem 2.1, we can find a polynomial pm such 1 that log |pm | < − log m on U ∩ E and that dm 1 log |pm (z0 )| > c − 1, dm L 1 log |pm | − u dλ < ε/2, dm where dm is some integer deg pm ≤ dm and L is the compact {z ∈ U : u(z) ≥ c}. After adding, if necessary, a homogeneous polynomial of degree dm to pm , we may assume that deg pm = dm . By setting δ = ec−1 we infer that λn {z ∈ L : |pm (z)| < δ dm } < ε/2. 8 It is now clear that pm , dm satisfy (i), (ii). It remains to show that (b) ⇒ (a). We write E = ∪m≥1 Km where Km is an increasing sequence of compact sets. Let z0 ∈ Cn \ E. Choose an increasing sequence of closed balls {Bm }m≥1 such that z0 ∈ B1 , ∪m≥1 Bm = Cn , Km ⊂ Bm ∀m ≥ 1. By (b), there exists δ > 0 such that for each m ≥ 1 there are a polynomial pm on Cn and an integer dm such that 1 1 1 log |pm (z0 )| > log δ, sup log |pm | < 0, sup log |pm | < − log m. dm Bm dm K m dm It follows that 1 u(z) = m≥1 2m d log |pm (z)| m defines a plurisubharmonic function on Cn and that satisfies u ≡ −∞ on E whereas u(z0 ) > −1. By Lemma 2.4 and since E is closed in Cn , we conclude that E is complete pluripolar in Cn . The proof is complete. Using Theorem 2.1 and the lines of the proof of Corollary 2.4, we easily prove the following characterization of closed complete pluripolar sets in an arbitrary pseudoconvex domain. Proposition 2.5. Let E be a closed subset of a pseudoconvex domain D in Cn . The following assertions are equivalent. (a) E is complete pluripolar in D. (b) For every relatively compact subdomain U of D, every z0 ∈ U \E and every ε > 0, there exists a constant δ > 0 such that we can find a sequence {pm }m≥1 of holomorphic functions on D and a sequence {dm }m≥1 of positive integers satisfying (i) |pm (z0 )| > δ dm ; (ii) ||pm ||U ∩E ≤ (1/m)dm , ||pm ||U < 1, λn {z ∈ U : |pm (z)| < δ dm } < ε. In the proof of Theorem 2.1, if we let K be a countable dense set of Cn then we obtain the following result whose proof we omitted. Corollary 2.6. Let u ∈ L(Cn ) be continuous. Then there exists a sequence {ϕm }m≥1 of polynomials with degree dm ≥ 1 in Cn such that 1 ϕm = log |ϕm | → u in L1loc (Cn ) and lim supm→∞ ϕ˜m = u on Cn . dm 9 References [1] T. Bloom, Some applications of the Robin functions to multivariable approximation theory, Journal of Approximation Theory 92 (1998), 1-21. [2] E. Bedford and A. Taylor, Plurisubharmonic functions with logarithmic singularities, Annales Inst. Fourier 38 (1988), 133-171. [3] J. Duval and N. Sibony, Polynomial convexity, rational convexity and currents, Duke Math. Journal, 79 (1995), 487-513. [4] N. Q. Dieu, Approximation of plurisubharmonic functions on bounded domains in Cn , Michigan Math. J. 54 (2006), no. 3, 697-711. [5] N. Q. Dieu and F. Wilkstr¨om, Jensen measures and approximation of plurisubharmonic functions, Michigan Math. J. 53 (2005), no. 3, 529-544. [6] A. Edigarian and J. Wiegerinck, The pluripolar hull of the graph of a holomorphic function with polar singularities, Indiana Math. 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Kieu Phuong Chi, Department of Mathematics, Vinh University,182 Le Duan, Vinh City, Vietnam E-mail address: kpchidhv@yahoo.com 10 ... consists of all plurisubharmonic functions Some refinements of the above result were given for the functions in L(Cn ) In fact, if u ∈ L(Cn ) then we are able to obtain approximation results in. .. pluripolar in Cn The proof is complete Using Theorem 2.1 and the lines of the proof of Corollary 2.4, we easily prove the following characterization of closed complete pluripolar sets in an arbitrary... Advance Study in Mathematics He wishes to express his gratitude to the institute for the support ¨ rmander’s theorem A variation of Ho The main result of this paper is the following theorem Theorem