International Symposium in Memory of Hua Loo Keng (held in August, 1988), vol. II.

Tangents of plurisubharmonic functions

Christer O. Kiselman

Contents:

1. Introduction

2. Basic estimates

3. Properties of tangent functions

4. Plurisubharmonic functions with prescribed tangents References

Resumo: Tanĝantoj de plursubharmonaj funkcioj

Oni studas lokajn ecojn de plursubharmonaj funkcioj per la nocio de tanĝanto kiu priskribas la konduton de la funkcio en ĉirkaŭaĵo de donita punkto. Estas montrite ke ekzistas plursubharmonaj funkcioj kun pluraj tanĝantoj, male al konjekto de Reese Harvey.

Abstract: Tangents of plurisubharmonic functions

Local properties of plurisubharmonic functions are studied by means of the notion of tangent which describes the behavior of the function near a given point. We show that there are plurisubharmonic functions with several tangents, disproving a conjecture of Reese Harvey.

1. Introduction

Let f be a plurisubharmonic function in an open set ω in C^{n}; we shall write this as f ∈ PSH (ω). If we want to study the behavior of f near a point x ∈ ω, it is natural to consider f(x + rz) for a small positive number r. However, if f takes the value −∞ at x, then f(x + rz) just tends to minus infinity as r → 0. So we subtract a constant to prevent this from happening: let us define, for any point x in ω such that f is not identically minus infinity near x, and any positive number r,

(1.1)  f_{x,r}(z) = f(x + rz) − sup_{x+rB} f,  |z| < \frac{1}{r}d_{ω}(x),

and let us write f_{r} = f_{0,r} for brevity. Here B is the open unit ball in C^{n} for the Euclidean metric, and d_{ω}(x) is the distance from x to the complement of ω. This means that we are looking at f with a microscope magnifying 1/r times, but have adjusted the level by an additive constant so that sup_{B} f_{x,r} = 0. Of the properties of the operation f ↦ f_{r} we may note the following: (af + b)_{r} = af_{r} if a and b are constants with a ≥ 0, b real; (f + g)_{r} ≥ f_{r} + g_{r}; and (f ∨ g)_{r} ≤ f_{r} ∨ g_{r}, where ∨ denotes the supremum.

Given any compact set K in C^{n}, the domain of f_{x,r} contains K if r is small. Moreover, as we shall see, the family (f_{x,r})_{0<r<δ} is bounded in L^{1}(K) if δ is a sufficiently small positive number. To every sequence (f_{x,r_{j}})_{j} there is a subsequence converging in L^{1}(K) and even a subsequence converging in Lloc^{1}(C^{n}). The limit must be a plurisubharmonic function. Let us agree to say that g ∈ PSH (C^{n}) is tangent to f at x ∈ ω if there is a sequence r_{j} → 0 such that f_{x,r_{j}} → g in Lloc^{1}(C^{n}) (or, which turns out to be the same thing, for the weak topology in D′(C^{n}); see Hörmander 1983, Theorem 4.1.9.b)). We shall denote by T_{x}(f) the set of all tangents, the tangent space of f at x.

Example 1.1. If f is the constant −∞, then T_{x}(f) = Ø. For f_{x,r} is never defined.

Example 1.2. If f is finite at x, then T_{x}(f) = {0}. To see this, note first that for |z| ≤ R,

f_{x,r}(z) ≤ sup_{x+rRB} f − sup_{x+rB} f → 0, r → 0,

so that lim sup_{r→0} f_{x,r} ≤ 0 everywhere, hence g ≤ 0 for all elements g ∈ T_{x}(f). On the other hand, f_{x,r}(0) = f(x) − sup_{x+rB} f → 0 in view of the upper semicontinuity of f, so that g must be identically zero.

As these two examples show, the tangent space can be of possible interest only if f takes the value −∞ at the point x but is not identically −∞ near x. The model example is this:

Example 1.3. If f = log |h| for a holomorphic function h which has a zero of order m ≥ 1 at 0, then we can write h = P + H where P is a homogeneous polynomial of degree m and H(z) = O(|z|^{m+1}), z → 0. With the notation a = sup_{B} |P | > 0 we can estimate f as follows:

sup_{rB} f ≤ log |ar^{m} + Cr^{m+1}| = log a + m log r + log(1 + Cr/a),

and similarly from below

sup_{rB} f ≥ log |ar^{m} − Cr^{m+1}| = log a + m log r + log(1 − Cr/a),

so that, as r tends to zero,

f_{r}(z) = f(rz) − sup_{rB} f = log |P (rz) + H(rz)| − sup_{rB} f =

= log |P (z) + r^{−m}H(rz)| + m log r − sup_{rB} f → log |P (z)/a|

pointwise. The convergence holds in fact also in Lloc^{1}, i. e., f_{r} → log |P/a| in Lloc^{1}(C^{n}). To see this, note that the convergence is uniform on compact sets which avoid the zeros of P, and that integrals over small sets can be estimated uniformly in the parameter r. So here T_{0}(f) = {log |P/a|}, reflecting, as a tangent should, the main term in the expansion of h.

Example 1.4. If f ∈ PSH (C^{n}) satisfies a homogeneity property

f(tz) = C log t + f(z), t > 0, z ∈ C^{n},

and if sup_{B} f = 0, then f_{r} = f for all r > 0 so that f is an eigenfunction for all operations f ↦ f_{r}. In this case the tangent space at the origin is T_{0}(f) = {f}. The homogeneity property mentioned is equivalent to the formally stronger

f(tz) = C log |t| + f(z), t ∈ C \ {0}, z ∈ C^{n};

see the proof of Proposition 3.1. Conversely it can be shown that these functions are the only ones that are eigenfunctions simultaneously for all r > 0.

The notion of a tangent has been defined in a more general setting. If T is a current we can define its push-forward (1/r)_{∗}T under the mapping x ↦ x/r. The current lim_{r→0}(1/r)_{∗}T, if this limit exists in a suitable topology, is called the tangent cone to T at 0; see Harvey 1977:332. To a plurisubharmonic function f we associate the current i∂ ¯∂f of type (1, 1) and bidimension (n − 1, n − 1); it is closed and (strongly and weakly) positive. Then (1/r)_{∗}(i∂ ¯∂f) = i∂ ¯∂f_{r}. Therefore the tangent cone to the positive closed current i∂ ¯∂f exists at 0 if and only if the tangent space T_{0}(f) of f is a singleton.

Harvey formulates as a conjecture that the tangent cone to a strongly positive closed current always exists (1977:332, Conjecture 1.32). This was so in Examples 1.2, 1.3 and 1.4. We shall see, however, that a plurisubharmonic function may have infinitely many tangents if n ≥ 2.

The tangent space T_{x}(f) is analogous to the limit set of a plurisubharmonic function of finite order studied by Sigurd^{−}sson 1986. Actually it is in many ways simpler, since the behavior of a plurisubharmonic function is more restricted near a point than at infinity. To give but one example, the logarithmic homogeneity is automatic here (see Proposition 3.1), whereas homogeneity of degree ρ has to be imposed as an extra and very restrictive condition in the case of limit sets of plurisubharmonic functions of finite order.

I am grateful to Ragnar Sigurd^{−}sson and Eric Bedford for helpful comments on an earlier version of this paper. And my most sincere thanks go to Chin-Huei Chang, Hsuan-Pei Lee and Jean-Pierre Demailly who saved me from publishing an erroneous lemma!

2. Basic estimates

If f is a plurisubharmonic function in an open subset ω of C^{n} and x is an arbitrary point of ω, we shall use two functions, u and U, to describe its behavior near x. We define for x ∈ ω, t < log d_{ω}(x),

u(x, t) = u_{f}(x, t) = ∫_{}_{z∈S} f(x + e^{t}z)

and

U(x, t) = U_{f}(x, t) = sup_{z∈S} f(x + e^{t}z).

Here S is the Euclidean unit sphere, and the barred integral sign indicates the mean value: in general

∫_{}_{A} f = ∫_{A} f/ ∫_{A} 1 provided 0 < ∫_{A} 1 < +∞.

So u(x, t) is the mean value of f over the sphere x+e^{t}S, and U(x, t) is the supremum of f over the same sphere; therefore obviously u ≤ U. To get an estimate in the opposite direction we shall use Harnack’s inequality which has the form

\frac{1 + |x|/r}{(1 − |x|/r)^{m−1}} h(0) ≤ h(x) ≤ \frac{1 − |x|/r}{(1 + |x|/r)^{m−1}} h(0)

for harmonic functions which satisfy h ≤ 0 in the ball of radius r in R^{m}. If f is subharmonic in a neighborhood of the ball e^{s}B̄ in C^{n} we can consider its harmonic majorant h there, which satsfies f(x) ≤ h(x) and

h(0) = ∫_{}_{z∈S} h(e^{s}z) = ∫_{}_{z∈S} f(e^{s}z) = u(0, s).

Therefore

U(0, t) = sup_{e^{t}S} f ≤ sup_{e^{t}S} h ≤ \frac{1 − e^{t−s}}{(1 + e^{t−s})^{2n−1}} u(0, s), t < s,

provided only f ≤ 0 in e^{s}B. If we apply this inequality to the function f −U(0, s) which is ≤ 0 in e^{s}B, we get, writing U(t) instead of U(0, t) for simplicity:

U(t) − U(s) ≤ \frac{1 − e^{t−s}}{(1 + e^{t−s})^{2n−1}} (u(s) − U(s)),

i. e.,

(2.1)  U(t) ≤ (1 − λ_{s−t})U(s) + λ_{s−t}u(s), t < s,

where λ_{s} is defined for s > 0 as

λ_{s} = \frac{1 − e^{−s}}{(1 + e^{−s})^{2n−1}} .

As a consequence of the maximum principle, u(x, t) and U(x, t) are increasing in t; by Hadamard’s three-circle theorem, they are convex functions of t. Therefore their slopes at −∞ exist:

(2.2)  ν_{f}(x) = lim_{t→−∞} \frac{u(x, t)}{t} and N_{f}(x) = lim_{t→−∞} \frac{U(x, t)}{t}

both exist. This follows from the fact that the slopes

\frac{u(x, t) − u(x, t_{0})}{t − t_{0}} and \frac{U(x, t) − U(x, t_{0})}{t − t_{0}}

are increasing in t. The first limit ν_{f}(x) is the Lelong number of f at x. The Lelong number is usually defined as a density of a measure (in the present case the (2n − 2)-dimensional density of the measure (2π)^{−1}△f at x). See e. g. Kiselman 1979 for the equivalence of the two definitions. The two limits in (2.2) are equal. In fact, since u ≤ U we immediately get ν_{f}(x) ≥ N_{f}(x). In the other direction it follows from (2.1), taking s = t + 1, that

U(t) ≤ (1 − λ_{1})U(t + 1) + λ_{1}u(t + 1),

whence

\frac{U(t)}{t} ≥ (1 − λ_{1})\frac{U(t + 1)}{t} + λ_{1} \frac{u(t + 1)}{t}, t < 0.

Letting t tend to −∞ we see that N_{f}(x) ≥ ν_{f}(x).

Lemma 2.1. Let f ∈ PSH (ω) be plurisubharmonic in an open set in C^{n}, let x ∈ ω be such that f is not identically minus infinity near x, and define f_{x,r} by (1.1). Let R be a positive number. Then the family (f_{x,r})_{0<r<δ} is bounded in L^{1}(RB) for some δ > 0.

Proof. We may assume without loss of generality that x = 0 and that f is defined and plurisubharmonic in a neighborhood of the closed unit ball in C^{n}. The function U, being convex and increasing, satisfies

(2.3)  ν_{f}(0)(s − t) ≤ U(s) − U(t) ≤ C(s − t), t ≤ s ≤ 0,

for some constant C; when t ≤ s ≪ 0 we even get

(2.4)  U(s) − U(t) ≤ (ν_{f}(0) + ε)(s − t)

for any preassigned ε > 0.

We can now estimate f_{r} from above in the ball RB as follows, using (2.3):

sup_{RB} f_{r} = U_{f}(log R + log r) − U_{f}(log r) ≤ C log^{+} R

for 0 < r ≤ min(1/R, 1). Hence (f_{r})_{0<r<δ} is bounded from above in RB by a constant if only δ ≤ min(1/R, 1).

There need of course not exist a pointwise bound from below, but we can estimate the mean value of f_{r} over a sphere. We note that (2.1) can be written

u(s) − U(s) ≥ \frac{1}{λ_{s−t}} (U(t) − U(s)), t < s,

so that, using (2.3) with s − t = 1, we get

u(s) − U(s) ≥ −\frac{C(s − t)}{λ_{s−t}} = −\frac{C}{λ_{1}} , s ≤ 0.

This shows that the mean value of f_{r} over e^{t}S is

u_{f_{r}}(t) = u_{f}(t + log r) − U_{f}(log r) ≥ U_{f}(t + log r) − U_{f}(log r) − \frac{C}{λ_{1}} ≥ −Ct^{−} − \frac{C}{λ_{1}}

for 0 < r ≤ min(e^{−t}, 1); the last inequality holds in view of (2.3). Hence (f_{r})_{0<r<δ} is bounded from below in L^{1}(RS) when δ ≤ min(1/R, 1). The integral over RB is

∫_{RB} f_{r} dλ = A_{2n−1} ∫_{0}^{R} u_{f_{r}}(log ρ)ρ^{2n−1} dρ = A_{2n−1} ∫_{−∞}^{log R} u_{f_{r}}(t)e^{2nt} dt ≥

≥ A_{2n−1} ∫_{−∞}^{log R}(−Ct^{−} − C/λ_{1})e^{2nt} dt = −C_{R};

here dλ is Lebesgue measure in C^{n} and A_{2n−1} is the area of the unit sphere S in C^{n}. The norm in L^{1}(RB), finally, is bounded since |f_{r}| ≤ 2 sup f_{r} − f_{r}, so that

∫_{RB} |f_{r}| dλ ≤ 2 sup_{RB} f_{r} ∫_{RB} 1 dλ − ∫_{RB} f_{r} dλ ≤ 2C log^{+} R ∫_{RB} 1 dλ + C_{R}.

Proposition 2.2. Let f ∈ PSH (ω) be plurisubharmonic in an open set in C^{n}, assume that f is not identically minus infinity near a given point x ∈ ω, and define f_{x,r} by (1.1). Let R be a positive number. Then the family (f_{x,r})_{0<r<δ} is relatively compact in L^{1}(RB) for some positive number δ. The tangent space T_{x}(f) is connected and compact

in Lloc^{1}(C^{n}).

Proof. We know from Lemma 2.1 that ∥f_{x,r}∥_{L^{1}(RB)} is bounded. We can apply Theorem 4.1.9.a) of Hörmander 1983 to deduce that every sequence (f_{x,r_{j}}) has a subsequence which converges in L^{1}(RB). Repeating this for every natural number R and taking a diagonal subsequence we get convergence in Lloc^{1}(C^{n}). The set of all limits obtained in this way must of course be closed, and the connectedness follows as for limit sets in Sigurd^{−}sson 1986:241.

3. Properties of tangent functions

Proposition 3.1. Let f ∈ PSH (ω), ω an open subset of C^{n}. Assume that f is not identically minus infinity near a point x of ω. Then every tangent function g ∈ T_{x}(f) satisfies the homogeneity relation

g(tz) = ν_{f}(x) log |t| + g(z), t ∈ C, z ∈ C^{n},

where ν_{f}(x) is the Lelong number of f at x. In particular, g is a plurisubharmonic function of minimal growth.

Corollary 3.2. In one dimension, T_{x}(f) is a singleton with the only function g(z) = ν_{f}(x) log |z| as element.

Proof of Proposition 3.1. We take x = 0 and write U(t) for U(0, t). From (2.4) we get

U_{f_{r}}(t) = U_{f}(t + log r) − U_{f}(log r) ≤ (ν_{f}(0) + ε)t, t ≥ 0, r < r_{ε}e^{−t}.

Using a sequence (r_{j}) such that f_{r_{j}} → g, we see that U_{g}(t) ≤ (ν_{f}(0) + ε)t for every t ≥ 0, and, since ε is arbitrary, U_{g}(t) ≤ ν_{f}(0)t for t ≥ 0. We may write this as

(3.1)  g(z) ≤ ν_{f}(0) log |z|

for |z| ≥ 1. Similarly,

U_{f_{r}}(t) = U_{f}(t + log r) − U_{f}(log r) ≤ ν_{f}(0)t, t ≤ 0,

for all sufficiently small r, and therefore U_{g}(t) ≤ ν_{f}(0)t for t ≤ 0, i. e., (3.1) holds also for |z| ≤ 1. Now fix z ∈ C^{n}. Since g is plurisubharmonic, the function h_{z}(t) = g(tz) − ν_{f}(0) log |t| is subharmonic in C \ {0}. But h_{z} is bounded from above in view of (3.1), so it can be extended to a subharmonic function in all of C. Moreover it is constant by Liouville’s theorem:

h_{z}(t) = g(tz) − ν_{f}(0) log |t| = h_{z}(1) = g(z).

This proves the proposition.

Knowing g is equivalent to knowing the open set Ω = {z ∈ C^{n}; g(z) < 0} and the number ν_{f}(0).

4. Plurisubharmonic functions with prescribed tangents

Theorem 4.1. Let M be a subset of PSH (C^{n}). Then M is the tangent space T_{x}(f) of some plurisubharmonic function f defined in a neighborhood of x if and only if the following four conditions hold:

1. Every element g ∈ M is homogeneous:

(4.1)  g(tz) = C log |t| + g(z), t ∈ C, z ∈ C^{n},

the constant C being equal to ν_{f}(x) by necessity;

2. Every element g ∈ M satisfies

(4.2)  sup_{B} g = 0;

3. M is closed in Lloc^{1}(C^{n}); and

4. M is connected for the topology induced by Lloc^{1}(C^{n}).

Proof. First let f be a given plurisubharmonic function and denote its tangent space at a point x by M = T_{x}(f). Property 1 was proved in Proposition 3.1, and property 2 is an easy consequence of the fact that every function f_{x,r} satisfies (4.2). Properties 3 and 4 were noted above in Proposition 2.2.

Now let M be a given set of plurisubharmonic functions satisfying the four properties. If M = Ø we take f as the constant −∞. If M is non-empty and C = 0, then M = {0} and we can take f as any finite constant or more generally as any plurisubharmonic function with ν_{f}(x) = 0. For the rest of the proof we shall suppose that M ≠ Ø and that C is positive. We shall construct f very much like Sigurd^{−}sson 1986: Theorem 1.2.1, ii). There is only a small difference in the proof due to a glueing procedure of a different kind.

Let (g_{j}) be a sequence of functions in M which is dense for the topology in L^{1}(B). Next let h_{j} be plurisubharmonic functions which are continuous on C^{n} \ {0} and satisfy (4.1) and (4.2) with the same constant C, and such that ∥h_{j} − g_{j}∥_{L^{1}(B)} → 0. We can obtain such functions h_{j} by convolving over the space of matrices as in Sigurd^{−}sson 1986:244, or over the unitary group as in Kiselman 1967:10 (such a convolution, as opposed to the usual smoothing by convolution on C^{n}, preserves the homogeneity (4.1)), and then adjust by a constant to satisfy (4.2). Therefore also the functions h_{j} are eigenfunctions for the operation (1.1): they satisfy (h_{j})_{r} = h_{j}. We shall define the function f as

(4.3)  f = sup_{j}(c_{j}h_{j} + n_{j}),

where n_{j} are constants to be determined, and where c_{j} are numbers tending to 1 and satisfying c_{j} > c_{j+1} > 1. These strict inequalities are essential for our construction. To be precise we shall choose (c_{j}) tending to one so fast that

∥c_{k}h_{k} − h_{k}∥_{L^{1}(kB)} = (c_{k} − 1)∥h_{k}∥_{L^{1}(kB)} → 0.

The idea is to choose the level constants n_{j} in such a way that in certain shells s_{k}/k ≤ |z| ≤ ks_{k}, the term of index k dominates the others: f = c_{k}h_{k} + n_{k} there. To be specific, let Z_{k} be the set where c_{k}h_{k} + n_{k} = f. Then Z_{k} shall be so wide that it contains the spherical shell s_{k}/k ≤ |z| ≤ ks_{k}. If this is so, then we can perform the operation f ↦ f_{s_{k}} on (4.3) (see (1.1) for the definition) to obtain

(4.4)  f_{s_{k}} = (c_{k}h_{k} + n_{k})_{s_{k}} = (c_{k}h_{k})_{s_{k}} = c_{k}h_{k}  for  \frac{1}{k} ≤ |z| ≤ k.

To determine the constants n_{j}, first choose n_{1} = 0 and s_{1} = 1. Next assume that n_{j} and s_{j} have been chosen for j = 1, 2, ..., k − 1. We then choose n_{k} so that

c_{k}h_{k}(z) + n_{k} ≤ c_{k−1}h_{k−1}(z) + n_{k−1}  when  |z| = \frac{s_{k−1}}{k − 1}.

This is possible since h_{k} and h_{k−1} are bounded on that sphere. We note that the term of index k will not influence f where |z| ≥ s_{k−1}/(k − 1). Next choose s_{k} so that

c_{k}h_{k}(z) + n_{k} ≥ c_{k−1}h_{k−1}(z) + n_{k−1}  when  |z| = ks_{k}.

This is possible since c_{k}h_{k} and c_{k−1}h_{k−1} are homogeneous of different degrees: 0 < c_{k}C < c_{k−1}C. As a consequence the term of index k − 1 in (4.3) will not influence f where |z| ≤ ks_{k}.

This procedure will define a function f satisfying (4.4): f_{s_{k}} = c_{k}h_{k} in kB \ k^{−1}B. Therefore the norm of f_{s_{k}} − c_{k}h_{k} in L^{1}(kB) is equal to its norm in L^{1}(k^{−1}B):

∥f_{s_{k}} − c_{k}h_{k}∥_{L^{1}(kB)} = ∥f_{s_{k}} − c_{k}h_{k}∥_{L^{1}(k^{−1}B)}.

This last quantity evidently tends to zero. Moreover, by the choice of c_{k}, we know that ∥c_{k}h_{k} − h_{k}∥_{L^{1}(kB)} → 0, so ∥f_{s_{k}} − h_{k}∥_{L^{1}(kB)} → 0, which implies that f_{s_{k}} − h_{k} → 0 in Lloc^{1}(C^{n}). To every given g ∈ M there is a subsequence of (h_{k}) which tends to g, so the corresponding subsequence of (f_{s_{k}}) tends to the same limit. Therefore g is tangent to f, and we have proved that T_{0}(f) contains M.

Example 4.2. To get a specific example of a function whose tangent space is not a singleton we can now take g_{j} equal to log |z| for j odd, and equal to log |z_{1}| ∨ ⋯ ∨ log |z_{n}| for j even—these functions are different if n ≥ 2. In the construction above we can take h_{j} = g_{j}, and it will yield a function whose tangent space contains both g_{0} and g_{1}. But it will also contain every function

G_{a}(z) = log |z_{1}| ∨ ⋯ ∨ log |z_{n}| ∨ (a + log |z|),

where a is a constant satisfying − log \sqrt{n} < a < 0. We note that G_{a} = g_{0} if a ≤ − log \sqrt{n} and that G_{a} = g_{1} if a = 0, so G_{a} describes, as a moves from − log \sqrt{n} to 0, a curve in PSH (C^{n}) connecting g_{0} to g_{1}. The tangent space in this case is precisely this curve.

Example 4.3. If we take g_{j}(z) = log |z_{1}| for j odd and g_{j}(z) = log |z_{2}| for j even we get a function whose tangent space contains all functions

G_{a,b}(z) = (log |z_{1}| + a) ∨ (log |z_{2}| + b),

where −∞ ≤ a, b ≤ 0 and a ∨ b = 0. Here G_{0,b} describes a curve from g_{1} to g_{1} ∨ g_{2} as b moves from −∞ to 0, and then G_{a,0} goes from g_{1} ∨ g_{2} to g_{2} as a moves from 0 to −∞. This time it is not possible to take h_{j} = g_{j}, at least if we want suprema of the type (4.3) which are locally finite in C^{n} \ {0}.

To complete the proof we must be a bit more careful to get T_{0}(f) as a subset of M. The technique to achieve this will be as in Sigurd^{−}sson 1986; specifically we shall use his Lemma 1.2.5 which we quote for reference:

Lemma 4.4. Let X be a compact connected metric space with metric d. Then there exists a sequence (x_{k}) in X such that its elements form a dense subset and d(x_{k+1}, x_{k}) tends (decreasingly) to zero.

In the given set M we shall use the metric induced by Lloc^{1}(C^{n}), which we can take as

d(f, g) = ∑_{j=1}^{∞} 2^{−j} min(1, ∥f − g∥_{L^{1}(jB)}).

Then ∥f − g∥_{L^{1}(jB)} < 1/j when d(f, g) < 2^{−j}/j. Using the lemma we can arrange that d(h_{k+1}, h_{k}) tends to zero, so that

∥h_{k+1} − h_{k}∥_{L^{1}(jB)} < 1/j

if k ≥ k_{j} for suitable indices k_{j}. We may assume k_{j} ≥ j. Defining R_{k} = j for k_{j} ≤ k < k_{j+1}, we may rewrite this as

(4.5)  ∥h_{k+1} − h_{k}∥_{L^{1}(R_{k}B)} < 1/R_{k} for k ≥ k_{1}.

We note that R_{k} ≤ k. Moreover R_{k} → +∞ as k → +∞, so that convergence with respect to all the seminorms ∥ · ∥_{L^{1}(R_{k}B)} is equivalent to convergence in the metric of

Lloc^{1}(C^{n}).

We have to prove that if (r_{j}) is a sequence tending to zero and such that (f_{r_{j}}) converges, then the limit of the latter sequence is in M. The construction gives us full control over (f_{r_{j}}) if r_{j} = s_{j}, the special radii selected above. We need to study the behavior of f_{r} for intermediate values of r. Then more than one term in (4.3) must be taken into account. However, since the s_{j} tend to zero faster than any geometric progression, it will not be necessary to consider more than two terms.

Let us study the function f in a spherical shell s_{k+1}/(k +1) ≤ |z| ≤ ks_{k} where only two terms may be of influence:

f = (c_{k+1}h_{k+1} + n_{k+1}) ∨ (c_{k}h_{k} + n_{k}).

Performing the operation f ↦ f_{r} for an r such that s_{k+1} ≤ r ≤ s_{k} we get

(4.6)  f_{r} = (c_{k+1}h_{k+1} + a) ∨ (c_{k}h_{k} + b)

in the shell

\frac{s_{k+1}}{(k + 1)r} ≤ |z| ≤ \frac{ks_{k}}{r} ,

where a and b are certain constants which must satisfy a ∨ b = 0 since sup_{B} f_{r} = 0 by definition.

The shell where (4.6) holds contains kB \ (k + 1)^{−1}B, for the outer and inner radii of the former are, respectively,

\frac{ks_{k}}{r} ≥ k, and \frac{s_{k+1}}{(k + 1)r} ≤ \frac{1}{k + 1}.

Therefore the norm in L^{1}(kB) of the difference between the two sides in (4.6) is equal to the norm in L^{1}((k + 1)^{−1}B) which tends to zero.

It remains to estimate the distance from the right-hand side of (4.6) to M. To do so we shall use the inequality

∥F ∨ (G + H) − F ∥_{L^{1}(A)} ≤ ∥G − F ∥_{L^{1}(A)} + ∥H ∨ 0∥_{L^{1}(A)}

which holds for arbitrary functions F, G, H ∈ L^{1}(A). We apply it to the supremum in (4.6), putting

A = A(k) = R_{k}B \ (k + 1)^{−1}B ⊂ kB,

where R_{k} is the radius occurring in (4.5),

F = c_{k}h_{k}, G = c_{k+1}h_{k+1}, H = a ≤ 0 if b = 0,

and

F = c_{k+1}h_{k+1}, G = c_{k}h_{k}, H = b ≤ 0 if a = 0.

We then get f_{r} = F ∨ (G + H) and H ∨ 0 = 0 in both cases, so

(4.7)  ∥f_{r} − c_{p}h_{p}∥_{L^{1}(A(k))} ≤ ∥c_{k+1}h_{k+1} − c_{k}h_{k}∥_{L^{1}(A(k))}

for p = k or p = k + 1. Given any sequence (r_{j}) tending to zero we can choose k_{j} such that s_{k_{j}+1} ≤ r_{j} ≤ s_{k_{j}}. Then (4.7) shows that

∥f_{r_{j}} − c_{p}h_{p}∥_{L^{1}(A(k_{j}))} ≤ ∥c_{k_{j}+1}h_{k_{j}+1} − c_{k_{j}}h_{k_{j}}∥_{L^{1}(A(k_{j}))}

for p = k_{j} or p = k_{j} + 1. We recall that, by our choice of c_{k},

∥c_{k}h_{k} − h_{k}∥_{L^{1}(R_{k}B)} ≤ ∥c_{k}h_{k} − h_{k}∥_{L^{1}(kB)} → 0

and that, by (4.5), the quantity ∥h_{k+1} − h_{k}∥_{L^{1}(R_{k}B)} tends to zero. This proves that f_{r_{j}} is as close as we wish to either h_{k_{j}} or h_{k_{j}+1} in the metric of Lloc^{1}(C^{n}), and therefore tends to an element of M: we have proved that T_{0}(f) is contained in M.

References

Harvey, Reese. 1977. Holomorphic chains and their boundaries. Several Complex Variables. Proc. Symp. Pure Math., vol. 30, part 1, 309–382. Providence, Rhode Island: Amer. Math. Soc.

Hörmander, Lars. 1983. The Analysis of Linear Partial Differential Operators I. Springer-Verlag.

Kiselman, Christer O. 1967. On entire functions of exponential type and indicators of analytic functionals. Acta Math. 117, 1–35.

Kiselman, Christer O. 1979. Densité des fonctions plurisousharmoniques. Bull. Soc. Math. France 107, 295–304.

Sigurd^{−}sson, Ragnar. 1986. Growth properties of analytic and plurisubharmonic functions of finite order. Math. Scand. 59, 235–304.

Author’s address: Uppsala University, Department of Mathematics,

Thunbergsvägen 3, S-752 38 Uppsala, Sweden.

Telephone: +46 18 18 32 16; +46 18 30 07 08 Electronic mail: MATCK @ SEUDAC21 . BITNET

Date: December 5, 1988