COMPACT OPERATORS AND NORMAL

FAMILIES OF SUBHARMONIC FUNCTIONS

Andriy A. Kondratyuk and Svyatoslav I. Tarasyuk

The properties of potential type compact operators are applied to the proof of the following theorem concerning normal families of subharmonic functions in the sense of Montel.

Let S^{m−1} be the unit sphere in ℝ^{m}, |S^{m−1}| its area. If m = 2 we assume 1 + \frac{1}{m−2} = +∞.

Theorem 1. Let F be a family of subharmonic in ℝ^{m}, m ≥ 2, functions u such that

sup_{u∈F} ∫_{S^{m−1}} |u(rx)| dS(x) < ∞

for every r ≥ 0. Then for every sequence {u_{n}} from F there exist a subharmonic in ℝ^{m} function v and a subsequence {u_{n_{j}}} such that for every r > 0 and every q, 1 ≤ q < 1 + \frac{1}{m−2}, we have

(1)  lim_{j→∞} ∫_{S^{m−1}} |u_{n_{j}}(rx) − v(rx)|^{q} dS(x) = 0.

For m = 2, q = 1 this result was obtained in [1].

To prove Theorem 1 we need some preliminary results.

Let K = \overline{K}_{R} be a closed ball of radius R. The sequence \{μ_{n}\} of positive measures on K is called weakly convergent to measure μ, if for every function g from C(K) we have

lim_{n→∞} ∫_{K} g dμ_{n} = ∫_{K} g dμ.

By standard arguments we obtain the following lemma.

Lemma. Let X_{j} be the Banach spaces, X_{1} →^{T} X_{2} →^{A_{n}} X_{3}, where T is a linear compact operator, {A_{n}} is a sequence of linear bounded operators convergent pointwise to zero and let F_{n} = A_{n}T. Then the sequence of norms ∥F_{n}∥ tends to zero.

We shall denote the Newton potential of measure μ_{n} by

P_{n}(y) = ∫_{K} \frac{dμ_{n}(η)}{|y − η|^{m−2}} , m ≥ 3.

The potential of measure μ is denoted by P (y).

Theorem 2. Let μ_{n} → μ weakly on \overline{K}_{R}. Then

(2)  lim_{n→∞} ∫_{S^{m−1}} |P_{n}(rx) − P (rx)|^{q} dS(x) = 0

for r > 0, 1 ≤ q < 1 + \frac{1}{m−2}.

Proof of Theorem 2. For r fixed, 0 < r < ∞, we define the linear operator T as follows

(T f)(η) = ∫_{S^{m−1}} \frac{f(x)}{|rx − η|^{m−2}} dS(x).

The q-degree of Newton kernel is integrable for q ∈ [1, 1 + \frac{1}{m−2}). So that by Sobolev-Kantorovich theorem [2], [3, p. 428] the operator T is a compact operator from L^{p}(S^{m−1}) into C(K) if 1/q + 1/p = 1. Using the Fubini theorem we have

∫_{S^{m−1}}[P_{n}(rx) − P (rx)]f(x) dS(x)
= ∫_{K} \{ ∫_{S^{m−1}} \frac{f(x)}{|rx − η|^{m−2}} dS(x)\}[dμ_{n}(η) − dμ(η)]
= ∫_{K}(T f)(η)[dμ_{n}(η) − dμ(η)] = A_{n}T f = F_{n}f.

On the other hand,

∫_{S^{m−1}} |P_{n}(rx) − P (rx)|^{q} dS(x)
= sup_{∥f∥_{Lp} ≤1} ∫_{S^{m−1}}[P_{n}(rx) − P (rx)]f(x) dS(x) = ∥F_{n}∥.

Applying Lemma we obtain (2). □

Proof of Theorem 1. Put

V_{1}(r) = sup_{u∈F} \frac{1}{|S^{m−1}|} ∫_{S^{m−1}} |u(rx)| dS(x), 0 ≤ r < ∞,

and consider a continuous majorant V of V_{1}, for example,

V (r) = ∫_{r}^{r+1} sup{V_{1}(τ): τ ≤ t} dt.

Let m=2. We may take u as a function of complex variable. Put

c_{k}(r, u) = \frac{1}{2π} ∫_{−π}^{π} e^{−ikθ}u(re^{iθ}) dθ, k ∈ ℤ.

By the Rubel-Taylor theorem [4], [5, p. 16], [6]

|c_{k}(r, u)| ≤ \frac{V (3r)}{|k| + 1}, u ∈ F, r > 0, k ∈ ℤ.

By the Anderson-Bearnstein theorem [1] for every sequences {u_{n}} from F exist a subsequence {u_{n_{j}}} and a subharmonic in ℂ function v such that

(3)  c_{k}(r, u_{n_{j}}) → c_{k}(r, v)

when j → ∞, r > 0, k ∈ ℤ.

In so doing, evidently,

(4)  |c_{k}(r, v)| ≤ \frac{V (3r)}{|k| + 1}, r > 0, k ∈ ℤ.

By the Hausdorff-Young theorem for q ≥ 2, 1/p + 1/q = 1, we have

m_{q}(r, u_{n} − v) = \{\frac{1}{2π} ∫_{−π}^{π} |u_{n_{j}}(re^{iθ}) − v(re^{iθ})|^{q} dθ\}^{1/q}

≤ { ∑_{k∈ℤ} |c_{k}(r, u_{n_{j}}) − c_{k}(r, v)|^{p}}^{1/p}.

The relation (1) for q ≥ 2 follows from this, (3) and (4). For q ∈ [1, 2) it follows in the same way if, moreover, we take into account that m_{q}(r, u) is increasing as a function of q.

Now suppose m ≥ 3. Let

N(r, u) = \frac{1}{|S^{m−1}|} ∫_{S^{m−1}} u(rx) dS(x).

If μ is the Riesz measure of the subharmonic function u then by classical Jensen’s formula,

N(2r, u) − N(r, u) = (m − 2) ∫_{r}^{2r} μ(\overline{K}_{t})\frac{dt}{t^{m−1}}
≥ μ(\overline{K}_{r})r^{m−2}(1 − 2^{2−m}).

Hence, for every sequence {μ_{n}} of Riesz measures associated with u_{n} ∈ F we have

μ_{n}(\overline{K}_{r}) ≤ \frac{r^{m−2}}{1 − 2^{2−m}} (V (2r) + V (r)).

The sequence {μ_{n}} is, consequently, bounded on compact sets from ℝ^{m}.

Using the Banach-Alaoglu theorem and diagonal process we find a measure μ and a subsequence \{μ_{n_{j}}\} weakly converging to μ in every ball \overline{K}_{l}, l ∈ ℕ. We denote {μ_{n_{j}}} by {μ_{n}}.

Let l be fixed. The function h_{n}(y) = u_{n}(y)−P_{n}(y), where P_{n}(y) denotes the Newton potential of μ_{n} in \overline{K}_{l}, is harmonic in K_{l}. Hence, for 0 < r < l we have

N(r, |h_{n}|) ≤ N(r, |u_{n}|) + N(r, |P_{n}|).

By virtue of (2) and condition N(r, |u_{n}|) ≤ V (r) we can write

sup_{n} N(r, |h_{n}|) < ∞

for every r ∈ (0, l). It follows from representation of h_{n} by Poisson integral that \{h_{n}\} is bounded sequence on compact subsets of \overline{K}_{l}. Using the compact principle we find a harmonic function h such that {h_{n}} converges uniformly to h on compact subsets of K_{l}. Denote v = h + P, where P is the Newton potential of μ. Using theorem 2 we get (1) for 0 < r < l.

If {u_{n}^{(1)}} is the subsequence for K_{1} and v^{(1)} is the corresponding limit function, {u_{n}^{2}}, v^{(2)} for K_{2}, then v^{(1)} coincides with v^{(2)} in K_{1} almost everywhere. Since v^{(1)} and v^{(2)} are subharmonic, they coincide everywhere in K_{1}. By recurrence, using diagonal process, we obtain (1) which is what had to be proved. □

References

[1] J. Anderson and A. Baernstein II, The size of the set on which a meromorphic function is large, Proc. London Math. Soc. 3 (1978), no. 36, 513–539.

[2] L. V. Kantorovich, On integral operators, Uspekhi Mat. Nauk 11 (1956), no. 2, 3–29. (Russian)

[3] L. V. Kantorovich and G. M. Akilov, Functional analysis, Nauka, Moscow, 1977. (Russian)

[4] L. A. Rubel and B. A. Taylor, A Fourier series method for meromorphic and entire functions, Bull. Soc. Math. France 96 (1968), 53–96.

[5] A. A. Kondratyuk, Fourier series and meromorphic functions, Lviv, 1988. (Russian)

[6] Ya. V. Vasylkiv, A criterion of completely regular growth of the δ-subharmonic function, Ukraïn. Mat. Zh. 37, no. 1, 8–13. (Russian)