����������� ������ 48 (1996), 15–21

UDK 517.57

���������� ������ ��� research paper

SUBHARMONIC BEHAVIOUR OF SMOOTH FUNCTIONS

Miroslav Pavlović

Abstract. We prove that |f|^{p}, p > 0, behaves like a subharmonic function if f is a C^{2}- function such that, for some constants K and K_{0},

|Δf(x)| ⩽ Kr^{−1} sup |∇f| + K_{0}r^{−2} sup |f|,

where the supremum is taken over B_{r}(x) = { z : |z − x| < r }. If in addition K_{0} = 0, then |∇f|^{p} has a similar property.

Throughout the paper we fix a positive integer n and denote by R^{n} the euclidean n-space. The euclidean ball of radius r, centered at x ∈ R^{n}, is denoted by B_{r}(x), and the unit ball is B = B_{1}(0). By dm we denote the Lebesgue measure in R^{n} normalized so that m(B) = 1, hence m(B_{r}(x)) = r^{n}. Throughout, G will denote a proper subdomain of R^{n}.

If f is a function harmonic in G, then the function |f|^{p} for p ⩾ 1 is subharmonic in G and therefore has the sub-mean-value property over balls. If p < 1, then |f|^{p} need not be subharmonic but, by a result of Hardy and Littlewood [4], Fefferman and Stein [3] and Kuran [6], there exists a constant K = K(n, p) < ∞ such that

|f(x)|^{p} ⩽ Kr^{−n} ∫_{B_{r}(x)} |f|^{p} dm

whenever B_{r}(x) ⊂ G. As observed in [1] and [7] a slight modification of Fefferman and Stein’s proof yields a more general result. To state it we define a class of functions with subharmonic behaviour.

The class sh(G). Let sh(G) denote teh class of non-negative, continuous functions u on G such that, for some constant K ⩾ 1,

u(x) ⩽ Kr^{−n} ∫_{B_{r}(x)} u dm

whenever B_{r}(x) ⊂ G.

Theorem A. If u ∈ sh(G), then u^{p} ∈ sh(G) for every p > 0.

A very short proof is in [8].

In [8] we discussed some simple sufficient conditions for a C^{1}-function f in order that |f| ∈ sh(G) or |∇f| ∈ sh(G). More precisely, |f| ∈ sh(G) if

|∇f(x)| ⩽ Kr^{−1} sup{ |f(z)| : z ∈ B_{r}(x) }  (1)

for some constant K ⩾ 0. And |∇f| ∈ sh(G) if

|∇f(x)| ⩽ Kr^{−1}ωf(x, r),  (2)

where ωf(x, r) is the oscillation of f over B_{r}(x). (In Section 2 we will state a vector variant of these results.)

In this paper we consider sufficient conditions for a C^{2}-function f to satisfy (1) or (2). The main result, Theorem 2, asserts that (1) is implied by

|Δf(x)| ⩽ Kr^{−1} sup_{B_{r}(x)} |∇f| + K_{0}r^{−2} sup_{B_{r}(x)} |f|,  (3)

while (2) is implied by (3,K_{0} = 0). (Here and elsewhere K, K_{0}, K_{1}, … denote constants independent of B_{r}(x) ⊂ G.)

Condition (3,K_{0} = 0) is satisfied if, for instance,

(i) f is harmonic, (ii) G is bounded and f is an eigenfunction of the ordinary Laplacian, (iii) G = B and f is an eigenfunction of the hyperbolic Laplacian.

Using Theorem 2 we prove that condition (3,K_{0} = 0) is satisfied if f is a polyharmonic function (Corollary 5).

1. Classes of smooth functions

For a sufficiently smooth function f: G → R^{k} let ∂_{j}f = ∂f/∂x_{j} (j = 1,…, n), whence ∂_{j}f(x) ∈ R^{k} for x ∈ G. The Laplacian Δ is defined by Δf = ∑_{j=1}^{n} ∂_{j}(∂_{j}f) and so Δf(x) ∈ R^{k} for x ∈ G. We use the symbol ∇f only in the case k = 1: ∇f = (∂_{1}f,…, ∂_{n}f). Thus if k = 1, then ∇f maps G into R^{n}. In the general case we denote by Df(x) the derivative of f at x ∈ G treated as a linear operator from R^{n} to R^{k}. If f is real valued, then ∥Df(x)∥ = |∇f(x)|, ∇(Δf) = Δ(∇f) and

∥D(∇f)(x)∥ ≍ { ∑_{i,j}(∂_{i}∂_{j}f(x))^{2}}^{1/2}, x ∈ G.

(We write A(x) ≍ B(x), x ∈ G, to denote that A(x)/B(x) is between two positive constants independent of x.)

The class HC^{1}(G). Let C^{1}(G) be the class of all C^{1}-functions defined in G and with values in some R^{k} (k is constant when f is fixed). The class HC^{1}(G) is the subclass of C^{1}(G) consisting of those f for which there is a constant K ⩾ 0 such that

∥Df(x)∥ ⩽ K sup{ |f(z)| : z ∈ B_{r}(x) }, B_{r}(x) ⊂ G.  (4)

Note that (4) is implied by

∥Df(x)∥ ⩽ K|f(x)|/δ_{G}(x),  (δ_{G}(x) = dist(x, ∂G))  (5)

which is a restriction on the growth of f and therefore is much stronger than (4). For example, if f > 0 is a real function on G = (0, +∞) ⊂ R^{1}, then (5) reduces to f′(x) ⩽ Kf(x)/x, which implies that f(x)/x^{K} is decreasing and x^{K}f(x) is increasing for x > 0.

Note also that HC^{1}(0, +∞) does not contain the function f(x) = sin x, which is seen by choosing x = 2sπ, r = 2sπ (s = 1, 2,…).

The class OC^{1}(G). This is the subclass of C^{1}(G) consisting of those f such that, for some constant K,

∥Df(x)∥ ⩽ Kr^{−1}ωf(x, r), B_{r}(x) ⊂ G,  (6)

where ωf(x, r) = sup{ |f(z) − f(x)| : z ∈ B_{r}(x) }. Clearly OC^{1}(G) ⊂ HC^{1}(G).

It was observed in [8] that every convex (or concave) function from C^{1}(G) belongs to OC^{1}(G). In particular the function f(x) = e^{x} is in HC^{1}(0, +∞) but f does not satisfy (5).

The class HC^{2}(G). This class consists of those f ∈ C^{2}(G) for which

|Δf(x)| ⩽ Kr^{−1} sup{ ∥Df(z)∥ : z ∈ B_{r}(x) }+K_{0}r^{−2} sup{ |f(z)| : z ∈ B_{r}(x) } (7)

for some constants K, K_{0}.

The condition |Δf(x)| ⩽ K∥Df(x)∥δ_{G}(x)^{−1} + K_{0}|f(x)|δ_{G}(x)^{−2} implies (7) with the same values of K and K_{0}.

The class OC^{2}(G). It consists of those f ∈ C^{2}(G) such that

|Δf(x)| ⩽ Kr^{−1} sup{ ∥Df(z)∥ : z ∈ B_{r}(x) }  (8)

for some constant K. In particular, OC^{2}(G) contains every function f for which

|Δf(x)| ⩽ K∥Df(x)∥/δ_{G}(x), x ∈ G.  (9)

Example 1. Condition (8) is satisfied if f is a harmonic function on an arbitrary domain G. Let f be an eigenfunction of Δ, i.e. Δf ≡ λf for some constant λ. Assuming that cl(rB) ⊂ G, where rB = B_{r}(0), we have

n ∫_{∂B} \frac{d}{dr} f(ry) dσ(y) = r^{1−n} ∫_{rB} Δf dm,

where dσ is the normalized surface measure on ∂B, which is a special case of Green’s formula. Hence

rΔf(0) = n ∫_{∂B} \frac{d}{dr} f(ry) dσ(y) − r^{1−n} ∫_{rB}(f − f(0)) dm,

and hence |Δf(0)| ⩽ nr^{−1} sup_{rB} |∇f| + r|λ| sup_{rB} |∇f|. Applying this to the function z ↦ f(x + z) we conclude that f ∈ OC^{1}(G) provided G is bounded. On the other hand, if f(x) = sin x_{1}, G = { x ∈ R^{n} : x_{1} > 0 }, then Δf = −f but f is not in OC^{2}(G).

Example 2. A function f ∈ C^{2}(B) is said to be hyperharmonic if Δ_{h}f ≡ 0, where

Δ_{h}f(x) = (1 − |x|^{2})^{2}[Δf(x) + 2(n − 2)(1 − |x|^{2})^{−1}x · ∇f(x)].

(x · y denotes the inner product in R^{n}.) It is clear that a hyperharmonic function satisfies (9) with K = 2(n − 2) and therefore belongs to OC^{2}(B). More generally, every eigenfunction of Δ_{h} belongs to OC^{2}(B), which can be proved by using the hyperbolic variant of Green’s formula. (See [5], where a complex hyperbolic analog of (8) was considered.)

2. Results

The following theorem was proved in [8] in the case of scalar functions. The proof of the vector variant is similar and we omit it.

Theorem 1. (a) If f ∈ HC^{1}(G), then |f| ∈ sh(G).

(b) If f ∈ OC^{1}(G), then the function x ↦ ∥Df(x)| belongs to sh(G).

Corollary 1. Let p > 0. A function f from C^{1}(G) belongs to HC^{1}(G) if and only if there is a constant K such that

|∇F (x)| ⩽ Kr^{−n−p} ∫_{B_{r}(x)} |f|^{p} dm, 0 < r < δ_{G}(x).

Let ω_{p}f(x, r) = {r^{−n} ∫_{B_{r}(x)} |f(z) − f(x)|^{p} dm(z)}^{1/p}.

Corollary 2. Let p > 0. A function f belongs to OC^{1}(G) if and only if |∇f(x)| ⩽ Kr^{−1}ω_{p}f(x, r), 0 < r < δ_{G}(x), for some constant K.

Proof. “If” part is trivial. Let f ∈ OC^{1}(G). Then f − c is in OC^{1}(G) ⊂ HC^{1}(G) for an arbitrary vector c. Hence, by Corollary 1,

∥Df(x)∥^{p} ⩽ K_{1}r^{−n−p} ∫_{B_{r}(x)} |f − c|^{p} dm.

As follows from [8], he constant K_{1} depends only on K from (6), p and n (not on c). The desired result now follows by taking c = f(x).

The main result of this paper is the following

Theorem 2. The following relations hold:

(a) HC^{2}(G) ⊂ HC^{1}(G) (b) OC^{2}(G) ⊂ OC^{1}(G).

Before proving the theorem we deduce some consequences.

Corollary 3. A function f ∈ C^{2}(G) belongs to HC^{2}(G) if and only if there is a constant K such that |Δf(x)| ⩽ Kr^{−2} sup{ |f(z)| : z ∈ B_{r}(x) }.

Proof. “If” part is trivial. “Only if” part is a consequence of Theorem 2(a).

Corollary 4. Let p > 0. For a function f ∈ C^{2}(G) the following assertions are equivalent:

(i) There is a constant K such that |Δf(x)| ⩽ Kr^{−2}ωf(x, r).

(ii) There is a constant K such that |Δf(x)| ⩽ Kr^{−2}ω_{p}f(x, r).

(iii) There is a constant K such that |Δf(x)|^{p} ⩽ Kr^{−n−p} ∫_{B_{r}(x)} ∥Df(z)∥^{p} dm(z).

(iv) f ∈ OC^{2}(G).

Proof. The implications (ii) ⇒ (i) ⇒ (iv) and (iii) ⇒ (iv) are trivial. The validity of implication (iv) ⇒ (iii) is easily derived from Theorem 2(b), Theorem 1(b) and Theorem A. That (iv) implies (ii) is deduced from Theorem 2(b) and Corollary 2.

As a further application of Theorem 2 we note a sufficient condition for a C^{3}-function to be in OC^{2}(G).

Theorem 3. A real valued C^{3}-function f belongs to OC^{2}(G) if there are constants K_{1} and K_{2} such that

|∇(Δf)(x)| ⩽ K_{1}r^{−1} sup_{B_{r}(x)} ∥D(∇f)∥ + K_{2}r^{−2} sup_{B_{r}(x)} |∇f|.  (10)

Proof. Since ∇(Δf) = Δ(∇f) condition (10) means that ∇f ∈ HC^{2}(G). Thus (10) implies ∇f ∈ HC^{1}(G), by Theorem 2(b), which means ∥D(∇f)(x)∥ ⩽ Kr^{−1} sup_{B_{r}(x)} |∇f| for some constant K. Since obviously |Δf| ⩽ const · ∥D(∇f)∥, it follows that f ∈ OC^{2}(G).

Corollary 5. A C^{4}-function f: G → R belongs to OC^{2}(G) if so does Δf. Consequently a C^{∞}-function f belongs to OC^{2}(G) if so does Δ^{k}f for some integer k. In particular every polyharmonic function of finite order belongs to OC^{2}. (A function f is polyharmonic if Δ^{k}f ≡ 0 for some integer k. For information see [2].)

Proof. Let Δf ∈ OC^{2}. Then Δf ∈ HC^{1}, by Theorem 2, i.e. |∇(Δf)(x)| ⩽ Kr^{−1} sup_{B_{r}(x)} |∇f|. Now the desired conclusion follows from Theorem 3.

Remark 1. As noted in the proof of Theorem 3 condition (10) means ∇f ∈ HC^{2}. By Corollary 3 condition (10) implies the existence of a constant K such that |∇(Δf)(x)| ⩽ Kr^{−2} sup_{B_{r}(x)} |∇f|, which is aparently stronger than (10).

Remark 2. It follows from the proofs of Theorem 3 and Corollary 5 that there hold the following implications:

Δf ∈ HC^{1} ⟹ ∇f ∈ OC^{2} ⟹ ∇f ∈ HC^{1} ⟹ f ∈ OC^{2}.

3. Proof of theorem 2.

The proof is based on the following consequence of Green’s formula.

Lemma 1. If f: B_{r}(x) → R^{k} is a C^{2}-function, then

∥Df(x)∥ ⩽ nr^{−1} sup_{B_{r}(x)} |f| + \frac{n}{n + 1}r sup_{B_{r}(x)} |Δf|.  (11)

Proof. In the case k = 1 a proof is in [2] (Proposition 3.1). If k > 1, we consider the functions u(z) = f(z) · ξ, ξ ∈ R^{k}, then use the formula ∇u(x) = Df(x)^{∗}ξ and choose ξ so that |ξ| = 1 and |Df(x)^{∗}ξ| = ∥Df(x)^{∗}∥ = ∥Df(x)∥. Then the result follows from the inequalities |u(z)| ⩽ |f(z)| and |Δu(z)| = |(Δf)(z)·ξ| ⩽ |Δf(z)|.

Lemma 2. Let F_{1}, F_{2} and F_{3} be nonnegative, continuous functions on G such that, for some constant K,

F_{1}(x)/K ⩽ r^{−1} sup_{B_{r}(x)} F_{2} + r sup_{B_{r}(x)} F_{3}  (12)

and

F_{3}(x)/K ⩽ r^{−1} sup_{B_{r}(x)} F_{1} + r^{−2} sup_{B_{r}(x)} F_{2}  (13)

whenever B_{r}(x) ⊂ G. Then there is a constant C = C(K) such that

F_{1}(x) ⩽ Cr^{−1} sup_{B_{r}(x)} F_{2}.  (14)

Proof. By translations the proof of (14) reduces to the case x = 0. Let cl(B_{ε}(0)) ⊂ G and F_{2} ⩽ 1 on B_{ε}(0). (In the general case we consider the functions F_{i}/A, where A is chosen so that F_{2}(z) ⩽ A for all z ∈ B_{ε}(0).) Choose x ∈ B_{ε}(0) so that F_{1}(y)(ε − |y|) ⩽ F_{1}(x)(ε − |x|) for all y ∈ B_{ε}(0). This implies that F_{1}(y) ⩽ 2F_{1}(x) for y ∈ B_{δ}(x), where δ = (ε − |x|)/2. Now we use the hypotheses to find y ∈ cl(B_{δ}(x)) so that

F_{1}(x)/K ⩽ r^{−1} + (Kr/t)F_{1}(y) + Krt^{−2}

for all r, t > 0 such that r + t = δ, which implies F_{1}(x)/K ⩽ r^{−1} + (2Kr/t)F_{1}(x) + Krt^{−2}. Now choose r, t so that r + t = δ and 2Kr/t = 1/2K, which implies that r = c_{1}(ε − |x|), t = c_{2}(ε − |x|) for some c_{i} = c_{i}(K), to obtain

F_{1}(x)/K ⩽ F_{1}(x)/2K + K_{1}(ε − |x|)^{−1},

where K_{1} = c_{1}^{−1}+c_{1}c_{2}^{−2}. Hence F_{1}(0)ε ⩽ F_{1}(x)(ε−|x|) ⩽ 2KK_{1}, and this concludes the proof.

Proof of Theorem 2. Let f satisfy (7). We may assume that K ⩾ n and K_{0} ⩾ n. Define functions

F_{1}(x) = ∥Df(x)∥, F_{2}(x) = |f(x)|, F_{3}(x) = |Δf(x)|.

Then (12) is sastisfied because of (11), and (13) is satisfied because of (7). Hence f ∈ HC^{1}(G), by Lemma 2. This proves assertion (a).

To prove (b) let f ∈ OC^{2}(G). Applying (a), together with its proof, to the functions f − c we find a constant K_{1} independent of x, r, c so that ∥Df(x)∥ ⩽ K_{1}r^{−1} sup_{B_{r}(x)} |f − c|. Finally we take c = f(x) to finish the proof.

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(received 14.09.1995.)

Faculty of Mathematics, P.O.B. 550, 11000 Beograd, Yugoslavia