Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 25, 2000, 457–465

GROWTH PROPERTIES OF SPHERICAL MEANS FOR MONOTONE BLD FUNCTIONS IN THE UNIT BALL

Yoshihiro Mizuta and Tetsu Shimomura

Hiroshima University, The Division of Mathematical and Information Sciences Faculty of Integrated Arts and Sciences, Higashi-Hiroshima 739-8521, Japan mizuta@mis.hiroshima-u.ac.jp Akashi National College of Technology, General Studies Nishioka Uozumi 674-8501, Japan; shimo@akashi.ac.jp

Abstract. Let u be a monotone (in the sense of Lebesgue) function on the unit ball B of R^{n} satisfying

∫_{B} |∇u(x)|^{p}(1 − |x|)^{α} dx < ∞,

where ∇ denotes the gradient, 1 < p < ∞ and −1 < α < p − 1. Then u has a boundary limit f(ξ) for almost every ξ ∈ ∂B, and u may be considered as a Dirichlet solution for the boundary function f. Our aim in this paper is to deal with growth properties of the spherical means

S_{q}(u_{r} − f) ≡ (∫_{∂B} |u(rξ) − f(ξ)|^{q} dS(ξ))^{1/q}.

In fact, we prove that

lim_{r→1−0}(1 − r)^{−ω}S_{q}(u_{r} − f) = 0

when p ≦ q ≦ ∞ and ω = (n − 1)/q − (n − p + α)/p > 0.

1. Introduction

If f is a pth summable function on the boundary S of the unit ball B in R^{n}, then we can find a (weak) Dirichlet solution u which is harmonic in B and

(1)  lim_{r→1−0} S_{p}(u_{r} − f) = 0,

where 1 ≦ p < ∞, u_{r}(z) = u(rz) for z ∈ S and

S_{p}(v) = (\frac{1}{σ_{n}} ∫_{S} |v(z)|^{p} dS(z))^{1/p}

with σ_{n} denoting the surface area of S. Furthermore, for functions f in some Lipschitz spaces Λ_{β}^{p,p}(S) (see Stein [16]), we can find Dirichlet solutions u in BLD

1991 Mathematics Subject Classification: Primary 31B25, 31B15.

(Beppo Levi and Deny) spaces for which (1) is satisfied. Conversely, if u is a harmonic function on B satisfying

(2)  ∫_{B} |∇u(x)|^{p}(1 − |x|)^{α} dx < ∞,

where 1 < p < ∞ and −1 < α < p − 1, then it is well known that u has nontangential limits f(ξ) at many boundary points ξ ∈ S and f belongs to a certain Lipschitz space (see Stein [16] and the first author [9], [10]). Of course, u is a Dirichlet solution for f.

In this paper, we are concerned with weighted limits of S_{q}(u_{r} − f) for monotone BLD functions u on B with boundary values f. We say that a continuous function u is monotone in an open set G, in the sense of Lebesgue, if both

max_{\overline{D}} u(x) = max_{∂D} u(x)  and  min_{\overline{D}} u(x) = min_{∂D} u(x)

hold for every relatively compact open set D with the closure \overline{D} ⊂ G (see [5]). Clearly, harmonic functions are monotone, and more generally, solutions of elliptic partial differential equations of second order and weak solutions for variational problems may be monotone. For these facts, see Gilbarg–Trudinger [2], Heinonen– Kilpeläinen–Martio [3], Reshetnyak [13], Serrin [14] and Vuorinen [17] and [18].

Our starting point is a result of Gardiner [1, Theorem 2] which states that if u is a Green potential in the unit ball B, then

lim inf_{r→1−0}(1 − r)^{(n−1)(1−1/q)}S_{q}(u_{r}) = 0

when (n − 3)/(n − 1) < 1/q ≦ (n − 2)/(n − 1) and q > 0.

Our first aim in this paper is to show the following result.

Theorem 1. Let u be a monotone function on B satisfying (2) with n−1 < p ≦ n + α. If p ≦ q < ∞ and

ω = \frac{n − 1}{q} − \frac{n − p + α}{p} > 0,

then

lim_{r→1−0}(1 − r)^{−ω}S_{q}(U_{r}) = 0,

where U_{r}(ξ) = u(rξ) − u(ξ) for ξ ∈ S.

The sharpness of the exponent will be discussed in the final section. We also find a BLD function u satisfying (2) and

lim sup_{r→1−0}(1 − r)^{−ω}S_{q}(U_{r}) = ∞,

when α < 0.

Corollary 1. Let u be a coordinate function of a quasiregular mapping on B satisfying (2). If n−1 < p ≦ n+α, p ≦ q < ∞ and ω = (n−1)/q−(n−p+α)/p > 0, then

lim_{r→1−0}(1 − r)^{−ω}S_{q}(U_{r}) = 0.

For the definition and basic properties of quasiregular mappings, we refer to [3], [13], and [17]. In particular, a coordinate function u = f_{i} of a quasiregular mapping f = (f_{1}, …, f_{n}): B → R^{n} is 𝒜 -harmonic (see [3, Theorem 14.39] and [13]) and monotone in B, so that Theorem 1 gives the present corollary.

It is well known that the coordinate functions of a bounded quasiconformal mapping on B have finite n-Dirichlet integral (see Vuorinen [18]), so that Corollary 1 gives the following result.

Corollary 2. Let u be a coordinate function of a bounded quasiconformal mapping on B. If n ≦ q < ∞, then

lim_{r→1−0}(1 − r)^{−(n−1)/q}S_{q}(U_{r}) = 0.

Next we are concerned with the case q = ∞. In order to give a general result, we consider a nondecreasing positive function φ on the interval [0, ∞) such that φ is of log-type, that is, there exists a positive constant M satisfying

φ(r^{2}) ≦ Mφ(r) for all r ≧ 0.

Set Φ_{p}(r) = r^{p}φ(r) for p > 1. Our next aim is to study the boundary behavior of monotone BLD functions u on B, which satisfy the weighted condition

(4)  ∫_{B} Φ_{p}(|∇u(x)|)(1 − |x|)^{α} dx < ∞.

Consider the function

κ(r) = [∫_{0}^{r}(t^{n−p+α}φ(t^{−1}))^{−1/(p−1)} \frac{dt}{t}]^{1−1/p}

for r > 0. We see (cf. [15, Lemma 2.4]) that if n − p + α < 0, then

κ(r) ∼ [r^{n−p+α}φ(r^{−1})]^{−1/p} as r → 0

and if n − p + α = 0 and φ(r) = (log(e + r))^{σ} with σ > p − 1, then

κ(r) ∼ [log(1/r)]^{(p−1−σ)/p} as r → 0.

Theorem 2. Let u be a monotone function on B satisfying (4) with −1 < α ≦ p − n. If κ(1) < ∞, then

lim_{r→1−0}[κ(1 − r)]^{−1}S_{∞}(U_{r}) = 0,

where U_{r}(ξ) = u(rξ) − u(ξ) for ξ ∈ S.

Corollary 3. Let u be a coordinate function of a quasiregular mapping on B satisfying (4) with −1 < α ≦ p − n. If κ(1) < ∞, then

lim_{r→1−0}[κ(1 − r)]^{−1}S_{∞}(U_{r}) = 0.

For related results, we also refer to Herron–Koskela [4], the first author [8],

[11] and the authors [12].

Finally we wish to express our deepest appreciation to the referee for his useful suggestions.

2. Proof of Theorem 1

Throughout this paper, let M denote various constants independent of the variables in question.

For a proof of Theorem 1, we need the following result, which gives an essential tool in treating monotone functions.

Lemma 1 (cf. [4], [6], [10]). Let p > n − 1. If u is a monotone BLD function on B(x_{0}, 2r), then

(5) |u(x) − u(y)|^{p} ≦ Mr^{p−n} ∫_{B(x_{0},2r)} |∇u(z)|^{p} dz  whenever x, y ∈ B(x_{0}, r).

Lemma 1 is a consequence of Sobolev’s theorem, so that the restriction p > n − 1 is needed; for a proof of Lemma 1, see for example [4, Lemma 7.1] or [10, Theorem 5.2, Chapter 8].

Now we are ready to prove Theorem 1, along the same lines as in the proof of [12, Theorem 2].

Proof of Theorem 1. Let u be a monotone function on B satisfying (2) with n − 1 < p ≦ n + α. If |s − t| ≦ r < \frac{1}{2}(1 − t), then Lemma 1 gives

\begin{aligned}|S_{q}(u_{s} − u_{t})| & = (\frac{1}{σ_{n}} ∫_{S} |u(sξ) − u(tξ)|^{q} dS(ξ))^{1/q} \\ & ≦ Mr^{(p−n)/p}(∫_{S}(∫_{B(tξ,2r)} |∇u(z)|^{p} dz)^{q/p} dS(ξ))^{1/q}, \end{aligned}

so that Minkowski’s inequality for integrals yields

|S_{q}(u_{s} − u_{t})| ≦ Mr^{(p−n)/p}(∫_{B(0,t+2r)−B(0,t−2r)} |∇u(z)|^{p}

× (∫_{{ξ∈S:|ξ−z/t|<2r/t}} dS(ξ))^{p/q} dz)^{1/p}

≦ Mr^{(p−n)/p}(2r/t)^{(n−1)/q}(∫_{B(0,t+2r)−B(0,t−2r)} |∇u(z)|^{p} dz)^{1/p}.

Growth properties of spherical means for monotone BLD functions 461 Let r_{j} = 2^{−j−1} , t_{j} = 1−r_{j−1} and A_{j} = B(0, 1−r_{j})−B(0, 1−3r_{j}) for j = 1, 2, …. For simplicity, set

ω = −(n − p + α)/p + (n − 1)/q > 0.

Then we find

|S_{q}(u_{t_{j}} − u_{r})| ≦ Mr_{j+1}^{ω}(∫_{A_{j}} |∇u(z)|^{p}(1 − |z|)^{α} dz)^{1/p}

for t_{j} ≦ r < t_{j} + r_{j+1} ,

|S_{q}(u_{r} − u_{s})| ≦ Mr_{j+2}^{ω}(∫_{A_{j}} |∇u(z)|^{p}(1 − |z|)^{α} dz)^{1/p}

for t_{j} + r_{j+1} ≦ r < s < t_{j} + r_{j+1} + r_{j+2} , and

|S_{q}(u_{s} − u_{t_{j+1}})| ≦ Mr_{j+2}^{ω}(∫_{A_{j+1}} |∇u(z)|^{p}(1 − |z|)^{α} dz)^{1/p}

for t_{j} + r_{j+1} + r_{j+2} ≦ s < t_{j+1} . Collecting these results, we have

\begin{aligned}|S_{q}(u_{t_{j}} − u_{r})| & ≦ Mr_{j}^{ω}(∫_{A_{j}} |∇u(z)|^{p}(1 − |z|)^{α} dz)^{1/p} \\ & + Mr_{j+1}^{ω}(∫_{A_{j+1}} |∇u(z)|^{p}(1 − |z|)^{α} dz)^{1/p} \end{aligned}

for t_{j} ≦ r < t_{j+1} . Hence it follows that

|S_{q}(u_{r} − u_{t_{j+m}})| ≦ M ∑_{l=j}^{j+m} r_{l}^{ω}(∫_{A_{l}} |∇u(z)|^{p}(1 − |z|)^{α} dz)^{1/p}

for t_{j} ≦ r < t_{j+m}. Since A_{l} ∩ A_{k} = ∅ for l ≧ k + 2, Hölder’s inequality gives

\begin{aligned}|S_{q}(u_{r} − u_{t_{j+m}})| & ≦ M(∑_{l=j}^{j+m} r_{l}^{p′ω})^{1/p′}(∑_{l=j}^{j+m} ∫_{A_{l}} |∇u(z)|^{p}(1 − |z|)^{α} dz)^{1/p} \\ & ≦ Mr_{j}^{ω}(∫_{B(0,1−r_{j+m})−B(0,1−3r_{j})} |∇u(z)|^{p}(1 − |z|)^{α} dz)^{1/p} \end{aligned}

for t_{j} ≦ r < t_{j+m}, where 1/p + 1/p′ = 1. Now, letting m → ∞, we establish

|S_{q}(U_{r})| ≦ M(1 − r)^{ω}(∫_{B−B(0,1−3r_{j})} |∇u(z)|^{p}(1 − |z|)^{α} dz)^{1/p}

for t_{j} ≦ r < t_{j+1} , which implies that

lim_{r→1}(1 − r)^{−ω}S_{q}(U_{r}) = 0,

as required.

3. Proof of Theorem 2

If B(x, 2r) ⊂ B, then, applying Lemma 1 and dividing the domain of integration into two parts

\begin{aligned}E_{1} & = \{z ∈ B(x, 2r) : |∇u(z)| > r^{−δ}\}, \\ E_{2} & = B(x, 2r) − E_{1}, \end{aligned}

we have

(6) |u(x) − u(y)|^{p} ≦ Mr^{(1−δ)p} + Mr^{p−n}[φ(r^{−1})]^{−1} ∫_{B(x,2r)} Φ_{p}(|∇u(z)|) dz

for y ∈ B(x, r), where M may depend on δ, 0 < δ < 1.

Let r_{j} = 2^{−j−1} , j = 0, 1, …. For x ∈ B, let x_{j} = (1 − 2r_{j})ξ with ξ = x/|x|. Then we find

|u(x_{j}) − u(x)| ≦ Mr_{j}^{1−δ} + Mr_{j}^{(p−n)/p}[φ(r_{j}^{−1})]^{−1/p}(∫_{B(x_{j},r_{j})} Φ_{p}(|∇u(z)|) dz)^{1/p}

when \frac{3}{2}r_{j} < ϱ(x) < \frac{5}{2}r_{j}, where ϱ(x) = |1 − |x||. Hölder’s inequality gives

|u(x_{j+m}) − u(x_{j})| ≦ M ∑_{l=j}^{j+m} r_{l}^{1−δ} + ∑_{l=j}^{j+m} r_{l}^{ω}[φ(r_{l}^{−1})]^{−1/p}

× (∫_{B(x_{l},r_{l})} Φ_{p}(|∇u(z)|)ϱ(z)^{α} dz)^{1/p}

≦ Mr_{j}^{1−δ} + M(∑_{l=j}^{j+m} r_{l}^{p′ω}[φ(r_{l}^{−1})]^{−p′/p})^{1/p′}

× (∑_{l=j}^{j+m} ∫_{B(x_{l},r_{l})} Φ_{p}(|∇u(z)|)ϱ(z)^{α} dz)^{1/p}

≦ Mr_{j}^{1−δ} + Mκ(r_{j})(∫_{B−B(0,1−3r_{j})} Φ_{p}(|∇u(z)|)ϱ(z)^{α} dz)^{1/p},

where ω = (p − n − α)/p. Since lim_{r→0} κ(r) = 0, we see that {u(x_{j})} is a Cauchy sequence. Thus (6) implies that lim_{r→1} u(rξ) exists and is finite for every ξ ∈ S. Now, letting m → ∞, we have

\begin{aligned}|U(x_{j})| & = |u(x_{j}) − u(ξ)| \\ & ≦ Mr_{j}^{1−δ} + Mκ(r_{j})(∫_{B−B(0,1−3r_{j})} Φ_{p}(|∇u(z)|)ϱ(z)^{α} dz)^{1/p}. \end{aligned}

Hence we establish

|U(x)| ≦ Mϱ(x)^{1−δ} + Mκ(ϱ(x))(∫_{B−B(0,1−3ϱ(x))} Φ_{p}(|∇u(z)|)ϱ(z)^{α} dz)^{1/p}.

Note that

κ(r) ≧ [∫_{r/2}^{r}(t^{n−p+α}φ(t^{−1}))^{−1/(p−1)} \frac{dt}{t}]^{1−1/p} ≧ [r^{n−p+α}φ(r^{−1})]^{−1/p}.

If we take δ so that (n − p + α)/p + 1 > δ > 0, then

lim_{r→0}[κ(r)]^{−1}r^{1−δ} = 0.

Now it follows that

lim sup_{|x|→1}[κ(ϱ(x))]^{−1}|U(x)| = 0,

as required.

The above proof also shows the following (see [7] and [9]).

Proposition 1. Let u be a monotone function on B satisfying (4) with −1 < α ≦ p − n. If κ(1) < ∞, then u has a finite nontangential limit at every ξ ∈ S.

Proposition 2. Let u be a monotone function on B satisfying (4) with n − 1 < p ≦ n + α. If κ(1) = ∞, then

lim_{r→1}(1 − r)^{−ω}[φ((1 − r)^{−1})]^{1/p}S_{q}(U_{r}) = 0

whenever p ≦ q < ∞ and ω = (n − 1)/q − (n − p + α)/p > 0.

4. Sharpness

1. The sharpness of the exponent −ω. Let −1 < α < p − 1. For δ > 0, consider the function u on B defined by

u(x) = (1 − |x|)^{1+a}|x − e|^{−b},

where a = δ − (α + 1)/p, b = (n − 1)/p and e = (1, 0, …, 0). Then the function u is monotone on B. To show this, let D be a relatively compact open set with the closure \overline{D} ⊂ B, and suppose u attains a maximum on \overline{D} at an interior point c ∈ D, and set

E = \{x ∈ \overline{B} : (1 − |x|)^{1+a} = u(c)|x − e|^{b}\}.

Since e ∈ E,

max_{\overline{D}} u(x) = max_{∂D} u(x)

holds. We also see that u attains a minimum on \overline{D} at a boundary point. Hence u is monotone on B. Further we have

∫_{B} |∇u(x)|^{p}ϱ(x)^{α} dx < ∞.

If k(x) = |x − e|^{−b}, then

S_{q}(u_{r}) = M(1 − r)^{1+a}S_{q}(k_{r}) ≧ M(1 − r)^{1+a+(1/q−1/p)(n−1)} = M(1 − r)^{ω+δ}.

This implies that the exponent −ω is sharp in Theorem 1.

2. The limits for BLD functions. Theorem 1 fails to hold for BLD functions, when α < 0 and (n − p + α)/p < (n − 1)/q < (n − p)/p.

For 0 < r_{j} < 1 and γ > 1, set

C_{j} = {x : |x − r_{j}e| ≦ (1 − r_{j})^{γ}},

where |e| = 1. We take {r_{j}} such that {C_{j}} are mutually disjoint. For φ ∈ C_{0}^{∞}(B(0, 1)) such that φ = 1 on B(0, \frac{1}{2}), define

u_{j}(x) = φ(|x − r_{j}e|/(1 − r_{j})^{γ})|x − r_{j}e|^{−a},

where

(7)  (n − p + α)/p < a < (n − p + α/γ)/p.

Then

∫_{B} |∇u_{j}(x)|^{p}(1 − |x|)^{α} dx ≦ M(1 − r_{j})^{α+γ(n−(a+1)p)}.

Note further that

S_{q}((u_{j})_{r_{j}}) ≧ c(1 − r_{j})^{γ(−a+(n−1)/q)}

with a positive constant c > 0, when a < (n − 1)/q. If we set u = ∑_{j} u_{j}, then

lim_{j→∞}(1 − r_{j})^{−ω}S_{q}(u_{r_{j}}) = ∞,

when a and γ are chosen so that

(8) (n − p + α/γ)/p + (1 − γ^{−1}){(n − 1)/q − (n − p)/p} < a < (n − 1)/q.

Growth properties of spherical means for monotone BLD functions 465

Now we suppose that a satisfies (7) and (8). Then, since α + γ(n − (a + 1)p) > 0, we can take {r_{j}} so that

∫_{B} |∇u(x)|^{p}(1 − |x|)^{α} dx < ∞.

Thus u satisfies (2) and

lim sup_{r→1}(1 − r)^{−ω}S_{q}(U_{r}) = ∞.

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Received 3 March 1999