Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 24, 1999, 45–60

BOUNDARY LIMITS OF SPHERICAL MEANS FOR BLD AND 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, Japan

Abstract. Our aim in this paper is to deal with the existence of boundary limits for BLD functions u 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. We consider the L^{q} -means over the spherical surfaces S(0, r) centered at the origin with radius r, and show that

lim inf_{r→1}(1 − r)^{(n−p+α)/p−(n−1)/q}(∫_{S(0,r)} |u(x)|^{q} dS(x))^{1/q} = 0

when q > 0 and (n − p − 1)/p(n − 1) < 1/q < (n − p + α)/p(n − 1). If u is in addition monotone in B in the sense of Lebesgue, then u is shown to have weighted boundary limit zero.

1. Introduction

Let R^{n} denote the n-dimensional Euclidean space. We use the notation B(x, r) to denote the open ball centered at x with radius r > 0, whose boundary is denoted by S(x, r). Consider the L^{q} -means over S(0, r) defined by

S_{q}(u, r) = (\frac{1}{|S(0, r)|} ∫_{S(0,r)} |u(x)|^{q} dS(x))^{1/q},

where |S(0, r)| denotes the surface area, which is written as |S(0, r)| = σ_{n}r^{n−1} ; in case q = ∞, S_{∞}(u, r) denotes the essential supremum of u over S(0, r). We note by Hölder’s inequality that S_{q}(u, r) is nondecreasing for q.

Let u be a Green potential in the unit ball B = B(0, 1). Gardiner [1, Theorem 2] showed that

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

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

when (n − 3)/(n − 1) < 1/q ≦ (n − 2)/(n − 1) and q > 0. This gives an extension of the result by Stoll [22] in the plane case, which states that

lim inf_{r→1}(1 − r)S_{∞}(u, r) = 0.

Recently Herron and Koskela [4, Theorem 7.3, Corollary 7.5] proved that

S_{∞}(u, r) ≦ M[log(2/(1 − r))]^{(n−1)/n}, 0 < r < 1,

with a positive constant M, when u is a monotone function on B with finite Dirichlet integral:

∫_{B} |∇u(x)|^{n} dx < ∞;

see the next section for the definition of monotone functions. We here note that harmonic functions are monotone, 𝒜 -harmonic functions and hence coordinate functions of quasiregular mappings are monotone (see [3] and [18]). Thus the class of monotone functions is considerably wide.

Our main aim in this paper is to establish the analogue of these results for BLD and monotone BLD functions u on B satisfying

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

where ϱ(x) = 1 − |x|, 1 < p < ∞ and −1 < α < p − 1. We first study weighted boundary limits of spherical L^{q} -means for BLD functions satisfying (1), and establish a result corresponding to [16, Theorem 2.1] given in half spaces.

If u is a monotone BLD function on B(x_{0}, 2r) and p > n − 1, then the key for our results is the fact that

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

see e.g. [4, Lemma 7.1], [6, Remark, p. 9] and, for the case p = n, [26, Section 16]. If u is harmonic, then (2) holds for p ≧ 1 by the mean value property, so that the condition p > n − 1 is not required for harmonic functions. Further we note that if p > n, then (2) holds for all BLD functions, on account of Sobolev’s theorem. Thus, if we restrict ourselves to monotone functions, then we have only to consider the case n − 1 < p ≦ n.

Related results are given by Gardiner [1], Stoll [22], [23], [24] and the first author [12], [13] and [16].

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

Boundary limits of spherical means for BLD and monotone BLD functions 47

2. Statement of results

If 1 < p < ∞, G is an open set in R^{n} and E ⊂ G, then the relative p-capacity is defined by

C_{p}(E; G) = inf ∫_{G} f(y)^{p} dy,

where the infimum is taken over all nonnegative measurable functions f on G such that

∫_{G} |x − y|^{1−n}f(y) dy ≧ 1 for every x ∈ E;

see [8] and [15] for the basic properties of p-capacity.

Following Ziemer [28], we say that a locally integrable function u is p-precise in G if

(i) ∫_{G} |∇u(x)|^{p} dx < ∞, where ∇ denotes the gradient; (ii) for every ε > 0 there exists an open set ω such that C_{p}(ω, G) < ε and u is

continuous as a function on G − ω.

According to Ohtsuka [17], we say that a function u is locally p-precise in G if it is p-precise in every relatively compact open subset of G.

We note that if u is locally p-precise in G, then u is partially differentiable almost everywhere on G and its spherical means over S(x, r) are well defined whenever S(x, r) ⊂ G, since a set of p-capacity zero has Hausdorff dimension at most n − p.

We first study the weighted boundary limits of spherical means for locally p-precise functions on B satisfying (1).

Theorem 1 (cf. [12, Theorem 2.1] and [16, Theorem 2.1]). Let u be a locally p-precise function on B satisfying (1) with −1 < α < p − 1. If p < q < ∞ and

\frac{n − p − 1}{p(n − 1)} < \frac{1}{q} < \frac{n − p + α}{p(n − 1)},

then

lim inf_{r→1}(1 − r)^{(n−p+α)/p−(n−1)/q}S_{q}(u, r) = 0.

The sharpness of the exponent will be discussed in the final section. For BLD functions in half spaces of R^{n}, Theorem 1 was already given by the first author [16, Theorem 2.1]; for the reader’s convenience, we give a proof of Theorem 1.

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 [18], Serrin [19], and Vuorinen [25], [26].

It will be seen that the existence of lower limit in Theorem 1 is derived as a consequence of fine limit argument on the line R^{1} . Next we show that the exceptional sets disappear for monotone functions.

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

\frac{1}{q} < \frac{n − p + α}{p(n − 1)},

then

lim_{r→1}(1 − r)^{(n−p+α)/p−(n−1)/q}S_{q}(u, r) = 0.

Corollary 1. Let u be a coordinate function of a quasiregular mapping on B satisfying (1). If n − 1 < p < n + α, p < q < ∞ and

\frac{1}{q} < \frac{n − p + α}{p(n − 1)},

then

lim_{r→1}(1 − r)^{(n−p+α)/p−(n−1)/q}S_{q}(u, r) = 0.

For the definition and basic properties of quasiregular mappings, we refer to [3], [18] and [25]. 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 monotone in B, so that Theorem 2 gives the present corollary.

In case 1/q = (n − p + α)/p(n − 1) > 0, one might expect that S_{q}(u, r) is bounded. In fact, we can show that this is true only in case 0 ≦ α < p−1 without assuming the monotonicity; see Remark 3 given below in the final section. We refer the reader to the result by Yamashita [27] who showed affirmatively the case p = 2 and α = 1 for harmonic functions. The case α = p − 1 remains open.

Finally we treat 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 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 final aim is to study the existence of weighted boundary limits of monotone BLD functions u on B, which satisfy

(3)  ∫_{B} Φ_{p}(|∇u(x)|)ϱ(x)^{α} dx < ∞,

where ϱ is as in (1). Consider the function

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

for 0 ≦ r ≦ 2^{−1} ; set κ(r) = κ(2^{−1}) for r > 2^{−1} . We see (cf. [20, 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 0 ≦ σ < p − 1, then

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

Theorem 3. Let u be a monotone function on B satisfying (3). If n − 1 < p ≦ n + α and κ(0) = ∞, then

lim_{|x|→1}[κ(ϱ(x))]^{−1}u(x) = 0.

In case φ ≡ 1, p = n and α = 0, Theorem 3 was proved by Herron–Koskela [4, Theorem 7.3, Corollary 7.5]. In view of [11, Theorem 1] and [16, Theorem 4.1], we see that if u is harmonic in B, then the conclusions of Theorems 2 and 3 remain true for p smaller than n − 1.

Corollary 2. Let u be a coordinate function of a quasiregular mapping on B satisfying (3). If n − 1 < p ≦ n + α and κ(0) = ∞, then

lim_{|x|→1}[κ(ϱ(x))]^{−1}u(x) = 0.

3. Preliminary lemmas

Throughout this paper, let ϱ(x) denote the distance of x ∈ R^{n} from the unit spherical surface S(0, 1), that is,

ϱ(x) = ||x| − 1|.

Further, let M denote various constants independent of the variables in question.

Recall the definition of relative p-capacity in the previous section. We write C_{p}(E) = 0 if

C_{p}(E ∩ G; G) = 0 for every bounded open set G.

We say that a property holds p-q.e. on G if the property holds for every x ∈ G except that in a set of p-capacity zero. In view of [13, Lemma 2.2], if E ⊂ B and C_{p}(E) = 0, then we can find a nonnegative measurable function f on B such that

∫_{B} f(y)^{p}ϱ(y)^{α} dy < ∞

and

∫_{B} |x − y|^{1−n}f(y) dy = ∞ for every x ∈ E.

Now we give several results which are used for the proof of Theorem 1.

Lemma 1. If u is a locally p-precise function on B satisfying (1) with −1 < α < p − 1, then it has an extension \overline{u} with compact support in R^{n} which is q -precise in R^{n} for 1 < q < min{p, p/(1 + α)} and satisfies

∫_{R^{n}} |∇\overline{u}(x)|^{p}ϱ(x)^{α} dx < ∞.

Proof. If 1 < q < p and q < p/(1 + α), then Hölder’s inequality gives

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

Hence we can find a q -precise extension \overline{u} to R^{n} by Stein [21, Chapter 5], or we may consider the inversion to define

\overline{u}(x) = u(x/|x|^{2}) for |x| > 1.

We may further assume that the extension \overline{u} vanishes outside B(0, 2), by considering χ\overline{u}, where χ is an infinitely differentiable function on R^{n} with compact support in B(0, 2).

We introduce Sobolev’s integral representation.

Lemma 2 (cf. [9]). Let 1 < q < ∞ and v be a q -precise function on R^{n} with compact support. Then

v(x) = c ∑_{j=1}^{n} ∫_{R^{n}} \frac{x_{j} − y_{j}}{|x − y|^{n}} \frac{∂v}{∂y_{j}} (y) dy

holds for q -q.e. on R^{n}, where c = |S(0, 1)|^{−1} .

Corollary 3. Let u be a locally p-precise function on B satisfying (1) with −1 < α < p − 1. Then

u(x) = c ∑_{j=1}^{n} ∫_{R^{n}} \frac{x_{j} − y_{j}}{|x − y|^{n}} \frac{∂\overline{u}}{∂y_{j}} (y) dy

holds for p-q.e. on B, where \overline{u} is an extension of u as in Lemma 1.

Lemma 3 (cf. [12, Lemma 2.1] and [13, Lemma 5.1]). If we set k_{y}(x) = |x − y|^{δ(1−n)} for fixed y and δ > 0, then

S_{q}(k_{y}, r) ≦ M \{ \begin{matrix}|y|^{−δ(n−1)} & if |y| ≧ 2r, \\ r^{−δ(n−1)} & if \frac{1}{2}r < |y| < 2r and 1/q > δ, \\ r^{−(n−1)/q}||y| − r|^{(1/q−δ)(n−1)} & if \frac{1}{2}r < |y| < 2r and 1/q < δ, \\ r^{−δ(n−1)}[log(2r/||y| − r|)]^{1/q} & if \frac{1}{2}r < |y| < 2r and 1/q = δ, \\ r^{−δ(n−1)} & if |y| ≦ \frac{1}{2}r. \end{matrix}

Corollary 4. If 1 < q < ∞, then

∫_{S(0,r)} |x − y|^{q(1−n)} dS(y) ≦ M||x| − r|^{−(n−1)(q−1)}

for every x ∈ R^{n}.

Boundary limits of spherical means for BLD and monotone BLD functions 51 Lemma 4. If −1 < β < 0 and 0 < (1 − n)q + n < −β, then

∫_{R^{n}} |x − y|^{q(1−n)}ϱ(y)^{β} dy ≦ Mϱ(x)^{q(1−n)+n+β}

for every x ∈ B.

Proof. In view of Corollary 4, we have

\begin{aligned}∫_{R^{n}} |x − y|^{q(1−n)}ϱ(y)^{β} dy & = ∫_{0}^{∞}(∫_{S(0,r)} |x − y|^{q(1−n)} dS(y))|1 − r|^{β} dr \\ & ≦ M ∫_{R^{1}} |r − |x||^{−(n−1)(q−1)}|1 − r|^{β} dr. \end{aligned}

Since 0 < −(n − 1)(q − 1) + 1 < 1 and 0 < β + 1 < 1 by our assumptions, the Riesz composition theorem yields

∫_{R^{n}} |x − y|^{q(1−n)}ϱ(y)^{β} dy ≦ Mϱ(x)^{−(n−1)(q−1)+β+1},

as required.

Lemma 5 (cf. [13, Corollary 5.1]). If μ is a finite measure on the real line R^{1} and 0 < d < 1, then

lim inf_{r→0} |r|^{d} ∫_{R^{1}} |r − t|^{−d} dμ(t) = μ({0}).

4. Proof of Theorem 1 Under the assumptions on p, α and q in Theorem 1, we can take (β, γ) such

that

α < β < p − 1, 0 < γ < 1,
p(n − 1)γ + p − n > 0,
p(n − 1)γ + p − n < β < p(n − 1)γ + α − p(n − 1)/q

and

\frac{1}{q} < γ < \frac{1}{q} + \frac{1}{p(n − 1)}.

In view of Lemma 1 and Corollary 3, we may assume that

|u(x)| ≦ ∫_{R^{n}} |x − y|^{1−n}f(y) dy

for every x ∈ B, where f is a nonnegative function on R^{n} which vanishes outside a bounded set and satisfies

∫_{R^{n}} f(y)^{p}ϱ(y)^{α} dy < ∞;

recall that ϱ(y) = ||y|−1|. Using Hölder’s inequality, we have with 1/p+1/p′ = 1

|u(x)| ≦ (∫_{R^{n}} |x − y|^{a(1−n)}ϱ(y)^{b} dy)^{1/p′}(∫_{R^{n}} |x − y|^{γ(1−n)p}f(y)^{p}ϱ(y)^{β} dy)^{1/p},

where a = (1 − γ)p′ and b = −βp′/p. Since −1 < b < 0 and

\frac{b}{a} < \frac{n}{a′} − 1 < 0, a′ = \frac{a}{a − 1},

Lemma 4 yields

|u(x)| ≦ Mϱ(x)^{(1−γ)(1−n)+n/p′−β/p}(∫_{R^{n}} |x − y|^{γ(1−n)p}f(y)^{p}ϱ(y)^{β} dy)^{1/p}.

In view of Minkowski’s inequality for integral we have

\begin{aligned}S_{q}(u, r) & ≦ M(1 − r)^{(1−γ)(1−n)+n/p′−β/p} \\ & × [∫_{R^{n}}(∫_{S(0,r)} |x − y|^{γ(1−n)q} dS(x))^{p/q}f(y)^{p}ϱ(y)^{β} dy]^{1/p} \end{aligned}

for 2^{−1} < r < 1. Since γq > 1, Corollary 4 gives

\begin{aligned}S_{q}(u, r) & ≦ M(1 − r)^{(1−γ)(1−n)+n/p′−β/p} \\ & × (∫_{R^{n}} ||y| − r|^{−(n−1)(γq−1)p/q}f(y)^{p}ϱ(y)^{β} dy)^{1/p}. \end{aligned}

For simplicity, set d = (n − 1)(γq − 1)p/q. Then we see that 0 < d < 1. Consider the function

K(s, t) = s^{pω}s^{p[(1−γ)(1−n)+n/p′−β/p]}|t − s|^{−d}t^{β−α}

for 0 ≦ s < 1 and 0 ≦ t < ∞, where we set

ω = (n − p + α)/p − (n − 1)/q.

Here note that

(1 − r)^{ω}S_{q}(u, r) ≦ M(∫_{R^{n}} K(1 − r, ϱ(y))f(y)^{p}ϱ(y)^{α} dy)^{1/p}.

Since ω + [(1 − γ)(1 − n) + n/p′ − β/p] > 0, we see that

lim_{s→0} K(s, t) = 0

for all fixed t > 0. If t ≧ \frac{3}{2}s, then

K(s, t) ≦ M(s/t)^{(n−1)γp+α−β−p(n−1)/q} ≦ M,

if 0 ≦ t ≦ \frac{1}{2}s, then

K(s, t) ≦ M(s/t)^{α−β} ≦ M

and if \frac{1}{2}s < t < \frac{3}{2}s, then

K(s, t) ≦ Ms^{d}|s − t|^{−d}.

Consequently, applying Lemma 5, we conclude that

lim inf_{r→1}(1 − r)^{ω}S_{q}(u, r) = 0.

Now the proof of Theorem 1 is completed.

5. Proof of Theorem 2

For a proof of Theorem 2, we need the following result, which gives an essential tool in treating monotone functions.

Lemma 6 (cf. [4, Lemma 7.1], [6, Remark, p. 9], [16, Section 16]). Let p > n − 1. If u is a monotone p-precise function on B(x_{0}, 2r), then

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

Lemma 6 is a consequence of Sobolev’s theorem, so that the restriction p > n − 1 is needed; for a proof of Lemma 6, see for example [4, Lemma 7.1] or [15, Theorem 5.2, Chapter 8].

Now we are ready to prove Theorem 2.

Proof of Theorem 2. Let u be a monotone function on B satisfying (1) with n − 1 < p < n + α. If |s − t| ≦ r < \frac{1}{2}(1 − t), then Lemma 6 yields

\begin{aligned}|S_{q}(u, s) − S_{q}(u, t)| & ≦ (\frac{1}{σ_{n}} ∫_{S(0,1)} |u(sξ) − u(tξ)|^{q} dS(ξ))^{1/q} \\ & ≦ Mr^{(p−n)/p}(∫_{S(0,1)}(∫_{B(tξ,2r)} |∇u(z)|^{p} dz)^{q/p} dS(ξ))^{1/q}, \end{aligned}

so that Minkowski’s inequality for integral yields

\begin{aligned}|S_{q}(u, s) − S_{q}(u, t)| & ≦ Mr^{(p−n)/p}(2r/t)^{(n−1)/q} \\ & × (∫_{B(0,t+2r)−B(0,t−2r)} |∇u(z)|^{p} dz)^{1/p}. \end{aligned}

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, …. As before, set

ω = (n − p + α)/p − (n − 1)/q > 0.

Then we find

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

for t_{j} ≦ r < t_{j} + r_{j+1} ,

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

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

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

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

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

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

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

Since A_{l} ∩ A_{k} = ∅ when l ≧ k + 2, Hölder’s inequality gives

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

More generally, if t_{j} ≦ r < 1, then we take m such that t_{j+m−1} ≦ r < t_{j+m}, and establish

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

which implies that

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

for all j. Therefore it follows that

lim_{r→1}(1 − r)^{ω}S_{q}(u, r) = 0,

as required.

6. Proof of Theorem 3

Let u be a monotone function on B satisfying (3) with n − 1 < p < n + α. If B(x, 2r) ⊂ B and 0 < δ < 1, then, applying Lemma 6 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

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

56 Yoshihiro Mizuta and Tetsu Shimomura Since φ(r^{−δ}) ≧ Mφ(r^{−1}) for r > 0, it follows that

(5) |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).

Let x_{0} = 0 and r_{j} = 2^{−j−1} , j = 0, 1, …. For ξ ∈ S(0, 1), let x_{j} = (1−2r_{j})ξ. Then we find with the aid of (5)

|u(x_{j}) − u(y_{1})|^{p} ≦ Mr_{j}^{(1−δ)p} + Mr_{j}^{p−n}[φ(r_{j}^{−1})]^{−1} ∫_{B(x_{j},r_{j})} Φ_{p}(|∇u(z)|) dz for y_{1} ∈ S(x_{j}, \frac{1}{2}r_{j}),

|u(y_{1}) − u(y_{2})|^{p} ≦ Mr_{j}^{(1−δ)p} + Mr_{j}^{p−n}[φ(r_{j}^{−1})]^{−1} ∫_{B(x_{j},r_{j})} Φ_{p}(|∇u(z)|) dz for y_{2} ∈ S(y_{1}, \frac{1}{4}r_{j}) and |u(y_{2}) − u(x_{j+1})|^{p} ≦ Mr_{j+1}^{(1−δ)p} + Mr_{j+1}^{p−n}[φ(r_{j+1}^{−1})]^{−1} ∫_{B(x_{j+1},r_{j+1})} Φ_{p}(|∇u(z)|) dz for y_{2} ∈ S(x_{j+1}, \frac{1}{2}r_{j+1}). Thus it follows that

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

so that

\begin{matrix}|u(x_{j+m}) − u(x_{j})| ≦ M ∑_{l=j}^{j+m} r_{l}^{1−δ} \\ + M ∑_{l=j}^{j+m} r_{l}^{(p−n−α)/p}[φ(r_{l}^{−1})]^{−1/p}(∫_{B(x_{l},r_{l})} Φ_{p}(|∇u(z)|)ϱ(z)^{α} dz)^{1/p}. \end{matrix}

Since B(x_{l}, r_{l}) ∩ B(x_{k}, r_{k}) = ∅ when l ≧ k + 2, Hölder’s inequality gives

|u(x_{j}) − u(x_{j+m})| ≦ Mr_{j}^{1−δ}

+ M(∑_{l=j}^{j+m} r_{l}^{p′(p−n−α)/p}[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+m})(∫_{B(0,1−r_{j+m})−B(0,1−3r_{j})} Φ_{p}(|∇u(z)|)ϱ(z)^{α} dz)^{1/p}.

If x ∈ B(x_{j+m}, r_{j+m}) with x_{j} = (1 − 2r_{j})x/|x|, then

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

which implies that

lim sup_{|x|→1}[κ(ϱ(x))]^{−1}|u(x)| ≦ M(∫_{B−B(0,1−3r_{j})} Φ_{p}(|∇u(z)|)ϱ(z)^{α} dz)^{1/p}

for all j. Therefore it follows that

lim_{|x|→1}[κ(ϱ(x))]^{−1}u(x) = 0,

as required.

7. Remarks

Remark 1. Let {e_{j}} be a sequence in B which tends to a boundary point. For a number a > 0 and a sequence {ε_{j}} of positive numbers, consider the function

u(x) = ∑_{j} ε_{j}|x − e_{j}|^{−a}.

If a < (n − p)/p, then we can choose {ε_{j}} such that

∫_{B} |∇u(x)|^{p} dx < ∞.

Further, if a > (n − 1)/q, then we have

S_{q}(u, |e_{j}|) = ∞.

This implies that the lower limit in Theorem 1 can not be replaced by the upper limit.

Remark 2. Let −1 < α < p − 1. For δ > 0, consider the function

f(y) = ||y| − 1|^{a}|y − e|^{−b},

where a = δ − (α + 1)/p, b = (n − 1)/p and e = (1, 0, …, 0). Then

∫_{B(2e,1)} f(y)^{p}ϱ(y)^{α} dy < ∞.

We consider the harmonic function u on B defined by

u(x) = ∫_{B(2e,1)}(y_{1} − x_{1})|x − y|^{−n}f(y) dy.

Then we apply [13, Lemmas 12.1 and 12.2] to establish

∫_{R^{n}} |∇u(x)|^{p}ϱ(x)^{α} dx < ∞

by considering Lipschitz transformations from neighborhoods of boundary points of B to half spaces. If x ∈ B, then

u(x) > ∫_{B(x^{∗},|x−e|/4)}(y_{1} − x_{1})|x − y|^{−n}f(y) dy > M|x − e|^{1+a−b},

where x^{∗} = (1 + \frac{1}{2}|x − e|)e. Hence, if k(x) = |x − e|^{1+a−b} and δ < (n − p + α)/p − (n − 1)/q, then

S_{q}(u, r) ≧ MS_{q}(k, r) ≧ M(1 − r)^{(p−n−α)/p+(n−1)/q+δ}.

This implies that the exponent (n − p + α)/p − (n − 1)/q is sharp in Theorems 1 and 2.

Remark 3. Let u be a locally p-precise function on B satisfying

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

We see that if 0 ≦ α < p − 1 and

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

then

(7)  S_{q}(u, r) ≦ M(∫_{R^{n}} |∇u(x)|^{p}ϱ(x)^{α} dx)^{1/p}.

Yamashita [27] derived the above inequality for harmonic functions u on B satisfying (6) with p = 2 and 0 ≦ α ≦ 1. In the hyperplane case, we refer to [16, Theorem 2.2], and the present result will be proved similarly. In fact, to prove (7), we apply Sobolev’s integral representation (Lemma 2 and Corollary 3) and write

u(x) = c ∑_{j=1}^{n} ∫_{R^{n}} \frac{x_{j} − y_{j}}{|x − y|^{n}} \frac{∂\overline{u}}{∂y_{j}} (y) dy.

Here we may assume that the extension \overline{u} vanishes outside B(0, 2). As in the proof of Theorem 2.2 of [16], we have by Hölder’s inequality

|u(x)| ≦ M ∫_{S(0,1)}(∫_{0}^{2} |x − ty^{∗}|^{(1−n)p′} ||t| − 1|^{−αp′/p}t^{n−1} dt)^{1/p′}

× (∫_{R^{1}} |∇\overline{u}(ty^{∗})|^{p}||t| − 1|^{α}t^{n−1} dt)^{1/p} dS(y^{∗})
≦ M ∫_{S(0,1)} |x^{∗} − y^{∗}|^{1−n+1/p′−α/p}

× (∫_{R^{1}} |∇\overline{u}(ty^{∗})|^{p}||t| − 1|^{α}t^{n−1} dt)^{1/p}dS(y^{∗}),

where x^{∗} = x/|x| and y^{∗} = y/|y|. Now it suffices to apply Sobolev’s inequality. The case α = p − 1 remains open.

Remark 4. Let u be a locally p-precise function on B satisfying (3). Note here that if

(8)  ∫_{0}^{1}[r^{n−p}φ(r^{−1})]^{−1/(p−1)} \frac{dr}{r} < ∞,

then u is continuous on B and satisfies (5) on the basis of [10, Lemma 3], so that the conclusions of Theorems 2 and 3 are also valid for u. If in addition

(9)  ∫_{0}^{1}[r^{n−p+α}φ(r^{−1})]^{−1/(p−1)} \frac{dr}{r} < ∞,

then u has a continuous extension to R^{n}, according to [10, Theorem 2]. For these facts, see also [13], [15] and [20].

References

[1] Gardiner, S.J.: Growth properties of pth means of potentials in the unit ball. - Proc. Amer. Math. Soc. 103, 1988, 861–869.

[2] Gilbarg, D., and N.S. Trudinger: Elliptic Partial Differential Equations of Second Order, Second Edition. - Springer-Verlag, 1983.

60 Yoshihiro Mizuta and Tetsu Shimomura

[3] Heinonen, J., T. Kilpeläinen and O. Martio: Nonlinear Potential Theory of Degenerate Elliptic Equations. - Clarendon Press, 1993.

[4] Herron, D.A., and P. Koskela: Conformal capacity and the quasihyperbolic metric. - Indiana Univ. Math. J. 45, 1996, 333–359.

[5] Lebesgue, H.: Sur le probléme de Dirichlet. - Rend. Circ. Mat. Palermo 24, 1907, 371–402.

[6] Manfredi, J.J., and E. Villamor: Traces of monotone Sobolev functions. - J. Geom. Anal. (to appear).

[7] Matsumoto, S., and Y. Mizuta: On the existence of tangential limits of monotone BLD functions. - Hiroshima Math. J. 26, 1996, 323–339.

[8] Meyers, N.G.: A theory of capacities for potentials in Lebesgue classes. - Math. Scand. 26, 1970, 255–292.

[9] Mizuta, Y.: Integral representations of Beppo Levi functions of higher order. - Hiroshima Math. J. 4, 1974, 375–396.

[10] Mizuta, Y.: Boundary limits of locally n-precise functions. - Hiroshima Math. J. 20, 1990, 109–126.

[11] Mizuta, Y.: On the existence of weighted boundary limits of harmonic functions. - Ann. Inst. Fourier (Grenoble) 40, 1990, 811–833.

[12] Mizuta, Y.: Spherical means of Beppo Levi functions. - Math. Nachr. 158, 1992, 241–262.

[13] Mizuta, Y.: Continuity properties of potentials and Beppo–Levi–Deny functions. - Hiroshima Math. J. 23, 1993, 79–153.

[14] Mizuta, Y.: Tangential limits of monotone Sobolev functions. - Ann. Acad. Sci. Fenn. Ser. A I Math. 20, 1995, 315–326.

[15] Mizuta, Y.: Potential Theory in Euclidean Spaces. - Gakkōtosho, Tokyo, 1996.

[16] Mizuta, Y.: Hyperplane means of potentials. - J. Math. Anal. Appl. 201, 1996, 226–246.

[17] Ohtsuka, M.: Extremal length and precise functions in 3-space. - Lecture Notes at Hiroshima University, 1972.

[18] Reshetnyak, Yu.G.: Space Mappings with Bounded Distortion. - Amer. Math. Soc. Transl. 73, 1989.

[19] Serrin, J.: Local behavior of solutions of quasi-linear equations. - Acta Math. 111, 1964, 247–302.

[20] Shimomura, T., and Y. Mizuta: Taylor expansion of Riesz potentials. - Hiroshima Math. J. 25, 1995, 323–339.

[21] Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. - Princeton Univ. Press, Princeton, 1970.

[22] Stoll, M.: Boundary limits of subharmonic functions in the unit disc. - Proc. Amer. Math. Soc. 93, 1985, 567–568.

[23] Stoll, M.: Rate of growth of pth means of invariant potentials in the unit ball of C^{n}. - J. Math. Anal. Appl. 143, 1989, 480–499.

[24] Stoll, M.: Rate of growth of pth means of invariant potentials in the unit ball of C^{n}, II. - J. Math. Anal. Appl. 165, 1992, 374–398.

[25] Vuorinen, M.: On functions with a finite or locally bounded Dirichlet integral. - Ann. Acad. Sci. Fenn. Ser. A I Math. 9, 1984, 177–193.

[26] Vuorinen, M.: Conformal Geometry and Quasiregular Mappings. - Lecture Notes in Math. 1319, Springer-Verlag, 1988.

[27] Yamashita, S.: Dirichlet-finite functions and harmonic functions. - Illinois J. Math. 25, 1981, 626–631.

[28] Ziemer, W.P.: Extremal length as a capacity. - Michigan Math. J. 17, 1970, 117–128.

Received 24 March 1997