Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 27, 2002, 205–213

ON LEBESGUE POINTS OF FUNCTIONS IN THE SOBOLEV CLASS W^{1,N}

Visa Latvala

University of Joensuu, Department of Mathematics P.O. Box 111, FI-80101 Joensuu, Finland; visa.latvala@joensuu.fi

Abstract. We present a method which appears to be useful in identifying the Lebesgue points for a function in the Sobolev class W^{1,n}. The key argument is applied for obtaining the uniqueness of asymptotic values, characterizing the approximate continuity, and for showing that functions satisfying a weak minimum principle possess Lebesgue points everywhere.

1. Introduction

This note deals with the pointwise behaviour of functions in the Sobolev space W_{loc}^{1,n}(R^{n}), n ≥ 2. The main ingredient of the paper is a method based on a somewhat non-standard combination of Sobolev type inequalities. This method, among others, implies that any function u ∈ W^{1,n}(B) defined in the unit ball B ⊂ R^{n} cannot have two different asymptotic values at the boundary point of B (Theorem 3.1). We have not been able to locate this general result in the existing literature. Of course, the question concerning asymptotic values has a long history and much stronger results are true if u is assumed to be, for example, monotone in the sense of Lebesgue, see e.g. [7], [8], [10] and the references therein.

We present two other applications of the key argument. Firstly, it is shown that for functions in W_{loc}^{1,n}(R^{n}) the points of approximate continuity coincide with the Lebesgue points (Theorem 3.4). The new feature here is that the approximate continuity implies the Lebesgue point property. Secondly, we prove that functions in W_{loc}^{1,n}(Ω) satisfying a weak minimum principle in an open set Ω ⊂ R^{n} possess Lebesgue points everywhere in Ω.

An elementary example shows that results do not hold in the same form for Sobolev functions u ∈ W^{1,p}(R^{n}), p < n. Another example shows that approximate continuity does not imply the n-fine continuity even for functions satisfying a weak minimum principle.

One should keep in mind that functions in W_{loc}^{1,n}(R^{n}) always possess Lebesgue points outside an n-polar set. Thus our results could be regarded as kind of minimal conclusions concerning the pointwise behaviour in the exceptional set.

2000 Mathematics Subject Classification: Primary 46E35, 31C25.

2. Definitions and auxiliary lemmas

Throughout n ≥ 2 and we use the notation B_{r} = B(x_{0}, r) for x_{0} ∈ R^{n} and r > 0. The boundary of the ball B(x_{0}, r) is denoted by ∂B(x_{0}, r) and the integral average of u ∈ L^{1}(B_{r}) over B_{r} is denoted by

u_{B_{r}} = ∫−_{B_{r}} u dx = \frac{1}{|B_{r}|} ∫_{B_{r}} u dx.

Here we integrate with respect to the n-dimensional Lebesgue measure | · |. The linear measure is denoted by | · |_{1} .

Let Ω ⊂ R^{n} be open. Then the Sobolev space W^{1,n}(Ω) consists of functions from the Lebesgue space L^{n}(Ω) for which |∇u| ∈ L^{n}(Ω). Here and elsewhere ∇u is the weak gradient of u. We write u ∈ W_{loc}^{1,n}(R^{n}) if u ∈ W^{1,n}(B) for all balls B ⊂ R^{n}.

Recall next the definitions of Lebesgue point and approximate continuity. Suppose that u: B(x_{0}, r) → [−∞, ∞] is integrable. Then x_{0} is called a Lebesgue point for u, if

lim_{r→0} ∫−_{B_{r}} |u − u(x_{0})| dx = 0.

The function u is called approximately continuous at x_{0} , if there is a set E ⊂ R^{n} with

lim_{r→0} \frac{|E ∩ B_{r}|}{|B_{r}|} = 1

such that the restriction u|_{E} is continuous at x_{0} .

The key argument is based on certain consequences of Poincaré and Sobolev inequalities. These consequences are well known for specialists but we give brief proofs for the reader’s convenience. Our first auxiliary lemma deals with functions vanishing on a large set.

Lemma 2.1. Let B ⊂ R^{n} be a ball and let u ∈ W^{1,n}(B). Suppose that

|{x ∈ B : |u(x)| > 0}| ≤ α|B|

for some 0 < α < 1. Then there is a constant c depending only on n and α such that

∫−_{B} |u|^{n} dx ≤ c ∫_{B} |∇u|^{n} dx.

Proof. The following argument is taken from [3, Lemma 2.8]. By the Minkowski and the Poincaré inequalities, we have

(∫_{B} |u|^{n} dx)^{1/n} ≤ c(|B| ∫_{B} |∇u|^{n} dx)^{1/n} + |u_{B}| |B|^{1/n}.

On Lebesgue points of functions in the Sobolev class W^{1,n} 207

By Hölder’s inequality,

|u_{B}| |B|^{1/n} ≤ |B|^{1/n−1} ∫_{{x∈B:|u(x)|>0}} |u| dx ≤ (∫_{B} |u|^{n} dx)^{1/n}α^{1−1/n},

and the assertion easily follows.

The other lemma concerns n-quasicontinuous representatives of Sobolev functions. Recall that u: Ω → [−∞, +∞] is n-quasicontinuous on an open set Ω ⊂ R^{n} if for every ε > 0 there is a set E ⊂ Ω such that C_{1,n}(E) < ε and the restriction of u to Ω \ E is continuous. In this paper C_{1,n} refers to the Sobolev n-capacity, see e.g. [1].

Lemma 2.2. Let x_{0} ∈ R^{n} and suppose that u ∈ W^{1,n}(R^{n}) is n-quasi-continuous. Then for any R > ε > 0 there is A_{ε} ⊂ [0, R] such that the linear measure of A_{ε} satisfies |A_{ε}|_{1} ≥ R − ε and

(osc u(∂B(x_{0}, r)))^{n} ≤ \frac{cr}{ε} ∫_{B(x_{0},R)} |∇u|^{n} dx

for all r ∈ A_{ε}. The constant c depends only on n.

Proof. A version of the Gehring oscillation lemma implies that u is continuous on ∂B_{r} = ∂B(x_{0}, r) and the estimate

(2.1)  (osc u(∂B_{r}))^{n} ≤ cr ∫_{∂B_{r}} |∇u|^{n} dσ

holds for a.e. r ∈]0, R[, see [4, Lemma 2.1]. Here c depends only on n. The assertion follows from this by elementary estimates. For r ∈]0, R], let φ(r) = ∫_{∂B_{r}} |∇u|^{n} dσ, where σ is the surface measure on the sphere ∂B_{r}. Recall that

∫_{B_{R}} |∇u|^{n} dx = ∫_{0}^{R} φ(r) dr.

For λ > 0, we have

|\{t ∈ [0, R] : φ(t) ≤ λ\}|_{1} ≥ R − \frac{1}{λ} ∫_{0}^{R} φ(r) dr.

Choosing λ = (1/ε) ∫_{0}^{R} φ(r) dr, A_{ε} = {t ∈ [0, R] : φ(t) ≤ λ}, we have |A_{ε}|_{1} ≥ R − ε and each r ∈ A_{ε} satisfies

∫_{∂B_{r}} |∇u|^{n} dσ = φ(r) ≤ λ = \frac{1}{ε} ∫_{0}^{R} φ(t) dt

assuming that λ > 0. Since the case λ = 0 is trivial, we conclude the assertion from (2.1).

3. Lebesgue points and approximate continuity

We show first that functions in W^{1,n}(B) cannot have two distinct asymptotic values at the boundary point x_{0} of B. Such a property is well known for quasiconformal functions and for monotone functions in the Sobolev space W^{1,n}(B); see [10, p. 181 and p. 189]. We state the result in general terms as follows.

Theorem 3.1. Let u ∈ W_{loc}^{1,n}(R^{n}) and let E ⊂ R^{n} with
(3.1)  β = lim inf_{R→0^{+}} \frac{|\{r ∈ (0, R) : ∂B(x_{0}, r) ∩ E ≠ ∅\}|_{1}}{R} > 0.

If u has a finite limit α as x → x_{0} in E, then

lim_{r→0^{+}} ∫−_{B(x_{0},r)} |u − α|^{n} dx = 0.

Proof. Define v ∈ W_{loc}^{1,n}(R^{n}) by v(x) = |u(x) − α| and denote

E_{ε} = {x ∈ Ω : v(x) ≤ ε}

for ε > 0. We divide the integral average of v^{n} over B_{r} into two parts by writing

∫−_{B_{r}} v^{n} dx = \frac{1}{|B_{r}|} ∫_{B_{r}∩E_{ε}} v^{n} dx + \frac{1}{|B_{r}|} ∫_{B_{r}\E_{ε}} v^{n} dx.

The first term on the right-hand side is trivially estimated as

\frac{1}{|B_{r}|} ∫_{B_{r}∩E_{ε}} v^{n} dx ≤ ε^{n}.

To estimate the second term we write

\begin{aligned}\frac{1}{|B_{r}|} ∫_{B_{r}\E_{ε}} v^{n} dx & ≤ \frac{2^{n−1}}{|B_{r}|} ∫_{B_{r}\E_{ε}}((v − ε)^{n} + ε^{n}) dx \\ & ≤ 2^{n−1} ∫−_{B_{r}}((v − ε)^{+})^{n} dx + 2^{n−1}ε^{n}. \end{aligned}

It is not hard to see that Lemma 2.2 implies together with (3.1) the inequality

|B_{r} ∩ E_{ε}| ≥ c|B_{r}|

for all r small enough. Here 0 < c < 1 depends only on n and β. Hence we may apply Lemma 2.1 and conclude that

∫−_{B_{r}}((v − ε)^{+})^{n} dx ≤ c ∫_{B_{r}} |∇v|^{n} dx

for all r small enough. It follows that

\frac{1}{|B_{r}|} ∫_{B_{r}\E_{ε}} v^{n} dx ≤ c ∫_{B_{r}} |∇v|^{n} dx + 2^{n−1}ε^{n}

for a constant c depending only on n and β. Consequently

∫−_{B_{r}} v^{n} dx ≤ ε^{n}(1 + 2^{n−1}) + c ∫_{B_{r}} |∇u|^{n} dx.

Letting r → 0, we obtain

lim sup_{r→0} ∫−_{B_{r}} v^{n} dx ≤ ε^{n}(1 + 2^{n−1}),

and the claim

lim_{r→0} ∫−_{B_{r}} |u − α|^{n} dx = 0

follows.

It is known that Theorem 3.1 does not hold for p < n. The following example is taken from [6].

Example 3.2. Define u in R^{2} by

u(z) = u(re^{iθ}) = \{ \begin{matrix}θ & for 0 ≤ θ ≤ \frac{1}{2}π, \\ \frac{1}{2}π & for \frac{1}{2}π ≤ θ ≤ π, \\ \frac{3}{2}π − θ & for π ≤ θ ≤ \frac{3}{2}π, \\ 0 & for \frac{3}{2}π ≤ θ ≤ 2π. \end{matrix}

Then u ∈ W_{loc}^{1,p}(R^{2}) for p < 2, but there is an infinite number of asymptotic values in the origin.

Theorem 3.1 easily implies that the points of approximate continuity coincide with the Lebesgue points for functions in W_{loc}^{1,n}(R^{n}).

Theorem 3.3. Let u ∈ W_{loc}^{1,n}(R^{n}) and let x_{0} ∈ R^{n}. Then the following are equivalent:

(i) lim_{r→0} ∫−_{B_{r}} u dx = u(x_{0}) < ∞,

(ii) lim_{r→0} ∫−_{B_{r}} |u − u(x_{0})|^{n} dx = 0,

(iii) app-lim_{x→x_{0}} u(x) = u(x_{0}) < ∞.

Proof. It is a trivial consequence of Hölder’s inequality that (ii) implies (i). Also, it is well known that (ii) implies (iii), see [11, p. 190]. We show first that (i) implies (ii). Suppose that

lim_{r→0} u_{B_{r}} = lim_{r→0} ∫−_{B_{r}} u dx = u(x_{0}).

By the Poincaré inequality,

\begin{aligned}∫−_{B_{r}} |u − u(x_{0})|^{n} dx & ≤ 2^{n−1} ∫−_{B_{r}} |u − u_{B_{r}}|^{n} dx + 2^{n−1} ∫−_{B_{r}} |u_{B_{r}} − u(x_{0})|^{n} dx \\ & ≤ C ∫_{B_{r}} |∇u|^{n} dx + 2^{n−1}|u_{B_{r}} − u(x_{0})|^{n}, \end{aligned}

and the claim (ii) easily follows. We conclude the assertion of the theorem by showing that (iii) implies (ii). Assume that

app-lim_{x→x_{0}} u(x) = u(x_{0}) < ∞.

By definition, there is a set E ⊂ R^{n} with the n-dimensional measure density 1 at x_{0} such that u → u(x_{0}) as x → x_{0} along the set E. It follows that

lim_{R→0} \frac{|\{r ∈ (0, R) : ∂B_{r} ∩ E ≠ ∅\}|_{1}}{R} = 1,

and (ii) holds by Theorem 3.1.

Remark 3.4. (a) Suppose that u ∈ L_{loc}^{1}(B(x_{0}, r)) is bounded. Then it is not hard to see that the approximate continuity implies the Lebesgue point property, see [9, Remark 6.7]. Vuorinen [9, Theorem 6.13] has given sufficient conditions for the case in which the approximate limit property implies the angular limit property for functions in the half-space Rn_{+} .

(b) Example 3.2 also shows that Theorem 3.3 does not hold for p < n. In fact, for u as in Example 3.2, the limit

lim_{r→0} ∫−_{B(0,r)} u dx

exists as a finite number. However, u is not approximately continuous in the origin. In particular, the origin is not a Lebesgue point for u.

We construct an example showing that the approximate continuity does not in general imply the n-fine continuity for functions in W^{1,n}(R^{n}), n > 2. For the notion of n-fine topology, see [1] and the references therein.

Example 3.5. For each j = 1, 2, …, let x_{j} = (2^{−j}, 0, …, 0), R_{j} = 2^{−2j−2} , r_{j} = 2^{−3j−3} , and define

u_{j}(x) = \{ \begin{matrix}\frac{log R_{j} − log |x − x_{j}|}{log R_{j} − log r_{j}} & for r_{j} < |x − x_{j}| < R_{j}, \\ 1 & for 0 ≤ |x − x_{j}| ≤ r_{j}, \\ 0 & for |x − x_{j}| ≥ R_{j}. \end{matrix}

The functions u_{j} are indeed extremal for the n-capacity of the spherical condenser (\overline{B} (x_{j}, r_{j}), B(x_{j}, R_{j})), see [1, pp. 35–36]. Define

u = ∑_{j=1}^{∞} u_{j}.

A computation then yields

∫_{B(x_{j},R_{j})} |∇u|^{n} dx = ω_{n−1}(log \frac{R_{j}}{r_{j}} )^{1−n} = ω_{n−1}((j + 1) log 2)^{1−n},

and therefore

∫_{R^{n}} |∇u|^{n} dx = ∑_{j=1}^{∞} ∫_{B(x_{j},R_{j})} |∇u_{j}|^{n} dx = ω_{n−1}(log 2)^{1−n} ∑_{j=2}^{∞} \frac{1}{j^{n−1}} < ∞

if only n ≥ 3. Here ω_{n−1} is the surface measure of ∂B(0, 1). Thus u ∈ W^{1,n}(R^{n}).

A straightforward calculation shows that the n-dimensional density of the set ⋃_{j=1}^{∞} B(x_{j}, R_{j}) is equal to zero in the origin. Hence u has the approximate limit zero in the origin. However, u does not have the n-fine limit zero in the origin. To see this, it is enough to prove that the set

E = ⋃_{j=1}^{∞} B(x_{j}, r_{j})

is not n-thin in the origin, [1, p. 234]. Equivalently, see [2, pp. 120–121], it is sufficient to show that

∑_{k=1}^{∞} C_{1,n}(E ∩ B(0, 2^{−k}))^{1/n−1} = ∞.

This follows from the estimate

C_{1,n}(B(x_{j}, r_{j}))^{1/n−1} ≥ c((log \frac{2}{r_{j}} )^{1−n})^{1/n−1} = c(log 2^{3j+4})^{−1}

= \frac{c}{(3j + 4) log 2},

since the constant c depends only on n, see [1, 2.41].

The function u in Example 3.5 satisfies the minimum principle in the Sobolev sense [6]. Hence Example 3.5 shows as well that the weak minimum principle is not enough to guarantee the pointwise n-fine continuity. On the other hand, the weak minimum principle does imply the pointwise approximate continuity. We finish the paper by proving this assertion. We again use the argument of Theorem 3.1 but also some facts about the n-fine topology are needed.

Theorem 3.6. Let Ω ⊂ R^{n} be open and suppose that u ∈ W_{loc}^{1,n}(Ω) is such that for each domain Ω′ ⋐ Ω the inequality m ≤ u holds a.e. in Ω′ whenever m ∈ R and (m − u)^{+} ∈ W_{0}^{1,n}(Ω′). Then u possesses Lebesgue points everywhere in Ω.

Proof. Take any n-quasicontinuous representative of u and define it pointwise in Ω by

(3.3)  u(z) = n-fine-lim inf_{x→z} u(x).

Then the resulting function u remains to be n-quasicontinuous [5, Theorem 2.145]. Denote v(x) = (u(x) − u(x_{0}))^{+} , E_{ε} = {x ∈ Ω : v(x) ≤ ε} for each ε > 0. We have the estimate

(3.4)  |B_{r} ∩ E_{ε}| ≥ \frac{1}{2}|B_{r}|

for all r small enough. This follows from Lemma 2.2 since the weak minimum principle implies the existence of a radius R > 0 such that

min v(∂B_{r}) < \frac{1}{2}ε

for all r < R. In fact, if there is a decreasing sequence r_{i} → 0, i = 0, 1, …, such that min v(∂B_{r_{i}}) ≥ \frac{1}{2}ε, we may apply the weak minimum principle in annuli together with the n-quasicontinuity to conclude that v ≥ \frac{1}{2}ε in B_{r_{0}} outside an npolar set. This contradicts (3.3). The estimate (3.4) allows us to apply Lemma 2.1 and proceed exactly the same way as in the proof of Theorem 3.1 in order to achieve

lim_{r→0} ∫−_{B_{r}}(u − u(x_{0}))^{+} dx = 0.

Hence it is enough to prove that

lim_{r→0} ∫−_{B_{r}}(u(x_{0}) − u)^{+} dx = 0.

To do this, observe first that u is locally bounded from below by the weak minimum principle. In fact, since u is n-quasicontinuous, there are lots of spheres ∂B(x_{0}, R) ⊂ Ω such that u is continuous on ∂B(x_{0}, R); see the proof of Lemma 2.2. Fix a number m < u(x_{0}) such that u ≥ m in a neighbourhood of x_{0} . By (3.3), for each ε > 0, there is an n-fine neighbourhood V_{ε} of x_{0} such that u ≥ u(x_{0})−ε in V_{ε}. We have

lim_{r→0} \frac{|B_{r} \ V_{ε}|}{|B_{r}|} = 0,

see e.g. [1, p. 230], and therefore

\begin{aligned}∫−_{B_{r}}(u(x_{0}) − u)^{+} dx & = \frac{1}{|B_{r}|} ∫_{B_{r}∩V_{ε}}(u(x_{0}) − u)^{+} dx + \frac{1}{|B_{r}|} ∫_{B_{r}\V_{ε}}(u(x_{0}) − u)^{+} dx \\ & ≤ ε\frac{|B_{r} ∩ V_{ε}|}{|B_{r}|} + (u(x_{0}) − m)\frac{|B_{r} \ V_{ε}|}{|B_{r}|} ≤ 2ε \end{aligned}

for all sufficiently small radii r. Thus x_{0} is a Lebesgue point for u.

References

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

[2] Heinonen, J., T. Kilpeläinen, and J. Malý: Connectedness in fine topologies. - Ann. Acad. Sci. Fenn. Ser. A I Math. 15, 1990, 107–123.

[3] Kinnunen, J., and N. Shanmugalingam: Quasi-minimizers on metric spaces. - Manuscript.

[4] Malý, J.: The area formula for W^{1,n} -mappings. - Comment. Math. Univ. Carolin. 35.2, 1994, 291–298.

[5] Malý, J., and W.P. Ziemer: Fine Regularity of Solutions of Elliptic Partial Differential Equations. - American Mathematical Society, Providence, 1997.

[6] Manfredi, J.J.: Monotone Sobolev functions. - J. Geom. Anal. 4, 1994, 393–402.

[7] Martio, O.: Equicontinuity theorem with an application to variational integrals. - Duke Math. J. 42, 1975, 569–581.

[8] Rickman, S.: Quasiregular Mappings. - Springer-Verlag, Berlin–Heidelberg, 1993.

[9] Vuorinen, M.: On the Harnack constant and the boundary behaviour of Harnack functions. - Ann. Acad. Sci. Fenn. Ser. A I Math. 7, 1982, 259–277.

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

[11] Ziemer, W.P.: Weakly Differentiable Functions. - Springer-Verlag, New York, 1989.

Received 5 January 2001