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CHAPTER 1
Weighted Sobolev spaces
In this first chapter we introduce the weighted Sobolev spaces H1,p(Ω; μ) and investigate their basic properties which are needed in chapters to come. Although many features of the unweighted theory axe retained, a somewhat different approach is mandatory.
We do not try to characterize those weights or measures which are admissible for our purposes. Instead, we elude the characterization problem in a customary way: the basic inequalities which are necessary for the development of the theory are included in the definition. The class of weights satisfying the given requirements is by no means restricted.
Throughout this book Ω will denote an open subset of Rn, n ≥ 2, and 1 < p < ∞.
1.1. p-admissible weights
Let w be a locally integrable, nonnegative function in Rn. Then a Radon measure μ is canonically associated with the weight w,
(1.2) μ(E) = ∫E w(x)dx.
Thus dμ(x) = w{x) dx, where dx is the n-dimensional Lebesgue measure. In what follows the weight w and the measure μ are identified via (1.2). We say that w (or μ) is p-admissible if the following four conditions are satisfied:
I 0 < w< ∞ almost everywhere in Rn and the measure μ is doubling, i.e. there is a constant CI > 0 such that
μ(2B) ≤ CIμ(B)
whenever B is a ball in Rn.
II If D is an open set and ψi [member of] C∞(D) is a sequence of functions such that [MATHEMATICAL EXPRESSION OMITTED], where v is a vector-valued measurable function in LP(D; μ; Rn), then v = 0.
III There are constants x > 1 and CIII > 0 such that
[MATHEMATICAL EXPRESSION OMITTED]
whenever B = B(x0, r) is a ball in Rn and ψ [member of] C∞0(B).
IV There is a constant CIV > 0 such that
[MATHEMATICAL EXPRESSION OMITTED]
whenever B = B(x0, r) is a ball in Rn and ψ [member of] C∞(B) is bounded.
Here
[MATHEMATICAL EXPRESSION OMITTED]
Convention.From now on, unless otherwise stated, we assume that μ is a p-admissible measure and dμ(x) = w(x) dx.
Let us make some remarks on conditions I-IV. It follows immediately from condition I that the measure μ and Lebesgue measure dx are mutually absolutely continuous, i.e. they have the same zero sets; so there is no need to specify the measure when using the ubiquitous expressions almost everywhere and almost every, both abbreviated a.e. Moreover, it easily follows from the doubling property that μ(Rn) = ∞.
Condition II guarantees that the gradient of a Sobolev function is well defined, a conclusion that cannot be expected in general (Fabes et al. 1982a, pp. 91-92).
Condition III is the weighted Sobolev embedding theorem or the weighted Sobolev inequality and condition IV is the weighted Poincaré inequality. The validity of these inequalities is crucial to the theory in this book.
The Lebesgue differentiation theorem holds: if f [member of] L1loc(Rn; μ), then for a.e. x in Rn
[MATHEMATICAL EXPRESSION OMITTED] (1.3)
For a proof, see Ziemer (1989, p. 14).
In general, if v is a measure and f is a v-integrable function on a set E with 0 < v(E) < ∞, we write the integral average of f on E as
[MATHEMATICAL EXPRESSION OMITTED]
For example, (1.3) is usually written as
[MATHEMATICAL EXPRESSION OMITTED]
The weighted Sobolev inequality III implies the following Poincaré type inequality. With an obvious abuse of terminology, in this book both condition IV and inequality (1.5) are referred to as the Poincaré inequality.
1.4. Poincaré inequality. If Ω is bounded, then
(1.5) [MATHEMATICAL EXPRESSION OMITTED]
for ψ [member of] C∞0(Ω).
Proof: Let x0 [member of] Ω. and write B = B(x0, diamΩ). If ψ [member of] C∞0(Ω), then the Hölder inequality and III imply
[MATHEMATICAL EXPRESSION OMITTED]
and the lemma follows.
Notation. Qualitatively, many properties of μ depend only on the constants which appear in conditions I, III, and IV. For short we write
cμ = (CI, κ, CIII, CIV).
Thus, saying that something depends on cμ means it depends on the above constants associated with μ.
1.6. Examples of p-admissible weights
Next we give some examples of p-admissible weights and show that a p-admissible weight is also q-admissible for all q greater than p.
The first example is the usual case when w = 1 and μ is Lebesgue measure. Then I is obvious, II is easy, and III is the ordinary Sobolev inequality which holds with
[MATHEMATICAL EXPRESSION OMITTED]
Moreover, for p< n we have that
(1.7) [MATHEMATICAL EXPRESSION OMITTED]
for ψ [member of] C∞0(Ω).
Condition IV is the classical Poincaré inequality; see, for instance, Chapter 7 in Gilbarg and Trudinger (1983).
For the second example consider the Muckenhoupt class Ap which consists of all nonnegative locally integrable functions w in Rn such that
[MATHEMATICAL EXPRESSION OMITTED]
where the supremum is taken over all balls B in Rn. If w belongs to Ap, then w is p-admissible; we emphasize that the index p is the same. The weight w is said to be in A1 if there is a constant c such that
[MATHEMATICAL EXPRESSION OMITTED]
for all balls B in Rn. Since A1 [subset] Ap whenever p > 1, an A1-weight is p-admissible for every p > 1.
We give the basic theory of Ap-weights in Chapter 15, where we also establish their p-admissibility.
The third example arises from the theory of quasiconformal mappings: if f: Rn ->Rn is a K-quasiconformal mapping and Jf(x) the determinant of its Jacobian matrix, then
w(x) = Jf(x)1-p/n
is p-admissible for 1 < p< n. This weight need not be in Ap. For instance, the function |x|δ is in Ap if and only if -n< δ < n(p - 1), but for the quasiconformal mapping
f(x) = x|x|γ, γ > -l,
Jf(x)1-p/n is comparable to |x|γ(n-p). Thus, if p < n, the function w(x) = |x|δ satisfies I-IV whenever δ > -n. The constants for μ depend only on n and δ. It follows from Theorem 1.8 that w(x) = |x|δ, δ > -n, is a p- admissible weight for all p > 1.
The above facts about quasiconformal mappings and admissible weights are proved in Chapter 15.
This discussion does not exhaust the body of admissible weights; there is a rapidly growing literature on weighted Sobolev and Poincaré inequalities. See Notes to this chapter.
1.8. Theorem.Suppose that w is a p-admissibie weight and q > p. Then w is q-admissible.
Proof: Condition I is trivial. Condition II follows by observing that
[MATHEMATICAL EXPRESSION OMITTED]
implies
[MATHEMATICAL EXPRESSION OMITTED]
for each G [??] D by Hölder's inequality.
Next we prove III. Let ψ [member of] C∞0(B), B = B(x0, r), and ψ = max(0, ψ). Then let
s = q/p >1
and note that the p-type inequality III holds for the function ψs this follows by approximation (see the proof of Lemma 1.11). Moreover, it suffices to verify the q-type inequality III for ψ. To do so, we combine
[MATHEMATICAL EXPRESSION OMITTED]
and
[MATHEMATICAL EXPRESSION OMITTED]
to obtain
[MATHEMATICAL EXPRESSION OMITTED]
as desired.
To verify inequality IV with p replaced by g, let ψ [member of] C∞0(B) be bounded. It suffices to find constants γ and C such that
[MATHEMATICAL EXPRESSION OMITTED]
this is due to the fact that
[MATHEMATICAL EXPRESSION OMITTED]
Again let s = q/p > 1 and write
v = max(ψ - γ, 0)s - max(γ - ψ, 0)s,
where γ is chosen so that
∫B v dμ = 0.
It is easily demonstrated (cf. Lemma 1.11) that the p-Poincaré inequality holds for v, that is
[MATHEMATICAL EXPRESSION OMITTED]
Since
|[nabla]v| = s|[nabla]ψ| |v|(s-1></s
and q = sp, Hölder's inequality yields
[MATHEMATICAL EXPRESSION OMITTED]
Finally, because |v|p = |ψ - γ|q, it follows that
[MATHEMATICAL EXPRESSION OMITTED]
as desired.
1.9. Sobolev spaces
For a function ψ [member of] C∞(Ω) we let
[MATHEMATICAL EXPRESSION OMITTED]
where, we recall, [nabla]ψ = ([partial derivative]1ψ ..., [partial derivative]nψ) is the gradient of ψ. The Sobolev space H1,p(Ω; μ) is defined to be the completion of
{ψ [member of] C∞(Ω): ||ψ||1,p< ∞}
with respect to the norm ||x||1,p. In other words, a function u is in H1,p(Ω; μ) if and only if u is in LP(Ω; μ) and there is a vector-valued function v in Lp(Ω; μ) = Lp(Ω; μ; Rn) such that for some sequence (ψi [member of] C∞(Ω)
[MATHEMATICAL EXPRESSION OMITTED]
and
[MATHEMATICAL EXPRESSION OMITTED]
as i -> ∞. The function v is called the gradient of u in H1,p(Ω; μ) and denoted by v = [nabla]u. Condition II implies that [nabla]u is a uniquely defined function in Lp(Ω; μ).
The space H1,p0(Ω; μ) is the closure of C∞(Ω) in H1,p(Ω; μ). It is clear that H1,p(Ω; μ) and H1,p0(Ω; μ) are Banach spaces under the norm ||x||1,p. Moreover, the norm ||x||1,p is uniformly convex and therefore the Sobolev spaces H1,p(Ω; μ) and H1,p0(Ω; μ) are reflexive (Yosida 1980, p. 127).
The corresponding local space H1,ploc(Ω; μ) is defined in the obvious manner: a function u is in H1,ploc(Ω; μ) if and only if u is in H1,p(Ω'; μ) for each open set Ω' [??] Ω. Note that for a function u [member of] H1,ploc(Ω; μ) the gradient [nabla]u is a well-defined function in Hploc(Ω; μ).
We alert the reader that the symbol [nabla]u stands for the gradient of u in a Sobolev space H1,ploc(Ω; μ); even for a C1-function u it is not a priori obvious that [nabla]u coincides with the usual gradient of u. We shall show later that they are equal (Lemma 1.11).
We also repeatedly invoke the Dirichlet spaces L1,p(Ω; μ) and L1,pO(Ω; μ):
[MATHEMATICAL EXPRESSION OMITTED]
and L1,p0(Ω; μ) is the closure of C∞(Ω) with respect to the seminorm
[MATHEMATICAL EXPRESSION OMITTED]
That is, L1,p0(Ω; μ) is the set of all functions u [member of] L1,p(Ω; μ) for which there exists a sequence ψj [member of] C∞(Ω) such that [nabla] ψj -> [nabla]u in Lp(Ω; μ).
As opposed to the standard Sobolev space H1,p(Ω; dx), an element in H1,p(Ω; μ) may have some peculiar features. For instance, a function in H1,p(Ω; μ) need not be locally integrable with respect to Lebesgue measure. To display a particular example, fix p > 1 and let w(x) = |x|p(n+1); then w is a p-admissible weight as discussed in Section 1.6. Now the function u(x) = |x|-n is in H1, plocRn; μ) and [nabla]u(x) = - nx|x|-n-2, but u is not locally integrable. In particular, there is no distribution in Rn that agrees with |x|-n in Rn \ {0}. The gradient of |x|-n above can be computed by using Lemma 1.11 and a truncation argument.
Sometimes a weighted Sobolev space is defined as the set of all locally Lebesgue integrable functions u such that u and its distributional gradient both belong to Lp(Ω; μ). Equipped with the norm ||u||1,p this produces a normed space which is not necessarily Banach as the example above shows. Consequently, this definition does not lead to the space H1,p(Ω; μ). However, if the weight w is in Ap, it can be shown that these two definitions give the same space (Kilpeläinen 1992b).
If we impose a mild additional condition on the weight w, each Sobolev function is a distribution. More precisely, if w1/(1-p) [member of] H1loc(Ω; dx), in particular if w [member of] Ap, then every Sobolev function u in H1,ploc(Ω; μ) is a distribution and [nabla]u is the distributional gradient of u; that is, u is locally Lebesgue integrable in Ω and
[MATHEMATICAL EXPRESSION OMITTED]
for all ψ [member of] C∞(Ω) and i = 1,2, ..., n. Here [partial derivative]iu is the ith coordinate of [nabla]u. To see this, first apply the Hölder inequality to u [member of] Lp(D; μ), D [??] Ω, and obtain
[MATHEMATICAL EXPRESSION OMITTED]
This implies that Lp(D; μ) is continuously embedded in L1(D; dx). Thus if (pj E C∞(Ω) converges in H1,p(Ω; μ) to u, then the sequences ψj and [partial derivative]iψj converge to u and [partial derivative]iu, respectively, in L1(D; dx) for all D [??] Ω, i = 1,2, ..., n. We have for all ψ [member of] C∞(Ω)
[MATHEMATICAL EXPRESSION OMITTED]
as j -> ∞. This proves that [nabla]u is the distributional gradient of u.
We prove in Lemma 1.11 that if u is a locally Lipschitz function in H1,p(Ω; μ), then [nabla]u is the distributional gradient of u.
1.10. Basic properties of Sobolev spaces
In the following few pages we demonstrate the basic properties of the Sobolev space H1,p(Ω; μ). The first fundamental fact to observe is that the Sobolev and Poincaré inequalities III, IV, and (1.5) hold for functions in H1,p0(B; μ), H1,p(B; μ) and H1,p0(Ω; μ), respectively.
(Continues…)
Excerpted from "Nonlinear Potential Theory of Degenerate Elliptic Equations"
by .
Copyright © 2006 Juha Heinonen, Tero Kilpeläinen, and Olli Martio.
Excerpted by permission of Dover Publications, Inc..
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