A Compact Embedding Theorem for Generalized Sobolev Spaces
aa r X i v : . [ m a t h . A P ] O c t Sept 24 A Compact Embedding Theoremfor Generalized Sobolev Spaces by Seng-Kee Chua, Scott Rodney and Richard L. Wheeden Abstract:
We give an elementary proof of a compact embedding theorem in abstract Sobolevspaces. The result is first presented in a general context and later specialized to the case ofdegenerate Sobolev spaces defined with respect to nonnegative quadratic forms on R n . Althoughour primary interest concerns degenerate quadratic forms, our result also applies to nondegener-ate cases, and we consider several such applications, including the classical Rellich-Kondrachovcompact embedding theorem and results for the class of s -John domains in R n , the latter forweights equal to powers of the distance to the boundary. We also derive a compactness resultfor Lebesgue spaces on quasimetric spaces unrelated to R n and possibly without any notion ofgradient. The main goal of this paper is to generalize the classical Rellich-Kondrachovtheorem concerning compact embedding of Sobolev spaces into Lebesgue spaces.Our principal result applies not only to the classical Sobolev spaces on open sets Ω ⊂ R n but also allows us to treat the degenerate Sobolev spaces defined in [SW2],and to obtain compact embedding of them into various L q (Ω) spaces. These de-generate Sobolev spaces are associated with quadratic forms Q ( x, ξ ) = ξ ′ Q ( x ) ξ , x ∈ Ω , ξ ∈ R n , which are nonnegative but may vanish identically in ξ for somevalues of x . Such quadratic forms and Sobolev spaces arise naturally in the studyof existence and regularity of weak solutions of some second order subelliptic lin-ear/quasilinear partial differential equations; see, e.g., [SW1, 2], [R1], [MRW],[RSW].The Rellich-Kondrachov theorem is frequently used to study the existence ofsolutions to elliptic equations, a famous example being subcritical and criticalYamabe equations, resulting in the solution of Yamabe’s problem; see [Y], [T], [A],[S]. Further applications lie in proving the existence of weak solutions to Dirichletproblems for elliptic equations with rough boundary data and coefficients; see [GT].In a sequel to this paper, we will apply our compact embedding results to studythe existence of solutions for some classes of degenerate equations. ept 24 w be a measure on a σ -algebra Σ of subsets of a set Ω , with Ω ∈ Σ . For < p ≤ ∞ , let L pw (Ω) denote the classof real-valued measurable functions f satisfying || f || L pw (Ω) < ∞ , where || f || L pw (Ω) = (cid:16) ´ Ω | f | p dw (cid:17) /p if p < ∞ and || f || L ∞ w (Ω) = ess sup Ω | f | , the essential supremumbeing taken with respect to w -measure. When dealing with generic functions in L pw (Ω) , we will not distinguish between functions which are equal a.e.- w . For E ∈ Σ , w ( E ) denotes the w -measure of E , and if < w ( E ) < ∞ then f E,w denotes the w -average of f over E : f E,w = w ( E ) ´ E f dw . Throughout the paper,positive constants will be denoted by C or c and their dependence on importantparameters will be indicated.For k ∈ N , let X (Ω) be a normed linear space of measurable R k -valued func-tions g defined on Ω with norm || g || X (Ω) . We assume that there is a subset Σ ⊂ Σ so that ( X (Ω) , Σ ) satisfies the following properties:(A) For any g ∈ X (Ω) and F ∈ Σ , the function g χ F ∈ X (Ω) , where χ F denotesthe characteristic function of F .( B p ) There are constants C , C , p satisfying ≤ C , C , p < ∞ so that if { F ℓ } isa finite collection of sets in Σ with X ℓ χ F ℓ ( x ) ≤ C for all x ∈ Ω , then X ℓ || g χ F ℓ || p X (Ω) ≤ C || g || p X (Ω) for all g ∈ X (Ω) .For ≤ N ≤ ∞ , we will often consider the product space L Nw (Ω) × X (Ω) .This is a normed linear space with norm || ( f, g ) || L Nw (Ω) × X (Ω) = || f || L Nw (Ω) + || g || X (Ω) . (1.1)A set S ⊂ L Nw (Ω) × X (Ω) will be called a bounded set in L Nw (Ω) × X (Ω) if sup ( f, g ) ∈S || ( f, g ) || L Nw (Ω) × X (Ω) < ∞ . Projection maps such as the one defined by π : ( f, g ) → f, ( f, g ) ∈ L Nw (Ω) × X (Ω) , (1.2) ept 24 w (Ω) < ∞ , then π ( L Nw (Ω) × X (Ω)) ⊂ L qw (Ω) if ≤ q ≤ N. Theorem 1.1.
Let w be a finite measure on a σ -algebra Σ of subsets of a set Ω ,with Ω ∈ Σ . Let ≤ p < ∞ , < N ≤ ∞ , X (Ω) be a normed linear spacesatisfying properties (A) and ( B p ) relative to a collection Σ ⊂ Σ , and let S be abounded set in L Nw (Ω) × X (Ω) .Suppose that S satisfies the following: given ǫ > , there are a finite numberof pairs { E ℓ , F ℓ } Jℓ =1 with E ℓ ∈ Σ and F ℓ ∈ Σ (the pairs and J may depend on ǫ )such that(i) w (cid:0) Ω \ ∪ ℓ E ℓ (cid:1) < ǫ and w ( E ℓ ) > ;(ii) { F ℓ } has bounded overlaps independent of ǫ with the same overlap constant asin ( B p ), i.e., J X ℓ =1 χ F ℓ ( x ) ≤ C , x ∈ Ω , (1.3) for C as in ( B p );(iii) for every ( f, g ) ∈ S , the local Poincaré-type inequality || f − f E ℓ ,w || L pw ( E ℓ ) ≤ ǫ || g χ F ℓ || X (Ω) (1.4) holds for each ( E ℓ , F ℓ ) .Let ˆ S be the set defined by ˆ S = (cid:8) f ∈ L Nw (Ω) : there exists { ( f j , g j ) } ∞ j =1 ⊂ S with f j → f a.e.- w (cid:9) . (1.5) Then ˆ S is compactly embedded in L qw (Ω) if ≤ q < N in the sense that for everysequence { f k } ⊂ ˆ S , there is a single subsequence { f k i } and a function f ∈ L Nw (Ω) such that f k i → f pointwise a.e.- w in Ω and in L qw (Ω) norm for ≤ q < N . Before proceeding with the proof of Theorem 1.1, we make several simpleobservations. First, in the definition of ˆ S , the property that f ∈ L Nw (Ω) fol-lows by Fatou’s lemma since the associated functions f j are bounded in L Nw (Ω) ,as S is bounded in L Nw (Ω) × X (Ω) by hypothesis. Fatou’s lemma also showsthat ˆ S is a bounded set in L Nw (Ω) . Moreover, since N > , if { f j } is boundedin L Nw (Ω) and f j → f a.e.- w , then ( f j ) E,w → f E,w for all E ∈ Σ ; in fact, inthis situation, by using Egorov’s theorem, we have ´ Ω f j ϕdw → ´ Ω f ϕdw for all ϕ ∈ L N ′ w (Ω) , /N + 1 /N ′ = 1 . ept 24 w ( E ℓ ) > in assumption (i) ensures that the aver-ages f E ℓ ,w in (1.4) are well-defined, it is not needed since we can discard any pair E ℓ , F ℓ with w ( E ℓ ) = 0 without affecting the inequality w (Ω \ ∪ E ℓ ) < ǫ or (1.3) and(1.4).Finally, since ˆ S contains the first component f of any pair ( f, g ) ∈ S , a simplecorollary of Theorem 1.1 is that the projection π defined in (1.2) is a compactmapping of S into L qw (Ω) , ≤ q < N , in the sense that for every sequence { ( f k , g k ) } ⊂ S , there is a subsequence { f k i } and a function f ∈ L Nw (Ω) such that f k i → f pointwise a.e.- w in Ω and in L qw (Ω) norm for ≤ q < N . Proof:
Let S satisfy the hypotheses and suppose { f k } k ∈ N ⊂ ˆ S . For each f k , usethe definition of ˆ S to choose a sequence { ( f jk , g jk ) } j ⊂ S with f jk → f k a.e.- w as j → ∞ . Since S is bounded in L Nw (Ω) × X (Ω) , there is M ∈ (0 , ∞ ) so that || ( f jk , g jk ) || L Nw (Ω) × X (Ω) ≤ M for all k and j . Also, as noted above, { f k } is boundedin L Nw (Ω) norm; in fact || f k || L Nw (Ω) ≤ M for the same constant M and all k .Since { f k } is bounded in L Nw (Ω) , then if < N < ∞ , it has a weakly convergentsubsequence, while if N = ∞ , it has a subsequence which converges in the weak-star topology. In either case, we relabel the subsequence as { f k } to preserve theindex. Fix ǫ > and let { E ℓ , F ℓ } Jℓ =1 satisfy the hypotheses of the theorem relativeto ǫ . Setting Ω ǫ = ∪ E ℓ , we have by assumption (i) that w (Ω \ Ω ǫ ) < ǫ. (1.6)Let us show that there is a positive constant C independent of ǫ so that X ℓ || f k − ( f k ) E ℓ ,w || pL pw ( E ℓ ) ≤ Cǫ p for all k . (1.7)Fix k and let ∆ denote the expression on the left side of (1.7). Since f jk − ( f jk ) E ℓ ,w → f k − ( f k ) E ℓ ,w a.e.- w as j → ∞ , Fatou’s lemma gives ∆ ≤ X ℓ lim inf j →∞ || f jk − ( f jk ) E ℓ ,w || pL pw ( E ℓ ) . Consequently, by using the Poincaré inequality (1.4) for S and superadditivity of lim inf , we obtain ∆ ≤ lim inf j →∞ X ℓ ǫ p || g jk χ F ℓ || p X (Ω) . ept 24 F ℓ have finite overlaps uniformly in ǫ , with the same overlapconstant C as in property ( B p ) of X (Ω) . Hence, by property ( B p ) applied to thelast expression together with boundedness of S , ∆ ≤ C ǫ p lim inf j →∞ || g jk || p X (Ω) ≤ C M p ǫ p . This proves (1.7) with C = C M p .Next note that ˆ Ω ǫ | f m − f k | p dw ≤ X ℓ ˆ E ℓ | f m − f k | p dw ≤ p − (cid:16) X ℓ ˆ E ℓ | f m − f k − ( f m − f k ) E ℓ ,w | p dw + X ℓ | ( f m − f k ) E ℓ ,w | p w ( E ℓ ) (cid:17) = 2 p − ( I + II ) . (1.8)We will estimate I and II separately. We have I ≤ p − X ℓ || f m − ( f m ) E ℓ ,w || pL pw ( E ℓ ) + X ℓ || f k − ( f k ) E ℓ ,w || pL pw ( E ℓ ) ! ≤ p − ( Cǫ p + Cǫ p ) = 2 p Cǫ p (1.9)by (1.7). To estimate II , first note that II = J X ℓ =1 | ( f m − f k ) E ℓ ,w | p w ( E ℓ ) = J X ℓ =1 w ( E l ) p − (cid:12)(cid:12)(cid:12) ˆ Ω ( f m − f k ) χ E ℓ dw (cid:12)(cid:12)(cid:12) p . Since w (Ω) < ∞ , each characteristic function χ E ℓ ∈ L N ′ w (Ω) , /N + 1 /N ′ = 1 (with N ′ = 1 if N = ∞ ). As { f k } converges weakly in L Nw (Ω) when < N < ∞ ,or converges in the weak-star sense when N = ∞ , then for m, k sufficiently largedepending on ǫ , and for all ≤ ℓ ≤ J , w ( E l ) p − (cid:12)(cid:12)(cid:12) ˆ Ω ( f m − f k ) χ E ℓ dw (cid:12)(cid:12)(cid:12) p ≤ ǫ p J . ept 24 II ≤ ǫ p for m, k sufficiently large depending on ǫ . Combining this estimatewith (1.8) and (1.9) shows that || f m − f k || L pw (Ω ǫ ) < Cǫ (1.10)for m, k sufficiently large and C = C ( M, C ) .Let us now show that { f k } is a Cauchy sequence in L w (Ω) . For m, k as in(1.10), Hölder’s inequality and the fact that || f k || L Nw (Ω) ≤ M for all k yield || f m − f k || L w (Ω) ≤ || f m − f k || L w (Ω ǫ ) + || f m − f k || L w (Ω \ Ω ǫ ) ≤ || f m − f k || L pw (Ω ǫ ) w (Ω ǫ ) p ′ + || f m − f k || L Nw (Ω \ Ω ǫ ) w (Ω \ Ω ǫ ) N ′ < Cǫw (Ω ǫ ) p ′ + 2 M w (Ω \ Ω ǫ ) N ′ < Cǫw (Ω) p ′ + 2 M ǫ N ′ by (1.6) . Since N ′ < ∞ , it follows that { f k } is Cauchy in L w (Ω) . Hence it has a subsequence(again denoted by { f k } ) that converges in L w (Ω) and pointwise a.e.- w in Ω to afunction f ∈ L w (Ω) . If N = ∞ , { f k } is bounded in L ∞ w (Ω) by hypothesis, so itspointwise limit f ∈ L ∞ w (Ω) . If N < ∞ , since { f k } is bounded in L Nw (Ω) , Fatou’sLemma implies that f ∈ L Nw (Ω) . This completes the proof in case q = 1 .For general q , we will use the same subsequence { f k } as above. Thus we onlyneed to show that { f k } converges in L qw (Ω) for < q < N . We will use Hölder’sinequality. Given q ∈ (1 , N ) , choose λ ∈ (0 , , namely λ = (cid:0) q − N (cid:1) / (cid:0) − N (cid:1) ,hence λ = 1 /q if N = ∞ , so that || f m − f k || L qw (Ω) ≤ || f m − f k || λL w (Ω) || f m − f k || − λL Nw (Ω) . (1.11)As before, || f k || L Nw (Ω) ≤ M , and therefore || f m − f k || − λL Nw (Ω) ≤ (2 M ) − λ , giving by(1.11) that { f k } is Cauchy in L qw (Ω) as it is Cauchy in L w (Ω) . This completes theproof of Theorem 1.1. (cid:3) A compact embedding result is also proved in [FSSC, Theorem 3.4] by usingPoincaré type estimates. However, Theorem 1.1 applies to situations not consid-ered in [FSSC] since it is not restricted to the context of Lipschitz vector fieldsin R n . Other abstract compact embedding results can be found in [HK1, Theo-rem 4] and [HK2, Theorem 8.1], including a version (see [HK1, Theorem 5]) forweighted Sobolev spaces with nonzero continuous weights, and a version in [HK2]for metric spaces with a single doubling measure. The proof in [HK1] assumesprior knowledge of the classical Rellich-Kondrachov compactness theorem (see e.g. ept 24 L Nw (Ω) to be precompact in L qw (Ω) , ≤ q < N ,without mentioning the sets { F ℓ } , the space X (Ω) , properties (A) and ( B p ), orconditions (1.3) and (1.4). We state this result in the next theorem. An applicationis given in §4. Theorem 1.2.
Let w be a finite measure on a σ -algebra Σ of subsets of a set Ω ,with Ω ∈ Σ . Let ≤ p < ∞ , < N ≤ ∞ and P be a bounded subset of L Nw (Ω) .Suppose there is a positive constant C so that for every ǫ > , there are a finitenumber of sets E ℓ ∈ Σ with(i) w (cid:0) Ω \ ∪ ℓ E ℓ (cid:1) < ǫ and w ( E ℓ ) > ;(ii) for every f ∈ P , X ℓ || f − f E ℓ ,w || pL pw ( E ℓ ) ≤ Cǫ p . (1.12) Let ˆ P = { f ∈ L Nw (Ω) : there exists { f j } ⊂ P with f j → f a.e. - w } . Then for every sequence { f k } ⊂ ˆ P , there is a single subsequence { f k i } and afunction f ∈ L Nw (Ω) such that f k i → f pointwise a.e.- w in Ω and in L qw (Ω) normfor ≤ q < N . Remark 1.3.
1. Given ǫ > , let { E ℓ } satisfy hypothesis (i) of Theorem 1.2.Hypothesis (ii) of Theorem 1.2 is clearly true for { E ℓ } if for every f ∈ P ,there are nonnegative constants { a ℓ } such that || f − f E ℓ ,w || L pw ( E ℓ ) ≤ ǫ a ℓ (1.13) and X a pℓ ≤ C (1.14) with C independent of f, ǫ . The constants { a ℓ } may vary with f and ǫ .2. Theorem 1.1 is a corollary of Theorem 1.2. To see why, suppose that thehypothesis of Theorem 1.1 holds. Define P by P = π ( S ) = { f : ( f, g ) ∈ S} .Let ǫ > and choose { ( E ℓ , F ℓ ) } as in Theorem 1.1. Given f ∈ P , choose any g such that ( f, g ) ∈ S and set a ℓ = || g χ F ℓ || X (Ω) for all ℓ . Then (1.4), (1.3)and property ( B p ) of X (Ω) imply (1.13) and (1.14). The preceding remarkshows that the hypothesis of Theorem 1.2 holds. The conclusion of Theorem1.1 now follows from Theorem 1.2. ept 24 Proof of Theorem 1.2:
Theorem 1.2 can be proved by checking through theproof of Theorem 1.1. In fact, the nature of hypothesis (1.12) allows simplificationof the proof. First recall that if f j → f a.e.- w and { f j } is bounded in L Nw (Ω) , then ( f j ) E,w → f E,w for every E ∈ Σ . Therefore, by the definition of ˆ P and Fatou’slemma, the truth of (1.12) for all f ∈ P implies its truth for all f ∈ ˆ P . Givena sequence { f k } in ˆ P , we follow the proof of Theorem 1.1 but no longer need tointroduce the { f jk } or prove (1.7) since (1.7) now follows from the fact that (1.12)holds for ˆ P . Further details are left to the reader.We close this section by listing an alternate version of Theorem 1.1 that we willuse in §3.4 when we consider local results. Theorem 1.4.
Let w be a measure (not necessarily finite) on a σ -algebra Σ ofsubsets of a set Ω , with Ω ∈ Σ . Let ≤ p < ∞ , < N ≤ ∞ , X (Ω) be a normedlinear space satisfying properties (A) and ( B p ) relative to a set Σ ⊂ Σ , and let S be a collection of pairs ( f, g ) such that f is Σ -measurable and g ∈ X (Ω) .Suppose that S satisfies the following conditions relative to a fixed set Ω ′ ∈ Σ (in particular Ω ′ ⊂ Ω ): for each ǫ = ǫ j = 1 /j with j ∈ N , there are a finite numberof pairs { E ǫℓ , F ǫℓ } ℓ with E ǫℓ ∈ Σ and F ǫℓ ∈ Σ such that(i) w (Ω ′ \ ∪ ℓ E ǫℓ ) = 0 and < w ( E ǫℓ ) < ∞ ;(ii) { F ǫℓ } ℓ has bounded overlaps independent of ǫ with the same overlap constantas in ( B p ), i.e., X ℓ χ F ǫℓ ( x ) ≤ C , x ∈ Ω , for C as in ( B p );(iii) for every ( f, g ) ∈ S , the local Poincaré-type inequality || f − f E ǫℓ ,w || L pw ( E ǫℓ ) ≤ ǫ || g χ F ǫℓ || X (Ω) holds for each ( E ǫℓ , F ǫℓ ) .Then for every sequence { ( f k , g k ) } in S with sup k h || f k || L Nw ( ∪ ℓ,j E /jℓ ) + || g k || X (Ω) i < ∞ , (1.15) there is a subsequence { f k i } of { f k } and a function f ∈ L Nw (Ω ′ ) such that f k i → f pointwise a.e.- w in Ω ′ and in L qw (Ω ′ ) norm for ≤ q ≤ p . If p < N , then also f k i → f in L qw (Ω ′ ) norm for ≤ q < N . ept 24 { E ǫℓ } willsatisfy Ω ′ ⊂ ∪ ℓ E ǫℓ for each ǫ , and consequently the condition in hypothesis (i) that w (Ω ′ \ ∪ ℓ E ǫℓ ) = 0 for each ǫ will be automatically true. Unlike Theorem 1.1, thevalue of q in Theorem 1.4 is always allowed to equal p . Although w (Ω) is notassumed to be finite in Theorem 1.4, w (Ω ′ ) < ∞ is true due to hypothesis (i)and the fact that the number of E ǫℓ is finite for each ǫ . As in Theorem 1.1, thehypothesis w ( E ǫℓ ) > is dispensible. Proof of Theorem 1.4:
The proof is like that of Theorem 1.1, with minorchanges and some simplifications. We work directly with the pairs ( f k , g k ) withoutconsidering approximations ( f jk , g jk ) . Due to the form of assumption (i) in Theorem1.4, neither the set Ω ǫ nor estimate (1.6) is now needed. Since w (Ω ′ \ ∪ ℓ E ǫℓ ) = 0 for each ǫ = 1 /j , we can replace Ω ǫ by Ω ′ in the proof, obtaining the estimate || f m − f k || L pw (Ω ′ ) < Cǫ (1.16)as an analogue of (1.10). In deriving (1.16), the weak and weak-star argumentsare guaranteed since by (1.15), sup k || f k || L Nw ( ∪ ℓ,j E /jℓ ) < ∞ . The main change in the proof comes by observing that the entire argument formerlyused to show that { f k } is Cauchy in L w (Ω) is no longer needed. In fact, (1.16)proves that { f k } is Cauchy in L pw (Ω ′ ) , and therefore it is also Cauchy in L qw (Ω ′ ) if ≤ q ≤ p since w (Ω ′ ) < ∞ . The first conclusion in Theorem 1.4 then follows. Toprove the second one, assuming that p, q < N , we use an analogue of (1.11) with Ω ′ in place of Ω and the same choice of λ , namely, || f m − f k || L qw (Ω ′ ) ≤ || f m − f k || λL w (Ω ′ ) || f m − f k || − λL Nw (Ω ′ ) . The desired conclusion then follows as before since we have already shown thatthe first factor on the right side tends to . Roughly speaking, a consequence of Theorem 1.1 is that a set of functions whichis bounded in L Nw (Ω) is precompact in L qw (Ω) for ≤ q < N if the gradients of thefunctions are bounded in an appropriate norm, and a local Poincaré inequality holds ept 24 L Nw (Ω) will be fulfilled if, for example,the functions satisfy a global Poincaré or Sobolev estimate with exponent N on theleft-hand side. In order to illustrate this principle more precisely, we first considerthe classical gradient operator and functions on R n with the standard Euclideanmetric. We include a simple way to see that the Rellich-Kondrachov compactnesstheorem follows from our results. Our derivation of this fact is different from thosein [AF] and [GT]; in particular, it avoids using the Arzelá-Ascoli theorem andregularization of functions by convolution. We also list compactness results for thespecial class of s -John domains in R n . In [HK1], the authors mention that suchresults follow from their development without giving specific statements. See also[HK2, Theorem 8.1]. We list results for degenerate quadratic forms and vectorfields in Section 3.We begin by proving a compact embedding result for some Sobolev spacesinvolving two measures. Let w be a measure on the Borel subsets of a fixed openset Ω ⊂ R n , and let µ be a measure on the σ -algebra of Lebesgue measurablesubsets of Ω . We also assume that µ is absolutely continuous with respect toLebesgue measure. If ≤ p < ∞ , let E pµ (Ω) denote the class of locally Lebesgueintegrable functions on Ω with distributional derivatives in L pµ (Ω) . If ≤ N ≤ ∞ ,we say that a set Y ⊂ L Nw (Ω) ∩ E pµ (Ω) (intersection of function spaces instead ofnormed spaces of equivalence classes) is bounded in L Nw (Ω) ∩ E pµ (Ω) if sup f ∈ Y (cid:8) || f || L Nw (Ω) + ||∇ f || L pµ (Ω) (cid:9) < ∞ . We use D to denote a generic open Euclidean ball. The radius and center of D will be denoted r ( D ) and x D , and if C is a positive constant, CD will denotethe ball concentric with D whose radius is Cr ( D ) . Theorem 2.1.
Let ˜Ω ⊂ Ω be open sets in R n . Let w be a Borel measure on Ω with w ( ˜Ω) = w (Ω) < ∞ and µ be a measure on the Lebesgue measurable sets in Ω which is absolutely continuous with respect to Lebesgue measure. Let ≤ p < ∞ , < N ≤ ∞ and S ⊂ L Nw (Ω) ∩ E pµ (Ω) , and suppose that for all ǫ > , there exists δ ǫ > such that k f − f D,w k L pw ( D ) ≤ ǫ k∇ f k L pµ ( D ) for all f ∈ S (2.1) and all Euclidean balls D with r ( D ) < δ ǫ and D ⊂ ˜Ω . Then for any sequence { f k } ⊂ S that is bounded in L Nw (Ω) ∩ E pµ (Ω) , there is a subsequence { f k i } anda function f ∈ L Nw (Ω) such that { f k i } → f pointwise a.e.- w in Ω and in L qw (Ω) norm for ≤ q < N . ept 24 ˜Ω and w with w ( ˜Ω) = w (Ω) < ∞ . For any two nonempty sets E , E ⊂ R n , let ρ ( E , E ) = inf {| x − y | : x ∈ E , y ∈ E } (2.2)denote the Euclidean distance between E and E . If x ∈ R n and E is a nonemptyset, we will write ρ ( x, E ) instead of ρ ( { x } , E ) . Let ˜Ω be an open subset of Ω . If Ω is bounded and Ω \ ˜Ω has Lebesgue measure , the measure w on Ω defined by dw = ρ ( x, R n \ ˜Ω) α dx clearly has the desired properties if α ≥ . The range of α canbe increased to α > − if Ω is a Lipschitz domain and Ω \ ˜Ω is a finite set. Indeed, if ∂ Ω is described in local coordinates x = ( x , . . . , x n ) by x n = F ( x , . . . , x n − ) with F Lipschitz, then the distance from x to ∂ Ω is equivalent to | x n − F ( x , . . . , x n − ) | ,and consequently the restriction α > − guarantees that w is finite near ∂ Ω byusing Fubini’s theorem; see also [C1, Remark 3.4(b)]. If Ω is bounded and Ω \ ˜Ω is finite, but with no restriction on ∂ Ω , the range can clearly be further increasedto α > − n for the measure ρ ( x, Ω \ ˜Ω) α dx . Also note that any w without pointmasses satisfies w ( ˜Ω) = w (Ω) if ˜Ω is obtained by deleting a countable subset of Ω . Proof of Theorem 2.1:
We will verify the hypotheses of Theorem 1.1. Let X (Ω) = (cid:8) g = ( g , . . . , g n ) : | g | = (cid:0) n X i =1 g i (cid:1) / ∈ L pµ (Ω) (cid:9) and || g || X (Ω) = || g || L pµ (Ω) . Then k∇ f k X (Ω) = k∇ f k L pµ (Ω) if f ∈ E pµ (Ω) .If f ∈ E pµ (Ω) , we may identify f with the pair ( f, ∇ f ) since the distributionalgradient ∇ f is uniquely determined by f up to a set of Lebesgue measure zero.Then L Nw (Ω) ∩ E pµ (Ω) can be viewed as a subset of L Nw (Ω) × X (Ω) . In Theorem 1.1,choose S to be the particular sequence { f k } ⊂ S in the hypothesis of Theorem2.1, and choose Σ to be the Lebesgue measurable subsets of Ω and Σ to be thecollection of balls D ⊂ Ω . Then hypotheses (A) and ( B p ) are valid with C = C for any C . Given ǫ > , since w ( ˜Ω) = w (Ω) < ∞ , there is a compact set K ⊂ ˜Ω with w (Ω \ K ) < ǫ . Let < δ ′ ǫ < ρ ( K, R n \ ˜Ω) (where ρ ( K, R n \ ˜Ω) is interpretedas ∞ if ˜Ω = R n ), let δ ǫ be as in (2.1), and fix r ǫ with < r ǫ < min { δ ǫ , δ ′ ǫ } . Byconsidering the triples of balls in a maximal collection of pairwise disjoint ballsof radius r ǫ / centered in K , we obtain a collection { E ǫℓ } ℓ of balls of radius r ǫ / which satisfy E ǫℓ ⊂ ˜Ω , have bounded overlaps with overlap constant independent ept 24 ǫ , and whose union covers K . Since K is compact, we may assume the collectionis finite. Also, w (cid:0) Ω \ ∪ ℓ E ǫℓ (cid:1) ≤ w (cid:0) Ω \ K (cid:1) < ǫ, and (1.4) holds with F ℓ = E ℓ = E ǫℓ by (2.1). Theorem 2.1 now follows fromTheorem 1.1 applied to Ω .In particular, we obtain the following result when w = µ is a Muckenhoupt A p ( R n ) weight, i.e., when dµ = dw = η dx where η ( x ) satisfies (cid:18) | D | ˆ D η dx (cid:19) (cid:18) | D | ˆ D η − / ( p − dx (cid:19) p − ≤ C if < p < ∞ and | D | − ´ D η dx ≤ C essinf D w if p = 1 for all Euclidean balls D , with C independent of D . As is well known, such a weight also satisfies theclassical doubling condition w ( D r ( x )) ≤ C (cid:16) rr ′ (cid:17) θ w ( D r ′ ( x )) , < r ′ < r < ∞ , (2.3)with θ ≥ np − ǫ for some ǫ > if p > , and with θ = n if p = 1 , where C and θ are independent of r, r ′ , x .We denote by W ,p,w (Ω) the weighted Sobolev space defined as all functionsin L pw (Ω) whose distributional gradient is in L pw (Ω) . Thus W ,p,w (Ω) = L pw (Ω) ∩ E pw (Ω) . If w (Ω) < ∞ , it follows that L Nw (Ω) ∩ E pw (Ω) ⊂ W ,p,w (Ω) when N ≥ p ,and that the opposite containment holds when N ≤ p . Theorem 2.2.
Let ≤ p < ∞ , w ∈ A p ( R n ) and Ω be an open set in R n with w (Ω) < ∞ . If < N ≤ ∞ , then any bounded subset of L Nw (Ω) ∩ E pw (Ω) isprecompact in L qw (Ω) if ≤ q < N . Consequently, if N > p and S is a subset of W ,p,w (Ω) with k f k L Nw (Ω) ≤ C ( k f k L pw (Ω) + k∇ f k L pw (Ω) ) for all f ∈ S , (2.4) then any set in S that is bounded in W ,p,w (Ω) is precompact in L qw (Ω) for ≤ q < N .If Ω is a John domain, there exists N > p ( N can be θp/ ( θ − p ) for some θ > p as described after (2.3)) such that W ,p,w (Ω) is compactly embedded in L qw (Ω) for ≤ q < N . In particular, the embedding of W ,p,w (Ω) into L pw (Ω) is compact when w ∈ A p ( R n ) and Ω is a John domain. ept 24 Remark 2.3.
When w = 1 and p < n , the choices N = np/ ( n − p ) and S = W ,p (Ω) guarantee (2.4) by the classical Sobolev inequality for functions in W ,p (Ω) (see e.g. [GT, Theorem 7.10]); here W ,p (Ω) denotes the closure in W ,p (Ω) of theclass of Lipschitz functions with compact support in Ω . Consequently, the classicalRellich-Kondrachov theorem giving the compact embedding of W ,p (Ω) in L q (Ω) for ≤ q < np/ ( n − p ) follows as a special case of the first part of Theorem 2.2. Proof.
We will apply Theorem 2.1 with w = µ . Fix p and w with ≤ p < ∞ and w ∈ A p ( R n ) . By [FKS], there is a constant C such that the weighted Poincaréinequality || f − f D,w || L pw ( D ) ≤ Cr ( D ) ||∇ f || L pw ( D ) , f ∈ C ∞ (Ω) , holds for all Euclidean balls D ⊂ Ω . Then since C ∞ (Ω) is dense in L Nw (Ω) ∩ E pw (Ω) if ≤ N < ∞ (see e.g. [Tur]), by fixing any ǫ > we obtain from Fatou’s lemmathat for all balls D ⊂ Ω with Cr ( D ) ≤ ǫ , || f − f D,w || L pw ( D ) ≤ ǫ ||∇ f || L pw ( D ) if f ∈ L Nw (Ω) ∩ E pw (Ω) . The same holds when N = ∞ since L ∞ w (Ω) = L ∞ (Ω) ⊂ L pw (Ω) due to the assump-tions w ∈ A p ( R n ) and w (Ω) < ∞ . With < N ≤ ∞ , the first statement of thetheorem now follows from Theorem 2.1, and the second statement is a corollary ofthe first one.Next, let Ω be a John domain. Choose θ > p so that w satisfies (2.3) and define N = θp/ ( θ − p ) . Then N > p and by [CW1, Theorem 1.8 (b) or Theorem 4.1], || f − f Ω ,w || L Nw (Ω) ≤ C ||∇ f || L pw (Ω) , ∀ f ∈ C ∞ (Ω) . Again, the inequality remains true for functions in W ,p,w (Ω) by density and Fa-tou’s lemma. It is now clear that (2.4) holds, and the last part of the theoremfollows.Our next example involves domains in R n which are more restricted. For special Ω , there are values N > such that k f k L N (Ω) ≤ C (cid:0) k f k L (Ω) + k∇ f k L p (Ω) (cid:1) (2.5)for all f ∈ L (Ω) ∩ E p (Ω) . Note that if Ω has finite Lebesgue measure, then W ,p (Ω) ⊂ L (Ω) ∩ E p (Ω) . As we will explain, (2.5) is true for some N > if Ω is an s -John domain in R n and ≤ s < pn − . Recall that for ≤ s < ∞ , a ept 24 Ω ⊂ R n is called an s -John domain with central point x ′ ∈ Ω iffor some constant c > and all x ∈ Ω with x = x ′ , there is a curve Γ : [0 , l ] → Ω so that Γ(0) = x, Γ( l ) = x ′ , | Γ( t ) − Γ( t ) | ≤ t − t for all [ t , t ] ⊂ [0 , l ] , and ρ (Γ( t ) , Ω c ) ≥ c t s for all t ∈ [0 , l ] .The terms -John domain and John domain are the same. When Ω is an s -Johndomain for some s ∈ [1 , p/ ( n − , it is shown in [KM], [CW1], [CW2] that(2.5) holds for all finite N with N ≥ s ( n − − p + 1 np (2.6)and for all f ∈ W ,p (Ω) without any support restrictions. Note that the right sideof (2.6) is strictly less than /p for such s , and consequently there are values N > p which satisfy (2.6). For N as in (2.6), the global estimate || f − f Ω || L N (Ω) ≤ C ||∇ f || L p (Ω) , f Ω = ˆ Ω f ( x ) dx/ | Ω | , (2.7)is shown to hold if f ∈ Lip loc (Ω) in [CW2], and then follows for all f ∈ L (Ω) ∩ E p (Ω) ; see the proof of Theorem 2.4 for related comments. Inequality (2.5) isclearly a consequence of (2.7).More generally, weighted versions of (2.7) hold for s -John domains and lead toweighted compactness results, as we now show. Let ≤ p < ∞ , and for real α and ρ ( x, Ω c ) as in (2.2), let L pρ α dx (Ω) be the class of Lebesgue measurable f on Ω with || f || L pραdx (Ω) = (cid:18) ˆ Ω | f ( x ) | p ρ ( x, Ω c ) α dx (cid:19) /p < ∞ . Theorem 2.4.
Suppose that ≤ s < ∞ and Ω is an s -John domain in R n . Let p, a, b satisfy ≤ p < ∞ , a ≥ , b ∈ R and b − a < p .(i) If n + a > s ( n − b ) − p + 1 , (2.8) then for any ≤ q < ∞ such that q > max (cid:26) p − n , s ( n − b ) − p + 1( n + a ) p (cid:27) , (2.9) ept 24 L ρ a dx (Ω) ∩ E pρ b dx (Ω) is compactly embedded in L qρ a dx (Ω) .(ii) If p > and n + ap > s ( n − b ) − p + 1 ≥ n + a, (2.10) then for any ≤ q < ∞ such that aq > max (cid:26) bp − , s ( n − b ) − p − n + 1 p (cid:27) , (2.11) L ρ a dx (Ω) ∩ E pρ b dx (Ω) is compactly embedded in L qρ a dx (Ω) . Remark 2.5.
1. If a = b = 0 , (2.8) is the same as s < pn − . If a = 0 ,(2.10) never holds.2. The requirement that b − a < p follows from (2.8) and (2.9) by consideringthe cases n − b ≥ and n − b < separately. Hence b − a < p automaticallly holds in part (i), but it is an assumption in part (ii). Also,(2.10) and (2.11) imply that q < p , and consequently that p > .3. Conditions (2.8) and (2.9) imply there exists N ∈ ( p, ∞ ) with q > N > max (cid:26) p − n , s ( n − b ) − p + 1( n + a ) p (cid:27) . (2.12) Conversely, (2.8) holds if there exists N ∈ ( p, ∞ ) so that (2.12) holds.4. Assumption (2.11) ensures that there exists N ∈ ( q, ∞ ) such that (2.11)holds with q replaced by N . Proof:
This result is also a consequence of Theorem 2.1, but we will deduce itfrom Theorem 1.1 by using arguments like those in the proofs of Theorems 2.1and 2.2. Fix a, b, p, q as in the hypothesis and denote ρ ( x ) = ρ ( x, Ω c ) . Choose w = ρ a dx and note that w (Ω) < ∞ since a ≥ and Ω is now bounded. Define X (Ω) = (cid:8) g = ( g , . . . , g n ) : | g | ∈ L pρ b dx (Ω) (cid:9) and || g || X (Ω) = || g || L pρadx (Ω) . Fix ǫ > and choose a compact set K ⊂ Ω with | Ω \ K | ρ a dx := ´ Ω \ K ρ a dx < ǫ . Also choose δ ′ ǫ with < δ ′ ǫ < ρ ( K, Ω c ) , where ρ ( K, Ω c ) is the Euclidean distance between K and Ω c . ept 24 D is a Euclidean ball with center x D ∈ K and r ( D ) < δ ′ ǫ , then D ⊂ Ω and ρ ( x ) is essentially constant on D ; in fact, for such D , ρ ( x D ) ≤ ρ ( x ) ≤ ρ ( x D ) , x ∈ D. We claim that for such D , the simple unweighted Poincaré estimate || f − f D || L p ( D ) ≤ Cr ( D ) ||∇ f || L p ( D ) , f ∈ Lip loc (Ω) , where f D = f D,dx , implies that for f ∈ Lip loc (Ω) , || f − f D,ρ a dx || L pρadx ( D ) ≤ ˜ C (cid:0) r ( D ) a − bp + diam(Ω) a − bp (cid:1) r ( D ) ||∇ f || L pρbdx ( D ) , (2.13)where f D,ρ a dx = ´ D f ρ a dx/ ´ D ρ a dx and ˜ C depends on C, a, b but is independent of
D, f . To show this, first note that for such D , since ρ ∼ ρ ( x D ) on D , the simplePoincaré estimate immediately gives || f − f D || L pρadx ( D ) ≤ ˜ Cρ ( x D ) a − bp r ( D ) ||∇ f || L pρbdx ( D ) , f ∈ Lip loc (Ω) , and then a similar estimate with f D replaced by f D,ρ a dx follows by standard argu-ments. Clearly (2.13) will now follow if we show that ρ ( x D ) a − bp ≤ r ( D ) a − bp + diam(Ω) a − bp for such D .However, this is clear since r ( D ) ≤ ρ ( x D ) ≤ diam (Ω) for D as above, and (2.13)is proved.We can now apply the weighted density result of [H], [HK1] to conclude that(2.13) holds for all f ∈ L ρ a dx (Ω) ∩ E pρ b dx (Ω) and all balls D with x D ∈ K and r ( D ) < δ ′ ǫ .Recall that a − bp + 1 > . Thus there exists r ǫ with < r ǫ < δ ′ ǫ and ˜ C (cid:0) r a − bp ǫ + diam(Ω) a − bp (cid:1) r ǫ < ǫ. Let Σ and Σ be as in the proof of Theorem 2.1, and let { E ℓ } ℓ = { F ℓ } ℓ be thetriples of balls in a maximal collection of pairwise disjoint balls centered in K with radius r ǫ . Then (2.13) and the choice of r ǫ give the desired version of (1.4),namely || f − f D,ρ a dx || L pρadx ( D ) ≤ ǫ ||∇ f || L pρbdx ( D ) ept 24 D = E ℓ and f ∈ L ρ a dx (Ω) ∩ E pρ b dx (Ω) . Next, use the last two parts of Remark2.5 to choose N ∈ ( q, ∞ ) so that either (2.9) or (2.11) holds with q there replacedby N . Every f ∈ L ρ a dx (Ω) ∩ E pρ b dx (Ω) then satisfies the global Poincaré estimate || f − f Ω ,ρ a dx || L Nρadx (Ω) ≤ C ||∇ f || L pρbdx (Ω) , f ∈ L ρ a dx (Ω) ∩ E pρ b dx (Ω) , (2.14)where f Ω ,ρ a dx = ´ Ω f ρ a dx/ ´ Ω ρ a dx . In fact, under the hypothesis of Theorem 2.4,this is proved for f ∈ Lip loc (Ω) ∩ L ρ a dx (Ω) ∩ E pρ b dx (Ω) in [CW2] for example, andthen follows for all f ∈ L ρ a dx (Ω) ∩ E pρ b dx (Ω) by the density result of [H], [HK1] andFatou’s lemma. By (2.14), || f || L Nρadx (Ω) ≤ C || f || L ρadx (Ω) + C ||∇ f || L pρbdx (Ω) for the same class of f . The remaining details of the proof are left to the reader.In passing, we mention that the role played by the distance function ρ ( x, Ω c ) in Theorem 2.4 can instead be played by ρ ( x ) = inf {| x − y | : y ∈ Ω } , x ∈ Ω , for certain Ω ⊂ Ω c ; see [CW2, Theorem 1.6] for a description of such Ω and therequired Poincaré estimate, and note that the density result in [HK1] holds forpositive continuous weights. In this section, Ω denotes a fixed open set in R n , possibly unbounded. For ( x, ξ ) ∈ Ω × R n , we consider a nonnegative quadratic form ξ ′ Q ( x ) ξ which maydegenerate, i.e., which may vanish for some ξ = 0 . Such quadratic forms occurnaturally in the context of subelliptic equations and give rise to degenerate Sobolevspaces as discussed below. Our goal is to apply Theorem 1.1 to obtain compactembedding of these degenerate spaces into Lebesgue spaces related to the gain inintegrability provided by Poincaré-Sobolev inequalities. The framework that wewill use contains the subelliptic one developed in [SW1, 2], where regularity theoryfor weak solutions of linear subelliptic equations of second order in divergence formis studied. ept 24 We now list some notation and assumptions that will be in force everywhere in §3even when not explicitly mentioned.
Definition 3.1.
A function d is called a finite symmetric quasimetric (or simplya quasimetric) on Ω if d : Ω × Ω → [0 , ∞ ) and there is a constant κ ≥ such thatfor all x, y, z ∈ Ω , d ( x, y ) = d ( y, x ) ,d ( x, y ) = 0 ⇐⇒ x = y, and d ( x, y ) ≤ κ [ d ( x, z ) + d ( z, y )] . (3.1)If d is a quasimetric on Ω , we refer to the pair (Ω , d ) as a quasimetric space.In some applications, d is closely related to Q ( x ) . For example, d is sometimeschosen to be the Carnot-Carathéodory control metric related to Q ; cf. [SW1].Given x ∈ Ω , r > , and a quasimetric d , the subset of Ω defined by B r ( x ) = { y ∈ Ω : d ( x, y ) < r } will be called the quasimetric d -ball centered at x of radius r . Note that every d -ball B = B r ( x ) satisfies B ⊂ Ω by definition.It is sometimes possible, and desirable in case the boundary of Ω is rough, tobe able to work only with d -balls that are deep inside Ω in the sense that theirEuclidean closures B lie in Ω . See part (ii) of Remark 3.6 for comments aboutbeing able to use such balls.Recall that D s ( x ) denotes the ordinary Euclidean ball of radius s centered at x . We always assume that d is related as follows to the standard Euclidean metric: ∀ x ∈ Ω and r > , ∃ s = s ( x, r ) > so that D s ( x ) ⊂ B r ( x ) . (3.2) Remark 3.2.
Condition (3.2) is clearly true if d -balls are open, and it is weakerthan the well-known condition of C. Fefferman and Phong stating that for eachcompact K ⊂ Ω , there are constants β, r > such that D r β ( x ) ⊂ B r ( x ) for all x ∈ K and < r < r . Throughout §3, Q ( x ) denotes a fixed Lebesgue measurable n × n nonnega-tive symmetric matrix on Ω and we assume that every d -ball B centered in Ω isLebesgue measurable. We will deal with three locally finite measures w, ν, µ onthe Lebesgue measurable subsets of Ω , each with a particular role. In §3.3, where ept 24 w (Ω) < ∞ but this assumptionis not required for the local results of §3.4. The measure µ is assumed to be abso-lutely continuous with respect to Lebesgue measure; the comment following (3.4)explains why this assumption is natural. In §3, we sometimes assume that w isabsolutely continuous with respect to ν , but we drop this assumption completelyin the Appendix.We do not require the existence of a doubling measure for the collection of d -balls, but we always assume that (Ω , d ) satisfies the weaker local geometric doublingproperty given in the next definition; see [HyM] for a global version. Definition 3.3.
A quasimetric space (Ω , d ) satisfies the local geometric doublingcondition if for every compact K ⊂ Ω , there exists δ ′ = δ ′ ( K ) > such that for all x ∈ K and all < r ′ < r < δ ′ , the number of disjoint d -balls of radius r ′ containedin B r ( x ) is at most a constant C r/r ′ depending on r/r ′ but not on K . W ,pν,µ (Ω , Q ) , W ,pν,µ, (Ω , Q ) We will define weighted degenerate Sobolev spaces by using an approach like theone in [SW2] for the unweighted case. We first define an appropriate space ofvectors, including vectors which will eventually play the role of gradients, wheresize is measured relative to the nonnegative quadratic form Q ( x, ξ ) = ξ ′ Q ( x ) ξ, ( x, ξ ) ∈ Ω × R n . For ≤ p < ∞ , consider the collection of measurable R n -valued functions ~g ( x ) =( g ( x ) , ..., g n ( x )) satisfying || ~g || L pµ (Ω ,Q ) = n ˆ Ω Q ( x, ~g ( x )) p dµ o p = n ˆ Ω | p Q ( x ) ~g ( x ) | p dµ o p < ∞ . (3.3)We identify any two functions ~g, ~h in the collection for which || ~g − ~h || L pµ (Ω ,Q ) = 0 .Then (3.3) defines a norm on the resulting space of equivalence classes. The form-weighted space L pµ (Ω , Q ) is defined to be the collection of these equivalence classes,with norm (3.3). By using methods similar to those in [SW2], it follows that L µ (Ω , Q ) is a Hilbert space and L pµ (Ω , Q ) is a Banach space for ≤ p < ∞ .Now consider the (possibly infinite) norm on Lip loc (Ω) defined by || f || W ,pν,µ (Ω ,Q ) = || f || L pν (Ω) + ||∇ f || L pµ (Ω ,Q ) . (3.4) ept 24 µ ( Z ) = 0 when Z hasLebesgue measure assures that ||∇ f || L pµ (Ω ,Q ) is well-defined if f ∈ Lip loc (Ω) ;in fact, for such f , the Rademacher-Stepanov theorem implies that ∇ f exists a.e.in Ω with respect to Lebesgue measure. Definition 3.4.
Let ≤ p < ∞ .1. The degenerate Sobolev space W ,pν,µ (Ω , Q ) is the completion under the norm(3.4) of the set Lip
Q,p (Ω) =
Lip
Q,p,ν,µ (Ω) = { f ∈ Lip loc (Ω) : || f || W ,pν,µ (Ω ,Q ) < ∞} .
2. The degenerate Sobolev space W ,pν,µ, (Ω , Q ) is the completion under the norm(3.4) of the set Lip
Q,p, (Ω) = Lip (Ω) ∩ Lip
Q,p (Ω) , where
Lip (Ω) denotes thecollection of Lipschitz functions with compact support in Ω . If Q ∈ L p/ loc (Ω) ,then Lip
Q,p, (Ω) = Lip (Ω) since ν and µ are locally finite. We now make some comments about W ,pν,µ (Ω , Q ) , most of which have analoguesfor W ,pν,µ, (Ω , Q ) . By definition, W ,pν,µ (Ω , Q ) is the Banach space of equivalenceclasses of Cauchy sequences of Lip
Q,p (Ω) functions with respect to the norm (3.4).Given a Cauchy sequence { f j } of Lip
Q,p (Ω) functions, we denote its equivalenceclass by [ { f j } ] . If { v j } ∈ [ { f j } ] , then { v j } is a Cauchy sequence in L pν (Ω) and {∇ v j } is a Cauchy sequence in L pµ (Ω , Q ) . Hence, there is a pair ( f, ~g ) ∈ L pν (Ω) × L pµ (Ω , Q ) so that || v j − f || L pν (Ω) → and ||∇ v j − ~g || L pµ (Ω ,Q ) → as j → ∞ . The pair ( f, ~g ) is uniquely determined by the equivalence class [ { f j } ] ,i.e., is independent of a particular { v j } ∈ [ { f j } ] . We will say that ( f, ~g ) is repre-sented by { v j } . We obtain a Banach space isomorphism J from W ,pν,µ (Ω , Q ) ontoa closed subspace W ,pν,µ (Ω , Q ) of L pν (Ω) × L pµ (Ω , Q ) by setting J ([ { f j } ]) = ( f, ~g ) . (3.5)We will often not distinguish between W ,pν,µ (Ω , Q ) and W ,pν,µ (Ω , Q ) . Similarly, W ,pν,µ, (Ω , Q ) will denote the image of W ,pν,µ, (Ω , Q ) under J , but we often considerthese spaces to be the same.It is important to think of a typical element of W ,pν,µ (Ω , Q ) , or W ,pν,µ (Ω , Q ) ,as a pair ( f, ~g ) as above, and not simply as the first component f . In fact, if ept 24 ( f, ~g ) ∈ W ,pν,µ (Ω , Q ) , the vector ~g may not be uniquely determined by f ; see [FKS,Section 2.1] for a well known example.If f ∈ Lip
Q,p (Ω) , then the pair ( f, ∇ f ) may be viewed as an element of W ,pν,µ (Ω , Q ) by identifying it with the equivalence class [ { f } ] corresponding to thesequence each of whose entries is f . When viewed as a class, ( f, ∇ f ) generallycontains pairs whose first components are not Lipschitz functions; for example, if f ∈ Lip
Q,p (Ω) and F is any function with F = f a.e.- ν , then ( f, ∇ f ) = ( F, ∇ f ) in W ,pν,µ (Ω , Q ) . However, in what follows, when we consider a pair ( f, ∇ f ) with f ∈ Lip
Q,p (Ω) , we will not adopt this point of view. Instead we will identify an f ∈ Lip
Q,p (Ω) with the single pair ( f, ∇ f ) whose first component is f (definedeverywhere in Ω ) and whose second component is ∇ f , which exists a.e. with re-spect to Lebesgue measure by the Rademacher-Stepanov theorem. This conventionlets us avoid assuming that w is absolutely continuous with respect to ν , written w << ν , in Poincaré-Sobolev estimates for Lip
Q,p (Ω) functions. We will reservethe notation H for subsets of Lip
Q,p (Ω) viewed in this way.On the other hand, W will denote various subsets of W ,pν,µ (Ω , Q ) with elementsviewed as equivalence classes. When our hypotheses are phrased in terms of such W , we will assume that w << ν in order to avoid technical difficulty associatedwith sets of measure ; see the comment after (3.18). In the Appendix, we dropthe assumption w << ν altogether.We will abuse the notation (3.4) by writing || ( f, ∇ f ) || W ,pν,µ (Ω ,Q ) = || f || L pν (Ω) + ||∇ f || L pµ (Ω ,Q ) , f ∈ Lip
Q,p (Ω) , (3.6)and we extend this to generic ( f, ~g ) ∈ W ,pν,µ (Ω , Q ) by writing || ( f, ~g ) || W ,pν,µ (Ω ,Q ) = || f || L pν (Ω) + || ~g || L pµ (Ω ,Q ) . (3.7) In this section, we state and prove compactness results which apply to the entireset Ω . Results which are more local are given in §3.4.In order to apply Theorem 1.1 in this setting, we will use the following versionof Poincaré’s inequality for d -balls. Definition 3.5.
Let ≤ p < ∞ , Lip
Q,p (Ω) be is as in Definition 3.4, and
H ⊂
Lip
Q,p (Ω) . We say that the
Poincaré property of order p holds for H if there is aconstant c ≥ so that for every ǫ > and every compact set K ⊂ Ω , there exists ept 24 δ = δ ( ǫ, K ) > such that for all f ∈ H and every d -ball B r ( y ) with y ∈ K and < r < δ , (cid:18) ˆ B r ( y ) | f − f B r ( y ) ,w | p dw (cid:19) p ≤ ǫ || ( f, ∇ f ) || W ,pν,µ ( B c r ( y ) ,Q ) . (3.8) Remark 3.6. (i) Inequality (3.8) is not of standard Poincaré form. A more typicalform is (cid:18) w ( B r ( y )) ˆ B r ( y ) | f − f B r ( y ) ,w | p dw (cid:19) p ≤ Cr µ ( B c r ( y )) ˆ B c r ( y ) | p Q ∇ f | p dµ ! p . (3.9) In [SW1, 2] and [R1], the unweighted version of (3.9) with p = 2 is used. Let ρ ( x, ∂ Ω) and ρ ( E, ∂ Ω) be as in (2.2). In [SW2], the unweighted form of (3.9) with p = 2 is assumed for all f ∈ Lip Q, (Ω) and all B r ( y ) with y ∈ Ω and < r <δ ρ ( y, ∂ Ω) for some δ ∈ (0 , independent of y, r . If K is a compact set in Ω ,this version would then hold for all B r ( y ) with y ∈ K and < r < δ ρ ( K, ∂ Ω) .For general p, w and µ , if for every compact K ⊂ Ω , (3.9) is valid for all B r ( y ) with y ∈ K and < r < δ ρ ( K, ∂ Ω) , then (3.8) follows easily provided lim r → (cid:26) sup y ∈ K r p w ( B r ( y )) µ ( B c r ( y )) (cid:27) = 0 (3.10) for every compact K ⊂ Ω . Note that (3.10) automatically holds if w = µ .If both (3.9) and (3.10) hold, then (3.8) is true for any choice of ν . In thissituation, one can pick ν = w in order to avoid technicalities encountered belowwhen w is not absolutely continuous with respect to ν .(ii) Especially when ∂ Ω is rough, it is simplest to deal only with d -balls B whichstay away from ∂ Ω , i.e., which satisfy B ⊂ Ω . (3.11) We can always assume this for the balls in (3.8) if the converse of (3.2) is alsotrue, namely if ∀ x ∈ Ω and r > , ∃ s = s ( r, x ) > such that B s ( x ) ⊂ D r ( x ) . (3.12) ept 24 To see why, let us first show that given a compact set K and an open set G with K ⊂ G ⊂ Ω , there exists t > so that B t ( y ) ⊂ G for all y ∈ K . Indeed,for such K and G , let t ′ = ρ ( K, G c ) . By (3.12), for each x ∈ K there exists r ( x ) > so that B r ( x ) ( x ) ⊂ D t ′ ( x ) . Further, by (3.2), there exists s ( x ) > so that D s ( x ) ( x ) ⊂ B r ( x ) / (2 κ ) ( x ) , where κ is as in (3.1). Since K is compact, we may choosefinite collections { B r i / (2 κ ) ( x i ) } and { D s i ( x i ) } with x i ∈ K , r i = r ( x i ) , s i = s ( x i ) ,and K ⊂ S D s i ( x i ) ⊂ S B r i / (2 κ ) ( x i ) . Now set t = min { r i / (2 κ ) } . Let y ∈ K andchoose i such that y ∈ B r i / (2 κ ) ( x i ) . By (3.1), B t ( y ) ⊂ B r i ( x i ) and consequently B t ( y ) ⊂ D t ′ ( x i ) . Since D t ′ ( x i ) ⊂ G , we obtain B t ( y ) ⊂ G for every y ∈ K , asdesired. In particular, B t ( y ) ⊂ Ω for all y ∈ K . Since the validity of (3.8) forsome δ = δ ( ǫ, K ) implies its validity for min { δ, t } , it follows that we may assume(3.11) for every B r ( y ) in (3.8) when (3.12) holds. Similarly, since the constant c in (3.8) is independent of K , we may assume as well that every B c r ( y ) in (3.8)has closure in Ω .(iii) We can often slightly weaken the assumption in Definition 3.5 that K isan arbitrary compact set in Ω . For example, in our results where w (Ω) < ∞ , itis generally enough to assume that for each ǫ > , there is a particular compact K with w (Ω \ K ) < ǫ such that (3.8) holds. However, in §3.4, where we do notassume w (Ω) < ∞ , it is convenient to keep the hypothesis that K is arbitrary. Given a set
H ⊂
Lip
Q,p (Ω) , define ˆ H = { f : there exists { f j } ⊂ H with f j → f a.e.- w } . (3.13)It will be useful later to note that if H is bounded in L Nw (Ω) for some N , then ˆ H is also bounded in L Nw (Ω) by Fatou’s lemma; in particular, every f ∈ ˆ H thenbelongs to L Nw (Ω) . See (3.15) for a relationship between ˆ H and the closure of H in W ,pν,µ (Ω , Q ) in case w << ν .We now state our simplest global result. Its proof is given after Corollary 3.11. Theorem 3.7.
Let the assumptions of §3.1 hold, w (Ω) < ∞ , ≤ p < ∞ , Lip Q,p (Ω) . Suppose that the Poincaré property of order p inDefinition 3.5 holds for H and that sup f ∈H (cid:8) || f || L Nw (Ω) + || f || L pν (Ω) + ||∇ f || L pµ (Ω ,Q ) (cid:9) < ∞ . (3.14) Then any sequence { f k } ⊂ ˆ H has a subsequence that converges in L qw (Ω) norm forevery ≤ q < N to a function belonging to L Nw (Ω) . ept 24 H ⊂ Lip Q,p (Ω) and ˆ H be as in (3.13). We reserve the notation H for theclosure of H in W ,pν,µ (Ω , Q ) , i.e., for the closure of the collection { ( f, ∇ f ) : f ∈ H} with respect to the norm (3.6). Elements of H are viewed as equivalence classes.If w << ν , then { f : there exists ~g such that ( f, ~g ) ∈ H} ⊂ ˆ H . (3.15)Indeed, if ( f, ~g ) ∈ H , there is a sequence { f j } ⊂ H such that ( f j , ∇ f j ) → ( f, ~g ) in W ,pν,µ (Ω , Q ) norm, and consequently f j → f in L pν (Ω) . By using a subsequence, wemay assume that f j → f pointwise a.e.- ν , and hence by absolute continuity that f j → f pointwise a.e.- w . This proves (3.15). In fact, it can be verified by usingEgorov’s theorem that { f : there exists { ( f j , ~g j ) } ⊂ H with f j → f a.e.- w } ⊂ ˆ H . (3.16)Theorem 3.7 and (3.15) immediately imply the following corollary. Corollary 3.8. Let the assumptions of §3.1 hold, w (Ω) < ∞ and w << ν . Let ≤ p < ∞ , < N ≤ ∞ , H ⊂ Lip Q,p (Ω) and H be the closure of H in W ,pν,µ (Ω , Q ) .Suppose that the Poincaré property of order p in Definition 3.5 holds for H andthat sup f ∈H n || f || L Nw (Ω) + || ( f, ∇ f ) || W ,pν,µ (Ω ,Q ) o < ∞ . (3.17) Then any sequence { f k } in { f : there exists ~g such that ( f, ~g ) ∈ H} has a subsequence that converges in L qw (Ω) norm for ≤ q < N to a function thatbelongs to L Nw (Ω) . Remark 3.9. Corollary 3.8 may be thought of as an analogue in the degeneratesetting of the Rellich-Kondrachov theorem since it contains this classical resultas a special case. To see why, set Q ( x ) = Id and w = ν = µ to be Lebesguemeasure. Then, given a bounded sequence { ( f k , ~g k ) } ⊂ W ,p (Ω) = W ,pdx,dx, (Ω , Q ) we may choose { f jk } ⊂ Lip (Ω) with ( f jk , ∇ f jk ) → ( f k , ~g k ) in W ,p (Ω) norm. Thus,setting H = { f jk } k ∈ N ,j>J k where each J k is chosen sufficiently large to preserveboundedness, the classical Sobolev inequality gives (3.17) with N = np/ ( n − p ) for ≤ p < n . The Rellich-Kondrachov theorem now follows from Corollary 3.8. ept 24 H is replaced by a set W ⊂ W ,pν,µ (Ω , Q ) with elements viewed as equivalence classes, assuming that w << ν .We then modify Definition 3.5 by replacing (3.8) with the analogous estimate (cid:18) ˆ B r ( y ) | f − f B r ( y ) ,w | p dw (cid:19) p ≤ ǫ || ( f, ~g ) || W ,pν,µ ( B c r ( y ) ,Q ) if ( f, ~g ) ∈ W . (3.18)The assumption w << ν guarantees that the left side of (3.18) does not changewhen the first component of a pair is arbitrarily altered in a set of ν -measure zero.If Poincaré’s inequality is known to hold for subsets of Lipschitz functions in theform (3.8), it can often be extended by approximation to the similar form (3.18)for subsets of W ,pν,µ (Ω , Q ) . Indeed, let us show without using weak convergencethat if w << ν and the Radon-Nikodym derivative dw/dν ∈ L p ′ ν (Ω) , /p + 1 /p ′ =1 , then (3.18) holds with W = W ,pν,µ (Ω , Q ) if (3.8) holds with H = Lip Q,p (Ω) .This follows easily from Fatou’s lemma since if ( f, ~g ) ∈ W ,pν,µ (Ω , Q ) and we choose { f j } ⊂ Lip Q,p (Ω) with ( f j , ∇ f j ) → ( f, ~g ) in W ,pν,µ (Ω , Q ) , then for any ball B , since f j → f in L pν (Ω) , we have ( f j ) B,w = 1 w ( B ) ˆ B f j dwdν dν → w ( B ) ˆ B f dwdν dν = f B,w . Of course we may also assume that f j → f a.e.- w by selecting a subsequence of { f j } which converges to f a.e.- ν . The same argument shows that if (3.18) holdsfor all pairs in any set W ⊂ W ,pν,µ (Ω , Q ) , then it also holds for pairs in the closure W of W in W ,pν,µ (Ω , Q ) . Moreover, if all balls B in question satisfy B ⊂ Ω (cf.(3.11)), then the assumption can clearly be weakened to dw/dν ∈ L p ′ ν,loc (Ω) . Aswe observed in Remark 3.6(ii), the balls in (3.8) can be assumed to satisfy (3.11)provided (3.12) is true.Analogues of Theorem 3.7 and Corollary 3.8 for a set W ⊂ W ,pν,µ (Ω , Q ) aregiven in the next result, which also includes the Rellich-Kondrachov theorem as aspecial case. Theorem 3.10. Let the assumptions of §3.1 hold, w (Ω) < ∞ and w << ν . Let ≤ p < ∞ , < N ≤ ∞ and W ⊂ W ,pν,µ (Ω , Q ) . Suppose that the Poincaré propertyin Definition 3.5 holds, but in the modified form given in (3.18), and that sup ( f,~g ) ∈W n || f || L Nw (Ω) + || ( f, ~g ) || W ,pν,µ (Ω ,Q ) o < ∞ . (3.19) ept 24 Let ˆ W = { f : there exists { ( f j , ~g j ) } ⊂ W with f j → f a.e. − w } . Then any sequence in ˆ W has a subsequence that converges in L qw (Ω) norm forevery ≤ q < N to a function belonging to L Nw (Ω) . In particular, if W denotesthe closure of W in W ,pν,µ (Ω , Q ) , then the same is true for any sequence in { f : there exists ~g such that ( f, ~g ) ∈ W} . As a corollary, we obtain a result for arbitrary sequences { ( f k , ~g k ) } which arebounded in W ,pν,µ (Ω , Q ) and whose first components { f k } are bounded in L Nw (Ω) . Corollary 3.11. Let the assumptions of §3.1 hold, w (Ω) < ∞ , w << ν , ≤ p < ∞ and < N ≤ ∞ . Suppose that the Poincaré property in Definition 3.5 holdsfor all of W ,pν,µ (Ω , Q ) , i.e., Definition 3.5 holds with (3.8) replaced by (3.18) for W = W ,pν,µ (Ω , Q ) . Then if { ( f k , ~g k ) } is any sequence in W ,pν,µ (Ω , Q ) such that sup k h || f k || L Nw (Ω) + || ( f k , ~g k ) || W ,pν,µ (Ω ,Q ) i < ∞ , there is a subsequence of { f k } that converges in L qw (Ω) norm for ≤ q < N to afunction belonging to L Nw (Ω) . If in addition dw/dν ∈ L p ′ ν (Ω) , /p + 1 /p ′ = 1 , theconclusion remains valid if the Poincaré property holds just for Lip Q,p (Ω) . In fact, the first conclusion in Corollary 3.11 follows by applying Theorem 3.10with W chosen to be the specific sequence { ( f k , ~g k ) } k in question, and the secondstatement follows from the first one and our observation above that (3.18) holdswith W = W ,pν,µ (Ω , Q ) if dw/dν ∈ L p ′ ν (Ω) , /p + 1 /p ′ = 1 , and if (3.8) holds with H = Lip Q,p (Ω) . Proofs of Theorems 3.7 and 3.10. We will concentrate on the proof of Theorem3.7. The proof of Theorem 3.10 is similar and omitted. We begin with a usefulcovering lemma. Lemma 3.12. Let the assumptions of §3.1 hold and w (Ω) < ∞ . Fix p ∈ [1 , ∞ ) and a set H ⊂ Lip Q,p (Ω) . Suppose the Poincaré property of order p in Definition3.5 holds for H , and let κ be as in (3.1) and c be as in (3.8). Then for every ǫ > , there are positive constants r = r ( ǫ, κ, c ) , M = M ( κ, c ) and a finite ept 24 collection { B r ( y k ) } k of d -balls, so that ( i ) w (cid:0) Ω \ [ k B r ( y k ) (cid:1) < ǫ, (3.20) ( ii ) X k χ B c r ( y k ) ( x ) ≤ M for all x ∈ Ω , (3.21) ( iii ) || f − f B r ( y k ) ,w || L pw ( B r ( y k )) ≤ ǫ || ( f, ∇ f ) || W ,pν,µ ( B c r ( y k ) ,Q ) (3.22) for all f ∈ H and all k . Note that M is independent of ǫ . Proof: We first recall the “swallowing” property of d -balls: There is a constant γ ≥ depending only on κ so that if x, y ∈ Ω , < r ≤ r < ∞ and B r ( x ) ∩ B r ( y ) = ∅ ,then B r ( x ) ⊂ B γr ( y ) . (3.23)Indeed, by [CW1, Observation 2.1], γ can be chosen to be κ + 2 κ .Fix ǫ > . Since w (Ω) < ∞ , there is a compact set K ⊂ Ω with w (Ω \ K ) < ǫ .Let δ ′ = δ ′ ( ǫ ) be as in Definition 3.3 for K , and let δ = δ ( ǫ ) be as in (3.8). Fix r with < r < min { δ, δ ′ / ( c γ ) } where c is as in (3.8). For each x ∈ K , use(3.2) to pick s ( x, r ) > so that D s ( x,r ) ( x ) ⊂ B r/γ ( x ) . Since K is compact, thereare finitely many points { x j } in K so that K ⊂ ∪ j B r/γ ( x j ) . Choose a maximalpairwise disjoint subcollection { B r/γ ( y k ) } of { B r/γ ( x j ) } . We will show that thecollection { B r ( y k ) } satisfies (3.20)–(3.22).To verify (3.20), it is enough to show that K ⊂ ∪ k B r ( y k ) . Let y ∈ K . Then y ∈ B r/γ ( x j ) for some x j . If x j = y k for some y k then y ∈ B r ( y k ) . If x j = y k forall y k , there exists y ℓ so that B r/γ ( y ℓ ) ∩ B r/γ ( x j ) = ∅ . Then B r/γ ( x j ) ⊂ B r ( y ℓ ) by(3.23), and so y ∈ B r ( y ℓ ) . In either case, we obtain y ∈ ∪ k B r ( y k ) as desired.To verify (3.21), suppose that { k i } Li =1 satisfies ∩ Li =1 B c r ( y k i ) = ∅ . Then by(3.23), B c r ( y k i ) ⊂ B c γr ( y k ) for ≤ i ≤ L . Since γ, c ≥ , we have B r/γ ( y k ) ⊂ B c r ( y k ) for all k , and consequently ∪ B r/γ ( y k i ) ⊂ ∪ B c r ( y k i ) ⊂ B c γr ( y k ) . By construction, { B r/γ ( y k ) } is pairwise disjoint in k . Since < r/γ < c γr < δ ′ ,the corresponding constant C in the definition of geometric doubling depends onlyon ( c γr ) / ( r/γ ) = c γ , i.e., C depends only on κ and c . Choosing M to be thisconstant, we obtain that L ≤ M as desired. The same argument shows that thecollection { B c r ( y k ) } has the stronger bounded intercept property with the samebound M , i.e., any ball in the collection intersects at most M − others. ept 24 < r < δ by construction. Hence (3.8)implies that for each k and all f ∈ H , || f − f B r ( y k ) ,w || L pw ( B r ( y k )) ≤ ǫ || ( f, ∇ f ) || W ,pν,µ ( B c r ( y k ) ,Q ) , (3.24)as required. This completes the proof of Lemma 3.12. (cid:3) The proof of Theorem 3.7 will be deduced from Theorem 1.1 by choosing X (Ω) = L pν (Ω) × L pµ (Ω , Q ) and considering the product space B N, X (Ω) = L Nw (Ω) × (cid:0) L pν (Ω) × L pµ (Ω , Q ) (cid:1) . We always choose Σ to be the Lebesgue measurable subsets of Ω and Σ = { B r ( x ) : r > , x ∈ Ω } . Note that X (Ω) and B N, X (Ω) are normed linear spaces (evenBanach spaces), and the norm in B N, X (Ω) is || ( h, ( f, ~g )) || B N, X (Ω) = || h || L Nw (Ω) + || f || L pν (Ω) + || ~g || L pµ (Ω ,Q ) . (3.25)The roles played in §1 by g and ( f, g ) are now played by ( f, ~g ) and ( h, ( f, ~g )) respectively.Let us verify properties (A) and ( B p ) in §1 with X (Ω) and Σ chosen asabove. To verify (A), fix B ∈ Σ and ( f, ~g ) ∈ X (Ω) . Clearly f χ B ∈ L pν (Ω) since f ∈ L pν (Ω) . Also, ˆ Ω (cid:16) ( ~gχ B ) ′ Q ( ~gχ B ) (cid:17) p dµ = ˆ B (cid:16) ~g ′ Q ( x ) ~g (cid:17) p dµ ≤ ˆ Ω (cid:16) ~g ′ Q ( x ) ~g (cid:17) p dµ < ∞ . Thus ( f, ~g ) χ B ∈ X (Ω) and property (A) is proved.To verify ( B p ), let { B l } be a finite collection of d -balls satisfying P l χ B l ( x ) ≤ C for all x ∈ Ω . Then if ( f, ~g ) ∈ X (Ω) , X l || ( f, ~g ) χ B l || p X (Ω) = X l (cid:0) || f χ B l || L pν (Ω) + || ~gχ B l || L pµ (Ω ,Q ) (cid:1) p ≤ p − X l (cid:16) || f χ B l || pL pν (Ω) + || ~gχ B l || p L pµ (Ω ,Q ) (cid:17) = 2 p − ˆ Ω | f | p X l χ B l ! dν + ˆ Ω ( ~g ′ Q~g ) p X l χ B l ! dµ ept 24 ≤ p − C (cid:16) || f || pL pν (Ω) + || ~g || p L pµ (Ω ,Q ) (cid:17) ≤ p C || ( f, ~g ) || p X (Ω) . This verifies ( B p ) with C chosen to be p C .The proof of Theorem 3.7 is now very simple. Let H satisfy its hypotheses andchoose S in Theorem 1.1 to be the set S = { ( f, ( f, ∇ f )) : f ∈ H} . Note that S is a bounded subset of B N, X (Ω) by hypothesis (3.14). Next, in or-der to choose the pairs { E ℓ , F ℓ } ℓ and verify conditions (i)–(iii) of Theorem 1.1(see (1.3) and (1.4)), we appeal to Lemma 3.12. Given ǫ > , let { E ℓ , F ℓ } ℓ = { B r ( y k ) , B c r ( y k ) } k where { y k } and r are as in Lemma 3.12. Then E ℓ , F ℓ ∈ Σ ,and conditions (i)–(iii) of Theorem 1.1 are guaranteed by Lemma 3.12. Finally, bynoting that the set ˆ H defined in (3.13) is the same as the set ˆ S defined in (1.5),the conclusion of Theorem 3.7 follows from Theorem 1.1. (cid:3) For special domains Ω and special choices of N , the boundedness assumption(3.14) (or (3.17)) can be weakened to sup f ∈H (cid:8) || f || L pν (Ω) + ||∇ f || L pµ (Ω ,Q ) (cid:9) = sup f ∈H || ( f, ∇ f ) || W ,pν,µ (Ω ,Q ) < ∞ . (3.26)This is clearly the case for any Ω and N for which there exists a global Sobolev-Poincaré estimate that bounds || f || L Nw (Ω) by || ( f, ∇ f ) || W ,pν,µ (Ω ,Q ) for all f ∈ H . Wenow formalize this situation assuming that w << ν . In the appendix, we considera case when w << ν fails.The form of the global Sobolev-Poincaré estimate we will use is given in thenext definition. It guarantees that (3.14) and (3.26) are the same when N = pσ . Definition 3.13. Let ≤ p < ∞ and H ⊂ Lip Q,p (Ω) . Then the global Sobolevproperty of order p holds for H if there are constants C > and σ > so that || f || L pσw (Ω) ≤ C || ( f, ∇ f ) || W ,pν,µ (Ω ,Q ) for all f ∈ H . (3.27)If w << ν , then (3.27) extends to ( f, ~g ) ∈ H . In fact, let ( f, ~g ) ∈ H and choose { f j } ⊂ H with ( f j , ∇ f j ) → ( f, ~g ) in W ,pν,µ (Ω , Q ) . Then f j → f in L pν (Ω) norm,and by choosing a subsequence we may assume that f j → f a.e.- ν . Hence f j → f a.e.- w because w << ν . Since each f j satisfies (3.27), it follows that || f || L pσw (Ω) ≤ C || ( f, ~g ) || W ,pν,µ (Ω ,Q ) if ( f, ~g ) ∈ H . (3.28) ept 24 H ⊂ Lip Q,p (Ω) and that w << ν , the same sequence { f j } as above is also boundedin L pσw (Ω) norm and so satisfies ( f j ) E,w → f E,w for measurable E by the sameweak convergence argument given after the statement of Theorem 1.1. Hence thePoincaré estimate in Definition 3.5 also extends to H in the same form as (3.18),with W there replaced by H , i.e., (cid:18) ˆ B r ( y ) | f − f B r ( y ) ,w | p dw (cid:19) p ≤ ǫ || ( f, ~g ) || W ,pν,µ ( B c r ( y ) ,Q ) if ( f, ~g ) ∈ H . (3.29)Hence, we immediately obtain the next result by choosing W = H and N = pσ inTheorem 3.10. Theorem 3.14. Let the assumptions of §3.1 hold, w (Ω) < ∞ and w << ν . Fix p ∈ [1 , ∞ ) and a set H ⊂ Lip Q,p (Ω) . Suppose the Poincaré and global Sobolevproperties of order p in Definitions 3.5 and 3.13 hold for H , and let σ be as in(3.27). If { ( f k , ~g k ) } is a sequence in H with sup k || ( f k , ~g k ) || W ,pν,µ (Ω ,Q ) < ∞ , (3.30) then { f k } has a subsequence which converges in L qw (Ω) for ≤ q < pσ , and thelimit of the subsequence belongs to L pσw (Ω) . A result for the entire space W ,pν,µ (Ω , Q ) follows by choosing H = Lip Q,p (Ω) inTheorem 3.14 or Corollary 3.8: Corollary 3.15. Suppose that the hypotheses of Theorem 3.14 hold for H = Lip Q,p (Ω) . If { ( f k , ~g k ) } ⊂ W ,pν,µ (Ω , Q ) and (3.30) is true then { f k } has a subse-quence which converges in L qw (Ω) for ≤ q < pσ , and the limit of the subsequencebelongs to L pσw (Ω) . See the Appendix for analogues of Theorem 3.14 and Corollary 3.15 withoutthe assumption w << ν . In this section, for general bounded measurable sets Ω ′ with Ω ′ ⊂ Ω , we study com-pact embedding of subsets of W ,pν,µ (Ω , Q ) into L qw (Ω ′ ) without assuming a globalSobolev estimate for Ω or Ω ′ and without assuming w (Ω) < ∞ . For some applica-tions, see the comment at the end of the section.The theorems below will assume a much weaker condition than the globalSobolev estimate (3.27), namely the following local estimate. ept 24 Definition 3.16. Let ≤ p < ∞ . We say that the local Sobolev property of order p holds if for some fixed constant σ > and every compact set K ⊂ Ω , there is aconstant r > so that for all d -balls B = B r ( y ) with y ∈ K and < r < r , || f || L pσw ( B ) ≤ C ( B ) || ( f, ∇ f ) || W ,pν,µ (Ω ,Q ) if f ∈ Lip ( B ) , (3.31) where C ( B ) is a positive constant independent of f . We will view any f ∈ Lip ( B ) as extended by to all of Ω . Remark 3.17. (i) A more standard assumption than (3.31) is a normalized in-equality that includes a factor r in the gradient term on the right side: (cid:18) w ( B r ( y )) ˆ B r ( y ) | f | pσ dw (cid:19) pσ ≤ C (cid:18) ν ( B r ( y )) ˆ B r ( y ) | f | p dν (cid:19) p + Cr (cid:18) µ ( B r ( y )) ˆ B r ( y ) | p Q ∇ f | p dµ (cid:19) p , (3.32) with C independent of r, y ; see e.g. [SW1] and [R1] in the unweighted case with p = 2 . Clearly (3.32) is a stronger requirement than (3.31).(ii) In the classical n -dimensional elliptic case for linear second order equationsin divergence form, Q satisfies c | ξ | ≤ Q ( x, ξ ) ≤ C | ξ | for some fixed constants c, C > and d is the standard Euclidean metric d ( x, y ) = | x − y | . For ≤ p For s ≥ , we say that the cutoff property of order s holds for µ if for each compact K ⊂ Ω , there exists δ = δ ( K ) > so that for every d-ball B r ( y ) with y ∈ K and < r < δ, there is a function φ ∈ Lip (Ω) and a constant γ = γ ( y, r ) ∈ (0 , r ) satisfying(i) ≤ φ ≤ in Ω ,(ii) supp φ ⊂ B r ( y ) and φ = 1 in B γ ( y ) ,(iii) ∇ φ ∈ L sµ (Ω , Q ) . Since µ is always assumed to be locally finite, the strongest form of Definition3.18, namely the version with s = ∞ , automatically holds if Q is locally boundedin Ω and (3.12) is true; recall that we always assume (3.2). To see why, fix acompact set K ⊂ Ω and consider B r ( y ) with y ∈ K and r < . Use (3.2) to choose ept 24 D ′ , D with common center y such that D ′ ⊂ D ⊂ B r ( y )( ⊂ Ω by definition ) . Construct a smooth function φ in Ω with support in D such that ≤ φ ≤ and φ = 1 on D ′ . By (3.12), there is γ > such that B γ ( y ) ⊂ D ′ .Then φ satisfies parts (i)-(iii) of Definition 3.18 with s = ∞ ; for (iii), we use thefact that ∇ φ has compact support in Ω together with local boundedness of Q andlocal finiteness of µ .To compensate for the lack of a global Sobolev estimate, given H ⊂ Lip Q,p (Ω) ,we will assume in conjunction with the cutoff property of some order s ≥ pσ ′ thatfor every compact set K ⊂ Ω , there exists δ = δ ( K ) > such that for every d -ball B with center in K and radius less than δ , there is a constant C ( B ) so that || f || L pt ′ µ ( B ) ≤ C ( B ) || ( f, ∇ f ) || W ,pν,µ (Ω ,Q ) if f ∈ H , (3.33)where t = s/p and /t + 1 /t ′ = 1 . Note that ≤ t ′ ≤ σ since s ≥ pσ ′ . Remark 3.19. Inequality (3.33) is different in nature from (3.31) even if t ′ = σ and w = µ since there is a restriction on supports in (3.31) but not in (3.33).However, (3.33) implies (3.31) when s = pσ ′ , w = µ and H contains all Lipschitzfunctions with support in any ball. On the other hand, (3.33) is often automaticif µ = ν . For example, as mentioned earlier, if Q is locally bounded and (3.12) istrue, then the cutoff property holds with s = ∞ , giving t = ∞ and t ′ = 1 . In thiscase, when µ = ν , the left side of (3.33) is clearly smaller than the right side (infact smaller than || f || L pν (Ω) ). We can now state our main local result. Theorem 3.20. Let the assumptions of §3.1 and condition (3.12) hold, and let w << ν . Fix p ∈ [1 , ∞ ) and suppose the Poincaré property of order p in Definition3.5 holds for a fixed set H ⊂ Lip Q,p (Ω) and the local Sobolev property of order p in Definition 3.16 holds. Assume the cutoff property of some order s ≥ pσ ′ is truefor µ , with σ as in (3.31), and that (3.33) holds for H with t = s/p . Then forevery { ( f k , ~g k ) } ⊂ H that is bounded in W ,pν,µ (Ω , Q ) norm, there is a subsequence { f k i } of { f k } and an f ∈ L pσw,loc (Ω) such that f k i → f pointwise a.e.- w in Ω and in L qw (Ω ′ ) norm for all ≤ q < pσ and every bounded measurable Ω ′ with Ω ′ ⊂ Ω . See the Appendix for a version of Theorem 3.20 without assuming w << ν .Recall that H = W ,pν,µ (Ω , Q ) if H = Lip Q,p (Ω) . In the important case when Q ∈ L ∞ loc (Ω) , Theorem 3.20 and Remark 3.19 immediately imply the next result. ept 24 Corollary 3.21. Let Q be locally bounded in Ω and suppose that (3.12) holds.Fix p ∈ [1 , ∞ ) , and with w = ν = µ , assume the Poincaré property of order p holds for Lip Q,p (Ω) and the local Sobolev property of order p holds. Then for everybounded sequence { ( f k , ~g k ) } ⊂ W ,pw,w (Ω , Q ) , there is a subsequence { f k i } of { f k } and a function f ∈ L pσw,loc (Ω) such that f k i → f pointwise a.e.- w in Ω and in L qw (Ω ′ ) norm, ≤ q < pσ , for every bounded measurable Ω ′ with Ω ′ ⊂ Ω . Proof of Theorem 3.20: We begin by using the cutoff property in Definition3.18 to construct a partition of unity relative to d -balls and compact subsets of Ω . Lemma 3.22. Fix Ω and s ≥ , and suppose the cutoff property of order s holdsfor µ . If K is a compact subset of Ω and r > , there is a finite collection of d -balls { B r ( y j ) } with y j ∈ K together with Lipschitz functions { ψ j } on Ω such that supp ψ j ⊂ B r ( y j ) and(a) K ⊂ [ j B r ( y j ) ,(b) ≤ ψ j ≤ in Ω for each j , and X j ψ j ( x ) = 1 for all x ∈ K ,(c) ∇ ψ j ∈ L sµ (Ω , Q ) for each j . Proof: The argument is an adaptation of one in [Ru] for the usual Euclideancase. The authors thank D. D. Monticelli for related discussions. Fix r > and acompact set K ⊂ Ω , and set β = min { δ/ , r } for δ = δ ( K ) as in Definition 3.18.Since β < δ , Definition 3.18 implies that for each y ∈ K , there exist γ ( y ) ∈ (0 , β ) and φ y ( x ) ∈ Lip (Ω) so that ≤ φ y ≤ in Ω , supp φ y ⊂ B β ( y )) , φ y = 1 in B γ ( y ) ( y ) and ∇ φ y ∈ L sµ (Ω , Q ) . The collection { B γ ( y ) ( y ) } y ∈ K covers K , so by (3.2) and thecompactness of K , there is a finite subcollection { B γ ( y j ) ( y j ) } mj =1 whose union covers K . Part (a) follows since γ ( y j ) < r . Next let φ j ( x ) = φ y j ( x ) and define { ψ j } mj =1 asfollows: set ψ = φ and ψ j = (1 − φ ) · · · (1 − φ j − ) φ j for j = 2 , .., m . Then each ψ j is a Lipschitz function in Ω , and supp φ j ⊂ B r ( y j ) since β < r . Also, ≤ ψ j ≤ in Ω and m X j =1 ψ j ( x ) = 1 − m Y j =1 (1 − φ j ( x )) , x ∈ Ω . If x ∈ K then x ∈ B γ ( y j ) ( y j ) for some j . Hence some φ j ( x ) = 1 and consequently P j ψ j ( x ) = 1 . This proves part (b). Lastly, we use Leibniz’s product rule tocompute ∇ ψ j and then apply Minkowski’s inequality j times to obtain part (c) ept 24 ∇ φ j ∈ L sµ (Ω , Q ) . (cid:3) The next lemma shows how the local Sobolev estimate (3.31) and Lemma 3.22lead to a local analogue of the global Sobolev estimate (3.27). Lemma 3.23. Let Ω ′ be a bounded measurable set with Ω ′ ⊂ Ω . Suppose that bothDefinition 3.16 and the cutoff property for µ of some order s ≥ pσ ′ hold, and alsothat (3.33) holds with t = s/p for a fixed set H ⊂ Lip loc (Ω) . Then there is a finiteconstant C (Ω ′ ) such that || f || L pσw (Ω ′ ) ≤ C (Ω ′ ) || ( f, ∇ f ) || W ,pν,µ (Ω ,Q ) if f ∈ H . (3.34) Proof: Let r be as in Definition 3.16 relative to the compact set Ω ′ ⊂ Ω , andlet δ be as in (3.33). Use Lemma 3.22 to cover Ω ′ by the union of a finite numberof d -balls { B j } each of radius smaller than min { r , δ } . Associated with this coveris a collection { ψ j } ⊂ Lip (Ω) with supp ψ j ⊂ B j , P j ψ j = 1 in Ω ′ , and ∇ ψ j ∈L sµ (Ω , Q ) . If f ∈ H , then || f || L pσw (Ω ′ ) = || f X j ψ j || L pσw (Ω ′ ) ≤ X j || ψ j f || L pσw ( B j ) . (3.35)Since ψ j f ∈ Lip ( B j ) , (3.31) and the product rule give || ψ j f || L pσw ( B j ) ≤ C ( B j ) || ( ψ j f, ∇ ( ψ j f )) || W ,pν,µ ( B j ,Q ) = C ( B j ) (cid:16) || ψ j f || L pν ( B j ) + || p Q ∇ ( ψ j f ) || L pµ ( B j ) (cid:17) ≤ C ( B j ) (cid:16) || ψ j f || L pν ( B j ) + || ψ j p Q ∇ f || L pµ ( B j ) + || f p Q ∇ ψ j || L pµ ( B j ) (cid:17) ≤ C ( B j ) (cid:16) || ( f, ∇ f ) || W ,pν,µ (Ω ,Q ) + || f p Q ∇ ψ j || L pµ ( B j ) (cid:17) , (3.36)where we have used | ψ j | ≤ . We will estimate the second term on the right of(3.36) by using (3.33). Recall that t = s/p ≥ σ ′ and /t + 1 /t ′ = 1 . Let C = max j || p Q ∇ ψ j || L sµ ( B j ) . By Hölder’s inequality and (3.33), || f p Q ∇ ψ j || L pµ ( B j ) ≤ || f || L pt ′ µ ( B j ) || p Q ∇ ψ j || L sµ ( B j ) ≤ CC ( B j ) || ( f, ∇ f ) || W ,pν,µ (Ω ,Q ) . (3.37) ept 24 || ψ j f || L pσw ( B j ) ≤ C ( B j ) (cid:0) CC ( B j ) (cid:1) || ( f, ∇ f ) || W ,pν,µ (Ω ,Q ) . By (3.35), for any f ∈ H , || f || L pσw (Ω ′ ) ≤ || ( f, ∇ f ) || W ,pν,µ (Ω ,Q ) X j C ( B j ) (cid:0) CC ( B j ) (cid:1) = C (Ω ′ ) || ( f, ∇ f ) || W ,pν,µ (Ω ,Q ) , which completes the proof of Lemma 3.23.Theorem 3.20 follows from Lemma 3.23 and Theorem 1.4. We will sketch theproof, omitting some familiar details. By choosing a sequence of compact setsincreasing to Ω and using a diagonalization argument, it is enough to prove theconclusion for a fixed measurable Ω ′ with compact closure Ω ′ in Ω . Fix such an Ω ′ and select a bounded open Ω ′′ with Ω ′ ⊂ Ω ′′ ⊂ Ω ′′ ⊂ Ω . For H as in Theorem 3.20,apply Lemma 3.23 to the set Ω ′′ to obtain || f || L pσw (Ω ′′ ) ≤ C (Ω ′′ ) || ( f, ∇ f ) || W ,pν,µ (Ω ,Q ) , f ∈ H . (3.38)By assumption, w << ν , so (3.38) extends to H in the form || f || L pσw (Ω ′′ ) ≤ C (Ω ′′ ) || ( f, ~g ) || W ,pν,µ (Ω ,Q ) , ( f, ~g ) ∈ H . (3.39)Let ǫ > . By hypothesis, H satisfies the Poincaré estimate (3.8) for balls B r ( y ) with y ∈ Ω ′ and r < δ ( ǫ, Ω ′ ) . Since the Euclidean distance between Ω ′ and ∂ Ω ′′ ispositive and we have assumed (3.12), we may also assume by Remark 3.6(ii) thatall such balls lie in the larger set Ω ′′ . Next we claim that (3.8) extends to H , i.e., (cid:18) ˆ B r ( y ) | f − f B r ( y ) ,w | p dw (cid:19) p ≤ ǫ || ( f, ~g ) || W ,pν,µ ( B c r ( y ) ,Q ) if ( f, ~g ) ∈ H , (3.40)for the same class of balls B r ( y ) . In fact, if ( f, ~g ) ∈ H and { f j } ⊂ H satisfies ( f j , ∇ f j ) → ( f, ~g ) in W ,pν,µ (Ω , Q ) norm, then there is a subsequence, still denoted { f j } , with f j → f a.e.- ν in Ω , and so with f j → f a.e.- w in Ω since w << ν .By (3.38), { f j } is bounded in L pσw (Ω ′′ ) . Hence, since the balls in (3.40) satisfy B r ( y ) ⊂ Ω ′′ , we obtain f jB r ( y ) ,w → f B r ( y ) ,w by our usual weak convergence argument,and (3.40) follows by Fatou’s lemma from its analogue (3.8) for the ( f j , ∇ f j ) . ept 24 { ( f k , ~g k ) } ⊂ H be bounded in W ,pν,µ (Ω , Q ) norm and apply Theorem1.4 with X (Ω) = L pν (Ω) × L pµ (Ω , Q ) to the set S defined by S = (cid:8)(cid:0) f k , ( f k , ~g k ) (cid:1)(cid:9) k , and with { ( E ǫℓ , F ǫℓ ) } ℓ chosen to be a finite number of pairs { ( B r ( y ℓ ) , B c r ( y ℓ ) } ℓ asin (3.40), but now with r fixed depending on ǫ , and with Ω ′ ⊂ ∪ ℓ B r ( y ℓ ) . Such afinite choice exists by (3.2) and the Heine-Borel theorem since Ω ′ is compact; cf.the proof of Lemma 3.12. Since Ω ′ is completely covered by ∪ ℓ E ǫℓ , assumption (i)of Theorem 1.4 is fulfilled. Moreover, the collection { F ǫℓ } has bounded overlapsuniformly in ǫ by the geometric doubling argument used to prove Lemma 3.12.Finally, (1.15) follows from (3.39) applied to the bounded sequence { ( f k , ~g k ) } since ∪ ℓ,ǫ E ǫℓ ⊂ Ω ′′ . Thus Theorem 1.4 implies that there is a subsequence { f k i } of { f k } and a function f ∈ L pσw (Ω ′ ) such that f k i → f a.e.- w in Ω ′ and in L qw (Ω ′ ) norm, ≤ q < pσ . This completes the proof of Theorem 3.20.For functions which are compactly supported in a fixed bounded measurable Ω ′ with Ω ′ ⊂ Ω , the proof of Theorem 3.20 can be modified to yield compactembedding into L qw (Ω ′ ) for the same Ω ′ without assuming (3.12). Of course wealways require (3.2). Given such Ω ′ and a set H ⊂ Lip Q,p, (Ω ′ ) , we may view H as a subset of Lip Q,p, (Ω) simply by extending functions in H to all of Ω as in Ω \ Ω ′ . In this way, the proof of Theorem 3.20 works without (3.12). For example,choosing H = Lip Q,p, (Ω ′ ) , we obtain Theorem 3.24. Let the assumptions of §3.1 hold and w << ν . Let Ω ′ be a boundedmeasurable set with Ω ′ ⊂ Ω . Fix p ∈ [1 , ∞ ) and suppose the Poincaré property oforder p in Definition 3.5 holds for Lip Q,p, (Ω ′ ) , with Lip Q,p, (Ω ′ ) viewed as a subsetof Lip Q,p, (Ω) using extension by , and suppose the local Sobolev property of order p in Definition 3.16 holds. Assume the cutoff property of some order s ≥ pσ ′ istrue for µ , with σ as in (3.31), and that (3.33) holds for Lip Q,p, (Ω ′ ) with t = s/p .Then for every sequence { ( f k , ~g k ) } ⊂ W ,pν,µ, (Ω ′ , Q ) which is bounded in W ,pν,µ (Ω ′ , Q ) norm, there is a subsequence { f k i } of { f k } and a function f ∈ L pσw (Ω ′ ) such that f k i → f pointwise a.e.- w in Ω ′ and in L qw (Ω ′ ) norm, ≤ q < pσ . The full force of the local Sobolev estimate in Definition 3.16 is not needed toprove Theorem 3.24. In fact, it is enough to assume that (3.31) holds only for ballscentered in the fixed compact set Ω ′ .The proof of Theorem 3.24 is like that of Theorem 3.20, working with the set Ω ′ that occurs in the hypotheses of Theorem 3.24. However, now (3.34) in theconclusion of Lemma 3.23 (with H = Lip Q,p, (Ω ′ ) ) remains valid if Ω ′ is replaced ept 24 Ω since every f ∈ Lip Q,p, (Ω ′ ) vanishes on Ω \ Ω ′ . The resultingestimate serves as a replacement for (3.38), so it is not necessary to demand thatthe E ǫℓ are subsets of a compact set Ω ′′ ⊂ Ω . Hence (3.12) is no longer required.Finally, the Poincaré estimate extends as usual to W ,pν,µ, (Ω ′ , Q ) (the closure of Lip Q,p, (Ω ′ )) , and due to support considerations, the E ǫℓ can be restricted to subsetsof Ω ′ by replacing E ǫℓ by E ǫℓ ∩ Ω ′ ; this guarantees w ( E ǫℓ ) < ∞ since w is locallyfinite by hypothesis.Recalling the comments made immediately after Definition 3.18 and in Remark3.19, we obtain a useful special case of Theorem 3.24: Corollary 3.25. Let the assumptions of §3.1 hold, Ω and Q be bounded, w = ν = µ and (3.12) be true. Let Ω ′ be a measurable set with Ω ′ ⊂ Ω . Fix p ∈ [1 , ∞ ) and suppose the Poincaré property of order p in Definition 3.5 holds for Lip Q,p, (Ω ′ ) and the local Sobolev property of order p in Definition 3.16 holds.Then for every { ( f k , ~g k ) } ⊂ W ,pν,µ, (Ω ′ , Q ) which is bounded in W ,pν,µ (Ω , Q ) norm,there is a subsequence { f k i } of { f k } and a function f ∈ L pσw (Ω ′ ) such that f k i → f pointwise a.e.- w in Ω ′ and in L qw (Ω ′ ) norm, ≤ q < pσ . In case p = 2 and all measures are Lebesgue measure, Corollary 3.25 is usedin [R1] to show existence of weak solutions to Dirichlet problems for some linearsubelliptic equations. It is also used in [R2] to derive the global Sobolev inequality || f || L σ (Ω ′ ) ≤ C (cid:16) ˆ Ω ′ | p Q ∇ f | dx (cid:17) / (3.41)for open Ω ′ with Ω ′ ⊂ Ω from the local estimate (3.32). L N in a quasimetric space In this section, we will consider the situation of an open set Ω in a topologicalspace X when X is also endowed with a quasimetric d . As there is no easy way todefine Sobolev spaces on general quasimetric spaces, this section concentrates onestablishing a simple criterion not directly related to Sobolev spaces ensuring thatbounded subsets of L Nw (Ω) are precompact in L qw (Ω) when ≤ q < N ≤ ∞ .We begin by further describing the setting for our result. The topology on X isexpressed in terms of a fixed collection T of subsets of X which may not be relatedto the quasimetric d . Thus when we say that a set O ⊂ X is open , we mean that ept 24 O ∈ T . Given an open Ω , we will assume each of the following: ( i ) ∀ x ∈ X and r > , the d -ball B r ( x ) = { y ∈ X : d ( x, y ) < r } is a Borel set; ( ii ) ∀ x ∈ X and r > , there is an open set O so that x ∈ O ⊂ B r ( x ) ; ( iii ) if X = Ω , then ∀ x ∈ Ω , d ( x, Ω c ) = inf { d ( x, y ) : y ∈ Ω c } > .Property ( ii ) serves as a substitute for (3.2).Unlike the situation in §3, d -balls centered in Ω may not be subsets of Ω unless X = Ω . However, we note the following fact. Remark 4.1. Properties ( ii ) and ( iii ) guarantee that for any compact set K ⊂ Ω ,there exists ε ( K ) > such that B r ( x ) ⊂ Ω if x ∈ K and r < ε ( K ) . In fact, firstnote that for any x ∈ Ω , ( iii ) implies that the d -ball B ( x ) with center x and radius r x = d ( x, Ω c ) / (2 κ ) lies in Ω . If K is a compact set in Ω , (ii) shows that K can becovered by a finite number of such balls { B ( x i ) } . With ε ( K ) chosen to be a suitablysmall multiple (depending on κ ) of min { r x i } , the remark then follows easily fromthe swallowing property of d -balls. Further, we assume that (Ω , d ) satisfies the local geometric doubling conditionin Definition 3.3, i.e., for each compact set K ⊂ Ω , there exists δ ′ ( K ) > suchthat for all x ∈ K and all < r ′ < r < δ ′ ( K ) , the number of disjoint d -balls ofcommon radius r ′ contained in B r ( x ) is at most a constant C r/r ′ depending on r/r ′ but not on K . We will choose δ ′ ( K ) ≤ ε ( K ) in the above.With this framework in force, we now state the main result of the section. Theorem 4.2. Let Ω ⊂ X be as above, and let w be a finite Borel measure on Ω such that given any ǫ > , there is a compact set K ⊂ Ω with w (Ω \ K ) < ǫ . Let ≤ p < ∞ and < N ≤ ∞ , and suppose S ⊂ L Nw (Ω) has the property that forany compact set K ⊂ Ω , there exists δ K > such that k f − f B,w k L pw ( B ) ≤ b ( f, B ) if f ∈ S and B = B r ( x ) , x ∈ K , < r < δ K , (4.1) where b ( f, B ) is a nonnegative ball set function. Further, suppose there is a con-stant c ≥ so that for every ǫ > and every compact set K ⊂ Ω , there exists ˜ δ ǫ,K > such that X B ∈F b ( f, B ) p ≤ ǫ p for all f ∈ S (4.2) for every finite family F = { B } of d -balls centered in K with common radius lessthan ˜ δ ǫ,K for which { c B } is a pairwise disjoint family of subsets of Ω . Then anysequence in S that is bounded in L Nw (Ω) has a subsequence that converges in L qw (Ω) for ≤ q < N to a function in L Nw (Ω) . ept 24 Proof. Let ǫ > and choose a compact set K ⊂ Ω with w (Ω \ K ) < ǫ .Next, for c ≥ , as in the proof of Lemma 3.12 there is a positive constant r = r ( ǫ, K, c ) < min { δ K , ˜ δ ǫ,K , δ ′ ( K ) , ε ( K ) / ( γc ) } (see (4.1),(4.2), Definition 3.3and Remark 4.1), where γ = κ +2 κ with κ as in (3.1), and a finite family { B r ( y k ) } k of d -balls centered in K satisfying K ⊂ ∪ k B r ( y k ) and whose dilates { B c r ( y k ) } k lie in Ω and have the bounded intercept property (with intercept constant M independent of ǫ ). Since { B c r ( y k ) } k has bounded intercepts with bound M , it canbe written as the union of at most M families of disjoint d -balls; see e.g. the proofof [CW1, Lemma 2.5]. By (4.2), we conclude that X k b ( f, B r ( y k )) p ≤ M ǫ p . Theorem 4.2 then follows immediately from Theorem 1.2; see also Remark 1.3(1).As an application of Theorem 4.2 we present a version of [HK2, Theorem 8.1]in the case p ≥ . Our version improves the one in [HK2] by allowing two differentmeasures and by relaxing the assumptions made about embedding and doubling.Furthermore, while the analogue in [HK2] of our (4.3) uses only the L w ( B ) norm onthe left side, it automatically self-improves to the L pw ( B ) norm due to the doublingassumption, with a further fixed enlargement of the ball c B on the right side; seee.g. [HK2, Theorem 5.1]. Corollary 4.3. Let X, d, Ω , w be as above, and let µ be a Borel measure on Ω . Fix ≤ p < ∞ , < N ≤ ∞ and c ≥ . Consider a sequence of pairs { ( f i , g i ) } ⊂ L Nw (Ω) × L pµ (Ω) such that for any compact set K ⊂ Ω , there exists ¯ δ K > with || f i − ( f i ) B,w || L pw ( B ) ≤ a ∗ ( B ) || g i || L pµ ( c B ) (4.3) for all i and all d -balls B centered in K with c B ⊂ Ω and r ( B ) < ¯ δ K , where a ∗ ( B ) is a non-negative ball set function satisfying lim r → n sup y ∈ K a ∗ ( B r ( y )) o = 0 . (4.4) Then if { f i } and { g i } are bounded in L Nw (Ω) and L pµ (Ω) respectively, { f i } has asubsequence converging in L qw (Ω) for ≤ q < N to a function belonging to L Nw (Ω) . Proof. Given ǫ > and compact set K ⊂ Ω , use (4.4) to choose r > so that a ∗ ( B r ) < ǫ/β for any d -ball B r centered in K with r < r , where β = ept 24 sup i || g i || L pµ (Ω) < ∞ . In Theorem 4.2, choose S = { f i } , δ K = δ K , b ( f i , B ) = a ∗ ( B ) || g i || L pµ ( c B ) and ˜ δ ǫ,K = min { δ K , δ ′ ( K ) , r , ε ( K ) /c } . If B is a d -ball with center in K and r ( B ) < ˜ δ ǫ,K , then c B ⊂ Ω . Hence, X B ∈F (cid:0) a ∗ ( B ) || g i || L pµ ( c B ) (cid:1) p ≤ ǫ p || g i || pL pµ (Ω) /β p ≤ ǫ p for every F as in Theorem 4.2. The conclusion now follows from Theorem 4.2. Remark 4.4. 1. The g i in (4.3) are usually the modulus of a fixed derivative ofthe corresponding f i , such as |∇ f i | when X is a Riemannian manifold. Moregenerally, g i may be the upper gradient of f i (see [Hei] for the definition).2. Theorem 4.2 can also be used to obtain an extension of Theorem 2.3 to s -Johndomains in quasimetric spaces; see [CW2, Theorem 1.6]. Here we briefly consider analogues of Theorem 3.14, Corollary 3.15 and Theorem3.20 without assuming w << ν , but adding the assumption that H is linear. Inthis case, (3.27) can be extended by continuity to obtain a bounded linear mapfrom H into L pσw (Ω) . Here, as always, H denotes the closure of { ( f, ∇ f ) : f ∈ H} in W ,pν,µ (Ω , Q ) . However, when w << ν fails, there is no natural way to obtainthe extension for every ( f, ~g ) ∈ H keeping the same f on the left side. In fact, let ( f, ~g ) ∈ H and choose { f j } ⊂ H with ( f j , ∇ f j ) → ( f, ~g ) in W ,pν,µ (Ω , Q ) . Linearityof H allows us to apply (3.27) to differences of the f j and conclude that { f j } is aCauchy sequence in L pσw (Ω) . Therefore f j → f ∗ in L pσw (Ω) for some f ∗ ∈ L pσw (Ω) ,and || f ∗ || L pσw (Ω) ≤ C || ( f, ~g ) || W ,pν,µ (Ω ,Q ) if ( f, ~g ) ∈ H . The function f ∗ is determined by ( f, ~g ) , i.e., f ∗ is independent of the particu-lar sequence { f j } ⊂ H above. Indeed, if { ˜ f j } is another sequence in H with ( ˜ f j , ∇ ˜ f j ) → ( f, ~g ) in W ,pν,µ (Ω , Q ) , and if ˜ f j → ˜ f ∗ in L pσw (Ω) , then by (3.27) andlinearity of H , || ˜ f j − f j || L pσw (Ω) ≤ C || ( ˜ f j − f j , ∇ ˜ f j − ∇ f j ) || W ,pν,µ (Ω ,Q ) → . ept 24 || ˜ f ∗ − f ∗ || L pσw (Ω) = 0 . Thus ( f, ~g ) determines f ∗ uniquely as anelement of L pσw (Ω) . Define a mapping T : H → L pσw (Ω) by setting T ( f, ~g ) = f ∗ . (5.1)Note that H is a linear set in W ,pν,µ (Ω , Q ) since H is linear, and that T is a boundedlinear map from H into L pσw (Ω) . Also note that T satisfies T ( f, ∇ f ) = f whenrestricted to those ( f, ∇ f ) with f ∈ H . Furthermore, if w << ν then T ( f, ~g ) = f for all ( f, ~g ) ∈ H , i.e., f ∗ = f a.e.- w for all ( f, ~g ) ∈ H . This follows since f j → f in L pν (Ω) norm and f j → f ∗ in L pσw (Ω) norm. In this appendix, where it is notassumed that w << ν , f ∗ plays a main role. One can find a function h such that h = f ∗ a.e.- w and h = f a.e.- ν , but as this fact is not needed, we omit its proof.An analogue of Theorem 3.14 is given in the next result. Theorem 5.1. Let all the assumptions of Theorem 3.14 hold except that now theset H is linear and we do not assume w << ν . Then the map T : H → L qw (Ω) defined in (5.1) is compact if ≤ q < pσ . Equivalently, if { ( f k , ~g k ) } is a sequence in H with sup k || ( f k , ~g k ) || W ,pν,µ (Ω ,Q ) < ∞ , then { f ∗ k } has a subsequence which convergesin L qw (Ω) for ≤ q < pσ , where f ∗ k = T ( f k , ~g k ) . Moreover, the limit of thesubsequence belongs to L pσw (Ω) . Proof: Let H satisfy the hypothesis of the theorem and let { ( f k , ~g k ) } ⊂ H bebounded in W ,pν,µ (Ω , Q ) . For each k , choose h k ∈ H so that || ( f k , ~g k ) − ( h k , ∇ h k ) || W ,pν,µ (Ω ,Q ) ≤ − k . (5.2)Set H = { h k } k ⊂ H . Then { ( h k , ∇ h k ) : h k ∈ H } is bounded in W ,pν,µ (Ω , Q ) .Further, (3.27) implies a version of (3.14), namely sup f ∈H n || f || L pσw (Ω) + || ( f, ∇ f ) || W ,pν,µ (Ω ,Q ) o < ∞ . Theorem 3.7 now applies to H with N = pσ and gives that any sequence in ˆ H has a subsequence which converges in L qw (Ω) norm for ≤ q < pσ to a functionbelonging to L pσw (Ω) . The sequence { h k } lies in ˆ H , as is easily seen by considering,for each fixed k , the constant sequence { f j } defined by f j = h k for all j . Weconclude that { h k } has a subsequence { h k l } converging in L qw (Ω) norm for ≤ q
Q,p (Ω) in Theorem 5.1 gives an analogue of Corollary 3.15: Corollary 5.2. Let the hypotheses of Theorem 5.1 hold for H = Lip Q,p (Ω) . Thenthe map T defined by (5.1) is a compact map of W ,pν,µ (Ω , Q ) into L qw (Ω) for ≤ q