An Improved Compact Embedding Theorem for Degenerate Sobolev Spaces
AAN IMPROVED COMPACT EMBEDDING THEOREM FOR DEGENERATESOBOLEV SPACES
DARIO D. MONTICELLI AND SCOTT RODNEY
Abstract.
This short note investigates the compact embedding of degenerate matrix weightedSobolev spaces into weighted Lebesgue spaces. The Sobolev spaces explored are defined as theabstract completion of Lipschitz functions in a bounded domain Ω with respect to the norm: (cid:107) f (cid:107) QH ,p ( v,µ ;Ω) = (cid:107) f (cid:107) L pv (Ω) + (cid:107)∇ f (cid:107) L pQ ( µ ;Ω) where the weight v is comparable to a power of the pointwise operator norm of the matrix valuedfunction Q = Q ( x ) in Ω. Following our main theorem, we give an explicit application wheredegeneracy is controlled through an ellipticity condition of the form w ( x ) | ξ | p ≤ ( ξ · Q ( x ) ξ ) p/ ≤ τ ( x ) | ξ | p for a pair of p -admissible weights. We also give explicit examples demonstrating the sharpness ofour hypotheses. Introduction
In the study of possibly degenerate elliptic partial differential equations of second order, exis-tence of weak solutions, be they variational or otherwise created, the compactness of embeddingsof degenerate matrix weighted Sobolev spaces into Lebesgue spaces plays an important role, seefor example [MR] and [CW]. Our work here improves the local results found in [CRW, section 3],specifically [CRW, corollary 3.20]. The perspective we take in this work was already thought ofin [R] and considers the problem in terms of Sobolev spaces constructed with respect to comple-tions of Lipschitz functions with respect to certain weighted/unweighted Lebesgue norms. Theadvantage of this is that we do not require any notion of a Myers-Serrin H = W result.The plan of this paper is as follows. In the introduction below, we describe our results in thecontext of the degenerate Sobolev spaces defined in Definition 1.1. The principal result of thispaper is Theorem 1.4 with others deduced as corollaries or consequences of slight modificationto the proof of Theorem 1.4. The geometric conditions guiding our work together with a carefuldescription of the required Sobolev and Poincar´e inequalities with related constructions are givenin section 2. Section 3 contains the proofs of our main results. In section 3 we give an example ofour main result applied in the context where degeneracy is encoded by a pair of weights admissiblein the sense of [ChW]. Section 4 contains a counter example demonstrating that in our setting,a global compact embedding may not hold under the hypotheses of local Sobolev and Poincar´einequalities.Recall that a normed linear space X is compactly embedded in a normed linear space Y if thereis a mapping P : X → Y so that given any bounded sequence { x i } i ⊂ X , the image sequence {P ( x i ) } i contains a convergent subsequence in Y . In the context of studying regularity of weak Date : March, 2018.1991
Mathematics Subject Classification.
Key words and phrases. degenerate Sobolev spaces, Sobolev inequality, p -Laplacian.D. D. Monticelli is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilit`a e le loro Ap-plicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and is supported by the PRIN-2015KB9WPT Grant “Variational methods with applications to problems in mathematical physics and geometry”.S. Rodney is supported by the Canadian NSERC Discovery Grant “Degenerate Elliptic Equations: Regularity ofweak solutions with applications”. a r X i v : . [ m a t h . A P ] A ug DARIO D. MONTICELLI AND SCOTT RODNEY solutions to possibly non-linear PDEs, the normed linear space X is the possibly degeneratematrix weighted Sobolev space QH ,p ( v, µ ; Ω). To be precise, let Ω be a domain (bounded openand connected subset) of R n and let µ be a regular measure on Ω absolutely continuous withrespect to Lebesgue measure. Fix also a µ -measurable function Q : Ω → S n (each matrix entryis µ -meas.) taking values in the collection S n of all non-negative definite self adjoint matrices.Given any 1 ≤ p < ∞ , we consider v associated to Q given by v ( x ) = (cid:107) Q ( x ) (cid:107) p/ where (cid:107) Q ( x ) (cid:107) op = sup | ξ | =1 , ξ ∈ R n | Q ( x ) ξ | is the operator norm of the matrix Q ( x ). We will use at all times the a weight v on Ω (that is, anonnegative locally integrable (w.r.t. µ ) function on Ω) such that there exists c > v ( x ) ≤ c v ( x ); x ∈ Ω . Consider now the collection
Lip (Ω) of all those locally Lipschitz functions with compact supportin Ω. We define a norm on this collection by setting (cid:107) f (cid:107) QH ,p ( v,µ ;Ω) = (cid:107) f (cid:107) L pv ( µ ;Ω) + (cid:107) (cid:112) Q ∇ f (cid:107) L p ( µ ;Ω) (1.1) = (cid:18) ˆ Ω | f | p v dµ (cid:19) /p + (cid:18) ˆ Ω (cid:12)(cid:12)(cid:12)(cid:112) Q ∇ f (cid:12)(cid:12)(cid:12) p dµ (cid:19) /p . Note that the norm is well defined for any f ∈ Lip (Ω) since √ Q ∈ L p ( µ ; Ω). Note that for such f , ∇ f exists µ -a.e. by the Rademacher-Stepanov theorem. Definition 1.1.
For ≤ p < ∞ , the Sobolev space QH ,p ( v, µ ; Ω) is defined as the completionof Lip (Ω) with respect to the norm (1.1) . The Sobolev space QH ,p ( v, µ ; Ω) is the completion of Lip Q (Ω) with respect to the same norm (1.1) where Lip Q (Ω) is the collection of all those locallyLipschitz functions f defined in Ω for which (cid:107) f (cid:107) QH ,p ( v,µ ;Ω) < ∞ . It is clear that QH ,p ( v, µ ; Ω) ⊂ QH ,p ( v, µ ; Ω) . Remark 1.2.
Note that (1.1) and the semi-norm (cid:18) ˆ Ω (cid:12)(cid:12)(cid:12)(cid:112) Q ∇ f (cid:12)(cid:12)(cid:12) p dµ (cid:19) /p are not in general equivalent on QH ,p ( v, µ ; Ω) unless a global Sobolev-type inequality of the form ˆ Ω | f | p vdµ ≤ C ˆ Ω (cid:12)(cid:12)(cid:12)(cid:112) Q ∇ f (cid:12)(cid:12)(cid:12) p dµ holds for every f ∈ Lip (Ω) . Although QH ,p ( v, µ ; Ω) is a collection of equivalence classes of Lip Q (Ω) sequences Cauchy withrespect to the norm (1.1) we will take the sound perspective of [CRW], [MRW], [MRW1], and[SW2] by identifying both QH ,p ( v, µ ; Ω) and QH ,p ( v, µ ; Ω) with a closed subset of L pv ( µ ; Ω) ×L pQ ( µ ; Ω) using a natural isometry; see [CRW] for more details. Remark 1.3.
The space L pQ ( µ ; Ω) is the collection of all µ -measurable vector valued functions (cid:126)k for which (cid:107) (cid:126)k (cid:107) L pQ ( µ ;Ω) = (cid:18) ˆ Ω (cid:12)(cid:12)(cid:12)(cid:112) Q ( x ) (cid:126)k ( x ) (cid:12)(cid:12)(cid:12) p dµ (cid:19) /p < ∞ In the case when µ is Lebesgue measure, completeness of L pQ ( µ ; Ω) is found in [CRR] with thecase p = 2 first treated in [SW2] . When µ is a general regular measure absolutely continuous withrespect to Lebesgue measure, completeness can be shown using the same techniques with minormodifications. N IMPROVED COMPACT EMBEDDING THEOREM FOR DEGENERATE SOBOLEV SPACES 3
With Remark 1.3, for the rest of this article we denote elements of QH ,p ( v, µ ; Ω) as pairs ( f, (cid:126)g )for which there is a sequence { f j } ∈ Lip Q (Ω) so that f j → f in L pv ( µ ; Ω) , and ∇ f j → (cid:126)g in L pQ ( µ ; Ω) . Our main results examine the compactness of the projection mapping π : QH ,p ( v, µ ; Ω) → L rv ( µ ; E ) defined by π ( (cid:126) u ) = π (( u, ∇ u )) = u for any E (cid:98) Ω. We now state our main theorem leaving the technical definitions to the nextsection.
Theorem 1.4.
Let < p < ∞ and ρ be a quasimetric defined in Ω whose open balls satisfyDefinition 2.1. Then, QH ,p ( v, µ ; Ω) is compactly embedded in L qv ( µ ; E ) for any E (cid:98) Ω and q ∈ [1 , pσ ) provided the pair (Ω , ρ ) satisfies each of the following. (1) (Ω , ρ ) admits a local Poincar´e inequality of order p (2) (Ω , ρ ) admits a local Sobolev property of order p and gain factor σ > . Given any open subdomain E (cid:98) Ω, extending Lipschitz functions with compact support in E byzero allows one to consider QH ,p ( v, µ ; E ) as a subspace of QH ,p ( v, µ ; Ω). This leads immediatelyto the following corollary of Theorem 1.4. Corollary 1.5.
Let < p < ∞ and E (cid:98) Ω . Then, QH ,p ( v, µ ; E ) is compactly embedded in L qv ( µ ; E ) for every q ∈ [1 , pσ ) provided the pair (Ω , ρ ) satisfies each of the following. (1) (Ω , ρ ) admits a local Poincar´e inequality of order p . (2) (Ω , ρ ) admits a local Sobolev property of order p and gain factor σ > . We also mention a result that is a consequence of the proof of Theorem 1.4; this will be clearlypointed out in section 3.
Theorem 1.6.
Assume the hypotheses of Theorem 1.4 omitting item (2). Then, QH ,p ( v, µ ; Ω) is compactly embedded in L qv ( µ ; Ω) for ≤ q < p . Further, given any E (cid:98) Ω , QH ,p ( v, µ ; Ω) iscompactly embedded in L qv ( µ ; E ) for ≤ q ≤ p . As a last result we give conditions similar to those of Theorem 1.4 under which one obtainscompact embedding on all of Ω, not only on E (cid:98) Ω. Theorem 1.7.
Let the hypotheses of Theorem 1.4 hold with the exception that we replace item(2) with (2*) (Ω , ρ ) admits a global Sobolev property of order p and gain σ > ; see Definition 2.7.Then both QH ,p ( v, µ ; Ω) and QH ,p ( v, µ ; Ω) are compactly embedded in L qv ( µ ; Ω) for any q ∈ [1 , pσ ) . While we will not prove this theorem explicitly, the result is gleaned from Remark 3.1 and aninterpolation inequality (cid:107) g (cid:107) L qv ( µ ;Ω) ≤ (cid:107) g (cid:107) λL v ( µ ;Ω) (cid:107) g (cid:107) − λL pσv ( µ ;Ω) valid for any g ∈ L pσv ( µ ; Ω).In section 4 we present an application of our results to degenerate Sobolev spaces where de-generacy is controlled by p -admissible weights in Ω. See section 4 for complete details. Theorem 1.8.
Fix a bounded domain Ω of R n . Let w ≤ τ be a pair of p -admissible weights,for some < p < + ∞ and let Q ( x ) be a non-negative definite matrix function that satisfies theellipticity condition w | ξ | p ≤ (cid:12)(cid:12)(cid:12)(cid:112) Q ( x ) ξ (cid:12)(cid:12)(cid:12) p ≤ τ | ξ | p . DARIO D. MONTICELLI AND SCOTT RODNEY
Then, there is a q > p so that QH ,p ( τ, dx ; E ) is compactly embedded in L rτ ( E ) for all ≤ r < q and E (cid:98) Ω . Further, QH ,p ( τ, dx ; Ω) is compactly embedded in L rτ ( E ) for ≤ r < q and any E (cid:98) Ω . Preliminaries
We begin this section by recalling the quasimetric structure upon which our result is built. Weassume there is a quasimetric ρ on Ω. That is, there is a κ ≥ x, y, z ∈ Ω( i ) ρ ( x, y ) ≥ x = y ( ii ) ρ ( x, y ) = ρ ( y, x )( iii ) ρ ( x, y ) ≤ κ ( ρ ( x, z ) + ρ ( z, y )) . (2.1)Given x ∈ Ω and r >
0, we denote the ρ -ball centered at x with radius r by B ( x, r ) = { y ∈ Ω : ρ ( x, y ) < r } . We require that ρ -balls are open which is equivalent to the conditionlim y → x ρ ( x, y ) = 0for every x ∈ Ω. Moreover, we require that for every x ∈ Ω there is a δ = δ ( x ) > B ( x, r ) ⊂ Ωfor any 0 < r < δ . As in [CRW], we will not need to assume that the family { B ( x, r ) } x ∈ Ω ,r> admits a doubling measure but we do require a local geometric doubling condition. Definition 2.1.
A quasimetric space (Ω , ρ ) is locally geometrically doubling if given any compactsubset K of Ω , there is a δ > so that < s ≤ r < δ and x ∈ K imply that B ( x, r ) may containat most C ( r/s ) centers of disjoint ρ -balls of radius s ; here C : (0 , ∞ ) → (0 , ∞ ) is independent of K . Remark 2.2.
This condition is weaker than the existence of a locally doubling measure for ρ -balls.We refer the reader to [HyM] for further details and discussions. The local geometric doubling condition is used to establish the following lemma giving coveringsof compact sets by ρ -balls with finite overlaps. We omit the proof here and point the reader tothe proof of [CRW, lemma 3.12]. Lemma 2.3.
Let K be a compact subset of Ω and c ≥ . Then, there are positive constants δ = δ ( K, κ, c ) and P = P ( κ, c ) so that for any < r < δ there is a finite collection of ρ -balls { B ( x j , r ) } Nj =1 , each centred in K , that satisfies (i) K ⊂ N (cid:91) j =1 B ( x j , r ) ⊂ N (cid:91) j =1 B ( x j , c r ) ⊂ Ω(ii) N (cid:88) j =1 χ B ( x j ,c r ) ( x ) ≤ P for any x ∈ N (cid:91) j =1 B ( x j , r )Associated to our collections of ρ -balls are the Sobolev and Poincar´e inequalities that form themain hypotheses of Theorem 1.4. Definition 2.4.
We say that (Ω , ρ ) supports a local Poincar´e property of order p if there is a c ≥ such that given any compact subset K of Ω and (cid:15) > , there is a δ > so that < r < δ and x ∈ K give (cid:107) f − f B ( x,r ) (cid:107) L pv ( µ ; B ( x,r )) < (cid:15) (cid:107) ( f, ∇ f ) (cid:107) QH ,p ( v,µ ; B ( x,c r )) (2.2) for any f ∈ Lip loc (Ω) , where f B = v ( B ) ´ B f v dµ . N IMPROVED COMPACT EMBEDDING THEOREM FOR DEGENERATE SOBOLEV SPACES 5
Remark 2.5.
Inequality (2.2) may feel unfamiliar. The reader may be more familiar with thestandard Q -weighted Poincar´e inequality: (cid:18) v ( B ) ˆ B | f − f B | p v dµ (cid:19) /p ≤ Cr (cid:18) µ ( c B ) ˆ c B (cid:12)(cid:12)(cid:12)(cid:112) Q ∇ f (cid:12)(cid:12)(cid:12) p dµ (cid:19) /p holding for any f ∈ Lip ( B ) . It is not difficult to see that this inequality is enough to ensureDefinition 2.4 provided lim r → sup x ∈ Ω (cid:20) r p v ( B ( x, r )) µ ( B ( x, C r )) (cid:21) = 0 . (2.3) Definition 2.6.
We say that (Ω , ρ ) supports a local Sobolev property of order p and gain σ ifthere is a σ ≥ so that given any compact K ⊂ Ω one can choose δ > with the property thatif B = B ( x, r ) is centred in K and of radius < r < δ then (cid:18) ˆ B | f | pσ v dµ (cid:19) /pσ ≤ C ( B ) (cid:107) ( f, ∇ f ) (cid:107) QH ,p ( v,µ ;Ω) (2.4) for every f ∈ Lip ( B ) . Definition 2.7.
We say that (Ω , ρ ) admits a global Sobolev inequality if there is a constant C > so that (cid:18) ˆ Ω | f | pσ v dµ (cid:19) /pσ ≤ C (cid:104) (cid:107) f (cid:107) L pv ( µ ;Ω) + (cid:107) (cid:112) Q ∇ f (cid:107) L p ( µ ;Ω) (cid:105) (2.5) for every f ∈ Lip (Ω) . Proofs
Proof of Theorem 1.4.
We consider first the case q = p . Fix an open set E (cid:98) Ω.Let { (cid:126) u n } n = { ( u n , ∇ u n ) } n be a bounded sequence in QH ,p ( v, µ ; Ω) with upper bound M andlet (cid:15) >
0. Given 0 < r < δ = min { δ , δ } , Lemma 2.3 provides a finite collection of ρ -balls { B ( x j , r ) } Nj =1 satisfying (i) and (ii) of Lemma 2.3. Further, for each 1 ≤ j ≤ N , we have (cid:107) f − f B ( x,r ) (cid:107) L pv ( µ ; B ( x,r )) < (cid:15) (cid:107) ( f, ∇ f ) (cid:107) QH ,p ( v,µ ; B ( x,c r )) for any f ∈ Lip loc (Ω) by Definition 2.4.In order to show { u n } is Cauchy in L pv ( µ ; E ) we estimate N (cid:88) j =1 ˆ B j | u m − u n | p v dµ ≤ C p N (cid:88) j =1 (cid:104) ˆ B j (cid:12)(cid:12) u m − u n − ( u n − u m ) B j (cid:12)(cid:12) p v dµ (3.1) + (cid:12)(cid:12) ( u m − u n ) B j (cid:12)(cid:12) p v ( B j ) (cid:105) = C p ( I + II )where B j = B ( x j , r ) and, for an integrable function g , g B = ffl B gv dµ is the v -average of g .We estimate I and II separately using different techniques. Beginning with I , we assume that r < δ = δ ( E ) and apply the Poincar´e inequality (2.2) to find I ≤ (cid:15) p N (cid:88) j =1 (cid:107) ( u n − u m , ∇ ( u m − u n )) (cid:107) pQH ,p ( v,µ ; B ( x j ,c r )) ≤ (cid:15) p P (cid:107) ( u n − u m , ∇ ( u m − u n )) (cid:107) pQH ,p ( v,µ ;Ω) (3.2) ≤ p P M p (cid:15) p DARIO D. MONTICELLI AND SCOTT RODNEY where P is the overlap constant for our collection as in Lemma 2.3.To estimate item II we use weak convergence. Indeed, since { u n } is a bounded sequence in L pv ( µ ; Ω), it admits a weakly convergent subsequence that we denote by { u n } to preserve theindex. As v ( B j ) is finite, the characteristic function χ B j ( x ) ∈ L p (cid:48) v ( µ ; Ω) for every j . Thus, thereis T ∈ N so that m, n ≥ T gives II ≤ N (cid:88) j =1 v − p ( B j ) (cid:12)(cid:12)(cid:12)(cid:12) ˆ Ω χ B j ( u m − u n ) v dµ (cid:12)(cid:12)(cid:12)(cid:12) p < (cid:15) p (3.3)Combining (3.2) and (3.3) with (3.1) we find (cid:107) u m − u n (cid:107) pL pv ( µ ; E ) < C(cid:15) p (1 + 2 p M p P )(3.4)when m, n ≥ T . This establishes convergence of our subsequence in L pv ( µ ; E ) and, by H¨older’sinequality, also in L qv ( µ ; E ) for 1 ≤ q ≤ p . This establishes Theorem 1.4 for the range 1 ≤ q ≤ p . Remark 3.1.
It is not difficult to now show that QH ,p ( v, µ ; Ω) is compactly embedded in L qv ( µ ; Ω) for the range ≤ q < p . Indeed, fix η > and assume that our set E satisfies v (Ω \ E ) < η . Then,for our subsequence { u n } constructed above, H¨older’s inequality and boundedness in QH ,p ( v, µ ; Ω) show that for any j, k ∈ N , (cid:107) u j − u k (cid:107) L v ( µ ;Ω) = (cid:107) u j − u k (cid:107) L v ( µ ; E ) + (cid:107) u j − u k (cid:107) L v ( µ ;Ω \ E ) ≤ (cid:107) u j − u k (cid:107) L pv ( µ ; E ) v ( E ) /p (cid:48) + (cid:107) u j − u k (cid:107) L pv ( µ ;Ω \ E ) v (Ω \ E ) /p (cid:48) ≤ (cid:107) u j − u k (cid:107) L pv ( µ ; E ) v ( E ) /p (cid:48) + 2 M η /p (cid:48) . (3.5) Since { u j } is Cauchy in L pv ( µ ; E ) , choosing j, k sufficiently large shows that { u n } is Cauchy in L v ( µ ; Ω) . That is, we have shown QH ,p ( v, µ ; Ω) is compactly embedded in L v ( µ ; Ω) . Interestingly,we also conclude the same for L qv ( µ ; Ω) when ≤ q < p through an appeal to H¨older’s inequality.Given < q < p , we may choose λ ∈ (0 , so that (cid:107) u j − u k (cid:107) L qv ( µ ;Ω) ≤ (cid:107) u j − u k (cid:107) λL v ( µ ;Ω) (cid:107) u j − u k (cid:107) − λL pv ( µ ;Ω) ≤ (2 M ) − λ (cid:107) u j − u k (cid:107) λL v ( µ ;Ω) . (3.6) This argument completes the proof of Theorem 1.6.
We now turn our attention to the range p < q < pσ . With 0 < r < δ as above, cover E withEuclidean balls D ( x, s ) where s = s ( x ) is chosen so that D ( x, s ) ⊂ B . By compactness, we mayselect { D ( x j , s j ) } N j =1 that covers E . Let { ϕ j } be a partition of unity subordinate to this coverand let f ∈ Lip loc (Ω). With D ( x j , s j ) ⊂ B j = B ( x j , r ) we see from the Sobolev inequality (2.4)that ˆ E | f | pσ v dµ ≤ C p (cid:88) j ˆ B j | f ϕ j | pσ v dµ ≤ C p (cid:88) j C ( B j ) (cid:34) ˆ B j | f | p v dµ + ˆ B j (cid:12)(cid:12)(cid:12)(cid:112) Q ∇ ( f ϕ j ) (cid:12)(cid:12)(cid:12) p dµ (cid:35) σ (3.7)since 0 ≤ ϕ j ( x ) ≤ j . The second term splits with integrand bounded above by C ∗ (cid:104)(cid:12)(cid:12)(cid:12)(cid:112) Q ∇ f (cid:12)(cid:12)(cid:12) p + | f | p v (cid:105) N IMPROVED COMPACT EMBEDDING THEOREM FOR DEGENERATE SOBOLEV SPACES 7 since v ≥ c − (cid:107) Q ( x ) (cid:107) p/ op and where C ∗ is a constant independent of f . Since the sum is finite, wefind a constant ˜ C = ˜ C ( E ) so that (cid:107) f (cid:107) L pσv ( µ ; E ) ≤ ˜ C (cid:107) ( f, ∇ f ) (cid:107) QH ,p ( v,µ ;Ω) (3.8)for any f ∈ Lip loc (Ω); by density also for any pair ( g, ∇ g ) ∈ QH ,p ( v, µ ; Ω). Thus, boundednessin L pσv ( µ ; E ) of our sequence { u n } is established. Given p < q < pσ , we may choose λ ∈ (0 ,
1) sothat (cid:107) u j − u k (cid:107) L qv ( µ ; E ) ≤ (cid:107) u j − u k (cid:107) λL pv ( µ ; E ) (cid:107) u j − u k (cid:107) − λL pσv ( µ ; E ) ≤ C (2 M ) − λ (cid:107) u j − u k (cid:107) λL pv ( µ ; E ) and we conclude that { u n } is Cauchy in L qv ( µ ; E ). This completes the proof of Theorem 1.4. (cid:3) Proof of Corollary 1.5.
Let E (cid:98) Ω and { (cid:126) u n } = { u n , ∇ u n } be a bounded sequence in QH ,p ( v, µ ; E ). Since each element (cid:126) u n may be viewed as an equivalence class of Cauchy sequencesof Lip ( E ) functions, we may choose a representative sequence { g nm } m ⊂ Lip ( E ) converging to u n in QH ,p ( v, µ ; E ) norm. For each m , set G nm = (cid:26) g nm ( x ) if x ∈ E x ∈ Ω \ E (3.9)The resulting sequence of extended functions { G nm } is Cauchy in QH ,p ( v, µ ; Ω) and convergesto (cid:126) w n = ( w n , ∇ w n ) ∈ QH ,p ( v, µ ; Ω) with (cid:107) u n − w n (cid:107) L pv ( µ ; E ) = (cid:107)∇ u n − ∇ w n (cid:107) L pQ ( µ ; E ) = (cid:107) (cid:126) u n − (cid:126) w n (cid:107) QH ,p ( v,µ ; E ) = 0 . From this we can also see that u n = w n in L pσv ( µ ; E ). Since our new sequence { (cid:126) w n } is bounded in QH ,p ( v, µ ; Ω), Theorem 1.4 provides a subsequence of { w n } (that we refer to as { w n } to preservethe index) that is Cauchy in L qv ( µ ; E ) for each q ∈ [1 , pσ ). Since (cid:107) u j − u k (cid:107) L qv ( µ ; E ) = (cid:107) w j − w k (cid:107) L qv ( µ ; E ) for every j, k and q ∈ [1 , pσ ], we find { u n } is Cauchy in L qv ( µ ; E ) for 1 ≤ q < pσ . We now concludethat QH ,p ( v, µ ; E ) is compactly embedded in L qv ( µ ; E ) for each 1 ≤ q < pσ completing the proofof Corollary 1.5. (cid:3) Application to Two Weight Degenerate Problems
As an application to Theorem 1.4, we present compact embeddings for Sobolev spaces withdegeneracy controlled by admissible weights. Given p >
1, two weights w ≤ τ on Ω are called p -admissible in Ω if each of the following conditions are met.(1) τ is doubling for the collection of Euclidean balls with center in Ω. That is, there is aconstant C so that given x ∈ Ω and r > τ ( D ( x, r )) = ˆ D ( x, r ) τ dz ≤ C ˆ D ( x,r ) τ dz = Cτ ( D ( x, r ))(2) w ∈ A p (Ω). For 1 < p < ∞ , the Muckenhoupt class of weights A p (Ω) is the collection ofall those non-negative functions ϕ ∈ L (Ω) for whichsup D (cid:18) | D | ˆ D ϕ dz (cid:19) (cid:18) | D | ˆ B ϕ − p dz (cid:19) p − < ∞ where the supremum is taken over all Euclidean balls D = D ( x , r ) = { x ∈ Ω : | x − x | C > q > p so that for 0 < s ≤ r and x ∈ Ω, sr (cid:18) τ ( D ( x, s )) τ ( D ( x, r )) (cid:19) /q ≤ C (cid:18) w ( D ( x, s ) w ( D ( x, r ) (cid:19) /p (4.1)where here for a weight ν , ν ( D ) = ´ D ν ( x ) dx . For the reader unfamiliar with such objects, powerweights τ ( x ) = | x | t form an excellent example category; [CMN] is also a good reference for thisdeep subject. [CMN] draws from [CW] and other classic works in the area to demonstrate thatfor admissible weights w ≤ τ there are constants C > , q > p so that for any Euclidean ball D = D ( x, r ) (cid:98) Ω one has(1) the local Poincar´e inequality (cid:18) τ ( D ) ˆ D | f − f D ; τ | q τ dx (cid:19) /q ≤ Cr (cid:18) w ( D ) ˆ D |∇ f | p w dx (cid:19) /p for any f ∈ Lip loc (Ω), and(2) the local Sobolev inequality (cid:18) τ ( D ) ˆ D | g | q τ dx (cid:19) /q ≤ Cr (cid:18) w ( D ) ˆ D |∇ g | p w dx (cid:19) /p for each g ∈ Lip ( D ).These inequalities are used to study second order degenerate elliptic problems ( Xu = ϕ ) withprincipal part of X given by a matrix weighted p -Laplacian: Lu = Div (cid:18)(cid:12)(cid:12)(cid:12)(cid:112) Q ∇ u (cid:12)(cid:12)(cid:12) p − Q ∇ u (cid:19) . The symmetric non-negative definite n × n matrix Q is assumed to satisfy the degenerate ellipticcondition w ( x ) | ξ | p ≤ (cid:12)(cid:12)(cid:12)(cid:112) Q ( x ) ξ (cid:12)(cid:12)(cid:12) p ≤ v ( x ) | ξ | p ≤ τ ( x ) | ξ | p , a.e. x ∈ Ω , ξ ∈ R n (4.2)where v = (cid:107) Q (cid:107) p/ op is the p th power of the operator norm of (cid:112) Q ( x ). Weak solution spaces forDirichlet and Neumann problems associated to such equations are the matrix weighted Sobolevspaces QH ,p ( τ, dx ; Ω) and QH ,p ( τ, dx ; Ω), as defined in § v ≡ τ . Because of the ellipticitycondition (4.2), we find the Sobolev and Poincar´e inequalities (2.4) and (2.2) of Definitions 2.6and 2.4. Indeed, to see that the Poincar´e holds, let f ∈ Lip loc (Ω), E a compact subset of Ωand fix a Euclidean ball D = D ( x, s ) with s < r = dist ( E, ∂ Ω). Using the two weight Poincar´eestimate we see (cid:18) ˆ D | f − f D | p τ dx (cid:19) /p ≤ τ ( D ) p (cid:18) τ ( D ) ˆ D | f − f D | q τ dx (cid:19) /q ≤ Cs τ ( D ) p w ( D ) p (cid:18) ˆ D | (cid:112) Q ∇ f | p dx (cid:19) /p . The balance condition (4.1) with r = r = dist( E, ∂ Ω) gives a positive constant C so that s (cid:34) τ ( D ) /q w ( D ) /p (cid:35) ≤ C for every x ∈ E . As a result, we see thatlim s → sup x ∈ E (cid:34) Cs τ ( D ) p w ( D ) p (cid:35) ≤ C lim s → sup x ∈ E τ ( D ( x, s )) q − pqp = 0 as q > p N IMPROVED COMPACT EMBEDDING THEOREM FOR DEGENERATE SOBOLEV SPACES 9 and we conclude that Definition 2.4 holds for Euclidean balls D ( x, r ) with c = 1. The argumentgiving Definition 2.6 is similar and left to the reader. This concludes the proof of Theorem 1.8.5. Failure of Global Compact Embedding Under Local Hypotheses In this section we give an example of a matrix function Q satisfying the conditions of Theorem1.4 on a specific domain of R n where the embedding π : QH ,p ( v, µ ; Ω) ⊂ QH ,p ( v, µ ; Ω) → L qv ( µ ; Ω)fails to be compact for any q > p > 1, where v ≡ dµ = dx in Ω. In particular, there is noembedding at all. A posteriori, this is because no global Sobolev inequality with gain σ > Example 1. Let n ≥ and set Ω = (0 , × ... × (0 , 1) = (0 , n ; the n -dimensional unit cube.Let p > , q > p and choose β ∈ [ q , p ) . Define Q ( x ) = Diag (cid:2) x , ..., (cid:3) . Since Q ( x ) is uniformly elliptic away from the boundary ∂ Ω , standard local Sobolev and Poincar´einequalities of order p hold on Euclidean balls contained in Ω , both with gain σ = nn − p when n > p or any σ > if n ≤ p .Consider the function u : Ω → R n defined by u ( x ) = (cid:40) ( x − β − ψ (ˆ x ) if < x < − β , if − β ≤ x < with gradient ∇ u ( x ) = (cid:40) ( − βx − β − ψ (ˆ x ) , ( x − β − ∇ ˆ x ψ (ˆ x )) if < x < − β , if − β ≤ x < where ˆ x = ( x , ..., x n ) ∈ (0 , n − and ψ ∈ C ∞ ((0 , n − ) .From these definitions it is easy to check that u ∈ L p (Ω) , (cid:12)(cid:12) √ Q ∇ u (cid:12)(cid:12) ∈ L p (Ω) and that u / ∈ L q (Ω) .We will now demonstrate a sequence of Lipschitz function with compact support in Ω that convergeto the pair ( u, ∇ u ) in the QH ,p (Ω) norm thus showing that ( u, ∇ u ) ∈ QH ,p (Ω) ⊂ QH ,p (Ω) .For j ∈ N , define the Lip (Ω) function (see Figure 1 for u j ( t ) in dimension n = 1 ) Figure 1. u j ( t ) u j ( x ) = if < x ≤ j , (cid:20)(cid:18)(cid:16) j (cid:17) β − (cid:19) j ( x − j ) (cid:21) ψ (ˆ x ) if j < x < j , ( x − β − ψ (ˆ x ) if j ≤ x < − β , if − β ≤ x < . (5.1) defined for j > β .It is not a difficult exercise to show that | u − u j | and (cid:12)(cid:12) √ Q ∇ ( u − u j ) (cid:12)(cid:12) converge to in L p (Ω) . Asa result, { u j } is a Cauchy sequence of Lip (Ω) functions in QH ,p (Ω) norm whose limit is ( u, ∇ u ) .Thus, QH ,p (Ω) (cid:54)⊂ L q (Ω) . That is, the embedding fails and is thus, obviously, not compact. Our next example demonstrates that the lack of a global Sobolev inequality a posteriori causesfailure of compact embedding of QH ,p (Ω) and of QH ,p (Ω) in L p (Ω), with Ω as above. In orderto describe this precisely we first appeal to a one dimensional version. Example 2. Fix I = [0 , , p > and set q ( t ) = t p . For large j ∈ N define the Lipschitz fucntionwith compact support in (0 , (see Figure 2) v j ( t ) = if < t ≤ n + n p +2 ,n p (cid:0) t − n (cid:1) p − n if n + n p +2 < t ≤ n ,n p (cid:0) n − t (cid:1) p − n if n < t ≤ n − n p +2 , if n − n p +2 < t ≤ v (cid:48) j ( t ) = < t ≤ n + n p +2 , p n p (cid:0) t − n (cid:1) p − if n + n p +2 < t ≤ n , − p n p (cid:0) n − t (cid:1) p − if n < t ≤ n − n p +2 , n − n p +2 < t ≤ . (5.3) Figure 2. v j ( t ) N IMPROVED COMPACT EMBEDDING THEOREM FOR DEGENERATE SOBOLEV SPACES 11 Clearly, for each j large enough ˆ v j ( t ) p dt ≤ , and using Jensen’s inequality and that v j ( t ) is concave in the interval ( n + n p +2 , n ) we see that ˆ v pj ( t ) dt = 2 ˆ n n + np +2 (cid:32) n p (cid:18) t − n (cid:19) p − n (cid:33) p dt ≥ (cid:32) ˆ n n + np +2 (cid:32) n p (cid:18) t − n (cid:19) p − n (cid:33) dt (cid:33) p (cid:18) n − n p +2 (cid:19) − p ≥ (cid:18) (cid:18) n − n p +2 (cid:19) (cid:18) n p − n (cid:19)(cid:19) p (cid:18) n − n p +2 (cid:19) − p → − p as j → ∞ . Then, the sequence { v j } is bounded in L p ( I ), it converges to zero pointwise everywhere in I , andit does not admit any subsequence converging in L p ( I ). Moreover, in case p (cid:54) = 2 there is a uniformconstant C = C ( p ) so that ˆ (cid:12)(cid:12)(cid:12)(cid:112) q ( t ) v (cid:48) j ( t ) (cid:12)(cid:12)(cid:12) p dt ≤ C ( p ) n (cid:34) ˆ n n + n p t p (cid:18) t − n (cid:19) − p dt + ˆ n − np +22 n t p (cid:18) n − t (cid:19) − p dt (cid:35) ≤ C ( p ) n − p (cid:34)(cid:18) t − n (cid:19) − p (cid:12)(cid:12)(cid:12)(cid:12) n n + n p − (cid:18) n − t (cid:19) − p (cid:12)(cid:12)(cid:12)(cid:12) n − np +22 n (cid:35) = C ( p ) n − p (cid:20) n − p − n − p (cid:21) ≤ C ( p ) < ∞ for every j .since p > 1. One can easily check the same for the case p = 2 using logarithms. Thus, thesequence is bounded in the one dimensional norm of qH ,p ( I ). Therefore, qH ,p ( I ) ⊂ qH ,p ( I )are continuously but not compactly embedded in L p ( I ).In the n -dimensional case a similar result holds when one chooses Ω = I n , Q ( x ) = Diag[ x p , , ..., ,v ≡ dµ = dx and the sequence of functions u j ( x ) = u j ( x , ˆ x ) = v j ( x ) ψ (ˆ x )for any ψ ∈ C ∞ ((0 , n − ) with ˆ x = ( x , ..., x n ). It is important to note here that no globalSobolev inequality of order p and gain σ > I n for the matrix Q . References [ChW] S. Chanillo and R. L. Wheeden, Weighted Poincar´e and Sobolev Inequalities and Estimates for WeightedPeano Maximal Functions, Amer. J. Math. 107 (5) (1985), pp. 1191-1226.[CMN] D. Cruz-Uribe, K. Moen, and V. Naibo, Regularity of solutions to degenerate p-Laplacian equations, J.Math. Anal. App. volume 401[1] (2013), pp. 458-478[CW] Seng-Kee Chua and R. L. Wheeden, Existence of weak solutions to degenerate p-Laplacian equations andintegral formulas, J. Diff. Eq. Volume 263[12] (2017), pp. 8186-8228.[CRW] Seng-Kee Chua, S. Rodney and R. L. Wheeden, A compact embedding theorem for generalized Sobolevspaces, Pacific J. Math. 265 (2013), pp. 17-57. [HyM] T. Hytnen and H. Martikainen, Non-homogeneous Tb theorem and random dyadic cubes on metric measurespaces, preprint, arXiv:0911.4387, 2009. CHECK!!![MR] D. D. Monticelli and S. Rodney, Existence and Spectral Theory for Weak Solutions of Neumann and DirichletProblems for Linear Degenerate Elliptic Operators with Rough Coefficients, J. Differential Equations Vol. 259[8](2015), pp. 4009-4044.[MRW] D. D. Monticelli, S. Rodney and R. L. Wheeden, Boundedness of weak solutions of degenerate quasilinearequations with rough coefficients, J. Diff. Int. Eq. 25 (2012), pp. 143–200.[MRW1] D. D. Monticelli, S. Rodney and R. L. Wheeden, Harnack’s inequality and H¨older continuity forweak solutions of degenerate quasilinear equations with rough coefficients, Nonlinear Analysis (2015),http://dx.doi.org/10.1016/j.na.2015.05.029.[R] S. Rodney, Existence of Weak Solutions of Linear Subelliptic Dirichlet Problems With Rough Coefficients,Can. J. Math. 64, no. 6 (2012), pp. 1395–1414.[SW2] E. T. Sawyer and R. L. Wheeden, Degenerate Sobolev spaces and regularity of subelliptic equations, Trans.Amer. Math. Soc., 362 (2010), pp. 1869–1906.[CRR] D. Cruz-Uribe (OFS), S. Rodney, and E. Rosta, Poincar´e Inequalities and Neumann Problems for the p -Laplacian, Bul. CMS. 2018. D. D. Monticelli, Dept. of Mathematics, Politecnico di Milano, 20133, Milano Italy E-mail address : [email protected] Scott Rodney, Dept. of Mathematics, Physics and Geology, Cape Breton University, Sydney, NSCanada E-mail address ::