Basic Properties of Sobolev Spaces
1.8.1 Main Result
LetAbe a subset of a Banach function space. The setAis called an algebra with respect to multiplication if there is a constant c > 0 such that the inclusionsu∈A, v∈Aimplyuv∈A anduv ≤cuv.
Note that the spaceWpl=Wpl(Rn) is not an algebra whenlp≤n, p >1 or l < n, p= 1. Indeed, if Wpl were an algebra, the following inequalities would occur:
uN1/N
Lp ≤uN1/N
Wpl ≤cuWpl,
whereuis an arbitrary function inWpl andN= 1,2, . . .. By letting N→ ∞, we can obtain
uL∞≤cuWpl.
Clearly, the last inequality is not true for the values p, l mentioned earlier.
Thus,Wplis generally not an algebra with respect to pointwise multiplication.
However, it is possible to describe the maximal algebra contained in Wpl. Namely, the following assertion holds.
Theorem. The subspaceWpl∩L∞ with the norm · Wpl+ · L∞ is the maximal algebra contained inWpl. In particular, the space Wpl is an algebra if lp > n,p >1 or if l≥n, p= 1.
We need an auxiliary multiplicative inequality to prove this theorem.
Lemma 1.Let1≤p≤ ∞andl≥2 be an integer. For anyu∈Wpl∩L∞ and any j= 1, . . . , l−1, the estimate
∇juLlp/j ≤cuL1−∞j/luj/lWl p
(1.8.1) holds with c independent ofu.
We shall deduce Lemma 1 with the aid of another auxiliary inequality.
Lemma 2.If u∈C0∞(Rn), p∈[1,∞),1≤q≤ ∞, and2r−1=p−1+q−1, then
∇uLr ≤c∇2u1/2Lpu1/2Lq , where cis a positive constant depending only on n.
Proof.It is sufficient to assumep >1 andq <∞. Then the extreme cases p= 1 orq=∞follow by lettingptend to 1 or qtend to∞.
We remark that the required inequality is a consequence of the one- dimensional estimate
|u|rdx≤cr0
|u|pdx 2pr
|u|qdx 2qr
(1.8.2) with 2r−1=p−1+q−1 andc0an absolute constant (the integration is taken overR1). Once this estimate has been established, the result follows by inte- grating with respect to the other variables and by applying H¨older’s inequality.
Note that for anyu∈C0∞(R1)
2c1uLr(i)≤ |i|1+r−1−p−1uLp(i)+|i|p−1−r−1−1uLq(i), (1.8.3) wherec1is an absolute positive constant,ian interval, and|i|its length. The last estimate follows from (1.1.18) and the H¨older inequality.
Inequality (1.8.2) is a consequence of the estimate
c1uLr(Δ)≤ u1/2Lpu1/2Lq, (1.8.4) where Δis an arbitrary interval inR1 of finite length. The last is verified as follows.
Fix a positive integerkand introduce the closed intervaliof length|Δ|/k with the same left point as Δ. Let us consider inequality (1.8.3) for this interval i. If the first summand on the right of the inequality is greater than the second, we put i1=i. In this case
cr1
i1
|u|rdx≤ |Δ|
k
1+r−r/p
R1|u|pdx rp
. (1.8.5)
Suppose the first term on the right of (1.8.3) is less than the second. We then increase the intervalileaving the left end point fixed until these two terms are
1.8 The Maximal Algebra inWp(Ω) 119 equal (clearly, the equality must take place for someiwith |i|<∞because 1 +r−1−p−1>0). Leti1denote the resulting interval. Then
cr1
i1
|u|rdx≤
i1
|u|pdx 2pr
i1
|u|qdx 2qr
. (1.8.6)
Putting the end point ofi1 to be the initial point of the next interval, repeat this process with the same k. We stop it when the closed finite intervals i1, i2, . . . (each of length at least|Δ|/k) form a covering of the intervalΔ. Note that the covering{i1, i2, . . .}contains at mostkelements, eachissupporting estimates (1.8.5) or (1.8.6) (with i1 replaced byis). Adding these estimates and applying H¨older’s inequality, one arrives at
cr1
Δ
|u|rdx≤k |Δ|
k
1+r−r/p
|u|pdx rp
+
|u|pdx 2pr
|u|qdx 2qr
.
Now (1.8.4) follows by letting k→ ∞ because 1 +r−r/p >1. The proof of
Lemma 2 is complete.
Proof of Lemma 1.It is sufficient to assumep <∞. First letu∈C0∞(Rn).
Ifaj =∇jupl/j, Lemma 2 implies a2j ≤caj−1aj+1 forj = 1, . . . , l−1. By induction on l, we obtainaj ≤ ca10−j/laj/ll , and Lemma 1 is established for smooth functions uwith compact support.
Turning to the general case u∈Wpl∩L∞, we first assume that suppuis bounded and consider a mollification uh ofu with radiush. Sinceuh∞ ≤ cu∞and by Lemma 1 applied touh we obtain
∇juhLlp/j ≤cuhj/lWl
pu1L−∞j/l.
Now passage to the limit as h→ 0 gives the required inequality for u with bounded support.
To conclude the proof of the lemma, we remove the assumption on the boundedness of suppu. To this end we introduce a cutoff functionη∈C0∞(B2) such that 0 ≤ η ≤ 1 and η|B1 = 1. Let ηk(x) = η(x/k), k = 1,2, . . .. An application of Lemma 1 to the functionuηk yields
∇juLlp/j(Bk)≤cuηkj/lWl
pu1L−∞j/l.
It remains to pass to the limit ask→ ∞to establish Lemma 1 in the general
case.
We now give a proof of the theorem stated previously.
Proof of Theorem.LetAbe an algebra contained inWpl.
As we have already seen at the beginning of the sectionA⊂L∞∩Wpl. To show that the spaceA =L∞∩Wpl is an algebra, we let u, v ∈A be arbitrary. Then
∇l(uv)
Lp ≤c
l
k=0
|∇ku| · |∇l−kv|
Lp
≤c
l
k=0
∇kuLlp/k∇l−kvLlp/(l−k)
by H¨older’s inequality. It follows from Lemma 1 that ∇l(uv)
Lp≤c
l
k=0
u(lL−∞k)/l∇luk/lLpvk/lL∞∇lv(lL−pk)/l,
and hence
∇l(uv)
Lp≤c
uL∞∇lvLp+vL∞∇luLp
, (1.8.7)
which does not exceedcuAvA. Thus,Ais an algebra and hence the space Wpl∩L∞ is the maximal algebra contained inWpl.
Iflp > n, p > 1 or l≥n, p= 1, then Wpl ⊂L∞ by Sobolev’s embedding theorem, and the spaceWplis an algebra for thesepandl. This concludes the
proof.
LetΩbe a domain inRn. We may ask whether the spaceWpl(Ω)∩L∞(Ω) is an algebra with respect to pointwise multiplication. Clearly, for l = 1 the answer is affirmative. Since Stein’s extension operator from a domainΩ∈C0,1 is continuous as an operator
Wpl(Ω)∩L∞(Ω)→Wpl Rn
∩L∞ Rn
,
the above question has the affirmative answer for finite sums of domains in C0,1. For example, Ω can be a bounded domain having the cone property.
However, it turns out that the space Wpl(Ω)∩L∞(Ω) is generally not an algebra.
1.8.2 The Space W22(Ω)∩L∞(Ω) Is Not Always a Banach Algebra Here we give an example of a bounded planar domain Ωsuch thatW22(Ω)∩ L∞(Ω) is not an algebra.
LetΩ be the union of the rectangle P ={(x, y) : x∈ (0,2), y ∈ (0,1)}, the squares Pk with edge length 2−k and the passages Sk of height 2−k and of width 2−αk, k= 1,2, . . .,α >1 (see Figs. 15and16). Define u= 0 onP,
1.8 The Maximal Algebra inWp(Ω) 121
Fig. 15. Fig. 16.
u(x, y) = 23k/2(y−1)2 onSk, k= 1,2, . . ., andu(x, y) = 2k/2(2(y−1)−2−k) onPk fork≥1. Straightforward calculations show that
∇2u2L2(Sk)= 22+(2−α)k, |u| ≤3, ∇2
u2
L2(Pk)= 8.
Thus, ifα >2, thenu∈W22(Ω)∩L∞(Ω), butu2∈/W22(Ω).
1.8.3 Comments to Sect.1.8
A general form of the inequality obtained in Lemma 1.8.1/1 is due to Gagliardo [300] and Nirenberg [640]. The proof of Lemma 1.8.1/2 follows the paper by Nirenberg [640] where it was also shown that L∞∩Wpl is an algebra.
A counterexample in Sect. 1.8.2 is taken from the paper by Maz’ya and Netrusov [572].