Multiplicative Inequalities

Basic Properties of Sobolev Spaces

1.4 Embedding Theorems of Sobolev Type

1.4.7 Multiplicative Inequalities

k

j=0

sup

B(x,1)

|∇ju| ≤CuV_p^l(B(x,1)).

Applying a dilation with coefficient, we obtain

k

j=0

^j sup

B(x,)

|∇ju| ≤c

l

i=0

^i−n/p∇iuLp(B(x,)). Therefore, forj= 0, . . . , k

sup

B(x,)

|∇ju| ≤c^l−j−n/p∇luL_p(B(x,))+C()u_Vp^l−1_(B(x,)) and thus

k

j=0

sup

G¯

|∇ju| ≤c ^l−k−n/puV_p^l(Rⁿ)+C()u_Vp^l−1_(G),

whereG is theneighborhood of ¯G. Sinceis an arbitrarily small number, it follows by the Lemma that the unit ball inV_p^l(Rⁿ) is compact inV_p^l−1(G).

Thus (c) is proved.

SinceV_p^l(B1) =W_p^l(B1) (see Corollary1.1.11), the last estimate is equivalent to

∇lwL_p(B₂)≤c

∇lwL_p(B₁)+wL_p(B₁)

.

Thus, applying a dilation, we obtain that the function v, mentioned in the statement of the Lemma, admits an extensionv∈C₀^l(B2) such that

∇lvLp(B2)≤c

∇lvLp(B)+⁻¹vLp(B)

. (1.4.38)

Let (l−k)p < n,p >1 orl−k≤n,p= 1. By Theorem1.4.4/1 we obtain

∇kvLt(μ,B2)≤cK^1/t∇lvLp(B2), (1.4.39) wheret=ps/(n−p(l−k)).

In the case (l−k)p = n, p > 1, we let p1 denote a number in [1, p), sufficiently close top. We putt=p1s/(n−p1(l−k)). Then, by Corollary1.4.1,

∇kvL_t(μ,B₂)≤cK^1/t∇lvL_p1(B₂)

≤cK^{1/tn/p1−n/p}∇lvLp(B2). (1.4.40) In the case (l−k)p > n,p≥1 we putt=∞. By Sobolev’s theorem

∇kvL_t(μ,B₂)≤c^l−k−n/p∇lvL_p(B₂). (1.4.41) Combining (1.4.39)–(1.4.41), we obtain

∇kvL_t(μ,B₂)≤c K^{1/tl−k−n/p+s/t}∇lvL_p(B₂). (1.4.42) By H¨older’s inequality,

∇kuL_q(μ,B)≤

μ(B)1/q−1/t

∇kvL_t(μ,B)

≤K^{1/q−1/ts(1/q−1/t)}∇kvLt(μ,B), which along with (1.4.42) gives

∇kvLt(μ,B2)≤cK^{1/qs/q+l−k−n/p}∇lvLp(B2).

Using (1.4.38), we complete the proof.

Theorem. 1. Let μ be a measure in Rⁿ which satisfies the condition (1.4.36)for somes∈[0, n]. Letp≥1 and letk,l be integers,0≤k≤l−1;

s > n−p(l−k) if p >1 and s≥n−l+k if p= 1. Then, for all u∈D, the estimate (1.4.35)holds, where C≤cK^1/q,n/p−l+k < s/q,q≥pand τ = (k−s/q+n/p)/l.

2. If (1.4.35)is valid for allu∈D, thenC≥c K^1/q.

Proof. According to the Lemma, for allx∈Rⁿ and >0,

∇kuL_q(μ,B(x,))

≤c K^{1/qs/q−n/p−k}

^l∇luLp(B(x,))+uLp(B(x,))

. (1.4.43) We fix an arbitrary0>0. If the first term on the right-hand side of (1.4.43) exceeds the second for =0, then we cover a pointx∈suppμby the ball B(x, ). Otherwise we increase until the first term becomes equal to the second. Then the pointxis covered by the ballB(x, ), where

=u^1/l_Lp(B(x,))∇lu⁻_Lp^1/l_(B(x,)). In both cases

∇ku^q_Lq(μ,B(x,))≤cK

^s₀^{−q(n/p−l+k)}∇lu^q_Lp(B(x,)) +∇lu^qτ_Lp(B(x,))u_L^q(1_p(B(x,))^−τ)

. (1.4.44) According to Theorem 1.2.1/1, we can select a subcovering {B⁽ⁱ⁾}i≥1 of finite multiplicity, depending only onn, from the covering{B(x, )}of suppμ.

Summing (1.4.44) over all ballsB⁽ⁱ⁾and noting that

i

a^α_ib^β_i ≤

i

a^α+β_i

α/(α+β) i

b^α+β_i

β/(α+β)

≤

i

ai

α i

bi

β

,

whereai,bi,α, andβ are positive numbersα+β≥1, we arrive at

∇kv^q_Lq(μ)≤cK(^q(n/p₀ ^−l+k)

i

∇lu^p_Lp(B(i))

q/p

+

i

∇lu^p_Lp(B(i))

τ q/p i

u^p_Lp(B(i))

(1−τ)q/p

. Since the multiplicity of the covering{B⁽ⁱ⁾}depends only onn, the right-hand side is majorized by

cK

^s₀^{−q(n/p−l+k)}∇lu^qL_q+∇lu^{τ q}L_pu⁽¹L_p^−τ)q

. Passing to the limit as₀→0, we complete the proof of case 1.

To prove case 2 it is sufficient to insert the functionu(x) = (y₁−x₁)^k× ϕ(⁻¹(x−y)), where ϕ ∈ D(B₂), ϕ = 1 on B₁, into (1.4.35). The result

follows.

Corollary 1.1. Let μbe a measure in Rⁿ such that K₁= sup

x∈Rⁿ,r∈(0,1)

r^−sμ

B(x, r)

<∞, (1.4.45) for some s∈[0, n]. Further let p≥1, let k and l be integers 0≤k ≤l−1;

s > n−p(l−k)if p >1 ands≥n−l+k ifp= 1. Then, for all u∈D,

∇kuLq(μ)≤C₁u^τ_Vl

pu¹_L⁻_p^τ, (1.4.46) where C1≤cK₁^1/q,n/p−l+k < s/q,q≥p, andτ = (k−s/p+n/p)/l.

2. If (1.4.46)is valid for allu∈D, thenC₁≥c K₁^1/q.

Proof. Let {Q⁽ⁱ⁾} denote a sequence of closed cubes with edge length 1 which forms a coordinate grid inRⁿ. LetO⁽ⁱ⁾be the center of the cubeQ⁽ⁱ⁾, O⁽⁰⁾=O, and let 2Q⁽ⁱ⁾be the concentric homothetic cube with edge length 2.

We put ηi(x) =η(x−O⁽ⁱ⁾),whereη∈C₀^∞(2Q⁽⁰⁾),η= 1 onQ⁽⁰⁾.

Applying the Theorem of the present subsection to the functionuηi and to the measure e→μ(e∩Q⁽ⁱ⁾), we obtain

∇k(uη_i)^p

L_q(μ)≤cK₁^p/q∇l(uη_i)^pτ

L_puη₁^p(1_Lp^−τ). Summing over iand using the inequality

ai

p/q

≤ a^p/q_i , whereai≥0, we arrive at (1.4.46).

The second assertion follows by insertion of the functionu, defined at the end of the proof of the Theorem, into (1.4.46).

The next assertion follows immediately from Corollary 1.

Corollary 2. Suppose there exists an extension operator which maps V_p^l(Ω) continuously into V_p^l(Rⁿ) and L_p(Ω) into L_p(Rⁿ) (for instance, Ω is a bounded domain of the class C^0,1). Further, let μ be a measure in Ω¯ satisfying (1.4.45), where s is a number subject to the same inequalities as in Corollary 1. Then for allu∈C^l(Ω)

∇kuL_q(μ,Ω)¯ ≤Cu^τV_p^l(Ω)u_L¹⁻_p(Ω)^τ , (1.4.47) wheren/p−l+k < s/q,q≥p≥1 andτ= (k−s/q+n/p)/l.

2.If for allu∈C^l( ¯Ω)the estimate(1.4.47)holds, then the measureμwith support in Ω¯ satisfies (1.4.45).

1.4.8 Comments to Sect.1.4

Theorem1.4.1/2 is due to D.R. Adams [2, 3]. The proof given above is bor- rowed from the paper by D.R. Adams [3]. The following analog of Corol- lary1.4.1was obtained by Maz’ya and Preobrazhenski [577] and will be proved in Sect. 11.9.

If 1< p < q,lp=n, then the best constantC in uLq(μ)≤CuW_p^l, u∈C₀^∞ is equivalent to

sup

x∈Rⁿ,r∈(0,1)

log2

r ^p−1_p

μ

Br(x)1/q

.

For μ = m_s, i.e., for the s-dimensional Lebesgue measure in R^s, inequal- ity (1.4.4) was proved by Sobolev [712] in the cases=nand by Il’in [394] in the cases < n. They used the integral representation (1.1.10) and the mul- tidimensional generalization of the following Hardy–Littlwood theorem (cf.

Hardy, Littlewood, and P´olya [351]).

If 1< p < q <∞andμ= 1−p⁻¹+q⁻¹, then the operator|x|^−μ∗f with f :R¹→R¹ mapsLp(R¹)continuously intoLq(R¹).

For one particular case, Lieb [496, 497] found an explicit expression for the norm of the operator|x|^−μ∗f,μ∈(0, n), acting on functions of nvariables.

His result can be written as the inequality

Rⁿ

Rⁿ

f(x)g(y)

|x−y|^μ dxdy

≤π¹²Γ((n−μ)/2) Γ(n−μ/2)

Γ(n/2) Γ(n)

^μ−n_n

fL 2n

2n−μgL 2n

2n−μ, (1.4.48) with the equality if and only if f and g are proportional to the function (|x−x₀|²+a²)^(μ−2n)/2, where a∈Randx₀∈Rⁿ.

Theorems1.4.2/1 and1.4.2/2 are due to the author [543, 548]. Inequality u(x)^n/(n−1)dx

(n−1)/n

≤Cn ∇u(x)dx (1.4.49) was proved independently by Gagliardo [299] and Nirenberg [641] using the same method, without discussion of the best value ofCn.

The proof based on the classical isoperimetric inequality (1.4.13) in Rⁿ, which gives the sharp constant (see (1.4.14)), was proposed simultaneously and independently by Federer and Fleming [273] and by Maz’ya [527].

Briefly, the proof by Gagliardo [299] and Nirenberg [640] runs as follows.

One notes that

u(x)≤2⁻¹

R

∂u

∂yi

(x₁, . . . , x_i−1, y_i, x_i+1, . . . , x_n) dy_i for 1≤i≤nand for allu∈C₀^∞. This yields

u(x)^n/(n−1)≤2^n/(1−n)

1≤i≤n

R

∂u

∂y_i dyi

1/(n−1)

.

Integrating successively with respect tox₁,x₂, and so on, and using the gen- eralized H¨older inequality

f₁· · ·f_n−1dμ

≤

1≤j≤n−1

f_jL_pj(μ)

with p₁=p₂ =· · · =p_n−1 =n−1 after every integration, we arrive at the inequality

uL_n/(n−1)≤2⁻¹

1≤i≤n

Rⁿ

∂u

∂xi

dx 1/n

. (1.4.50)

(Note that (1.4.50) is equivalent to the isoperimetric inequality m_n(g)n−1

≤2⁻¹

1≤i≤n

∂g

cos(ν, x_i)ds,

which can be proved by a straightforward modification of the proof of Theo- rem1.4.2/1.)

By the inequality between the geometric and arithmetic means, we obtain the estimate

uL_n/(n−1) ≤(2n)⁻¹

Rⁿ1≤i≤n

∂u

∂xi

dx. (1.4.51)

Its optimality is checked by a sequence of mollifications of the characteristic function of the cube{x: 0≤xi ≤1}.

Obviously, (1.4.51) implies (1.4.49) withCn= (2n^1/2)⁻¹, but this value of Cn is not the best possible.

It is worth mentioning that Gagliardo’s paper [299] contains a more general argument based on the same idea which leads to the embedding ofW_p^l(Ω) to Lq(Ωs), whereΩsis ans-dimensional surface situated in ¯Ω.

Gromov [325] gave a proof of (1.4.49) where the integration is taken over a normed n-dimensional space X. The value Cn = n⁻¹ in Gromov’s proof is the best possible provided the unit ball in X has volume 1. This proof is based on the so-called increasing triangular mappings which were apparently introduced to Convex Geometry by Knothe [437], who used them to obtain some generalizations of the geometric Brunn–Minkowski inequality. Such a mapping transports a given probability measure on the Euclidean space to another one, and under mild regularity assumptions, it is defined in a unique

way. These mappings have a simple description in terms of conditional prob- abilities, and were apparently known in Probability Theory before Knothe’s work.

In time it became clear that triangular mappings may be used to obtain various geometric and analytic inequalities. Bourgain [137] applied them to prove Khinchin-type (i.e., reverse H¨older) inequalities for polynomials of a bounded degree over high-dimensional convex bodies, with constants that are dimension free.

There is a discussion of this method in Bobkov [111, 112], where triangular mappings were used to study geometric inequalities of dilation type.

Using wavelet decompositions, weak estimates, and interpolation, Cohen, DeVore, Petrushev, and Xu [208] forn= 2, and Cohen, Meyer and Oru [209]

forn≥2, obtained the following improvement of (1.4.49):

uL n

n−1 ≤C∇u⁽ⁿ_L1^−1)/nu^1/n_B , (1.4.52) where B is the distributional Besov space B_∞¹⁻_,ⁿ_∞. One equivalent norm in B_∞¹⁻_,∞ⁿ,

f →sup

t>0

t^(n−1)/2PtfL_∞,

whereP_tis the heat semigroup onRⁿ, was used by Ledoux [485] in his direct semigroup argument leading to (1.4.52).

Another powerful method for proving Sobolev-type inequalities is based upon symmetrization of functions (it will be demonstrated in Sects. 2.3.5 and 2.3.8) was developed in different directions during the last 40 years. In particular, it led to generalizations and refinements of those inequalities for the so-calledrearrangement invariant spaces: Klimov [426, 427, 430]; Mossino [619]; Kolyada [443, 444], Talenti [742, 743]; Klimov and Panasenko [436];

Edmunds, Kerman, and Pick [253]; Bastero, M. Milman, and Ruiz [76]; M.

Milman and Pustylnik [607]; Cianchi [197]; Kerman and Pick [418, 419]; Mar- tin and M. Milman [518–522]; Martin, M. Milman, and Pustylnik [524]; Pick [659, 660]; Cianchi, Kerman, and Pick [205]; and Cianchi and Pick [207], et al.

Using symmetrization methods, Martin and M. Milman [517] showed that forα <0,andf ∈(W₁¹+W_∞¹)∩B_∞^α_,∞,

f^∗∗(s)≤c(n, α)

|∇f|^∗∗(s)_1+|α|^|α|

f_B^1+|α|α¹

∞,∞, where h^∗∗(t) = ¹_tt

0h^∗(s) ds. This gives another approach to (1.4.52) and other inequalities of a similar nature.

Although the constant in (1.4.14) is the best possible, it can be improved by restricting the class of admissible functions in this inequality. For example, since for anyN-gonΩN ⊂R²the isoperimetric inequality

s(∂ΩN)2

≥(4/N) tan(π/N)m2(ΩN)

is valid (see [714]) then duplicating the proof of Theorem 1.4.2/1 we obtain the following assertion.

Letu_N be a function onR²with compact support, whose graph is a polygon with N sides. Then

(4/N) tan(π/N)

R²|u_N|²dx≤

R²|∇u_N|dx 2

.

Lemma 1.4.3 is a special case of a result due to D.R. Adams [2]. Theo- rem1.4.3was proved by the author [551].

Theorem 1.4.5for μ =ms is the classical Sobolev theorem (see Sobolev [712, 713]) with supplements due to Il’in [394], Gagliardo [299], Niren- berg [640], and Morrey [612]. Here we stated this theorem in the form pre- sented by Gagliardo [299].

The continuity of functions in W_p¹(Ω) for p > 2, n = 2, was proved by Tonelli [754].

To Remark1.4.5we add that ifn=p(l−k),l > k,p >1, the inequality

Ω

exp

c|∇ku(x)| uV_p^l(Ω)

p/(p−1)

dx≤c0 (1.4.53)

holds with positive constantscandc₀, as shown for the first time by Yudovich in 1961 [809]. (See also Pohozhaev [662] and Trudinger [762]. Concerning the best value ofc0in inequalities of type (1.4.53) see Comments to Chap. 11.)

The estimate (1.4.32) is contained in the paper by Morrey [612]. Lem- ma 1.4.6 is the classical lemma due to Rellich [672]. Theorem 1.4.6/2 was proved by Kondrashov [447] forp >1 and by Gagliardo [299] forp= 1.

In connection with the estimate (1.4.35) we note that multiplicative in- equalities of the form

∇juLq ≤c∇lu^τL_pu¹L⁻_r^τ

and their modifications are well known (see Il’in [393] and Ehrling [257]).

Their general form is due to Gagliardo [300] and Nirenberg [640] (see also Solonnikov [717]). The papers by Gagliardo [300] and Nirenberg [640] contain the following theorem.

Theorem 1.Let Ωbe a bounded domain having the cone property and let uσ=

Ω

|u|^σdx 1/σ

forσ >0. Then

∇juq ≤c

∇lup+ur

τ

u¹r^−τ, (1.4.54) where p≥1, 1/q=j/n+τ(1/p−1/n) + (1−τ)/r for all τ∈[j/l,1]unless 1 < p <∞ andl−j−n/p is a nonnegative integer when (1.4.54)holds for τ ∈[j/l,1).

In the paper by Nirenberg [641] the stated result is supplemented by the following assertion.

Theorem 2. Letσ <0,s= [−n/σ],−α=s+n/σand let uσ= sup|∇su| forα= 0, us= [∇su]α forα >0, where

[f]α= sup

x=y

|x−y|^−αf(x)−f(y).

Further, let 1/r =−β/n, β >0. Then (1.4.54) holds for β ≤j < l and for allτ∈[(j−β)/(l−β),1], except the case mentioned in Theorem 1.

The proof is reduced to derivation of the inequality

I

u^(i)qdx≤c

I

u^(l)pdx+ [u]^p_β

[u]^q_β^−p for functions of the variablexon a unit intervalI.