SPACES AND GENERAL INEQUALITIES

Syafiq Johar syafi[email protected]

Contents

1 Definitions 1 1.1 Ck and H¨olderSpaces ...... 1 1.2 Sobolev Spaces ...... 2

2 Sobolev Inequalities 4 2.1 Case: 1 ≤ p < d ...... 4 2.2 Case: d < p < ∞ ...... 5 2.3 General Sobolev inequalities ...... 5 2.4 Consequences: Sobolev Embeddings ...... 5 2.5 Graphical Representation of Sobolev Embeddings ...... 6 2.6 Compactness and Rellich-Kondrachov Theorem ...... 7

3 Other Inequalities 7 3.1 Poincar´e’sInequality ...... 7 3.2 Difference Quotients ...... 7 3.3 Other Useful Inequalities ...... 8

1 Definitions

1.1 Ck and H¨olderSpaces

Definition 1.1 (Ck Spaces). The space Ck(Ω) is the set of all k-times continuously differentiable func- tions such that the following norm is finite:

X α ||u||Ck(Ω) = sup |D (x)|, x∈Ω |α|≤k

α k for which D is just the usual . The space (C (Ω), || · ||Ck(Ω)) is a Banach space. Some special sets:

• C0(Ω) and C∞(Ω) are the set of all bounded continuous and smooth functions with bounded on Ω respectively,

• Ck(Ω) is set of elements of C∞(Ω) with bounded derivatives up to order k,

k k k k • Cloc(Ω) is the set of all C (ω) for all ω compact subsets of Ω. C (Ω) $ Cloc(Ω). For example, x k ∗ k ∗ e ∈ Cloc(R) for all k ∈ N but not in C (R) for any k ∈ N .

k k • Cc (Ω) is the subset of C (Ω) with compact supports.

1 Definition 1.2 (Ck,γ Spaces). We define the γ-H¨olderseminorm of u :Ω → R for γ ∈ (0, 1] to be |u(x) − u(y)| 0,γ |u|C (Ω) = sup γ , x,y∈Ω |x − y| x6=y and we define the k, γ-H¨oldernorm of u as

X α ||u||Ck,γ (Ω) = ||u||Ck (Ω) + |D u|C0,γ (Ω). |α|=k

k The k, γ-H¨olderspace is the subset of Cb (Ω) defined by

k,γ k C (Ω) = {u : ||u||Ck,γ (Ω) < ∞} ⊂ C (Ω).

This space with the corresponding k, γ-H¨oldernorm is also a Banach space.

Remark 1.1. For H¨olderspaces, we only consider the case for which 0 < γ ≤ 1. The cases for γ > 1 are uninteresting because any function in Ck,γ (Ω) for γ > 1 are functions which are constant on each connected component of Ω. C0,1(Ω) is called the set of Lipschitz continuous functions.

Lemma 1.1. [TT] For Ω a bounded subset of Rd and k ∈ N∗, we have the following inclusions:

Ck+1(Ω) ⊂ Ck,β(Ω) ⊂ Ck,α(Ω) ⊂ Ck(Ω) for 0 < α ≤ β ≤ 1

1.2 Sobolev Spaces

Definition 1.3 (Lp Spaces). The Lp(Ω) space for 1 ≤ p < ∞ is the space of functions for which their p-th power is Lebesgue integrable over Ω i.e.

( 1 )   p p p L (Ω) = u : ||u||Lp(Ω) = |u| dµ < ∞ . ˆΩ

The L∞ space is defined as the space of functions with finite essential supremum i.e.

p L (Ω) = {u : ||u||L∞(Ω) = ess supΩ|u| < ∞},

where ess sup is defined to be

ess supΩ|u| = inf{C ≥ 0 : |u(x)| ≤ C for a.e. x ∈ Ω}.

Remark 1.2. We usually think of the class of Lp(Ω) functions to be up to equivalence. In other words, two functions u and v of Lp(Ω) are the “same” function in Lp(Ω) if they are equal almost everywhere.

Remark 1.3. Equivalently, the Lp(Ω) space can also be defined as the completion of the C∞(Ω) space

with respect to the || · ||Lp(Ω) norm. Lemma 1.2. For Ω a set with finite measure (i.e. µ(Ω) < ∞), we have the following inclusion:

Lq(Ω) ⊂ Lp(Ω) for 1 ≤ p ≤ q ≤ ∞

There are more functions in Lp spaces than continuous functions and thus they are the natural ob- jects to study in PDEs as candidate solutions. We would like to extend the differentiability of continuous functions to Lp functions. However, Lp functions are not necessarily continuous and thus not differ- entiable! Thus, we need a weaker notion of differentiation which is called the weak derivative. Using weak derivatives, we can define spaces which are analogous to Ck spaces, which we call the Sobolev spaces. This would then allow us to combine Lemmas 1.1 and 1.2 for more general inclusions.

2 Definition 1.4 (). The Sobolev space W k,p(Ω) consists of all locally summable functions u :Ω → R such that for each multi-index α with |α| ≤ k, Dαu exists in the weak sense and is in Lp(Ω). Definition 1.5. If u ∈ W k,p(Ω), we define its norm to be:  P α p 1/p  |α≤k Ω |D u| dx if 1 ≤ p < ∞, ||u||W k,p(Ω) = P ´ α  |α|≤k ess supΩ|D u| if p = ∞.

The Sobolev spaces W k,p(Ω) for each k ∈ N∗ and 1 ≤ p ≤ ∞ with this norm are Banach spaces. Clearly, W 0,p(Ω) = Lp(Ω). Remark 1.4. Equivalently, the Sobolev space W k,p(Ω) can also be defined as the completion of the ∞ space C (Ω) with respect to the || · ||W k,p(Ω) norm.

k,p ∞ k,p Definition 1.6. We denote by W0 (Ω) the closure of Cc (Ω) in W (Ω).

Lemma 1.3. Analogous to Lemma 1.2, for Ω a set of finite measure and k ∈ N∗, we have the inclusion: W k,q(Ω) ⊂ W k,p(Ω) for 1 ≤ p ≤ q ≤ ∞

k,p k,p k ∞ Remark 1.5. The spaces W (Ω), Wloc (Ω), C (Ω) and C (Ω) are all separable (i.e. there is a countable dense subset for each space, which are the polynomials with rational coefficients).

Theorem 1.1. For each k ∈ N and 1 ≤ p ≤ ∞, the Sobolev space W k,p(Ω) is a Banach space. For p = 2, the Sobolev space W k,2(Ω) is a and denoted Hk(Ω) with the obvious inner product. Usually, during computations, we might need to take derivatives. Since W k,p(Ω) only have weak derivatives, it is useful to use the density of smooth functions in this space and take limits: Theorem 1.2 (Global approximation by smooth functions). Assume Ω is bounded and ∂Ω is C0,1. k,p ∞ k,p Suppose as well that u ∈ W (Ω) for some 1 ≤ p < ∞. Then there exist functions um ∈ Cc (Ω)∩W (Ω) k,p such that um → u in W (Ω).

d ∞ d If we replace Ω with and open subset of R , the above still holds. The closure of Cc (R ) in the W k,∞(Rd) norm is contained in Ck+1(Rd) which is smaller and not dense in W k,∞(Rd) [TT]. Thus, the previous approximation does not hold for p = ∞. Theorem 1.3 (Extension Theorem). [ELC] Assume Ω is bounded and ∂Ω is C0,1. Let Ω0 be a bounded open set such that Ω0 ⊂⊂ Ω. Then, there exists a bounded linear operator E : W 1,p(Ω) → W 1,p(Rd) for 1 ≤ p ≤ ∞ such that for each u ∈ W 1,p(Ω): (i) Eu = u a.e. in Ω.

(ii) Eu has within Ω0.

0 1,p n 1,p (iii) ||Eu||W (R ) ≤ C(p, Ω, Ω )||u||W (Ω). Theorem 1.4 (Trace Theorem). [ELC] Assume Ω is bounded and ∂Ω is C0,1. Then, there exists a bounded linear operator T : W 1,p(Ω) → Lp(∂Ω) for 1 ≤ p ≤ ∞ such that:

1,p (i) T u = u|∂Ω if u ∈ W (Ω) ∩ C(Ω).¯

1,p (ii) ||T u||Lp(∂Ω) ≤ C(p, Ω)||u||W 1,p(Ω) for each u ∈ W (Ω). Theorem 1.5 (Trace-zero functions in W 1,p). Assume Ω is bounded and ∂Ω is C0,1. Suppose further- more that u ∈ W 1,p(Ω). Then:

1,p u ∈ W0 (Ω) ⇐⇒ T u = 0 on ∂Ω

3 2 Sobolev Inequalities

2.1 Case: 1 ≤ p < d

Definition 2.1. If 1 ≤ p < d, the Sobolev conjugate of p is defined as:

1 1 1 dp = − ⇒ p∗ = > p p∗ p d d − p

1 n Theorem 2.1 (Gagliardo-Nirenberg-). Assume 1 ≤ p < d. Then, for all u ∈ Cc (R ):

||u|| p∗ d ≤ C(p, d)||Du|| p d L (R ) L (R ) Remark 2.1. Why are the numbers p and p∗ very specific? This is to ensure that the inequality is invariant by scaling the space variable. For example say x = λy for some constant λ > 0, then by changing the integration with respect to the x variable to the y variable, we have:

x d p∗ u = λ ||u(y)||Lp∗ ( d), p∗ R λ L (R) x d−p Du = λ p ||Du(y)||Lp( d). p R λ L (R) Thus, to ensure that the inequality is invariant under scaling of the space variable, the constants must match up. Otherwise, you can find some large enough λ such that the Gagliardo-Nirenberg-Sobolev d d−p ∗ dp inequality is violated. Thus, we must have: p∗ = p ⇒ p = d−p . The following are two important theorems in Sobolev inequalities:

Theorem 2.2 (Estimates for W 1,p for 1 ≤ p < d). [ELC] [LL] Let Ω be a bounded open subset of Rd ∗ and suppose that ∂Ω is C0,1. Assume 1 ≤ p < d and u ∈ W 1,p(Ω). Then, u ∈ Lp (Ω) with the estimate:

||u||Lp∗ (Ω) ≤ C(p, d, Ω)||u||W 1,p(Ω)

1,p d Theorem 2.3 (Estimates for W0 for 1 ≤ p < d). [ELC] [LL] Let Ω be a bounded open subset of R 0,1 1,p q ∗ and suppose that ∂Ω is C . Assume 1 ≤ p < d and u ∈ W0 (Ω). Then, u ∈ L (Ω) for each p ≤ q ≤ p with the estimate: ∗ ||u||Lq (Ω) ≤ C(p, d, q, Ω)||Du||Lp(Ω) for all p ≤ q ≤ p In particular, for all 1 ≤ p < ∞, we have the Poincar´einequality:

||u||Lp(Ω) ≤ C(p, d, Ω)||Du||Lp(Ω) Remark 2.2. Here are some remarks regarding Theorems 2.2 and 2.3: 1. Notice the difference between the inequalities in Theorem 2.2 and Theorem 2.3. If u is vanishing on the boundary, we only require the Lp norm of ∇u on the RHS. However, if not, we require the full W 1,p norm of u in the RHS. The reason for this is because we can choose a constant non-negative

function over Ω thus ||u||Lp∗ (Ω) > 0 but ||∇u||Lp(Ω) = 0. 2. For a compact closed Riemannian manifold, Theorem 2.3 does not hold by the above argument.

3. We can also get a Poincar´etype inequality for u ∈ W 1,p(Ω) which is called the Poincar´e-Wirtinger inequality which will be explained in Theorem 3.1 in Chapter 3.

1,p d 1,p d 4. Since any function u ∈ W (R ) = W0 (R ) for 1 ≤ p < ∞ is such that |u|, |∇u| → 0 as |x| → ∞, we can extend Theorem 2.3 from Ω to the whole of Rd by density of compactly supported smooth functions in the space Lp(Rd) [DCF] . 5. If |Ω| < ∞, we can omit the condition p ≤ q in Theorem 2.3 by virtue of Lemma 1.2.

4 2.2 Case: d < p < ∞

Theorem 2.4 (Morrey’s inequality). [ELC] Assume d < p < ∞. Then for all u ∈ C1(Rd) ∩ Lp(Rd), u d is H¨oldercontinuous with the estimate for γ = 1 − p :

||u|| 0,γ d ≤ C(p, d)||u|| 1,p d C (R ) W (R )

Thus, for all k ∈ N∗, if u ∈ W k+1,p(Rn), then u ∈ Ck,γ (Rn) after being redefined on a set of measure 0. k,p d r,γ d d In general, W (R ) ⊂ C (R ) such that γ = k − r − p .

2.3 General Sobolev inequalities

Here we denote a more general Sobolev conjugate:  dp ∗  d−kp if kp < d, pk = ∞ if kp ≥ d.

∗ ∗ Note that the Sobolev conjugate defined in Definition 2.1, p = p1.

Theorem 2.5 (General Sobolev inequalities). [ELC] [DCF] Let Ω be a bounded open subset of Rd with a C0,1 boundary. Assume u ∈ W k,p(Ω).

d q ∗ (i) If k < p , then u ∈ L (Ω) where p ≤ q ≤ pk. We also have the estimate:

||u||Lq (Ω) ≤ C(k, p, d, Ω)||u||W k,p(Ω)

d q (ii) If k = p , then u ∈ L (Ω) for 1 < p ≤ q < ∞. We also have the estimate:

||u||Lq (Ω) ≤ C(k, p, d, Ω)||u||W k,p(Ω)

d d k−b p c−1,γ ¯ (iii) If k > p , then u ∈ C (Ω) for some γ ∈ (0, 1), where:  j d k d d  − + 1 if ∈/ Z 0 < γ < p p p d  1 if p ∈ Z. We also have the estimate:

k,p ||u|| k− d −1,γ ≤ C(k, p, d, γ, Ω)||u||W (Ω) C b p c (Ω)¯

This theorem also holds if Ω ⊂ Rd is any open set (not necessarily bounded) by density of compactly supported functions in the space W k,p(Rd) for 1 ≤ p < ∞.

2.4 Consequences: Sobolev Embeddings

The inequalities in the previous subsection allows us to rank and compare the Sobolev spaces W k,p(Ω) where Ω and open subset of Rd with ∂Ω that is C0,1. We can also compare Sobolev spaces with H¨older spaces.

Theorem 2.6. We have the following inclusions:

k,p q ∗ W (Ω) ⊂ L (Ω) for p ≤ q ≤ pk (require q < ∞ for kp = d).

Note that if |Ω| < ∞, then we can omit the p ≤ q condition by Lemma 1.3.

5 Theorem 2.7. By inductively using Theorem 2.6, we have that for an open subset Ω of Rd (not necessarily bounded), we have:

k,p l,q ∗ W (Ω) ⊂ W (Ω) if 0 ≤ l ≤ k and 1 ≤ p ≤ q < ∞ s.t. p(k − l) ≤ d and p ≤ q ≤ pk−l

Theorem 2.8. [TT] More generally, for an open subset Ω of Rd (not necessarily bounded), we have:  0 ≤ l ≤ k and 1 < p < q ≤ ∞ s.t. d − k ≤ d − l, W k,p(Ω) ⊂ W l,q(Ω) if p q d d with : q < ∞ OR p − k < q − l.

Moreover, for the case kp > d, things get even nicer as we can relate Sobolev spaces with the space of continuous and H¨oldercontinuous functions.

Theorem 2.9. For j ∈ N∗, we have the following inclusions: W k+j,p(Ω) ⊂ Cj,γ (Ω) ⊂ Cj(Ω) for kp > d for some 0 < γ < 1.

Thus, we have that if kp > d, W k,p(Ω) ⊂ C0(Ω) (so we can choose continuous representatives for each element of W k,p(Ω)).

2.5 Graphical Representation of Sobolev Embeddings

A quick way of summarising the Sobolev embeddings is to consider the following (adapted from [CL]). Suppose we want to know in which W k,p(Ω) spaces does a given W k0,p0 (Ω) space is embedded in. Draw 1 k,p a horizontal axes for the value of p where p is the power in the W and a vertical axis for the quantity k which is the number of weak derivatives in W k,p space.

1 1. Fix the range of p. If we consider Theorem 2.7, we consider the p ∈ (0, 1] i.e. 1 ≤ p < ∞. For Theorem 2.8, we consider the interval [0, 1) i.e. 1 < p ≤ ∞. In the example below, we consider the latter.

1 2. From the point ( , k0) draw a line of gradient d, which we call Ld(k0, p0). This line is given by p0 the equation: d d Ld(k0, p0): − k0 = − k. p0 p

3. If Ld(k0, p0) intersects the k-axis at some k ≥ 0, we ignore that point of intersection.

4. Thus, the space W k0,p0 (Ω) is embedded in the space W k,p(Ω) in the shaded region:

  k0,p0 1 k W ∼ , k0 Ld(k0, p0) . p0 . Ck0−1,1(Ω) ( Ck0 (Ω) For 0 < γ < 1,Ck0−1,γ (Ω)

Ck0−1(Ω) This box k . − . d . p is shaded < k 0 if |Ω| < ∞ d − k = k0 − p d 0 p 0 1 p 1 = 0 1 1 ∞ p0

Figure 1: Visualising the embeddings in Theorem 2.8 graphically.

6 Remark 2.3. When p = ∞, the points lying on the line Ld(p0, k0) itself are not guaranteed to be contained in W d,p. Although, some might do! For example, W d,1 ⊂ L∞ but W 1,d 6⊂ L∞ (e.g. the 1 1,d unbounded function u(x) = log log(1 + |x| ) ∈ W ).

2.6 Compactness and Rellich-Kondrachov Theorem

Definition 2.2. Let X and Y be Banach spaces and X ⊂ Y . We say that X is compactly embedded in Y i.e. X ⊂⊂ Y provided that:

(i) ||u||Y ≤ C||u||X for u ∈ X.

∞ (ii) Each bounded sequence in X is precompact in Y i.e. if {uk}k=1 is a bounded sequence in X, then ∞ some subsequence {ukj }j=1 converges to some limit u in Y .

Theorem 2.10 (Rellich-Kondrachov Compactness Theorem). [ELC] Assume Ω is a bounded open subset of Rd and ∂Ω is C1. Suppose that 1 ≤ p < d, then, for all q ∈ [1, p∗):

W 1,p(Ω) ⊂⊂ Lq(Ω).

Rellich-Kondrachov is useful for applying Banach-Alaoglu theorem and the following theorem:

Theorem 2.11. If T : X → Y is a compact operator, then any weakly converging sequence in X converges strongly in Y .

Remark 2.4. As a brief summary, Rellich-Kondrachov says that if Ω is bounded with C0,1 boundary ∗ ∗ and the inequalities in Theorems 2.6 and 2.8 for k, l, p, q, pk, ql are strict (except for the inequality 0 ≤ l), then the embeddings are compact.

3 Other Inequalities

3.1 Poincar´e’sInequality

Theorem 3.1 (Poincar´e-Wirtingerinequality). We denote the average of u over Ω as Ω u dy. Let Ω be a bounded connected open subset of Rd with a C0,1 boundary ∂Ω. Assume 1 ≤ p < ∞ffl. Then, for all u ∈ W 1,p(Ω), we have the following estimate for 1 ≤ q ≤ p∗:

||u − u dy||Lq (Ω) ≤ C(p, d, Ω)||Du||Lp(Ω) Ω

In particular,

||u − u dy||Lp(Ω) ≤ C(p, d, Ω)||Du||Lp(Ω) Ω

3.2 Difference Quotients

Definition 3.1. Assume u :Ω → R is a locally summable function and Ω0 ⊂⊂ Ω. The i-th difference quotient for i = 1, 2, . . . , d of size h for x ∈ Ω0 and h ∈ R such that 0 < |h| < dist(Ω0, ∂Ω) is: u(x + he ) − u(x) Dhu(x) = i . i h

h h h h We denote D u = (D1 u, D2 u, . . . , Dd u).

7 Theorem 3.2 (Difference quotients and weak derivatives). [ELC] For each Ω0 ⊂⊂ Ω and all 0 < |h| < 1 0 2 dist(Ω , ∂Ω): (i) Suppose that 1 ≤ p < ∞ and u ∈ W 1,p(Ω), we have the following estimate:

h ||D u||Lp(Ω0) ≤ C||Du||Lp(Ω)

p 0 h 1,p 0 (ii) Assume 1 < p < ∞, u ∈ L (Ω ) and there exists C such that ||D u||Lp(Ω0) ≤ C, then u ∈ W (Ω )

with ||Du||Lp(Ω0) ≤ C.

3.3 Other Useful Inequalities

Theorem 3.3 (Cauchy’s inequality with ε). For a, b > 0 and ε > 0, we have: b2 ab ≤ εa2 + . 4ε 1 1 Theorem 3.4 (Young’s inequality). [ELC] Let 1 < p, q < ∞ such that p + q = 1. Then, for a, b > 0 and ε > 0: ap bq ab ≤ + , p q p q − q −1 ab ≤ εa + C()b where C(ε) = (εp) p q .

Pm 1 Theorem 3.5 (General H¨older’sinequality). [ELC] Let 1 ≤ p1, p2, . . . , pm ≤ ∞ such that = 1 k=1 pk p and assume uk ∈ L k (Ω) for k = 1, 2, . . . , m. Then: m Y |u u ··· u | dµ(x) ≤ ||u || p . ˆ 1 2 m k L k (Ω) Ω k=1 1 1 In particular, if p + q = 1, then:

|fg| dµ(x) ≤ ||f||Lp(Ω)||g||Lq (Ω). ˆΩ p 1 Theorem 3.6 (Interpolation inequality for L norms). [ELC] Assume 1 ≤ s ≤ r ≤ t ≤ ∞ and r = θ 1−θ s t r s + t . Suppose also u ∈ L (Ω) ∩ L (Ω). Then, u ∈ L (Ω) and:

θ (1−θ) ||u||Lr (Ω) ≤ ||u||Ls(Ω)||u||Lt(Ω). Theorem 3.7 (Minkowski’s inequality). [LL] Suppose that (Ω, µ(x)) and (Γ, ν(y)) are σ-finite measure spaces and f(x, y) is jointly measurable, then

1 1   p  p   p f(x, y)) dν(y) dµ(x) ≤ f(x, y)p dµ(x) dν(y) ˆΩ ˆΓ ˆΓ ˆΩ Remark 3.1. The statement abive is more general than what most books would call Minkowski’s inequality. If we pick (Γ, ν(y)) to be a finite discrete set of n elements with counting measure, then we would get:

||f1(x) + ··· fn(x)||Lp(Ω) ≤ ||f1||Lp(Ω) + ··· + ||fn||Lp(Ω), which is the more familiar form of Minkowski’s inequality.

Theorem 3.8 (Another inequality by Young). [LL] Assume that f ∈ Lp(Rd) and g ∈ Lq(Rd) such that 1 1 1 p + q = 1 + r . Then we have

||f ∗ g|| r d ≤ C||f|| p d ||g|| q d , L (R ) L (R ) L (R ) where ∗ denotes the convolution of two functions.

8 References

[CL] Chen, L. Sobolev Spaces and Elliptic Equations. Accessed: 14 November 2016. Webpage: http: //www.math.uci.edu/~chenlong/226/Ch1Space.pdf.

[DCF] Demengel, F. and Demengel, G. Functional Spaces for the Theory of Elliptic Partial Differential Equations. Springer-Verlag, London (2012).

[ELC] Evans, L.C. Partial Differential Equations. American Mathematical Society, Providence (2002).

[LL] Lieb, E.H, and Loss, M.. Analysis. American Mathematical Society, Providence (2001).

[TT] Tao, T. 245C, Notes 4: Sobolev Spaces. Accessed: 14 November 2016. Webpage: https: //terrytao.wordpress.com/2009/04/30/245c-notes-4-sobolev-spaces/.

[WYW] Wu, Z., Yin, J. and Wang, C. Elliptic and Parabolic Equations. World Scientific Publishing, Singapore (2006).

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