$L^p$ Space and Sobolev Spaces

1 $L^p$ Space

1.1 Density

Theorem 1.1 (Density of simple functions) Let $(X,\mathscr{A},\mu)$ be a measure space, and let $p$ satisfy $1 \leq p \leq +\infty$. Then the simple functions in $L^p(X,\mathscr{A},\mu)$ form a dense subspace of $L^p(X,\mathscr{A},\mu)$.

Lemma 1.2 (Approximation of measurable function by simple function) If $f : X \rightarrow [0, \infty]$ is a positive measurable function, then there is a monotone increasing sequence of positive simple functions $\varphi_n : X → [0, \infty)$ with $\varphi_1 ≤ \varphi_2 ≤ · · · ≤ \varphi_n ≤ . . .$ such that $\varphi_n \rightarrow f$ pointwise. If $f$ is bounded, then $\varphi_n \rightarrow f$ uniformly.

Proof of Lemma 1.2 For each $n \in \mathbb N$, we divide the interval $[0, n]$ into $n2^{n}$ subintervals of width $1/2^n$

$$I_{k,n} = (k2^{−n} ,(k + 1)2^{−n}], k = 0, 1, . . . , n2^{n }-1,$$

and let $E_n = (n, ∞]$. Define

$$E_{k,n} = f^{-1} (I_{k,n}), F_n = f ^{−1} (J_n).$$

Let $\varphi_n(x) = n\chi_{E_n} (x) + \sum\limits_{k=0}^{n2^n-1}\dfrac{k}{2^n}\chi_{E_{k,n}(x)}$. Then $\varphi_n(x)\leq\varphi_{n+1}(x)\leq f(x)$. We show pointwise convergence in the following.

(1) For $x$ such that $f(x)< M$ if $x\geq M$, then for $n>M$, $f(x)-\varphi_n(x)<\dfrac{1}{2^n}$.

(2) For $x$ such that $f(x)=+\infty$, $lim_{n\rightarrow \infty}\varphi_n(x)=+\infty$.

Proof of Theorem 1. It is sufficient to prove that we can approximate a positive function $f : X → [0, ∞]$ by simple functions.

First suppose that $f \in L_p (X)$ where $1 \leq p < \infty$. Then, from Theorem 1.1, there is an increasing sequence of simple functions $\{\varphi_n\}$ such that $\varphi_n \rightarrow f$ pointwise. Since $|\varphi_n(x)|\leq |f(x)|$, $\varphi_n\in L_p(X)$. Also, $|f − \varphi_n|^{p} ≤ |f|^{p}$ suggests that $|f − \varphi_n|^{p}$ is integrable. As we have $\varphi_n\rightarrow f$ pointwise, $|f − \varphi_n|^{p}\rightarrow 0$ pointwise. Using dominated convergence theorem,

$$\int|f − \varphi_n|^pd\mu\rightarrow 0$$

Secondly, if $f\in L^{\infty}(X)$, then we may choose a representative of $f$ that is bounded. According to Theorem 1, there is a sequence of simple functions that converges uniformly to $f$, and therefore in $L^{\infty}(X)$.

Theorem 1.3 (Density of continuous functions) The space $C_c(\mathbb R_n)$ is dense in $L^1 (\mathbb R_n)$. (could be extended to $L^p$ for $p\in[1,\infty)$)

Proof (1). Using Borel regularity of Lebesgue measure and Urysohn’s lemma.

Proof (2). Using Lusin’s theorem.

1.2 Completeness

Theorem 1.4 (Riesz–Fischer theorem) $L^p (X)$ is complete for $p\in[1,\infty]$.

Proof. We only prove this for $p\in[1,\infty)$.

(1) Let $f_{n_0}$ be a cauchy sequence in $L^p(X)$. There exists a subsequence $\{f_n\}$ such that

$$\sum\limits_{k=1}^{\infty}||f_{k+1}-f_k||_p<\infty$$

Let

$$g_n=|f_1|+\sum\limits_{k=1}^{n-1}|f_{k+1}-f_k|$$

For each $x\in X$, $\{g_n(x)\} \uparrow$, and thus $g(x)=limg_n(x)$ (in the sense of $[0,\infty]$). By monotone convergence theorem and the fact that $g_n^p$ is a monotone sequence, $\int g^p_nd\mu\rightarrow\int gd\mu$. Since

$$||g_n||_p\leq||f_1||_p+\sum\limits_{k=1}^{\infty}||f_{k+1}-f_k||_p<\infty$$

$g\in L^p(X)$ and $g$ is almost everywhere finite. We also have

$$g_m(x)-g_n(x)\geq |f_m(x)-f_n(x)| \text{ for } m>n \tag{1}$$

(2) Since $g$ is almost everywhere finite and $g(x)=limg_n(x)$, $\mathop{lim}\limits_{m,n\rightarrow\infty} g_m(x)-g_n(x)=0$ holds almost everywhere, and so do $\mathop{lim}\limits_{m,n\rightarrow\infty} f_m(x)-f_n(x)=0$. Thus $\{f_n(x)\}$ forms a cauchy sequence almost everywhere $\Rightarrow$ $f_n(x)$ has finite limit almost everywhere, which is denoted as $f(x)$. Let $m\rightarrow\infty$,

$$|f(x)-f_n(x)|\leq g(x)-g_n(x)\leq g(x), a.e.$$

$|f(x)|\leq f_n(x)+g(x)\Rightarrow f(x)\in L^p(X)$. Also,

$$|f(x)-f_n(x)|^p\leq [g(x)]^p \Rightarrow |f(x)-f_n(x)|^p\in L(X)$$

According to Dominated convergence theorem, $||f(x)-f_n(x)||_p\rightarrow 0$.

2 Sobolev Spaces

2.1 Weak Derivative

Definition 2.1 (Distributional derivative) For $f ∈ L^1_{loc}(\Omega)$ and $\alpha \in \mathbb{N}^n$ , the α-distributional partial derivative $D^{\alpha}f \in \mathscr{D}’ (Ω)$ is defined as

$$\left <D^{\alpha} f, \varphi \right>:= (−1)^{|\alpha|} \int_{\Omega} fD^{\alpha}ϕ\varphi dx, \forall \varphi \in \mathscr D(Ω).$$

Definition 2.2 (Weak derivative) For $f ∈ L^1_{loc}(\Omega)$ and $\alpha \in \mathbb{N}^n$ , we say $f$ has α-weak derivative if its distributional derivative $D^{\alpha}f \in L_{loc}^1(\Omega), i.e. $ there exists $v\in L_{loc}^1(\Omega)$ s.t.

$$\int_{\Omega}v\varphi dx= (−1)^{|\alpha|} \int_ Ω fD^{\alpha}ϕ\varphi dx, , \forall \varphi \in \mathscr D(Ω).$$

Definition 2.3 (Sobolev spaces). For $1 \leq p \leq\infty$, we define,

$$W^{k,p}(\Omega) = \{f \in L^p (\Omega) : D^{\alpha} f \in (\Omega) ,\forall |\alpha| \leq k\}$$

with the norm

$$|| f||_{k,p}= \left(\sum\limits_{|\alpha|\leq k}\int_{\Omega}|D^{\alpha }f | ^ p \right )^{\frac{1}{p}}$$

2.2 Density Results of Different Mollifications

Definition 2.4 (Standard mollification)

Theorem 2.1 Let $f \in W^{k,p}(\Omega), 1 \leq p < \infty$ and $f_{\epsilon} = η ∗ f : \Omega \rightarrow \mathbb {R}$. Then

$$f_{\epsilon} \mathop{\longrightarrow}\limits_{}^{\epsilon\rightarrow 0} f \text{ in } L^p(\Omega)\\ f_{\epsilon} \mathop{\longrightarrow}\limits_{}^{\epsilon\rightarrow 0} f \text{ in } W^{k,p}(K) ,\forall K\subset\Omega$$

The result is the same for mollification from inside.

Proposition 2.2 $C_{0}^{\infty}(\Omega)$ is not dense in $W^{k,p}(\Omega)$.

Proof. Under some conditions like $C^1$ boundary, $\tau_0 u=\mathop{lim}\limits_{k\rightarrow\infty}u_k|_{\partial u}=0$. (Theorem 5.8)

Definition 2.5 Following Proposition 2.3, we denote by $W^{k,p}_0(\Omega)$ the closure of $C_{0}^{\infty}(\Omega)$ in $W^{k,p}(\Omega)$.

Theorem 2.3 (Meyers-Serrin’s theorem) $C^{\infty}(\Omega)\cap W^{k,p}(\Omega)$ is dense in $W^{k,p}(\Omega)$.

Definition 2.6 (translated mollification)

Theorem 2.4 Let $\Omega = \mathbb R^n_ {+}$ and $1 \leq p < \infty$. Let $\hat{f_{\epsilon}}$ be translated mollification. Then

$$\hat f_{\epsilon} \mathop{\longrightarrow}\limits_{}^{\epsilon\rightarrow 0} f \text{ in } L^p(\Omega)\\ \hat f_{\epsilon} \mathop{\longrightarrow}\limits_{}^{\epsilon\rightarrow 0} f \text{ in } W^{k,p}(\Omega)$$

Also, $\hat f_{\epsilon}\in C_0^{\infty} (\overline\Omega)$.

Theorem 2.5 If $\Omega$ satisfies the segment condition, then $C^{\infty} (\overline\Omega)$ is dense in $W^{k,p}(\Omega)$, for $1 \leq p < \infty$. [Theorem 3 in Section 5.3.3 in Evan’s book]

2.3 Traces

Deal with the boundary: Since $L^p$ space has no restrictions for boundary (null set), it is meaningless to discuss the solution in $L^p$ space. Thus, we turn to $W^{k,p}(\Omega)$. The way to proceed is the following: since $C^{\infty}(\overline\Omega)$ is dense in $W^{k,p}(\Omega)$ [Theorem 2.5], given $f \in W^{k,p}(\Omega)$ we can find a sequence $f_k \in C^{\infty}(Ω)$ s.t.

$$f_k\rightarrow f \text{ in }W^{k,p}(\Omega)$$

For each $f_k$, the trace $f_k|\partial\Omega$ is uniquely defined, so that we define the “trace of $f$ on ∂Ω” as $\mathop{lim}\limits_{k\rightarrow \infty} f_k|\partial\Omega$. The crucial question is whether such limit exists and with respect to which topology.

Theorem 2.5 (Trace theorem in $\mathbb R^n_+$). There exists a bounded linear operator $τ_0 : W^{1,p}(\mathbb R^n_+) \rightarrow L^p (\partial \mathbb R^n_+), 1 \leq p < \infty$, s.t. $\tau_0u = u|\partial {\mathbb R_n^+}$ if $u \in W^{1,p}(\mathbb R^n_+) ∩ C_0 (\overline{\mathbb R^+_n})$.

Theorem 2.6 (Trace theorem in $\Omega$ ). Let $\Omega$ be a domain with bounded boundary $\partial\Omega$ of class $C^1$. There exists a bounded linear operator $τ_0 : W^{1,p}(\Omega) \rightarrow L^p (\partial\Omega), 1 \leq p < \infty$, s.t. $\tau_0u = u|\partial {\Omega}$ if $u \in W^{1,p}(\Omega) ∩ C_0 (\overline{\Omega})$.

There are several corollaries:

• Integration by part.

• Poincare inequality: Let $\Omega \subset \mathbb R^n$ be a bounded domain. Then there exists a constant $C_P>0$ such that

  $$\Vert f\Vert_{L^p(\Omega)} \leq C_P \Vert Df\Vert_{L^p(\Omega)}, ∀f \in W^{1,p}_0(\Omega).$$