7 \(L^p\) Spaces

Key takeaways:

Convexity: properties of convex functions, Young’s inequality, and Jensen’s inequality.
Construction of \(L^p\) as a Banach space: Use Holder’s and Minkowski’s inequalities to establish the Banach property; dual of \(L^p\).

Main results:

Jensen’s inequality (theorem 7.1): generalizalized property of convexity w.r.t. integration.
\(L^p\) space is a Banach space that consists of equivalence classes of a.e. equal functions.
- For \(p\geq q\geq 1\) and \(\mu(\Omega)<\infty\), we have \(L^{p} \subset L^q\).
The convexity of \(x\mapsto |x|^{p>1}\) and monotonicity in \(p\) implies:
- Normed vector space properties of \(L^p(\Omega)\) holds for \(1\leq p\leq \infty\). (proposition 7.3).
- Young’s inequality (proposition 7.2): holds for \(1<p<\infty\); it is essentially the quantitative statement of convexity,
- Holder’s inequality (proposition 7.4): holds for \(1\leq p\leq \infty\), can be saturated by pointwise-saturating Young’s inequality for \(p\neq 1, \infty\).
- Inclusion: \(0<p<q<\infty\) and finite \(\mu(\Omega)\) implies \(L^p \supset L^q\) (proposition 7.5).
- Variational formula: \(\|f\|_p = \sup_{\|g\|_{\bar p}\leq 1} |f\cdot g|\); holds for \(1\leq p < \infty\) generally and \(p=\infty\) for \(\sigma\)-finite \(\mu\) (proposition 7.6).
- Minkowski’s inequality (theorem 7.2) states the convexity of the \(L_p\) norm and holds for \(1\leq p\leq \infty\).
Young and Holder’s inequality are the main tools for applying convexity. Young’s inequality does not hold for \(p=1, \infty\). Holder’s inequality hold for \(p\in [1, \infty]\) although saturation conditions for \(p=1, \infty\) does not come from pointwise saturation of Young’s inequality.
Fatou’s lemma 4.2 is a great tool for demonstrating \(p\)-norm convergence using pointwise convergence.

Convexity

Definition 7.1 (convex set and function) A convex set \(K\subset \mathbb R^n\) is closed under \((x, y)\mapsto \lambda x + \bar y\) for all \(\lambda \in [0, 1]\). A convex function on a convex set \(K\) is a real-valued function satisfying \[ f(\lambda x + \bar \lambda y) \leq \lambda f(x) + \bar \lambda f(y) \] the function \(f\) is strictly convex at \(x\) if the inequality is strict for all \(y\).

Proposition 7.1 If \(K\) is an open set, then a convex function on \(K\) is continuous.

Definition 7.2 (support plane) The support plane to a graph of \(f:K\to \mathbb R\) at \(x\in K\subset \mathbb R^n\) is a plane in \(\mathbb R^{n+1}\) that touches \((x, f(x))\) and that nowhere lies above the graph; if the support plane at \(x\) is unique, then it is the tangent plane.

For example, there is no unique support plane at \(x=0\) for \(|x|\). For convex \(f\), the existence of a tangent plane is equivalent to differentiability. In particular, the tangent plane for \(y=f(x)\) at \(x_0\) has expression \[ y - f(x_0) = \nabla f(x_0) \cdot (y - x) \] Thus for general convex functions, for each \(x\) there exists a vector \(V\in \mathbb R^n\) such that \[ f(y) \geq f(x) + V\cdot (y - x) \tag{7.1} \]

Definition 7.3 (Hölder conjugate) The Hölder conjugate of \(p>1\) is given by the equation \[ p^{-1} + q^{-1} = 1 \] We also denote \(q=\bar p\). It is oftentimes more convenient to note that \[ p = \dfrac q {q - 1} \iff q = \dfrac p {p - 1} \]

Proposition 7.2 (Young's inequality) Given Hölder conjugates \(p, q>1\), \[ ab \leq \dfrac{a^p}{p} + \dfrac{b^q}{q}, \quad \forall a, b\geq 0 \] Equality is saturated when \(a^p = b^q\).

Proof

Consider the convex function \(f(a) = \dfrac{a^p}{p}\). Its Legendre transform is \[\begin{align} g(b) &= \sup_a \left(ab - \dfrac{a^p}{p}\right) = a_*b - \dfrac{a_*^p}{p},\quad a_* = b^{1/(p-1)} \\ &= b^{1 + 1/(p-1)} - \dfrac 1 p b^{p/(p-1)} = b^q - \dfrac 1 p b^q = \dfrac{b^q}{q} \end{align}\] Young’s inequality follows from the variational characterization of the Legendre transform. Equality is saturated when \(a=b^{1/(p-1)} \iff a^p = b^q\).

Definition 7.4 (average of a measurable function) Assuming \(\mu(\Omega)\) is finite, the expectation (average) of a measurable function \(f:\Omega\to \mathbb R\) is defined as \[ \langle f\rangle= \dfrac 1 {\mu(\Omega)} \int_\Omega f\, d\mu \]

Theorem 7.1 (Jensen's inequality) Given a convex function \(J:\mathbb R\to \mathbb R\) and measurable \(f:\Omega\to \mathbb R\) with \(\mu(\Omega)<\infty\), then

The negative part \((J\circ f)_-\) is in \(L^1(\Omega)\) (recall the decomposition \(f=f_+ - f_-\).
\(\langle J\circ f\rangle\geq J(\langle f\rangle)\); when \(J\) is strictly convex at \(\langle f\rangle\), there is equality iff \(f\) is constant.

Proof idea: integrate the supporting vector equation (7.1).

Proof

Fixing \(\langle f\rangle\), there exists a constant \(V\in \mathbb R\) such that \[ J(\forall t)\geq J(\langle f\rangle) + V(t - \langle f\rangle) \tag{7.2} \] This bounds the negative component (but not the positive one!) \[ [J\circ f]_-(x) \leq |J(\langle f\rangle)| + |V||\langle f\rangle| + |V| |f(x)| \] using \(\mu(\Omega)<\infty\), this bounded function is integrable. To obtain (2), integrate (7.2) for all \(t=f(x)\) \[ \langle J \circ f\rangle\geq J(\langle f\rangle) - V \langle f\rangle+ V\langle f\rangle \] with \(J\) strictly convex, the inequality is saturated when \(\langle J\circ f\rangle= J(\langle f\rangle)\).

\(L^p\) as a Banach space

Definition 7.5 (Lp space, norm) Given a positive (nonnegative) measure \(\mu\) and \(1\leq p<\infty\), the space \(L^p(\Omega, d\mu)\) is the space of complex-valued functions \(f:\Omega\to \mathbb C\) such that:

\(f\) is \(\mu\)-measurable.
\(|f|^p\) is \(\mu\)-integrable.

The measure \(d\mu\) is usually omitted if understood. Given \(f\in L^p(\Omega)\), its norm is \[ \|f\|_p = \left[\int_\Omega |f(x)|^p \mu(dx)\right]^{1/p} \] This definition is extended to \(L^\infty(\Omega, d\mu)\) as the space of \(\mu\)-a.e. bounded functions, with \[ \|f\|_\infty = \inf \{ K: \mu(\{x\in \Omega: |f(x)|>k\}) = 0 \} \] The \(\infty\)-norm is also called the essential supremum of \(|f|\).

Proposition 7.3 (normed vector space) \(L^p(\Omega, d\mu)\) is a vector space, and the \(L^p\) norm satisfies for \(1\leq p\leq \infty\):

Linearity: \(\|\lambda f\|_p = |\lambda| \|f\|_p\).
Nonnegativity: \(\|f\|_p\geq 0\) with equality iff \(f=0\) for \(\mu\)-a.e.
Triangle inequality: \(\|f+g\|_p \leq \|f_p\| + \|g\|_p\).

Proof

Vector space properties: closure under scalar multiplication is apparant. Closure under addition follows from the convexity of \(|x|^p\). \[ \left\|\dfrac{\alpha + \beta}{2}\right\|^p \leq \dfrac{|\alpha|^p + |\beta|^p}{2} \] which follows from the convexity of \(|x|^p\).
Norm properties: linearity follows from definition. For nonnegativity, \(\|f\|_p = 0 \iff \|f^p\|_1 = 0 \iff f=0\), where the second and third equalities are to be understood as \(\mu\)-a.e. Triangle inequality is proved by Minkowski’s inequality 7.2.

Proposition 7.4 (Hölder's inequality) Given measurable \(f, g\) on \((\Omega, \Sigma, \mu)\) and \(p\in [1, \infty]\) \[ \|fg\|_1 \leq \|f\|_p \|g\|_{\bar p} \]

Proof idea: normalize \(f, g\), then apply Young’s inequality.

Proof

The cases \(p=1\) or \(\infty\) is straightforward, so consider \(p>1\). Consider the nontrivial case where \(0< \|f\|_p, \|g\|_{\bar p} < \infty\). First assume \(\|f||_p = \|g\|_{\bar p}=1\), then using Young’s inequality (proposition 7.2): \[\begin{align} \int |fg|\, d\mu &\leq \int \dfrac 1 p |f|^p + \dfrac 1 {\bar p} \|g\|^{\bar p}\, d\mu = \dfrac 1 {p} + \dfrac 1 {\bar p} = 1 \end{align}\] This completes the proof in the special case; for unnormalized \(f, g\), apply the homogeneity of the norms.

Proposition 7.5 If \(\mu(\Omega)<\infty\), then for \(0<p<q<\infty\), we have \[ \|f\|_p \leq \mu(\Omega)^{(q-p)/pq} \|f\|_q \] and \(L^q \subset L^p\).

Proof

Apply Holder’s inequality to \(r=q/p>1\): \[\begin{align} \|f\|_p^p &= \||f|^p\|_1 \leq \| |f|^p\|_{q/p} \|1\|_{\overline{q/p}} = \|f\|_q^p \mu(\Omega)^{(q-p)/q} \\ \|f\|_p &\leq \mu(\Omega)^{(q-p)/pq} \|f\|_q \end{align}\]

Proposition 7.6 (variational formula for p-norm) Given \(1\leq p<\infty\) and \(f\in L^p\) \[ \|f\|_p = \sup_{\|g\|_{\bar p} \leq 1} \|fg\|_1 \] This holds for \(p=\infty\) assuming \(\sigma\)-finite \(\mu\).

Proof

If \(\|f\|_p=0\) then both sides are zero trivially. The \(p=1\) case holds trivially, so first consider \(1 < p < \infty\), One side of Holder’s inequality gives \[ \left|\int fg\, d\mu \right| \leq \|fg\|_1 \leq \|f\|_p \] To prove the other direction, one can choose \(\|g\|\) which saturates the inequality above by (1) matching the sign of \(f\) and (2) saturate Young’s inequality by \(g(x)^{\bar p} = f(x)^p\). Note how the converse breaks down for \(p=\infty\), in which case Holder’s inequality is not implied by Young’s.

Remark (counterexample). For \(p=\infty\), consider the non \(\sigma\)-finite measure on \(X=\{b\}\) with \(\mu(\emptyset)=0, \mu(\{b\})=\infty\). Then \(L^1=\{0\}\). For the constant function \(f=1\), we have \(\|f\|_\infty = 1\) (recall definition 7.5 carefully).

Theorem 7.2 (Minkowski's inequality) Given \(1\leq p \leq \infty\) and \(f, g\in L^p\), \(\|f+g\|_p \leq \|f\|_p + \|g\|_p\).

Proof

For arbitrary \(h\in L^{\bar p}\) with \(\|h\|_{\bar p}\leq 1\), we have \[ \left|\int (f+g) h\, d\mu \right| \leq (\|f\|_p + \|g\|_p) \|h\|_{\bar p} \leq \|f\|_p + \|g\|_p \] Take supremum over \(h\) and invoke the variational formula (proposition 7.6) to obtain the desired result.

Proposition 7.7 (Lp is a Banach space) Every Cauchy sequence in \(L^p\) converges.

Drop to a subsequence, use Minkowski’s inequality to argue absolute convergence, next use Fatou’s lemma to establish convergence in \(p\)-norm from a.e pointwise convergence.

Proof

First consider \(1\leq p<\infty\). By proposition 1.7, without loss of generality drop to a subsequence \(f_j\) (which can be defined inductively using the Cauchy condition) and define \(f_0=0\) such that \[ \sum_{k=1}^\infty \|f_k - f_{k-1}\|_p < \infty \]

Conditions for the dominated convergence theorem: we aim to show the integrability of \[ g_m = \sum_{k=1}^m |f_k - f_{k-1}| \to g \in L^p \] note that \(g_m < g\). This follows by Minkowski’s inequality (theorem 7.2) and the definition of our subsequence we have \[ \|g\|_p^p = \int |g|^p\, d\mu = \lim_{m\to \infty} |g_m|^p\, d\mu \] The second equality uses the monotone convergence theorem 4.7.
Next define the partial sum \[ f_m = \sum_{k=1}^m f_k - f_{k-1} \] We have \(|f_m| \leq g_m\), the absolute convergent sequence \(g_m(x)\) (for a.e. \(x\)) also implies the convergence of \(f_m(x)\to f(x)\) a.e. Then \(|f(x)|\leq g(x)\implies f\in L^p\).
Finally, use Fatou’s lemma 4.2 to demonstrate convergence in \(p\)-norm: \[\begin{align} \|f_k - f\|_p = \left( \int |f_k - f|^p\, d\mu \right) \leq \liminf_{j\to \infty} \left( \int |f_k - f_j|^p\, d\mu \right) = \liminf_{j\to \infty} \|f_k - f_j\|_p \end{align}\]

Duality

Recall that the dual space \((L^p)^*\) consists of bounded linear operators on \(L^p\) and is a Banach space. We will show that \((L^p)^*\) under the operator norm is equivalent to \(L^{\bar p}\) under the \(\bar p\)-norm. Also recall the weak topology and convergence 6.5.

Theorem 7.3 (Riesz representation theorem) The dual space of \((L^p)\) for \(1< p < \infty\) is \(L^{\bar p}\); every linear functional \(L\in L^p(\Omega)^*\) has the form, for some unique \(v\in L^{\bar p}\), \[ L(g) = \int_\Omega v(x)g(x)\, d\mu \] This is true for \(p=1\) if \((\Omega, \Sigma, \mu)\) is \(\sigma\)-finite. The identification \((L^p)^* \cong L^{\bar p}\) is further norm-preserving: \[ \|L\| = \|v\|_q \]