Section 3.6 Integrable functions
The theory of \(L^p\) spaces can be developed over any measure space. For simplicity, and also because this is often the case in applications, we will restrict our discussion to open sets of \(\bR^n\) with the Lebesgue measure. This includes also the case of manifolds, since they inherit Lebesgue measure via charts.Subsection 3.6.1 \(L^p(K)\)
We consider here two cases of compact sets \(K\text{:}\) the closure \(\overline{\Omega}\) of a bounded open set \(\Omega\) and a compact manifold, with or without boundary, such as \(\bS^1\) or the \(n\)-torus \(\bT^n\text{.}\) In either case, the Lebesgue measure of \(K\) is finite and the \(L^p\) norm is well-defined over \(C^\infty(K)\) for every \(p\geq1\text{.}\) Hence, one can repeat in this case the discussion make in Section 3.1. Namely, one can define the Riemann integral of smooth functions over \(K\) and then define the set of functions Lebesgue-integrable over \(K\) as the completion of \(C^\infty(K)\) under the \(L^1\) norm
\begin{equation*}
\|f\|_{L^1(K)} = \int_K|f(x)dx.
\end{equation*}
Similarly one defines the spaces \(L^p(K)\) as the completion of \(C^\infty(K)\) in the \(L^p\) norm and prove that \(L^p(K)\) contains all Lesbesgue-integrable functions with finite \(L^p(K)\) norm.Subsection 3.6.2 \(L^p_{loc}(\Omega)\)
We consider now integrable spaces over an open set \(\Omega\subset\bR^n\text{.}\) The main difference is that, in this case, the \(L^1\) norm is not well-defined over \(C^\infty(\Omega)\) since functions may be unbounded and the integral can diverge. In this case, therefore, the more appropriate space to start from is not \(C^\infty(\Omega)\) but rather \(C^\infty_c(\Omega)\text{.}\) The "integrable analogues" of \(C^0(\Omega)\) are the spaces \(L^p_{loc}( \Omega)\) of the functions that are "\(L^p\)-locally integrable" on \(\Omega\text{,}\) namely those functions \(f\) such that, for every compact \(K\subset\Omega\text{,}\) \(f\in L^p(K)\text{.}\) For instance, when \(\Omega=\bR^n\text{,}\) \(1\not\in L^p(\Omega)\) for any finite \(p\) while, on the contrary, \(1\in L^p_{loc}(\Omega)\) for every \(p>0\text{.}\) As usual, we work in the simple case \(\Omega=\bR^n\text{.}\) We start by defining the class of locally Lebesgue-integrable functions as the completion of \(C^\infty_c(\bR^n)\) under the distance function
\begin{equation*}
d(f,g)=\sum_{k=1}^\infty\frac{1}{2^k}\frac{\|f-g\|_{L^1(Q_k)}}{1+\|f-g\|_{L^1(Q_k)}}.
\end{equation*}
We denote this completion by \(L^1_{loc}(\bR^n)\text{.}\) One can prove that this Frechet space is not Banach, namely no norm can generate this topology, just like it happens in case of \(C^k(\bR^n)\text{.}\) Now, for \(p>1\) we consider the similar distance funtions
\begin{equation*}
d(f,g)=\sum_{k=1}^\infty\frac{1}{2^k}\frac{\|f-g\|_{L^p(Q_k)}}{1+\|f-g\|_{L^p(Q_k)}}.
\end{equation*}
and denote by \(L^p_{loc}(\bR^n)\) the completion of \(C^\infty_c(\bR^n)\) under this distance. Theorem 3.6.1.
\begin{equation*}
L^p_{loc}(\bR^n) = \{ f\in L^1_{loc}(\bR^n)\,:\;\|f\|_{L^p(Q_k)}<\infty\text{ for all }k=1,2,\dots\}.
\end{equation*}
Proof.
Theorem 3.6.2.
Let \(f\in L^p_{loc}(\bR^n)\text{,}\) let \(\rho\in C^\infty(\bR^n)\) be a mollifier and set \(f_h=\rho_h\star f\text{.}\) Then \(f_h\to f\) in \(L^p_{loc}(\bR^n)\text{.}\)Proof.
\begin{equation*}
K_1\subset K_2\subset\dots\subset\Omega\text{ such that }\cup_{i=1}^\infty K_i=\Omega\text{,}
\end{equation*}
defines distance functions
\begin{equation*}
d(f,g)=\sum_{k=1}^\infty\frac{1}{2^k}\frac{\|f-g\|_{L^p(K_k)}}{1+\|f-g\|_{L^p(K_k)}}
\end{equation*}
and applies the same arguments above. Then
\begin{equation*}
\dots\supset L^{1/2}_{loc}\supset\dots\supset L^1_{loc}\supset\dots\supset L^2_{loc}\supset\dots\supset L^\infty_{loc}\supset C^0\supset C^1\supset \dots\supset C^\infty,
\end{equation*}
where all these spaces are Frechet. The fact that \(L^p_{loc}(\Omega)\supset L^q_{loc}(\Omega)\) for \(p<q\) is due to the Holder inequality. Indeed, let \(r=p/q\text{,}\) \(f=|h|^q\) and \(g=1\text{.}\) Then \(r'=p/(p-q)\) and
\begin{equation*}
\|\,1\cdot |h|^q\,\|_{L^1(K)} \leq \|\,|h|^q\,\|_{L^r(K)} \|1\|_{L^{r'}(K)} \leq c\|h\|^q_{L^p(K)},
\end{equation*}
so that
\begin{equation*}
\|h\|_{L^q(K)} \leq c'\|h\|_{L^p(K)}.
\end{equation*}
Hence, if \(f\in L^p(K)\text{,}\) then \(f\in L^q(K)\) and the injection of \(L^p(K)\) in \(L^q(K)\) is continuous. Notice, finally, that \(L^p_{loc}(K)=L^p(K)\) for every compact set \(K\text{.}\)Subsection 3.6.3 \(L^p(\Omega)\)
Similarly to how we did in case of the integrable functions over the interval \([0,1]\text{,}\) we define \(L^1(\Omega)\) to be the closure of \(C^\infty_c(\Omega)\) under the norm
\begin{equation*}
\|f\|_{L^1(\Omega)} = \int_\Omega\|f(x)|\,dx
\end{equation*}
As we discussed in Section 3.1, this defines the class of Lesbegue-integrable functions on \(\Omega\) and, via the norm, the value of the corresponding Lebesgue integral. The same argument highlighted in Section 3.1 shows that the closure \(L^p(\Omega)\) of \(C^\infty_c(\Omega)\) under the norm
\begin{equation*}
\|f\|_{L^p(\Omega)} = \left[\int_\Omega\|f(x)|^p\,dx\right]^{1/p}
\end{equation*}
is the space of all functions with finite \(L^p\) norm. When \(\Omega\) is bounded, the Lebesgue measure of \(\Omega\) is finite and things work precisely as in case of the interval \([0,1]\text{.}\) In particular, \(L^p(\Omega)\subset L^q(\Omega)\) for \(p>q\) and the "rainbow of functions" holds. When \(\Omega\) is unbounded, in particular when \(\Omega=\bR^n\text{,}\) then, for every \(p\neq q\text{,}\) there are functions in \(L^p(\Omega)\) that are not \(L^q(\Omega)\) and viceversa.Subsection 3.6.4 \(L^\infty\) spaces
The essential supremum of a function \(f(x)\) is defined as
\begin{equation*}
\|f(x)\|_\infty := \mbox{ess sup}_{\Omega} |f(x)| = \inf \bigl\{\sup_{x\in S} |f(x) |: S\subset\bar\Omega, \mbox{ and } \Omega\backslash S \mbox{ has measure zero }\bigr\}.
\end{equation*}
Example 3.6.3.
\begin{equation*}
f(x) = \left\{\begin{array}{cc} 2 \amp \mbox{ for } x=0\\
1 \amp \mbox{ else }\end{array} \right.
\end{equation*}
has
\begin{equation*}
\mbox{ess sup}_{[-1,1]} |f(x) | =1 \neq \sup_{[-1,1]} |f(x) | =2 .
\end{equation*}
Example 3.6.4.
\begin{equation*}
\mbox{ess sup}_{\Omega} |f(x)| = \sup_{\Omega} |f(x)|.
\end{equation*}
In this sense we have
\begin{equation*}
\|f\|_{L^\infty}=\|f\|_{C^0}=\|f\|_\infty.
\end{equation*}
Theorem 3.6.5.
Let \(\Omega\) have finite volume, then
\begin{equation*}
\|f\|_\infty = \lim_{p\to\infty} \|f\|_p
\end{equation*}
and if \(\|f\|_p\leq K\) for all \(p\text{,}\) then \(\|f\|_\infty \leq K\text{.}\)Proof.
\begin{equation*}
\|f\|_p =\left(\int_\Omega|f|^p dx \right)^{\frac{1}{p}} \leq \|f\|_\infty \left(\int_\Omega 1^p dx\right)^{\frac{1}{p}} = |\Omega|^{\frac{1}{p}} \|f\|_\infty,
\end{equation*}
which implies that
\begin{equation*}
\limsup_{p\to\infty} \|f\|_p \leq \|f\|_\infty.
\end{equation*}
Moreover, for each \(\ep>0\) there exists a set \(A\) of non-zero measure such that
\begin{equation*}
|f(x)| \geq \|f\|_\infty - \ep, \qquad \mbox{for all}\quad x\in A.
\end{equation*}
Therefore
\begin{equation*}
\int_\Omega |f(x)|^p dx \geq \int_A|f(x)|^p dx \geq |A|\left(\|f\|_\infty - \ep \right)^p,
\end{equation*}
which implies
\begin{equation*}
\|f\|_p \geq |A|^{\frac{1}{p}} \left(\|f\|_\infty - \ep \right).
\end{equation*}
We obtain in the limit that
\begin{equation*}
\liminf_{p\to\infty} \|f\|_p \geq \|f\|_\infty,
\end{equation*}
proving our claim.Theorem 3.6.6.
\(L^\infty(\Omega)\) is complete, i.e. a Banach space.Proof.
\begin{equation*}
|f_n(x) |\leq \|f_n\|_\infty \qquad \mbox{a.e.}
\end{equation*}
and
\begin{equation*}
|f_n(x)-f_m(x) |\leq \|f_n-f_m\|_\infty \qquad \mbox{a.e.}
\end{equation*}
Hence for a.e. \(x\in\Omega\) the set \(\{f_n(x)\}\) is a Cauchy sequence in \(\RR\text{,}\) which has a limit
\begin{equation*}
f_n(x) \to f(x) \qquad \mbox{a.e. in} \quad \Omega.
\end{equation*}
Hence we find a well defined function \(f(x)\text{,}\) which based on the above estimates, is in \(L^\infty(\Omega)\text{.}\)