For \(1\le p\le\infty\) and \(1/p+1/q=1\), Hölder’s inequality \(\norm{fg}_1\le\norm{f}_p\norm{g}_q\) and Minkowski’s inequality \(\norm{f+g}_p\le\norm{f}_p+\norm{g}_p\) make \(L^p\) a normed space, and by the Riesz–Fischer theorem it is complete. On a finite measure space \(L^q\subseteq L^p\) for \(p\le q\); on \(\RR\) there is no inclusion between \(L^p\) and \(L^q\) for \(p\ne q\), as \(x^{-a}\chi_{(0,1)}\) and \(x^{-a}\chi_{(1,\infty)}\) show for suitable \(a>0\). The norm \(\norm{f}_\infty\) is the essential supremum of \(\abs{f}\), and if \(f\in L^r\) for some \(r<\infty\), then \(\norm{f}_p\to\norm{f}_\infty\) as \(p\to\infty\). Young’s inequality: \(\norm{f*g}_r\le\norm{f}_p\norm{g}_q\) for \(1\le p,q,r\le\infty\) with \(1+1/r=1/p+1/q\). For an approximate identity \((\varphi_\varepsilon)\) in \(L^1(\RR^n)\) and \(f\in L^p(\RR^n)\) with \(1\le p<\infty\), \(\varphi_\varepsilon*f\to f\) in \(L^p\), while no \(g\in L^1(\RR^n)\) satisfies \(g*f=f\) for all \(f\in L^1(\RR^n)\). For \(1\le p<\infty\) and \(f\in L^p(\RR^n)\), \(\norm{f(\cdot+h)-f}_p\to0\) as \(h\to0\).
Essential supremum norm and essentially bounded functions (Tag D-3PVRB)
Let \((X,\mcm,\mu)\) be a measure space and let \(f\colon X\to\CC\) be measurable. The essential supremum of \(\abs{f}\) is \[ \norm{f}_\infty \coloneqq \inf \theset{\alpha \geq 0 \suchthat \mu\qty{\theset{x\in X \suchthat \abs{f(x)} \geq \alpha}} = 0} \in [0,\infty], \] with \(\inf\emptyset=\infty\). The function \(f\) is essentially bounded if there exists a real number \(c\) such that \(\mu\qty{\theset{x\in X \suchthat \abs{f(x)} > c}} = 0\), equivalently \(\norm{f}_\infty < \infty\).
When \(\norm{f}_\infty<\infty\), the set \(\theset{\abs{f}>\norm{f}_\infty}=\bigcup_{k\geq 1}\theset{\abs{f}\geq\norm{f}_\infty+1/k}\) is a countable union of null sets, so \(\abs{f(x)} \leq \norm{f}_\infty\) for almost every \(x\), and \(\norm{f}_\infty\) is the smallest constant with this property.
$L^\infty$ (Tag D-MFDEJ)
Let \((X,\mcm,\mu)\) be a measure space. The space \(L^\infty(X)\) is \[ L^\infty(X) \coloneqq \theset{f\colon X\to \CC \suchthat f \text{ is measurable and essentially bounded}} = \theset{f\colon X\to \CC \suchthat f \text{ is measurable and } \norm{f}_\infty < \infty}, \] where \(\norm{f}_\infty\) is the essential supremum of \(\abs{f}\).
Hölder's inequality (Tag PR-7BGSE)
Let \((X,\mcm,\mu)\) be a measure space, let \(1\le p,q\le\infty\) satisfy \(\frac 1 p + \frac 1 q = 1\), and let \(f,g\colon X\to\CC\) be measurable. Then \[ \norm{f g}_{1} \leq \norm{f}_{p} \norm{g}_{q} . \] For \(1<p,q<\infty\), this reads \[ \int_X \abs{fg}\dmu \leq \qty{\int_X \abs{f}^p\dmu}^{1/p} \qty{\int_X \abs{g}^q\dmu}^{1/q} . \]
Minkowski's inequality (Tag PR-XUVZY)
Let \((X,\mcm,\mu)\) be a measure space, let \(1\leq p<\infty\), and let \(f,g\in L^p(\mu)\). Then \(f+g\in L^p(\mu)\) and \[ \norm{f+g}_p \leq \norm{f}_p+ \norm{g}_p . \]
$L^p$ spaces are Banach spaces (Tag PR-L35O7)
Let \((X,\mcm,\mu)\) be a measure space and let \(1\leq p\leq\infty\). Then \(L^p(X,\mu)\), whose elements are classes of functions modulo equality \(\mu\)-almost everywhere, with the norm \(\norm{\cdot}_p\), is a Banach space.
First let \(1\leq p<\infty\), and let \((f_n)\) be Cauchy in \(L^p\). Choose a subsequence \((f_{n_k})\) such that \[ \norm{f_{n_{k+1}}-f_{n_k}}_p\leq 2^{-k} \qquad(k\geq1). \] Set \[ g_k\coloneqq\abs{f_{n_{k+1}}-f_{n_k}}, \qquad S_N\coloneqq\sum_{k=1}^N g_k. \] Minkowski’s inequality gives \[ \norm{S_N}_p\leq\sum_{k=1}^N2^{-k}\leq1. \] Since \(S_N\uparrow S\coloneqq\sum_{k\geq1}g_k\), the monotone convergence theorem yields \[ \int_X S^p\dmu =\lim_{N\to\infty}\int_X S_N^p\dmu \leq1. \] Thus \(S<\infty\) almost everywhere. Outside a null set, the series \[ \sum_{k\geq1}\big(f_{n_{k+1}}(x)-f_{n_k}(x)\big) \] converges absolutely; there, define \(f(x)\) so that this sum equals \(f(x)-f_{n_1}(x)\), and put \(f\coloneqq0\) on the exceptional null set.
For every \(k\) and \(N>k\), \[ \abs{f_{n_N}-f_{n_k}} \leq\sum_{j=k}^{N-1}g_j. \] Letting \(N\to\infty\) and applying Fatou’s lemma together with Minkowski’s inequality gives \[ \norm{f-f_{n_k}}_p \leq\sum_{j=k}^{\infty}\norm{g_j}_p \leq\sum_{j=k}^{\infty}2^{-j} =2^{1-k}. \] Hence \(f_{n_k}\to f\) in \(L^p\). Given \(\varepsilon>0\), choose \(k\) with \(\norm{f_{n_k}-f}_p<\varepsilon/2\) and \(n_k\geq N\), where \(N\) is such that \(\norm{f_n-f_m}_p<\varepsilon/2\) for all \(m,n\geq N\). Then \(\norm{f_n-f}_p<\varepsilon\) for all \(n\geq N\), so \(f_n\to f\) in \(L^p\).
Now let \(p=\infty\), and again choose a subsequence with \[ \norm{f_{n_{k+1}}-f_{n_k}}_\infty\leq2^{-k}. \] For each \(k\) there is a null set \(Z_k\) outside which \[ \abs{f_{n_{k+1}}-f_{n_k}}\leq2^{-k}. \] Outside the null set \(Z\coloneqq\bigcup_kZ_k\), the series of successive differences converges uniformly by the Weierstrass \(M\)-test. Hence \((f_{n_k})\) converges uniformly on \(X\setminus Z\) to a bounded measurable function \(f\), extended by \(0\) on \(Z\), and \[ \norm{f-f_{n_k}}_\infty\leq2^{1-k}. \] As in the case \(p<\infty\), since \((f_n)\) is Cauchy, \(f_n\to f\) in \(L^\infty\).
Inclusions among $L^p$ spaces (Tag FT-GQRV2)
Let \(1\le p< q\le\infty\).
If \((X,\mcm,\mu)\) is a measure space with \(\mu(X)<\infty\), then \(L^q(X)\subseteq L^p(X)\), and \(\norm{f}_p\le\mu(X)^{\frac1p-\frac1q}\norm{f}_q\) for \(f\in L^q(X)\), with \(\frac1\infty\coloneqq0\). In particular \(L^\infty(X) \subseteq L^2(X) \subseteq L^1(X)\).
For counting measure on \(\ZZ\), \(\ell^p(\ZZ)\subseteq\ell^q(\ZZ)\), and \(\norm{a}_q\le\norm{a}_p\) for \(a\in\ell^p(\ZZ)\). In particular \(\ell^1(\ZZ) \subseteq \ell^2(\ZZ) \subseteq \ell^\infty(\ZZ)\).
For Lebesgue measure on \(\RR^n\), neither \(L^p(\RR^n)\subseteq L^q(\RR^n)\) nor \(L^q(\RR^n)\subseteq L^p(\RR^n)\) holds.
For \(q=\infty\), \(\int_X\abs{f}^p\le\mu(X)\norm{f}_\infty^p\). For \(q<\infty\), apply Hölder’s inequality with exponents \(r=q/p\) and \(r'=q/(q-p)\) to \(\abs{f}^p\cdot1\): \[ \int_X\abs{f}^p\dmu\le\qty{\int_X\abs{f}^q\dmu}^{p/q}\mu(X)^{1-p/q}, \] and take \(p\)-th roots.
If \(\norm{a}_p=1\), then \(\abs{a_k}\le1\) for all \(k\), so \(\norm{a}_\infty\le1\) and, for \(q<\infty\), \(\sum_k\abs{a_k}^q\le\sum_k\abs{a_k}^p=1\). The general case follows by scaling \(a\) by \(1/\norm{a}_p\).
Let \(f_a(x)\coloneqq\abs{x}^{-a}\one_{\abs{x}\le1}\) and \(g_a(x)\coloneqq\abs{x}^{-a}\one_{\abs{x}>1}\). In polar coordinates, \(f_a\in L^r(\RR^n)\) if and only if \(ar<n\) (for \(r<\infty\)), and \(g_a\in L^r(\RR^n)\) if and only if \(ar>n\). If \(q<\infty\), choose \(a\) with \(n/q\le a<n/p\): then \(f_a\in L^p\setminus L^q\), and choosing \(a\) with \(n/q<a\le n/p\) gives \(g_a\in L^q\setminus L^p\). If \(q=\infty\), then \(f_a\in L^p\setminus L^\infty\) for \(0<a<n/p\), and the constant function \(1\) lies in \(L^\infty\setminus L^p\).
Continuity of translation in $L^1$ (Tag T-BFOJT)
For \(h\in\RR^n\) and \(f\in L^1(\RR^n)\) let \((\tau_hf)(x)\coloneqq f(x-h)\). For every \(f\in L^1(\RR^n)\), \[ \lim_{h\to 0}\norm{\tau_h f - f}_1 = 0 . \]
Each \(\tau_h\) is a linear isometry of \(L^1(\RR^n)\), so the family \(\theset{\tau_h\suchthat h\in\RR^n}\) is equicontinuous as a family of maps \(L^1(\RR^n)\to L^1(\RR^n)\). Together with the pointwise convergence \(\tau_hf\to f\) of the theorem, this makes the convergence uniform on compact subsets \(K\subseteq L^1(\RR^n)\): \[ \lim_{h\to0}\sup_{f\in K}\norm{\tau_hf-f}_1=0 . \]
Convolution of functions on $\RR^n$ (Tag D-TS42Y)
Let \(f,g\colon\RR^n\to\CC\) be measurable. The convolution of \(f\) and \(g\) is the function \[ (f * g)(x) \coloneqq \int_{\RR^n} f(x-y)\, g(y) \, dy, \] defined at every \(x\in\RR^n\) for which the integral converges absolutely.
Young's convolution inequality (Tag PR-2C3SZ)
Let \(1\le p,q,r\le\infty\) satisfy \[ \frac1r = \frac1p + \frac1q - 1 . \] If \(f\in L^p(\RR^n)\) and \(g\in L^q(\RR^n)\), then the convolution \(f\ast g\) is defined almost everywhere, \(f\ast g\in L^r(\RR^n)\), and \[ \norm{f \ast g}_{r} \leq \norm{f}_{p} \norm{g}_{q} \] [@Fol13].
The following are the cases \(p=q=r=1\); \(p=1\) and \(q=r\); \(r=\infty\) with \(\frac1p+\frac1q=1\); and \(p=q=2\), \(r=\infty\): \[ \begin{aligned} \norm{f\ast g}_1 & \leq \norm{f}_1 \norm{g}_1, \\ \norm{f\ast g}_q & \leq \norm{f}_1 \norm{g}_q, \\ \norm{f\ast g}_\infty & \leq \norm{f}_p \norm{g}_q, \\ \norm{f\ast g}_\infty & \leq \norm{f}_2 \norm{g}_2. \end{aligned} \]
Approximations to the identity (Tag D-ARQFC)
Let \(\phi\in L^1(\RR^n)\) with \(\int_{\RR^n}\phi = 1\), and for \(t>0\) let \(\phi_t(x)\coloneqq t^{-n}\phi(t^{-1}x)\) be its dilations. Then \(\phi\), or the family \((\phi_t)_{t>0}\), is called an approximate identity.
Let \(\phi\in L^1(\RR^n)\) with \(\int\phi=1\). Then:
\(\int \phi_t = \int\phi = 1\) and \(\norm{\phi_t}_1 = \norm{\phi}_1\) for every \(t>0\); in particular \(\norm{\phi_t}_1=1\) when \(\phi\geq 0\).
\(\sup_{t>0} \norm{\phi_t}_1 < \infty\).
For every \(h>0\), \(\lim_{t\to 0^+} \int_{\abs{x} \geq h} \abs{\phi_t(x)}\dx = 0\).
The substitution \(x=ty\), \(dx=t^n\,dy\), gives \(\int\phi_t=\int\phi\) and \(\norm{\phi_t}_1=\norm{\phi}_1\), which proves the first two items. The same substitution gives \[ \int_{\abs{x}\geq h}\abs{\phi_t(x)}\dx=\int_{\abs{y}\geq h/t}\abs{\phi(y)}\,dy, \] and the right side tends to \(0\) as \(t\to 0^+\) by dominated convergence, with dominating function \(\abs{\phi}\in L^1\).
Convolution with an approximate identity converges in $L^1$ (Tag T-HHFGB)
Let \(\phi\in L^1(\RR^n)\) with \(\int_{\RR^n}\phi=1\), let \(\phi_t(x)\coloneqq t^{-n}\phi(\inverseof{t} x)\) for \(t>0\) be the corresponding approximate identity, and let \(f\in L^1(\RR^n)\). Then \[ \lim_{t\to 0^+}\norm{f * \phi_{t} - f}_1 = 0 . \]