Translation continuity and normalised local approximate identities on an LCA group

(Reduction to a $\sigma$-compact essential support.) For $n\ge 1$ the Borel set $E_n:=\{\mathrm{\mid }f\mathrm{\mid }>1\mathrm{/}n\}$ has $m_G(E_n)\le n^p\mathrm{\parallel }f{\mathrm{\parallel }}_p^p<+\mathrm{\infty }$; by [F1] choose an open $U_n\supseteq E_n$ with $m_G(U_n)<+\mathrm{\infty }$ and compact $K_{n,j}\subseteq U_n$ with $m_G(K_{n,j})\to m_G(U_n)$. Then $S:={\bigcup }_{n,j}{K}_{n,j}$ is $\sigma$-compact of $\sigma$-finite measure, $m_G(\{\mathrm{\mid }f\mathrm{\mid }>0\}\setminus S)=0$, and replacing $f$ by $f\cdot {\mathbf{1}}_S$ changes it at most on a null set, hence changes neither the class in $L^p$ nor any norm. For the product calculations below we additionally use an a.e. pointwise limit of real or complex $C_c$ approximants supplied by [F2] as representative, zero where the sequence fails to converge. The convolution-algebra supplier's product-measurability argument applies verbatim to $L^p$ approximants and to their compact-support unions. Thus we may assume $f=0$ outside a $\sigma$-compact set $S$ and the kernels $f(x-y)$ below are product measurable.

(Translation is an isometry.) For $f\in L^p$ and $x\in G$, the substitution $t\mapsto t+x$ preserves $m_G$ and therefore $\mathrm{\parallel }{T}_{x}f{\mathrm{\parallel }}_p=\mathrm{\parallel }f{\mathrm{\parallel }}_p$; the map is linear and respects a.e. equality, so it acts on the quotient $L^p$.

(Convolution with a compactly supported kernel.) Let $u\in C_c(G;\mathbb{C})$ (including real kernels) and assume $f$ vanishes outside the $\sigma$-compact set $S$. The function $(y,x)\mapsto \mathrm{\mid }u(y)\mathrm{\mid }\mathrm{\mid }f(x-y){\mathrm{\mid }}^p$ is supported in $\mathrm{supp}u\times (S+\mathrm{supp}u)$, a $\sigma$-finite product by [F1], so Tonelli's theorem gives ${\int }_G{\int }_G\mathrm{\mid }u(y)\mathrm{\mid }\mathrm{\mid }f(x-y){\mathrm{\mid }}^{p}d{m}_G(y)d{m}_G(x)=\mathrm{\parallel }u{\mathrm{\parallel }}_1\mathrm{\parallel }f{\mathrm{\parallel }}_p^p<+\mathrm{\infty }$ after the substitution $x\mapsto x+y$; by Hölder's inequality with the finite measure $\mathrm{\mid }u\mathrm{\mid }d{m}_G$ this makes ${\int }_G\mathrm{\mid }u(y)\mathrm{\mid }\mathrm{\mid }f(x-y)\mathrm{\mid }d{m}_G(y)$ finite for $m_G$-a.e. $x$. For those $x$ the two defining integrals agree, $(u\ast f)(x)={\int }_{G}u(y)f(x-y)d{m}_G(y)={\int }_{G}f(z)u(x-z)d{m}_G(z)=(f\ast u)(x)$, by inversion followed by translation in the substitution $z=x-y$. Applying Minkowski's integral inequality [F3] to $F(x,y):=u(y)f(x-y)$ on the $\sigma$-finite product yields $\mathrm{\parallel }u\ast f{\mathrm{\parallel }}_p=\parallel {\int }_{G}u(y)f(\cdot -y)d{m}_G(y){\parallel }_p\le {\int }_G\mathrm{\mid }u(y)\mathrm{\mid }\mathrm{\parallel }{T}_{y}f{\mathrm{\parallel }}_{p}d{m}_G(y)=\mathrm{\parallel }u{\mathrm{\parallel }}_1\mathrm{\parallel }f{\mathrm{\parallel }}_p\mathrm{.}$

(Norm continuity of translation.) Fix $f\in L^p$ and $\epsilon >0$. By [F2] choose $f_0\in C_c(G;\mathbb{F})$ for the scalar field $\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$, with $\mathrm{\parallel }f-f_0{\mathrm{\parallel }}_p<\epsilon \mathrm{/}3$. If $f_0=0$, the isometry gives $\mathrm{\parallel }{T}_{z}f-f{\mathrm{\parallel }}_p<2\epsilon \mathrm{/}3$ for every $z$. Otherwise put $K_0:=\mathrm{supp}{f}_0$; its compact thickening below has positive finite measure. Choose a compact symmetric identity neighbourhood $W$ and a compact symmetric neighbourhood $E\subseteq W$ so small that $\mathrm{\mid }{f}_0(t-z)-f_0(t)\mathrm{\mid }<\epsilon \mathrm{/}(3m_G(K_0+W{)}^{1\mathrm{/}p})$ for all $t\in G$ and $z\in E$; this is possible by the uniform-continuity argument on the compact set $K_0+W$: cover $K_0+W$ by finitely many translates $y_j+V_j$ on which $f_0$ varies by less than the bound, and intersect the corresponding symmetric neighbourhoods of $0$. Then $\mathrm{\parallel }{T}_z{f}_0-f_0{\mathrm{\parallel }}_p<\epsilon \mathrm{/}3$ for $z\in E$, and step 1.2 gives, for every $z\in E$, $\mathrm{\parallel }{T}_{z}f-f{\mathrm{\parallel }}_p\le \mathrm{\parallel }{T}_z(f-f_0){\mathrm{\parallel }}_p+\mathrm{\parallel }{T}_z{f}_0-f_0{\mathrm{\parallel }}_p+\mathrm{\parallel }{f}_0-f{\mathrm{\parallel }}_p<2\epsilon \mathrm{/}3+\epsilon \mathrm{/}3=\epsilon \mathrm{.}$ Hence $x\mapsto T_{x}f$ is norm-continuous at $0$, and at every $x_0$ by $T_{x+x_0}=T_x{T}_{x_0}$ and step 1.2.

(The weighted-average estimate.) Let $u\in C_c(G)$ satisfy $u\ge 0$ and ${\int }_{G}ud{m}_G=1$. Since ${\int }_{G}u(y)d{m}_G(y)=1$, for a.e. $x$ $(u\ast f)(x)-f(x)={\int }_{G}u(y)(f(x-y)-f(x))d{m}_G(y),$ and Minkowski's inequality [F3] applied to $G(x,y):=u(y)(f(x-y)-f(x))$ on $\mathrm{supp}u\times (S\cup (S+\mathrm{supp}u))$, a $\sigma$-finite product containing both terms, gives $\mathrm{\parallel }u\ast f-f{\mathrm{\parallel }}_p\le {\int }_{G}u(y)\mathrm{\parallel }{T}_{y}f-f{\mathrm{\parallel }}_{p}d{m}_G(y)\le {\sup }_{y\in \mathrm{supp}u}\mathrm{\parallel }{T}_{y}f-f{\mathrm{\parallel }}_p\mathrm{.}$

(Normalised local approximate identities.) Let $U$ be an identity neighbourhood. By continuity of addition at $0$ choose a symmetric open $V$ with $V+V\subseteq U$, Then $\bar{V}\subseteq V+V\subseteq U$: for $x\in \bar{V}$, the open set $x+V$ meets $V$, so $x\in V-V=V+V$. By [F4] choose $v\in C_c(G)$ with $0\le v\le 1$, $v(0)=1$, and $v=0$ outside $V$; hence $\mathrm{supp}v\subseteq \bar{V}\subseteq U$. Then $w(x):=v(x)v(-x)$ is symmetric, nonnegative, compactly supported in $\bar{V}\subseteq U$, and $w(0)=1$, so $c:={\int }_{G}wd{m}_G>0$ by [F1]; set $u_U:=c^{-1}w$. Then $u_U\ge 0$ is symmetric with ${\int }_G{u}_{U}d{m}_G=1$ and $\mathrm{supp}{u}_U\subseteq \bar{V}\subseteq U$. Given $f\in L^p$ and $\epsilon >0$, step 2.2 provides a symmetric identity neighbourhood $E$ with $\mathrm{\parallel }{T}_{y}f-f{\mathrm{\parallel }}_p<\epsilon$ for every $y\in E$. For every identity neighbourhood $U\subseteq E$ and every admissible kernel $u_U$, one has $\mathrm{supp}{u}_U\subseteq U\subseteq E$, so step 3.1 yields $\mathrm{\parallel }{u}_U\ast f-f{\mathrm{\parallel }}_p<\epsilon$, and by step 2.1 also $\mathrm{\parallel }{u}_U\ast f{\mathrm{\parallel }}_p\le \mathrm{\parallel }f{\mathrm{\parallel }}_p$ and $\mathrm{\parallel }f\ast u_U{\mathrm{\parallel }}_p=\mathrm{\parallel }{u}_U\ast f{\mathrm{\parallel }}_p\le \mathrm{\parallel }f{\mathrm{\parallel }}_p$; this bound is uniform over admissible kernels. The set of all pairs $(U,u)$ is directed by shrinking $U$: two pairs have a common later pair by constructing a kernel inside their intersected neighbourhood. Thus both limits hold for this net without choosing kernels simultaneously. For $p=1$ it is a contractive two-sided approximate identity of $A=L^1(G,m_G)$.

Steps 1.2, 2.2 and 4.1 prove the isometry, the norm continuity, the existence of the symmetric normalised cutoffs and both approximate-identity limits with their contractive bounds.