Continuity of Translation and Dilation on $L^p$ spaces Let us consider any $f \in L^p(U)$, where $U \subset \mathbb R^n$ is open, and $1 < p < \infty$. We know the translation operator $f(x) \mapsto f(x+a)$ and the dilation operator $f(x) \mapsto f(s x)$, where $a \in \mathbb R^n$ and $s \in \mathbb R^n$. **Questions:** i) Under which conditions on $U$ and $p$ are translation and dilation bounded, when $a$ and $s$ are fixed, and what is the norm? ii) Conversely, when $f$ is fixed, under what conditions are translation and dilation continuous in the parameters, and how good is that continuity (like locally Hölder)?

It seems the following.

I will consider that $s\in\Bbb R$, not in $\Bbb R^n$, because I don’t understand what is $sx\in\Bbb R^n$

The correctness of the definition of the operator imposes restrictions on the set $U$: $U+a\subset U$ or $sU\subset U$. Or we may for each function $f\in L^p(U)$ consider its extension in $L^p(\Bbb R^n)$ such that $f|(\Bbb R^n\setminus U)\equiv 0$ and to deal with this extension.

i) The translation operator $T_a$ preserves the norm:

$$\| f\|^p=\int_U f(x)^pd\mu=\int_{\Bbb R^n} f(x)^pd\mu=\int_U f(x+a)^pd\mu=\|T_a f\|^p.$$

The dilation operator $D_s$ has norm $|s|^{-n/p}$:

$$\|f\|^p=\int_U f(x)^pd\mu=\int_{\Bbb R^n} f(x)^pd\mu=|s|^{-n}\int_{\Bbb R^n} f(sx)^pd\mu=|s|^{-n}\|D_sf\|^p.$$

ii) I hope that similarly to answers to a question pointed by @saz, we can show that $\lim_{r \to r_0} \|T_rf−T_{r_0}f\|_{L_p} =0$ and $\lim_{s \to s_0} \|D_sf−D_{s_0}f\|_{L_p}=0$.