Let $H^s$, $0\leq s<\infty$ be the $L^2$ based Sobolev spaces such that

$$ \hat{f}(\xi)(1+|\xi|^2)^{s/2} \in L^2. $$

Let $r_1,r_2,p_1,p_2>0$ be given parameters. Assume that a linear operator $T$ satisfies that $$ T:H^{r_1}\rightarrow H^{r_2} $$ with operator norm $$ \|T\|_{H^{r_1},H^{r_2}}=C_1 $$ and $$ T:H^{p_1}\rightarrow H^{{p_2}} $$ with operator norm $$ \|T\|_{H^{p_1},H^{p_2}}=C_2. $$

**Question 1**I would like to have a precise reference for the following result (that I'm confident it's true)

**Lemma**: Assume $r_2=r_1$ and $p_2=p_1$. Assume also that $T$ satisfies the assumptions above and let
$$
s_1=\theta r_1+(1-\theta)p_1.
$$

Then

$$ T:H^{s_1}\rightarrow H^{{s_1}} $$ with operator norm $$ \|T\|_{H^{s_1},H^{s_1}}\leq C_1^\theta C_2^{1-\theta}. $$

**Question 2** Is there some result in the case $r_1\neq r_2$, $p_1\neq p_2$.?

Thanks!!

an equalityfor the operator norm, or only an estimate? $\endgroup$