Suppose we have an operator $T$ such that $||Tf||_s \leq C_s ||f||_s$ for all $f \in H^s(M)$ and $s \in \mathbb{N} \cup \{0\}$, where $M$ is a Riemannian manifold. Can we say from here that $||Tf||_s \leq C_s ||f||_s$ for all $f \in H^s(M)$ and $s \geq 0$ ?
I was trying to approach it in the following two ways but stuck in both:
For any $s \geq 0$ such that $s$ is not an integer, we can bound $s$ in between nonnegative integers $s_1$ and $s_2$ where $s_1 \leq s \leq s_2$. Then we have $\infty > ||f||_{s_1} \geq ||f||_{s} \geq ||f||_{s_2}$ for all $f \in H^{s_1}(M)$. We have $||Tf||_{s_1} \leq C_{s_1} ||f||_{s_1}$ and $||Tf||_{s_2} \leq C_{s_2} ||f||_{s_2}$. By Sobolev interpolation we can say $$||Tf||_s \leq k_1 ||Tf||_{s_1} + k_2 ||Tf||_{s_2} \leq k_1 C_{s_1} ||f||_{s_1} + k_2 C_{s_2} ||f||_{s_2}.$$ But then how to proceed further?
Let $s' > s$ be an integer. For an $f \in H^{s}(M)$ we can find a sequence $\{f_j\}_{j \geq 1}$ of compactly supported functions/forms in $M$ such that $f_j \to f$ in $H^s(M)$. Since $f_j$ s are compactly supported, we can say $||Tf_j||_{s'} \leq C_{s'} ||f_j||_{s'} $ for all $j$. By Rellich's lemma we know the inclusion of $H^{s'}(M)$ into $H^s(M)$ is a compact embedding. But then what ?