I am reading J. Duoandikoetxea's Fourier Analysis and on a few occasions have encountered the description of an operator $T$ being "bounded on $L^p$". Such an example of this terminology can be found at the top of page 46 where the strong maximal function $M_s$ is described by the quote:
Then $M_s$ is bounded on $L^p(\mathbb{R}^n)$, $p>1$.
I had implicitly assumed that this means the operator is bounded as an operator from $L^p$ to $L^p$ (meaning there exists $C>0$ such that $\lVert Tf\rVert_p\leq C\lVert f\rVert_p$), but I am questioning this now.