I consider $d \geq 2$ and the domain $ D = [0,1]^d$. I consider a function $f : D \to \mathbb{R}$ that is $k$ times continuously differentiable. [$k$ can be specified as large as possible.] I am wondering if it is possible to use the Sobolev embedding theorem to bound the supremum of the function $f$ using $L^p$ norms of its derivatives. So my question is:
Are there values of $k \in \mathbb{N}^*$ and $1 \leq p < \infty$ so that there exists a finite constant $C$, depending only on $d$, so that the following is true? $$ \sup_{t \in D} |f(t)| \leq C \sum_{i_1,...,i_d \in \mathbb{N}; 1_1+...+i_d \leq k} \int_{D} \left| \frac{\partial^{i_1}}{\partial t_1^{i_1}}...\frac{\partial^{i_d}}{\partial t_d^{i_d}} f(t) \right|^p dt. $$
In particular, do the values (k=1,p=d) or (k=d,p=1) work? [I think that I have read in another question in this website that (k=1,p=d) does not work, but now I am confused.]
Thanks.