$L\log L$ and $L^p$ embedding

Question

My question is a simple one: I am aware of the embedding $L^p(\Omega)\in L\log L(\Omega)$ for finite measure spaces, with constant $\frac{cp}{p-1}$. Does this embedding hold on for instance, the whole of $\mathcal{R}^n$, and what is the constant in the norm estimates? Would it blow up as $|\Omega|\rightarrow\infty$ and $p\rightarrow 1$ as well? Thanks in advance.