This article by Neil S. Trudinger presents two theorems concerning the embedding of certain spaces defined in terms of $L_p$ norms into Orlicz spaces. Specifically, the results include: 1. The space $W^{1}_{1,1}(\Omega)$, consisting of strongly differentiable functions with derivatives in the Morrey space $L_{1,1}(\Omega)$, can be continuously embedded in the Orlicz space $L_{\phi}(\Omega)$ where $\phi(t) = e^{|t|} - |t| - 1$. 2. The Sobolev spaces $W^k_{p}(\Omega)$, where $n = kp$, can be continuously embedded in the Orlicz space $L_{\phi}(\Omega)$ where $\phi(t) = e^{|t|^{1/(n-1)}} - 1$. Both results are shown to be sharp in a certain sense. The first result leads to a simplified proof of a weak form of a measure-theoretic result by John and Nirenberg, which is used to establish Harnack inequalities for weak solutions of elliptic equations. The second result fills a gap in the well-known Sobolev embedding theorems and has applications to eigenvalue or non-uniqueness problems for nonlinear elliptic equations. The article also includes preliminaries on Sobolev spaces, Morrey spaces, and Orlicz spaces, as well as two lemmas used in the proofs. The proofs of the embedding theorems are detailed, and the sharpness of the results is demonstrated through specific examples. Finally, the article discusses the implications of these embedding theorems for eigenvalue problems in nonlinear elliptic equations, extending previous work by Berger and Browder.This article by Neil S. Trudinger presents two theorems concerning the embedding of certain spaces defined in terms of $L_p$ norms into Orlicz spaces. Specifically, the results include: 1. The space $W^{1}_{1,1}(\Omega)$, consisting of strongly differentiable functions with derivatives in the Morrey space $L_{1,1}(\Omega)$, can be continuously embedded in the Orlicz space $L_{\phi}(\Omega)$ where $\phi(t) = e^{|t|} - |t| - 1$. 2. The Sobolev spaces $W^k_{p}(\Omega)$, where $n = kp$, can be continuously embedded in the Orlicz space $L_{\phi}(\Omega)$ where $\phi(t) = e^{|t|^{1/(n-1)}} - 1$. Both results are shown to be sharp in a certain sense. The first result leads to a simplified proof of a weak form of a measure-theoretic result by John and Nirenberg, which is used to establish Harnack inequalities for weak solutions of elliptic equations. The second result fills a gap in the well-known Sobolev embedding theorems and has applications to eigenvalue or non-uniqueness problems for nonlinear elliptic equations. The article also includes preliminaries on Sobolev spaces, Morrey spaces, and Orlicz spaces, as well as two lemmas used in the proofs. The proofs of the embedding theorems are detailed, and the sharpness of the results is demonstrated through specific examples. Finally, the article discusses the implications of these embedding theorems for eigenvalue problems in nonlinear elliptic equations, extending previous work by Berger and Browder.