This paper by Neil S. Trudinger discusses two embedding theorems for certain function spaces into Orlicz spaces. The first theorem shows that the space $ W_{1,1}^{1}(\Omega) $, consisting of 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 $. The second theorem demonstrates that Sobolev spaces $ W_{p}^{k}(\Omega) $, where $ n = kp $, can be continuously embedded in the same Orlicz space with $ \phi(t) = e^{|t|^{n}/(n-1)} - 1 $. Both results are sharp in a certain sense. The first result is used to simplify a weak form of a measure-theoretic result by John and Nirenberg, which is then used by Moser, Serrin, and the author to establish Harnack inequalities for weak solutions of elliptic equations. These results do not depend on the John-Nirenberg lemma. The second result fills a gap in the well-known Sobolev embedding theorems and has applications to partial differential equations, particularly in eigenvalue and non-uniqueness problems for nonlinear elliptic equations. The paper also presents preliminary definitions of Sobolev spaces, Morrey spaces, and Orlicz spaces. It includes two lemmas that are used in the proofs of the embedding theorems. The first lemma provides an estimate for functions in $ W_{1}^{1}(\Omega) $, while the second is a Poincaré-type lemma. The embedding theorems are proven using techniques involving Hölder's inequality and the properties of Orlicz spaces. The first theorem is shown to be sharp, and the second is used to generalize results on eigenvalue problems for nonlinear elliptic equations. The paper also discusses the implications of these results for the study of eigenvalues and eigenfunctions of such equations, and references several related works by other authors.This paper by Neil S. Trudinger discusses two embedding theorems for certain function spaces into Orlicz spaces. The first theorem shows that the space $ W_{1,1}^{1}(\Omega) $, consisting of 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 $. The second theorem demonstrates that Sobolev spaces $ W_{p}^{k}(\Omega) $, where $ n = kp $, can be continuously embedded in the same Orlicz space with $ \phi(t) = e^{|t|^{n}/(n-1)} - 1 $. Both results are sharp in a certain sense. The first result is used to simplify a weak form of a measure-theoretic result by John and Nirenberg, which is then used by Moser, Serrin, and the author to establish Harnack inequalities for weak solutions of elliptic equations. These results do not depend on the John-Nirenberg lemma. The second result fills a gap in the well-known Sobolev embedding theorems and has applications to partial differential equations, particularly in eigenvalue and non-uniqueness problems for nonlinear elliptic equations. The paper also presents preliminary definitions of Sobolev spaces, Morrey spaces, and Orlicz spaces. It includes two lemmas that are used in the proofs of the embedding theorems. The first lemma provides an estimate for functions in $ W_{1}^{1}(\Omega) $, while the second is a Poincaré-type lemma. The embedding theorems are proven using techniques involving Hölder's inequality and the properties of Orlicz spaces. The first theorem is shown to be sharp, and the second is used to generalize results on eigenvalue problems for nonlinear elliptic equations. The paper also discusses the implications of these results for the study of eigenvalues and eigenfunctions of such equations, and references several related works by other authors.