Sobolev’s original definition of his spaces $L^{m,p}\left(\Omega \right)$ is revisited. It only assumed that $\Omega \subseteq {ℝ}^n$ is a domain. With elementary methods, essentially based on Poincare’s inequality for balls (or cubes), the existence of intermediate derivates of functions $u\in L^{m,p}\left(\Omega \right)$ with respect to appropriate norms, and equivalence of these norms is proved.

Let $\Omega$ be a bounded open set in ${ℝ}^n$, $n\ge 2$. In a well-known paper Indiana Univ. Math. J., 20, 1077–1092 (1971) Moser found the smallest value of $K$ such that $$sup\left\{{\int }_{\Omega }exp\left({\left(\frac{\left|f\left(x\right)\right|}{K}\right)}^{n/(n-1)}\right):f\in W_0^{1,n}\left(\Omega \right),{\parallel \nabla f\parallel }_{L^n}\le 1\right\}<\infty .$$ We extend this result to the situation in which the underlying space $L^n$ is replaced by the generalized Zygmund space $L^n{log}^{n-1}L{log}^{\alpha }logL$$(\alpha <...$

We survey results from the paper [CPS] in which we developed a new sharp iteration method and applied it to show that the optimal Sobolev embeddings of any order can be derived from isoperimetric inequalities. We prove thereby that the well-known link between first-order Sobolev embeddings and isoperimetric inequalities translates to embeddings of any order, a fact that had not been known before. We show a general reduction principle that reduces Sobolev type inequalities of any order involving...

We prove an approximation lemma on (stratified) homogeneous groups that allows one to approximate a function in the non-isotropic Sobolev space ${\dot{NL}}^{1,Q}$ by $L^{\infty }$ functions, generalizing a result of Bourgain–Brezis. We then use this to obtain a Gagliardo–Nirenberg inequality for $$ on the Heisenberg group ${ℍ}^n$.

A weak molecule condition is given for the Triebel-Lizorkin spaces Ḟ_p^{α,q}, with 0 < α < 1 and 1 < p, q < ∞. As an easy corollary, one may deduce, by atomic-molecular methods, a Triebel-Lizorkin space "T1" Theorem of Han and Sawyer, and Han, Jawerth, Taibleson and Weiss, for Calderón-Zygmund kernels K(x,y) which are not assumed to satisfy any regularity condition in the y variable.

The classical Whitney extension theorem states that every function in Lip$(\beta ,F)$, $F\subset {\mathbf{R}}^n$, $F$ closed, $k\&lt;\beta \le k+1$, $k$ a non-negative integer, can be extended to a function in Lip$(\beta ,{\mathbf{R}}^n)$. Her Lip$(\beta ,F)$ stands for the class of functions which on $F$ have continuous partial derivatives up to order $k$ satisfying certain Lipschitz conditions in the supremum norm. We formulate and prove a similar theorem in the $L^p$-norm.The restrictions to ${\mathbf{R}}^d$, $d\&lt;n$, of the Bessel potential spaces in ${\mathbf{R}}^n$ and the Besov or generalized Lipschitz spaces in ${\mathbf{R}}^n$ have been...

A non-homogeneous Hardy-like inequality has recently been found to be closely related to the knowledge of the lowest eigenvalue of a large class of Dirac operators in the gap of their continuous spectrum.