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 a sharp pointwise estimate of the nonincreasing rearrangement of the fractional maximal function of ⨍, $M_{\gamma }⨍$, by an expression involving the nonincreasing rearrangement of ⨍. This estimate is used to obtain necessary and sufficient conditions for the boundedness of $M_{\gamma }$ between classical Lorentz spaces.

Let $L$ be an Archimedean Riesz space with a weak order unit $u$. A sufficient condition under which Dedekind [$\sigma$-]completeness of the principal ideal $A_u$ can be lifted to $L$ is given (Lemma). This yields a concise proof of two theorems of Luxemburg and Zaanen concerning projection properties of $C\left(X\right)$-spaces. Similar results are obtained for the Riesz spaces $B_n\left(T\right)$, $n=1,2,\cdots$, of all functions of the $n$th Baire class on a metric space $T$.

It follows easily from a result of Lindenstrauss that, for any real twodimensional subspace v of L¹, the relative projection constant λ(v;L¹) of v equals its (absolute) projection constant $\lambda \left(v\right)=su{p}_X\lambda (v;X)$. The purpose of this paper is to recapture this result by exhibiting a simple formula for a subspace V contained in $L^{\infty }\left(\nu \right)$ and isometric to v and a projection $P_{\infty }$ from C ⊕ V onto V such that $\parallel P_{\infty }\parallel =\parallel P₁\parallel$, where P₁ is a minimal projection from L¹(ν) onto v. Specifically, if $P₁={\sum }_{i=1}^2{U}_i\otimes v_i$, then $P_{\infty }={\sum }_{i=1}^2{u}_i\otimes V_i$, where $d{V}_i=2v_{i}d\nu$ and $d{U}_i=-2u_{i}d\nu$.

The paper presents a simple proof of Proposition 8 of [2], based on a new and simple description of isometries between CD 0-spaces.

There are necessary conditions for a point x from the unit sphere to be a denting point of the unit ball of Orlicz spaces equipped with the Orlicz norm generated by arbitrary Orlicz functions. In contrast to results in [12, 17, 16], we present also examples of Orlicz spaces in which strongly extreme points of the unit ball are not denting points.

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 ${ℍ}^n$.