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.

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 $d{U}_i=-2u_{i}d\nu$.

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 investigate the behaviour of Fourier coefficients with respect to the system of ultraspherical polynomials. This leads us to the study of the “boundary” Lorentz space ${ℒ}_{\lambda }$ corresponding to the left endpoint of the mean convergence interval. The ultraspherical coefficients $c{ₙ}^{\left(\lambda \right)}\left(f\right)$ of ${ℒ}_{\lambda }$-functions turn out to behave like the Fourier coefficients of functions in the real Hardy space ReH¹. Namely, we prove that for any $f\in {ℒ}_{\lambda }$ the series ${\sum }_{n=1}^{\infty }c{ₙ}^{\left(\lambda \right)}\left(f\right)cosn\theta$ is the Fourier series of some function φ ∈ ReH¹ with ${\left|\right|\phi \left|\right|}_{ReH¹}{\le c\left|\right|f\left|\right|}_{{ℒ}_{\lambda }}$.

We give conditions on pairs of weights which are necessary and sufficient for the operator $T\left(f\right)=K*f$ to be a weak type mapping of one weighted Lorentz space in another one. The kernel $K$ is an anisotropic radial decreasing function.

We review the main facts that are behind a unified construction for the commutator theorem of the main interpolation methods.