We define and analyze Toeplitz operators whose symbols are the elements of the complex quantum plane, a non-commutative, infinite dimensional algebra. In particular, the symbols do not come from an algebra of functions. The process of forming operators from non-commuting symbols can be considered as a second quantization. To do this we construct a reproducing kernel associated with the quantum plane. We also discuss the commutation relations of creation and annihilation operators which are defined...

Let X be a completely regular Hausdorff space, E and F be Banach spaces. Let Cb(X, E) be the space of all E-valued bounded, continuous functions on X, equipped with the strict topology β. We develop the Riemman-Stieltjes-type Integral representation theory of (β, || · ||F) -continuous operators T : Cb(X, E) → F with respect to the representing Borel operator measures. For X being a k-space, we characterize strongly bounded (β, || · ||F)-continuous operators T : Cb(X, E) → F. As an application, we...

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.