We discuss boundedness and compactness properties of the embedding $M_{\Lambda }^1\subset L^1\left(\mu \right)$, where $M_{\Lambda }^1$ is the closed linear span of the monomials $x^{{\lambda }_n}$ in $L^1\left([0,1]\right)$ and $\mu$ is a finite positive Borel measure on the interval $[0,1]$. In particular, we introduce a class of “sublinear” measures and provide a rather complete solution of the embedding problem for the class of quasilacunary sequences $\Lambda$. Finally, we show how one can recapture some of Al Alam’s results on boundedness and the essential norm of weighted composition operators from $M_{\Lambda }^1$...

Let $K\subset {ℝ}^m$ ($m\ge 2$) be a compact set; assume that each ball centered on the boundary $B$ of $K$ meets $K$ in a set of positive Lebesgue measure. Let $C_0^{\left(1\right)}$ be the class of all continuously differentiable real-valued functions with compact support in ${ℝ}^m$ and denote by ${\sigma }_m$ the area of the unit sphere in ${ℝ}^m$. With each $\varphi \in C_0^{\left(1\right)}$ we associate the function $$W_K\varphi \left(z\right)=\frac{1}{{\sigma }_m}\underset{{ℝ}^m\setminus K}{\to }\int \mathrm{g}rad\varphi \left(x\right)·\frac{z-x}{{|z-x|}^m}x$$ of the variable $z\in K$ (which is continuous in $K$ and harmonic in $K\setminus B$). $W_K\varphi$ depends only on the restriction ${\varphi |}_B$ of $\varphi$ to the boundary $B$ of $K$. This gives rise to a linear operator $W_K$ acting from...

We characterize all the extreme points of the unit ball in the space of trilinear forms on the Hilbert space ${ℂ}^2$. This answers a question posed by R. Grząślewicz and K. John [7], who solved the corresponding problem for the real Hilbert space ${ℝ}^2$. As an application we determine the best constant in the inequality between the Hilbert-Schmidt norm and the norm of trilinear forms.