Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Compactness and weak-star continuity of derivations on weighted
  convolution algebras   Let $\omega$ be a continuous weight on $\mathbb R^+$ and let $L^1(\omega)$ be
the corresponding convolution algebra. By results of Gr{\o}nb{\ae}k and Bade &
Dales the continuous derivations from $L^1(\omega)$ to its dual space
$L^{\infty}(1/\omega)$ are exactly the maps of the form
$$(D_{\phi}f)(t)=\int_0^{\infty}f(s)\,\frac{s}{t+s}\,\phi(t+s)\,ds\qquad\text{($t\in\mathbb
R^+$ and $f\in L^1(\omega)$)}$$ for some $\phi\in L^{\infty}(1/\omega)$. Also,
every $D_{\phi}$ has a unique extension to a continuous derivation
$\bar{D}_{\phi}:M(\omega)\to L^{\infty}(1/\omega)$ from the corresponding
measure algebra. We show that a certain condition on $\phi$ implies that
$\bar{D}_{\phi}$ is weak-star continuous. The condition holds for instance if
$\phi\in L_0^{\infty}(1/\omega)$. We also provide examples of functions $\phi$
for which $\bar{D}_{\phi}$ is not weak-star continuous. Similarly, we show that
$D_{\phi}$ and $\bar{D}_{\phi}$ are compact under certain conditions on $\phi$.
For instance this holds if $\phi\in C_0(1/\omega)$ with $\phi(0)=0$. Finally,
we give various examples of functions $\phi$ for which $D_{\phi}$ and
$\bar{D}_{\phi}$ are not compact.

26
The categories of this paper are: bayes-an
5
The categories of this paper are: gr-qc
28
The categories of this paper are: alg-geom
30
The categories of this paper are: q-alg
9
The categories of this paper are: nlin
11
The categories of this paper are: hep-lat
7
The categories of this paper are: hep-th
25
The categories of this paper are: patt-sol
36
The categories of this paper are: cmp-lg
22
The categories of this paper are: chao-dyn
19
The categories of this paper are: eess
10
The categories of this paper are: q-bio
16
The categories of this paper are: stat
37
The categories of this paper are: supr-con
33
The categories of this paper are: plasm-ph
13
The categories of this paper are: quant-ph