### Absolute continuity and hyponormal operators.

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We investigate the convergence behavior of the family of double sine integrals of the form ${\int}_{0}^{\infty}{\int}_{0}^{\infty}f(x,y)sinuxsinvydxdy$, where (u,v) ∈ ℝ²₊:= ℝ₊ × ℝ₊, ℝ₊:= (0,∞), and f: ℝ²₊ → ℂ is a locally absolutely continuous function satisfying certain generalized monotonicity conditions. We give sufficient conditions for the uniform convergence of the remainder integrals ${\int}_{a\u2081}^{b\u2081}{\int}_{a\u2082}^{b\u2082}$ to zero in (u,v) ∈ ℝ²₊ as maxa₁,a₂ → ∞ and ${b}_{j}>{a}_{j}\ge 0$, j = 1,2 (called uniform convergence in the regular sense). This implies the uniform convergence of the partial integrals...

Let ${\mathcal{A}}_{\beta}$ be the Beurling algebra with weight $(1+|n|{)}^{\beta}$ on the unit circle $\mathbb{T}$ and, for a closed set $E\subseteq \mathbb{T}$, let ${J}_{{\mathcal{A}}_{\beta}}\left(E\right)=\{f\in {\mathcal{A}}_{\beta}:f=0\phantom{\rule{0.166667em}{0ex}}\text{on}\phantom{\rule{4pt}{0ex}}\text{a}\phantom{\rule{4pt}{0ex}}\text{neighbourhood}\phantom{\rule{4pt}{0ex}}\text{of}\phantom{\rule{0.166667em}{0ex}}E\}$. We prove that, for $\beta \>\frac{1}{2}$, there exists a closed set $E\subseteq \mathbb{T}$ of measure zero such that the quotient algebra ${\mathcal{A}}_{\beta}/\stackrel{\u203e}{{J}_{{\mathcal{A}}_{\beta}}\left(E\right)}$ is not generated by its idempotents, thus contrasting a result of Zouakia. Furthermore, for the Lipschitz algebras ${\lambda}_{\gamma}$ and the algebra $\mathcal{A}\mathcal{C}$ of absolutely continuous functions on $\mathbb{T}$, we characterize the closed sets $E\subseteq \mathbb{T}$ for which the restriction algebras ${\lambda}_{\gamma}\left(E\right)$ and $\mathcal{A}\mathcal{C}\left(E\right)$ are generated by their idempotents.

We study the relation between standard ideals of the convolution Sobolev algebra ${\u208a}^{\left(n\right)}\left(t\u207f\right)$ and the convolution Beurling algebra L¹((1+t)ⁿ) on the half-line (0,∞). In particular it is proved that all closed ideals in ${\u208a}^{\left(n\right)}\left(t\u207f\right)$ with compact and countable hull are standard.

The note is related to a recently published paper J. M. Park, J. J. Oh, C.-G. Park, D. H. Lee: The AP-Denjoy and AP-Henstock integrals. Czech. Math. J. 57 (2007), 689–696, which concerns a descriptive characterization of the approximate Kurzweil-Henstock integral. We bring to attention known results which are stronger than those contained in the aforementioned work. We show that some of them can be formulated in terms of a derivation basis defined by a local system of which the approximate basis...