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We prove that the Hilbert matrix operator $H:A^2_{\ln^{\alpha}} \to A^2$ is bounded when $\alpha\geq 2$ and that $H$ does not map $A^2_{\ln^{\alpha}} $ into $A^2$ when $\alpha<2$, solving an open problem posed by B. Karapetrovi\'c and M. Jevti\'c. The boundedness when $\alpha=2$ is shown as a corollary to the more gene...
We examine the generalized Hilbert (matrix) operators $$ H_{g,\gamma} : (a_n) \mapsto \sum_{n=1}^{\infty} \bigg(\frac{k}{n}\bigg)^{\gamma} \frac{g_k a_n}{n+k} $$ on the $\ell^p$ spaces, $1<p<\infty$, where $g=(g_n)$ is a sequence and $-1/p<\gamma<1-1/p$. Given a partition of the natural numbers $\bigcup_j I_j = \mathbb...
Let $g$ be analytic in the unit disc and consider the generalized Hilbert operator $$ \mathcal{H}_g(f)(z)=\int_0^1 f(t)g'(tz)\, dt. $$ The boundedness of $\mathcal H_g$ on $H^p$ is characterized by the mean Lipschitz condition $g\in\Lambda\left(p,\frac{1}{p}\right)$ when $1<p\leq2$, while the problem remains open for $...
We prove that if a real-valued function $f\in L^1$ on the complex unit circle has a gap of width at least $\pi$ in its essential range, then $\exp(\tilde f)$ is not integrable, where $\tilde f$ is the conjugate function. More generally, the exponential function can be replaced by any nonnegative convex function $Y$ sat...
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