In this paper we consider a class of integral operators on $L^2(0,\infty)$ that are unitarily equivalent to little Hankel operators between weighted Bergman spaces. We calculate the norms of such integral operators and as a by-product obtain a generalization of the Hardy-Hilbert’s integral inequality. We also consider the discrete version of the inequality which give the norms of the companion matrices of certain generalized Bergman-Hilbert matrices. These results are then generalized to vector valued case and operator valued case.

We generalize results of Fan and Zhang [6] on absolute continuity and singularity of the golden Markov geometric series to nonuniform stochastic series given by arbitrary Markov process. In addition we describe an application of these results in fractal geometry.

In this paper, by using the decomposition theorem for weak Hardy spaces, we will obtain the boundedness properties of some integral operators with variable kernels on these spaces, under some Dini type conditions imposed on the variable kernel $\Omega(x,z)$.

We consider a Gause type predator-prey system with functional response given by $θ(x)=\arctan(ax), where $a \gt 0$, and give a counterexample to the criterion given in Attili and Mallak [Commun. Math. Anal. 1:33-40(2006)] for the nonexistence of limit cycles. When this criterion is satisfied, instead this system can have a locally asymptotically stable coexistence equilibrium surrounded by at least two limit cycles.

In this paper we show the existence of one-dimensional solitons (travelling waves of finite energy) for a generalized nonlinear dispersive equation modeling the deformations of a hyperelastic compressible plate. From the Hamiltonian structure for such equation we find the natural space for the travelling wave solutions and characterize travelling waves variationally as minimizers of an energy functional under a suitable constraint. Our approach involves the Lions's Concentration-Compactness Lemma.

The aim of this paper is to prove new quantitative uncertainty principles for the Fourier transform connected with the spherical mean operator. The first of these results is an extension of the Donoho and Stark's uncertainty principle. The second result extends the Heisenberg-Pauli-Weyl uncertainty principle. From these two results we deduce a continuous-time principle for the $L^p$ theory, when $1 \lt p \le 2$.