We establish an asymptotic property of Toeplitz operators on the Dirichlet space and on a weighted Bergman space. Additionally, we determine a sufficient condition for a self-commutator inequality to hold in a specific case.
We study Li-Yorke and distributional chaos for weighted shifts on directed trees. We give a complete characterization of Li-Yorke chaotic weighted shifts on rooted directed trees. Also we show some sufficient conditions for distributional chaos of these operators. Our results generalize the previous results on Li-Yorke and distributionally chaotic unilateral and bilateral weighted shifts. It also has relations with some papers in which other important dynamical properties were investigated for weighted shifts on trees.
We provide a complete proof of a sharp Gagliardo-Nirenberg-type inequality on the unit interval, in its additive-invariant form inf (c is an element of R )|| f - c || L infinity <= (3/2)(1/3)( inf (c is an element of R )|| f - c || (L2))(2/3 ) ( )|| f' || (1/3)(L infinity) which is motivated by the quantitative stability analysis in optimal transport theory. The proof employs the one-dimensional area formula together with rather elementary calculations.
We apply proof mining techniques to obtain quantitative and qualitative results on asymptotic and T-asymptotic regularity for the inexact generalized Halpern iteration, a viscosity-type extension of an iteration recently studied by Kanzow and Shehu. Specializing our results to the Kanzow-Shehu iteration and the sequential averaging method (SAM) yields analogous results for these iterations. Furthermore, we compute rates of (T-)asymptotic regularity for particular choices of the parameter sequences, and for one of them, we obtain linear rates as an application of a lemma due to Sabach and Shtern.
In this paper, we consider the Phi-Laplacian problem with Dirichlet boundary condition, -div(Phi(|del u|del u/del u) = lambda g(.)Phi(u) in Omega,lambda is an element of R and u|partial derivative Omega and u|(partial derivative Omega) = 0. The term Phi is a real odd and increasing homeomorphism, g is a nontrivial, nonnegative function in L infinity(Omega) and Omega subset of R-N is abounded connected domain. We examine the asymptotic behavior of sequences of eigenvalues of the differential equation. The treatment is based solely on the asymptotic homogeneity of Phi plus an additional condition on the domain (the segment property). We choose a sequence of eigenfunctions tending either to zero or infinity (in the sense of the norm). The core result of these notes shows that the liminf of the associated sequence of eigenvalues coincides with the first eigenvalue of the usual p-Laplace operator, and that the weak-* limit of the corresponding eigenfunctions is an associated ground state.
