We establish an upper bound for the norm of a fractal Hilbert transform in the space of Hölder analytic functions in the sense of Douglis.

Let $\{Q_{n,\lambda}^{(\alpha)}\}_{n\ge 0}$ be the sequence of monic orthogonal polynomials with respect the Gegenbauer-Sobolev inner product $$\langle f,g\rangle s:=\int_{-1}^1 f(x)g(x)(1-x^2)^{\alpha-\frac{1}{2}} dx+\lambda \int_{-1}^1 f'(x)g'(x)(1-x^2)^{\alpha-\frac{1}{2}}dx,$$ where $\alpha \gt -\frac{1}{2}$ and $\lambda \ge 0$. In this paper we use a recent result due to B.D. Bojanov and N. Naidenov [3], in order to study the maximization of a local extremum of the $k$th derivative $\frac{d^k}{dx^k}$ in $[-M_{n,\lambda},M_{n,\lambda}]$, where $M_{n,\lambda}$ is a suitable value such that all zeros of the polynomial $Q_{n,\lambda}^{(\alpha)}$ are contained in $[-M_{n,\lambda},M_{n,\lambda}]$ and the function $\left|Q_{n,\lambda}^{(\alpha)}\right|$ attains its maximal value at the end-points of such interval. Also, some illustrative numerical examples are presented.

The aim of this work is to study the essential ascent and the related essential ascent spectrum of closed unbounded operators on a Banach space. Our approach is based on the concept of paracomplete subspaces of Banach spaces. We prove an unbounded spectral mapping theorem for the ascent spectrum and the essential ascent spectrum. A characterization of closed unbounded operators with finite essential ascent as direct sum of a suitable operators is proved. The new notion of a-essential index for closed unbounded operators with finite essential ascent is introduced. We also give some perturbations results for such operators. This paper extends some results proved in [1] to closed unbounded operators.

Recent work of Bui, Duong and Yan in [1] defined Besov spaces associated with a certain operator $L$ under the weak assumption that $L$ generates an analytic semigroup $e^{-tL}$ with Poisson kernel bounds on $L^2({\mathcal X})$ where ${\mathcal X}$ is a (possibly non-doubling) quasi-metric space of polynomial upper bound on volume growth. This note aims to extend Theorem 5.12 in [1], the decomposition of Besov spaces associated with Schrödinger operators, to more general $\alpha$, $p$, $q$.

A classical result due to A. P. Calderón states that Bessel potential spaces and Sobolev spaces defined on the same Lebesgue space and of the same integer order are isomorphic. We show that this result remains true when we replace Lebesgue spaces by some particular subspaces of Wiener amalgam space.

In this paper, the authors use the discrete Littlewood-Paley-Stein theory to introduce weighted multi-parameter Triebel-Lizorkin and Besov spaces associated with non-isotropic flag singular integrals under a rather weak weight condition $(w\in A_\infty)$. They also obtain the boundedness of flag singular integrals on these spaces.

Recent work of Bui, Duong and Yan in [2] defined Besov spaces associated with a certain operator $L$ under the weak assumption that $L$ generates an analytic semigroup $e^{-tL}$ with Poisson kernel bounds on $L^2({\mathcal X})$ where ${\mathcal X}$ is a (possibly non-doubling) quasi-metric space of polynomial upper bound on volume growth. This note aims to extend certain results in [2] to a more general setting when the underlying space can have different dimensions at $0$ and infinity. For example, we make some extensions to the Besov norm equivalence result in Proposition 4.4 of [2], such as to more general class of functions with suitable decay at $0$ and infinity, and to non-integer $k\geq 1$.