Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0705.4161

Riesz Bases of Root Vectors of Indefinite Sturm-Liouville problems with eigenparameter dependent boundary conditions, I

Paul Binding·Branko Ćurgus
View original source ↗Read source document ↗Export metadata ↓

Abstract

We consider a regular indefinite Sturm-Liouville problem with two self-adjoint boundary conditions, one being affinely dependent on the eigenparameter. We give sufficient conditions under which a basis of each root subspace for this Sturm-Liouville problem can be selected so that the union of all these bases constitutes a Riesz basis of a corresponding weighted Hilbert space.

Explore related subjects

math.CAmath.FA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Paul Binding, Branko Ćurgus. 2007-05-29. Riesz Bases of Root Vectors of Indefinite Sturm-Liouville problems with eigenparameter dependent boundary conditions, I. https://arxiv.org/abs/0705.4161

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Yet Another Table of Integrals

This collection of sums and integrals has been harvested from the mathematical and physical literature in unstructured ways. Its main use is backtracking the original sources whenever an integral of the reader's application resembles one of the items in the collection.

math.CA↗
arXiv2026

Hausdorff content $A_p$ weights: weighted maximal inequalities and fundamental structural properties

Let $\mathcal H_{\infty}^δ$ denote the Hausdorff content of dimension $δ\in(0,n]$ defined on subsets of $\mathbb R^n$. The principal problem, considered in this paper, is to characterize the non-negative function $w$ for which the weighted $L^p$ inequality with $p\in(1,\infty)$ and the weighted weak $L^1$ inequality on Hardy-Littlewood maximal operators associated with Hausdorff content hold true. To achieve this, we introduce a class of Hausdorff content $A_p$ weights, denoted as $\mathcal A_{p,δ}$, and then show that, for any $p\in(1,\infty)$ and $δ\in(0,n]$, the weighted $L^p$ maximal inequality holds true if and only if $w\in\mathcal A_{p,δ}$, and the weighted weak $L^1$ maximal inequality holds true if and only if $w\in\mathcal A_{1,δ}$ by proposing a novel approach. As the second objective, applying this approach, the seminal structural properties of classical Muckenhoupt $A_p$ weights, such as the reverse Hölder inequality [R. R. Coifman and C. Fefferman, Studia Math. 51 (1974), 241-250], the self-improving property [B. Muckenhoupt, Trans. Amer. Math. Soc. 165 (1972), 207-226], and Jones' factorization theorem [P. W. Jones, Ann. of Math. (2) 111 (1980), 511-530], are all established for $\mathcal A_{p,δ}$. The main novelties of this paper lies in: (i) we discover $\mathcal A_{p,δ}$ weights with rich structural properties, that increase strictly on dimension exponent and hence reduce to the classical $A_p$ class at the endpoint dimension, forming proper subsets of it for smaller dimension exponent; (ii) by proposing a novel approach, this work naturally extends the classical theory beyond measure-theoretic frameworks, while also provides novel insights even when reduced to the classical $A_p$ weight case.

math.CA↗
arXiv2026

Directional maximal operators in the plane

This monograph investigates the Lebesgue boundedness of planar directional maximal operators $D_Ω$. These are maximal averages of functions over line segments in $\mathbb R^2$ whose slopes lie in a specified set $Ω\subseteq\mathbb R$. A large body of work has identified a geometric property of $Ω$, called finite-order lacunarity, as a key factor in ensuring that $D_Ω$ is Lebesgue bounded. While several variations of this notion exist, they all centre on the distribution of gaps in $Ω$. Building on earlier work, an article of Bateman(2009) asserted a dichotomy for such operators. Namely, $D_Ω$ is bounded on $L^p$ for all $p\in (1,\infty)$ precisely when the slope set $Ω$ is finite-order lacunary, or equivalently, when $Ω$ does not admit Kakeya-type sets. Conversely, sublacunary direction sets $Ω$ admit Kakeya-like phenomena, implying that $D_Ω$ is unbounded on $L^p$ for all $p\in [1,\infty)$. Recent work of Hagelstein, Radillo-Murguia, and Stokolos(2024) identified a gap in the proof of this assertion and produced counterexamples for which the separation mechanism underlying that proof fails, demonstrating the need for a corrected framework. We establish the corrected characterization by introducing a new notion of admissible finite-order lacunarity that faithfully reflects the combinatorial structure of the direction set. This leads to a tree-theoretic characterization in terms of finite splitting number and provides the foundation for new geometric and probabilistic constructions establishing the equivalence between finite-order lacunarity, the absence of Kakeya-type sets, and the boundedness of directional maximal operators. The resulting framework not only resolves the gap in the earlier proof, but also identifies admissible finite-order lacunarity as the structural invariant governing these phenomena.

math.CA↗