1,720 research outputs found
The sharp A(p) constant for weights in a reverse-Holder class
No full textCoifman and Fefferman established that the class of Muckenhoupt weights is equivalent to the class of weights satisfying the "reverse Holder inequality". In a recent paper V. Vasyunin [17] presented a proof of the reverse Holder inequality with sharp constants for the weights satisfying the usual Muckenhoupt condition. In this paper we present the inverse, that is, we use the Bellman function technique to find the sharp A(p) constants for weights in a reverse-Holder class on an interval; we also find the sharp constants for the higher-integrability result of Gehring [7].Additionally, we find sharp bounds for the A(p) constants of reverse-Holder-class weights defined on rectangles in R-n, as well as bounds on the A(p) constants for reverse-Holder weights defined on cubes in R-n, without claiming the sharpness.</p
Signaturenverzeichnis abendländischer und Musikhandschriften der ehem. Preussischen Staatsbibliothek
No full textGehring P., Gebhardt W. Signaturenverzeichnis abendländischer und Musikhandschriften der ehem. Preussischen Staatsbibliothek. In: Scriptorium, Tome 13 n°1, 1959. pp. 127-130
Signaturenverzeichnis abendländischer und Musikhandschriften der ehem. Preussischen Staatsbibliothek
No full textGehring P., Gebhardt W. Signaturenverzeichnis abendländischer und Musikhandschriften der ehem. Preussischen Staatsbibliothek. In: Scriptorium, Tome 13 n°1, 1959. pp. 127-130
The ubiquitous quasidisk
No full textThis book focuses on gathering the numerous properties and many different connections with various topics in geometric function theory that quasidisks possess. A quasidisk is the image of a disk under a quasiconformal mapping of the Riemann sphere. In 1981 Frederick W. Gehring gave a short course of six lectures on this topic in Montreal and his lecture notes "Characteristic Properties of Quasidisks" were published by the University Press of the University of Montreal. The notes became quite popular and within the next decade the number of characterizing properties of quasidisks and their ramifications increased tremendously. In the late 1990s Gehring and Hag decided to write an expanded version of the Montreal notes. At three times the size of the original notes, it turned into much more than just an extended version. New topics include two-sided criteria. The text will be a valuable resource for current and future researchers in various branches of analysis and geometry, and with its clear and elegant exposition the book can also serve as a text for a graduate course on selected topics in function theory. Frederick W. Gehring (1925-2012) was a leading figure in the theory of quasiconformal mappings for over fifty years. He received numerous awards and shared his passion for mathematics generously by mentoring twenty-nine Ph.D. students and more than forty postdoctoral fellows. Kari Hag received her Ph.D. under Gehring's direction in 1972 and worked with him on the present text for more than a decade
A fractional Gehring lemma, with applications to nonlocal equations
No full textTo Carlo Sbordone on his 65th birthday.This paper reports the content of a talk given by the second-named author at the Accademia dei Lincei on November 26, 2013.International audienceWe describe a fractional version of the classical Gehring lemma. As a consequence, new self-improving regularity properties of solutions to integrodifferential equations emerge
Molecular insights into the eye evolution of bivalvian molluscs
No full textThe intention of my PhD project was to gain more insights into eye evolution and to provide further evidence for the recently proposed idea that all eye-types found in eumetazoans derive from a common Pax6-dependent proto-type eye (Gehring and Ikeo, 1999). To do so, we decided to focus on eyes found in bivalves. Two main reasons prompted us to investigate the molecular basis of bivalvian eye formation. In the first place, all major eye-types, the compound eye, consisting of numerous ommatidia, the camera eye with a single lens and the mirror eye with a reflecting mirror in the back of the eye, are found in bivalves. Hence, the occurrence of different eye-types within the same phylogenetic class makes it very unlikely that these eyes arose as independent formations during evolution. A more elegant alternative is to assume that the compound-, camera-, and mirror eyes of clams evolved monophyletically from a common ancestral precursor. The second reason why we decided to investigate bivalvian eyes is their unusual anatomical position, the edge of the mantle. So far, molecular data and most prominently Pax6 expression were exclusively gathered from “cerebral eyes” of bilaterians, with the only exception of the non-cerebral Hesse eyecups of the lancelet, which by the way do not show any Pax6 expression (Glardon et al., 1998). In this study we focused on two bivalvian species, Arca noae and Pecten maximus, representing the compound eye-type and the mirror eye-type, respectively. We isolated two genes, Pax6 and Six1/2, known to be high up in the genetic regulatory cascade of eye development, from Arca and Pecten. Our expression studies of Pax6 and Six1/2 support the idea that these two genes are necessary for the formation of the olfactory system throughout the animal kingdom. In contrast, we could not assign Pax6 and Six1/2 expression to the visual system with absolute certainty. In a second project, we isolated three opsin genes, one from Arca and two opsin genes from Pecten. A Go-coupled opsin was isolated from Pecten which was shown to be exclusively expressed in the rhabdomeric photoreceptor cells of the proximal retina. The second opsin gene isolated from Pecten and the opsin gene from Arca were shown to be expressed in various tissues, suggesting a putative role in the photic regulation of peripheral circadian clocks. Moreover, phylogenetic analysis indicate that each of these two opsin genes may constitute a novel opsin subfamily
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