scholarly journals On the Global Gaussian Lipschitz Space

2017 ◽  
Vol 60 (3) ◽  
pp. 707-720 ◽  
Author(s):  
Liguang Liu ◽  
Peter Sjögren
Keyword(s):  
Euclidean Space ◽  
Preceding Paper ◽  
Lipschitz Space ◽  
Continuity Condition ◽  
Poisson Integral ◽  
Boundedness Condition ◽  
Bounded Functions ◽  
Similar Inequality ◽  
Logarithmic Growth

AbstractIt is well known that the standard Lipschitz space in Euclidean space, with exponent α ∈ (0, 1), can be characterized by means of the inequality , where is the Poisson integral of the function f. There are two cases: one can either assume that the functions in the space are bounded, or one can not make such an assumption. In the setting of the Ornstein–Uhlenbeck semigroup in ℝn, Gatto and Urbina defined a Lipschitz space by means of a similar inequality for the Ornstein–Uhlenbeck Poisson integral, considering bounded functions. In a preceding paper, the authors characterized that space by means of a Lipschitz-type continuity condition. The present paper defines a Lipschitz space in the same setting in a similar way, but now without the boundedness condition. Our main result says that this space can also be described by a continuity condition. The functions in this space turn out to have at most logarithmic growth at infinity.

scholarly journals Quantum differentiability of essentially bounded functions on Euclidean space

2017 ◽  
Vol 273 (7) ◽  
pp. 2353-2387 ◽  
Author(s):  
Steven Lord ◽  
Edward McDonald ◽  
Fedor Sukochev ◽  
Dmitry Zanin
Keyword(s):  
Euclidean Space ◽  
Bounded Functions

scholarly journals On the Superstability of Lobačevskiǐ’s Functional Equations with Involution

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Jaeyoung Chung ◽  
Bogeun Lee ◽  
Misuk Ha
Keyword(s):  
Functional Equation ◽  
Euclidean Space ◽  
Functional Equations ◽  
Direct Consequence ◽  
Functional Inequality ◽  
Commutative Group ◽  
Bounded Functions

LetGbe a uniquely2-divisible commutative group and letf,g:G→Candσ:G→Gbe an involution. In this paper, generalizing the superstability of Lobačevskiǐ’s functional equation, we considerf(x+σy)/22-g(x)f(y)≤ψ(x)orψ(y)for allx,y∈G, whereψ:G→R+. As a direct consequence, we find a weaker condition for the functionsfsatisfying the Lobačevskiǐ functional inequality to be unbounded, which refines the result of Găvrută and shows the behaviors of bounded functions satisfying the inequality. We also give various examples with explicit involutions on Euclidean space.

scholarly journals Modified Poisson Integral and Green Potential on a Half-Space

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Lei Qiao
Keyword(s):  
Euclidean Space ◽  
Half Space ◽  
Analytic Functions ◽  
Harmonic Functions ◽  
Poisson Integral ◽  
Dimensional Euclidean Space ◽  
Superharmonic Functions ◽  
Growth Properties ◽  
Green Potential

We discuss the behavior at infinity of modified Poisson integral and Green potential on a half-space of then-dimensional Euclidean space, which generalizes the growth properties of analytic functions, harmonic functions and superharmonic functions.

The Poisson Integral of a Singular Measure

1983 ◽  
Vol 35 (4) ◽  
pp. 735-749 ◽  
Author(s):  
Patrick Ahern
Keyword(s):  
Lower Bound ◽  
Euclidean Space ◽  
Lebesgue Measure ◽  
Open Subset ◽  
Borel Measure ◽  
The Other ◽  
Poisson Integral ◽  
Singular Measure ◽  
Almost Everywhere ◽  
Image Position

Let σ be a finite positive singular Borel measure defined on Euclidean space RN. For w ∈ RN and y > 0, its Poisson integral is defined by the formulawhere CN is chosen so thatSince σ is singular, almost everywhere with respect to Lebesgue measure on RN. On the other hand, almost everywhere dσ. It follows that for all sufficiently small y,is a non-empty open subset of RN. If σ has compact support then |Ey| → 0 as y → 0, where |Ey| denotes the Lebesgue measure of Ey. In this paper we give a lower bound on the rate at which |Ey| may go to zero. The lower bound depends on the smoothness of the measure; the smoother the measure, the more slowly |Ey| may approach 0.

Logarithmic growth and Frobenius filtrations for solutions of p-adic differential equations

2009 ◽  
Vol 8 (3) ◽  
pp. 465-505 ◽  
Author(s):  
Bruno Chiarellotto ◽  
Nobuo Tsuzuki
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Local Field ◽  
Differential Module ◽  
Frobenius Structures ◽  
Bounded Functions ◽  
Series Development ◽  
Rank 2 ◽  
Logarithmic Growth

AbstractFor a ∇-module M over the ring K[[x]]0 of bounded functions over a p-adic local field K we define the notion of special and generic log-growth filtrations on the base of the power series development of the solutions and horizontal sections. Moreover, if M also admits a Frobenius structure then it is endowed with generic and special Frobenius slope filtrations. We will show that in the case of M a ϕ–∇-module of rank 2, the Frobenius polygon for M and the log-growth polygon for its dual, Mv, coincide, this is proved by showing explicit relationships between the filtrations. This will lead us to formulate some conjectural links between the behaviours of the filtrations arising from the log-growth and Frobenius structures of the differential module. This coincidence between the two polygons was only known for the hypergeometric cases by Dwork.

Morphological Characterization of Mycobacteriophage R1

1974 ◽  
Vol 32 ◽  
pp. 254-255
Author(s):  
B. L. Soloff ◽  
T. A. Rado
Keyword(s):  
Carbon Source ◽  
Stationary Phase ◽  
Growth Phase ◽  
Minimal Medium ◽  
Morphological Characterization ◽  
Logarithmic Growth Phase ◽  
Complete Lysis ◽  
Lysogenic Culture ◽  
Logarithmic Growth

Mycobacteriophage R1 was originally isolated from a lysogenic culture of M. butyricum. The virus was propagated on a leucine-requiring derivative of M. smegmatis, 607 leu−, isolated by nitrosoguanidine mutagenesis of typestrain ATCC 607. Growth was accomplished in a minimal medium containing glycerol and glucose as carbon source and enriched by the addition of 80 μg/ ml L-leucine. Bacteria in early logarithmic growth phase were infected with virus at a multiplicity of 5, and incubated with aeration for 8 hours. The partially lysed suspension was diluted 1:10 in growth medium and incubated for a further 8 hours. This permitted stationary phase cells to re-enter logarithmic growth and resulted in complete lysis of the culture.

THE EFFECT OF GONADECTOMY ON THE RESPONSE OF THE PITUITARY I. C. S. H.-CONTENT TO STEROID SEX HORMONES

1960 ◽  
Vol XXXV (II) ◽  
pp. 245-252 ◽  
Author(s):  
G. P. van Rees ◽  
F. J. A. Paesi
Keyword(s):  
Sex Hormones ◽  
Sex Steroids ◽  
Low Dose ◽  
Testosterone Propionate ◽  
Preceding Paper ◽  
Direct Consequence ◽  
Two Factors ◽  
Reflex Stimulation ◽  
Steroid Sex Hormones ◽  
Stimulation Of

ABSTRACT In the experiments reported in this paper the hypothesis that the decrease in the pituitary I. C. S. H.-content, which occurs after administration of steroid sex hormones in gonadectomized animals, is counteracted by a reflex stimulation of the hypophysis initiated by the operation has been investigated. If treatment with a low dose of testosterone propionate (100 μg) was started immediately after castration, the resulting decrease in the pituitary I. C. S. H.-content became more marked if the reflex stimulation of the hypophysis had been prevented. If, however, two months were allowed to elapse before the beginning of treatment, the presence or absence of this reflex was no longer of importance for the effect of testosterone propionate on the pituitary I. C. S. H.-content. And yet, in this case too, the decrease in the pituitary I. C. S. H.-content by testosterone propionate was less than in intact animals (see preceding paper). Hence this decrease appears to be counteracted by two factors: one rapidly occurring and short lasting, resulting from a reflex elicited by gonadectomy; the other gradually increasing in potency and possibly a direct consequence of the continued absence of pituitary inhibiting sex steroids.

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