scholarly journals Fuglede’s theorem in generalized Orlicz–Sobolev spaces

2021 ◽  
Author(s):  
Jonne Juusti
Keyword(s):  
Sobolev Spaces ◽  
Phase Growth ◽  
Absolutely Continuous ◽  
Double Phase ◽  
Special Cases

AbstractIn this paper, we show that Orlicz–Sobolev spaces $$W^{1,\varphi }(\varOmega )$$ W 1 , φ ( Ω ) can be characterized with the ACL- and ACC-characterizations. ACL stands for absolutely continuous on lines and ACC for absolutely continuous on curves. Our results hold under the assumptions that $$C^1(\varOmega )$$ C 1 ( Ω ) functions are dense in $$W^{1,\varphi }(\varOmega )$$ W 1 , φ ( Ω ) , and $$\varphi (x,\beta ) \ge 1$$ φ ( x , β ) ≥ 1 for some $$\beta > 0$$ β > 0 and almost every $$x \in \varOmega $$ x ∈ Ω . The results are new even in the special cases of Orlicz and double phase growth.

scholarly journals Global continuity and higher integrability of a minimizer of an obstacle problem under generalized Orlicz growth conditions

2019 ◽  
Author(s):  
Arttu Karppinen
Keyword(s):  
Obstacle Problem ◽  
Variable Exponent ◽  
Polynomial Growth ◽  
Growth Conditions ◽  
Phase Growth ◽  
Higher Integrability ◽  
Double Phase ◽  
Special Cases

AbstractWe prove continuity up to the boundary of the minimizer of an obstacle problem and higher integrability of its gradient under generalized Orlicz growth. The result recovers similar results obtained in the special cases of polynomial growth, variable exponent growth and produces new results for Orlicz and double phase growth.

scholarly journals Applications of (a,b)-continued fraction transformations

2011 ◽  
Vol 32 (2) ◽  
pp. 739-761 ◽  
Author(s):  
SVETLANA KATOK ◽  
ILIE UGARCOVICI
Keyword(s):  
Continued Fraction ◽  
Natural Extension ◽  
Absolutely Continuous ◽  
Coding Sequences ◽  
One Dimensional ◽  
Sofic Shift ◽  
Absolutely Continuous Invariant Measure ◽  
Special Cases ◽  
The One ◽  
Absolutely Continuous Invariant

AbstractWe describe a general method of arithmetic coding of geodesics on the modular surface based on the study of one-dimensional Gauss-like maps associated to a two-parameter family of continued fractions introduced in [Katok and Ugarcovici. Structure of attractors for (a,b)-continued fraction transformations.J. Modern Dynamics4(2010), 637–691]. The finite rectangular structure of the attractors of the natural extension maps and the corresponding ‘reduction theory’ play an essential role. In special cases, when an (a,b)-expansion admits a so-called ‘dual’, the coding sequences are obtained by juxtaposition of the boundary expansions of the fixed points, and the set of coding sequences is a countable sofic shift. We also prove that the natural extension maps are Bernoulli shifts and compute the density of the absolutely continuous invariant measure and the measure-theoretic entropy of the one-dimensional map.

scholarly journals Diagonal sections of copulas, multivariate conditional hazard rates and distributions of order statistics for minimally stable lifetimes

2021 ◽  
Vol 9 (1) ◽  
pp. 394-423
Author(s):  
Rachele Foschi ◽  
Giovanna Nappo ◽  
Fabio L. Spizzichino
Keyword(s):  
Order Statistics ◽  
Hazard Rate ◽  
Joint Distribution ◽  
Survival Function ◽  
Hazard Rates ◽  
Archimedean Copulas ◽  
Absolutely Continuous ◽  
Marginal Distributions ◽  
Special Cases ◽  
Survival Copula

Abstract As a motivating problem, we aim to study some special aspects of the marginal distributions of the order statistics for exchangeable and (more generally) for minimally stable non-negative random variables T 1, ..., Tr. In any case, we assume that T 1, ..., Tr are identically distributed, with a common survival function ̄G and their survival copula is denoted by K. The diagonal sections of K, along with ̄G, are possible tools to describe the information needed to recover the laws of order statistics. When attention is restricted to the absolutely continuous case, such a joint distribution can be described in terms of the associated multivariate conditional hazard rate (m.c.h.r.) functions. We then study the distributions of the order statistics of T 1, ..., Tr also in terms of the system of the m.c.h.r. functions. We compare and, in a sense, we combine the two different approaches in order to obtain different detailed formulas and to analyze some probabilistic aspects for the distributions of interest. This study also leads us to compare the two cases of exchangeable and minimally stable variables both in terms of copulas and of m.c.h.r. functions. The paper concludes with the analysis of two remarkable special cases of stochastic dependence, namely Archimedean copulas and load sharing models. This analysis will allow us to provide some illustrative examples, and some discussion about peculiar aspects of our results.

A Product Formula and a Non-Negative Poisson Kernel for Racah-Wilson Polynomials

1980 ◽  
Vol 32 (6) ◽  
pp. 1501-1517 ◽  
Author(s):  
Mizan Rahman
Keyword(s):  
Orthogonal Polynomials ◽  
Series Representation ◽  
General System ◽  
Continuous Measure ◽  
Absolutely Continuous ◽  
Angular Momenta ◽  
Special Cases ◽  
Absolutely Continuous Measure ◽  
Wilson Polynomials ◽  
Special Case

Physicists have long been using Racah's [7] 6-j symbols as a representation for the addition coefficients of three angular momenta. Racah himself discovered a series representation of the 6-j symbol which can be expressed as a balanced 4F3 series of argument 1, that is, a generalized hypergeometric function such that the sum of the 3 denominator parameters exceeds that of the 4 numerator parameters by 1. What Racah does not seem to have realized or, perhaps, cared to investigate, is that his 4F3 functions, with variables and parameters suitably identified, form a system of orthogonal polynomials in a discrete variable. The orthogonality of 6-j symbols as an orthogonality of 4F3 polynomials was recognized much later by Biedenharn et al. [3] in some special cases. Recently J. Wilson [13, 14] introduced a very general system of orthogonal polynomials expressible as balanced 4F3 functions of argument 1 orthogonal with respect to an absolutely continuous measure and/or a discrete weight function. Wilson's polynomials contain Racah's 6-j symbols as a special case. These polynomials might rightfully be credited to Wilson alone, but justice might be better served if we call them Racah-Wilson polynomials.

scholarly journals A direct proof of Sobolev embeddings for quasi-homogeneous Lizorkin–Triebel spaces with mixed norms

2007 ◽  
Vol 5 (2) ◽  
pp. 183-198 ◽  
Author(s):  
Jon Johnsen ◽  
Winfried Sickel
Keyword(s):  
Sobolev Spaces ◽  
Direct Proof ◽  
Embedding Theorem ◽  
Sobolev Embedding ◽  
Homogeneous Type ◽  
Sobolev Embedding Theorem ◽  
Mixed Norms ◽  
Sobolev Embeddings ◽  
Special Cases ◽  
Anisotropic Spaces

The article deals with a simplified proof of the Sobolev embedding theorem for Lizorkin–Triebel spaces (that contain theLp-Sobolev spacesHpsas special cases). The method extends to a proof of the corresponding fact for general Lizorkin–Triebel spaces based on mixedLp-norms. In this context a Nikol' skij–Plancherel–Polya inequality for sequences of functions satisfying a geometric rectangle condition is proved. The results extend also to anisotropic spaces of the quasi-homogeneous type.

scholarly journals An absolute continuity for positive operators on Banach lattices

1991 ◽  
Vol 14 (1) ◽  
pp. 83-92
Author(s):  
W. Feldman ◽  
C. Piston ◽  
Calvin E. Piston
Keyword(s):  
Banach Lattice ◽  
Absolute Continuity ◽  
Compact Operators ◽  
Positive Operators ◽  
Banach Lattices ◽  
Absolutely Continuous ◽  
Special Cases

For positive operators on a Banach lattice, absolute contnuity conditions are considered. An operator absolutely continuous with respect toTis compared to sums of compositions ofTtogether with orthomorphisms or in special cases projections. Consequences For compact operators on functions spacesC(X)are considered.

Characterizations of Besov-Type and Triebel–Lizorkin–Type Spaces via Averages on Balls

2017 ◽  
Vol 60 (3) ◽  
pp. 655-672 ◽  
Author(s):  
Ciqiang Zhuo ◽  
Winfried Sickel ◽  
Dachun Yang ◽  
Wen Yuan
Keyword(s):  
Sobolev Spaces ◽  
Morrey Spaces ◽  
Independent Interest ◽  
Special Cases ◽  
Type Spaces ◽  
Besov Type Spaces

AbstractLet ℓ ∊ ℕ and α > (§, 2ℓ). In this article, the authors establish equivalent characterizations of Besov-type spaces, Triebel–Lizorkin-type spaces, and Besov–Morrey spaces via the sequence {ƒ-Bl,2-kƒ}k consisting of the diòerence between f and the ball average Bl,2-kƒ. These results lead to the introduction of Besov-type spaces, Triebel–Lizorkin-type spaces, and Besov–Morrey spaceswith any positive smoothness order onmetricmeasure spaces. As special cases, the authors obtain a new characterization ofMorrey–Sobolev spaces and Qα spaces with ∈ > (0, 1), which are of independent interest.

scholarly journals KERNEL ESTIMATION WHEN DENSITY MAY NOT EXIST

2008 ◽  
Vol 24 (3) ◽  
pp. 696-725 ◽  
Author(s):  
Victoria Zinde-Walsh
Keyword(s):  
Empirical Studies ◽  
Kernel Estimation ◽  
Kernel Estimator ◽  
Generalized Functions ◽  
Continuous Variables ◽  
Asymptotic Results ◽  
Absolutely Continuous ◽  
Conditional Mean ◽  
Special Cases ◽  
Nonparametric Kernel

Nonparametric kernel estimation of density and conditional mean is widely used, but many of the pointwise and global asymptotic results for the estimators are not available unless the density is continuous and appropriately smooth; in kernel estimation for discrete-continuous cases smoothness is required for the continuous variables. Nonsmooth density and mass points in distributions arise in various situations that are examined in empirical studies; some examples and explanations are discussed in the paper. Generally, any distribution function consists of absolutely continuous, discrete, and singular components, but only a few special cases of nonparametric estimation involving singularity have been examined in the literature, and asymptotic theory under the general setup has not been developed. In this paper the asymptotic process for the kernel estimator is examined by means of the generalized functions and generalized random processes approach; it provides a unified theory because density and its derivatives can be defined as generalized functions for any distribution, including cases with singular components. The limit process for the kernel estimator of density is fully characterized in terms of a generalized Gaussian process. Asymptotic results for the Nadaraya–Watson conditional mean estimator are also provided.

scholarly journals Hölder Regularity of Quasiminimizers to Generalized Orlicz Functional on the Heisenberg Group

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Junli Zhang ◽  
Pengcheng Niu
Keyword(s):  
Heisenberg Group ◽  
Maximal Function ◽  
Riesz Potential ◽  
Hölder Regularity ◽  
Phase Growth ◽  
Moser Iteration ◽  
Covering Lemma ◽  
Exponential Functional ◽  
Double Phase ◽  
Holder Regularity

In this paper, we apply De Giorgi-Moser iteration to establish the Hölder regularity of quasiminimizers to generalized Orlicz functional on the Heisenberg group by using the Riesz potential, maximal function, Calderón-Zygmund decomposition, and covering Lemma on the context of the Heisenberg Group. The functional includes the p -Laplace functional on the Heisenberg group which has been studied and the variable exponential functional and the double phase growth functional on the Heisenberg group that have not been studied.

scholarly journals Existence and Nonexistence of a Solution for a Nonlinear p(x)-Elliptic Problem with Right-Hand Side Measure

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Elhoussine Azroul ◽  
Abdelkrim Barbara ◽  
Hicham Redwane
Keyword(s):  
Total Variation ◽  
Nonlinear Problem ◽  
Sobolev Spaces ◽  
Elliptic Problem ◽  
Radon Measure ◽  
Variable Exponents ◽  
Absolutely Continuous ◽  
Borel Set ◽  
Right Hand ◽  
Existence And Nonexistence

We discuss the existence and nonexistence of solution of a nonlinear problem p(x)-elliptic-div(a(x,∇u))+g(x,u,∇u)=μ, where μ is a Radon measure with bounded total variation, by considering the Sobolev spaces with variable exponents. This study is done in two cases: (i) μ is absolutely continuous with respect to p(x)-capacity. and (ii) μ is concentrated on a Borel set of null p(x)-capacity.

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