scholarly journals Continuous solutions to two iterative functional equations

2021 ◽  
Author(s):  
Karol Baron
Keyword(s):  
Probability Measure ◽  
Functional Equations ◽  
Continuous Functions ◽  
Continuous Solutions ◽  
Hölder Continuous ◽  
Hölder Continuous Functions

AbstractBased on iteration of random-valued functions we study the problem of solvability in the class of continuous and Hölder continuous functions $$\varphi $$ φ of the equations $$\begin{aligned} \varphi (x)=F(x)-\int _{\Omega }\varphi \big (f(x,\omega )\big )P(d\omega ),\\ \varphi (x)=F(x)+\int _{\Omega }\varphi \big (f(x,\omega )\big )P(d\omega ), \end{aligned}$$ φ ( x ) = F ( x ) - ∫ Ω φ ( f ( x , ω ) ) P ( d ω ) , φ ( x ) = F ( x ) + ∫ Ω φ ( f ( x , ω ) ) P ( d ω ) , where P is a probability measure on a $$\sigma $$ σ -algebra of subsets of $$\Omega $$ Ω .

scholarly journals Yang–Mills Measure on the Two-Dimensional Torus as a Random Distribution

2019 ◽  
Vol 372 (3) ◽  
pp. 1027-1058
Author(s):  
Ilya Chevyrev
Keyword(s):  
Random Distribution ◽  
Random Variable ◽  
Continuous Functions ◽  
Small Scale ◽  
Dimensional Torus ◽  
Hölder Continuous ◽  
Line Segments ◽  
Yang Mills ◽  
Hölder Continuous Functions ◽  
On Line

Abstract We introduce a space of distributional 1-forms $$\Omega ^1_\alpha $$Ωα1 on the torus $$\mathbf {T}^2$$T2 for which holonomies along axis paths are well-defined and induce Hölder continuous functions on line segments. We show that there exists an $$\Omega ^1_\alpha $$Ωα1-valued random variable A for which Wilson loop observables of axis paths coincide in law with the corresponding observables under the Yang–Mills measure in the sense of Lévy (Mem Am Math Soc 166(790), 2003). It holds furthermore that $$\Omega ^1_\alpha $$Ωα1 embeds into the Hölder–Besov space $$\mathcal {C}^{\alpha -1}$$Cα-1 for all $$\alpha \in (0,1)$$α∈(0,1), so that A has the correct small scale regularity expected from perturbation theory. Our method is based on a Landau-type gauge applied to lattice approximations.

scholarly journals NONCLASSICAL SPECTRAL ASYMPTOTICS AND DIXMIER TRACES: FROM CIRCLES TO CONTACT MANIFOLDS

2017 ◽  
Vol 5 ◽  
Author(s):  
HEIKO GIMPERLEIN ◽  
MAGNUS GOFFENG
Keyword(s):  
Multiplication Operator ◽  
Continuous Functions ◽  
Spectral Asymptotics ◽  
Hölder Continuous ◽  
Contact Manifolds ◽  
Integral Formulas ◽  
Weyl Law ◽  
Wide Range ◽  
Hölder Continuous Functions ◽  
Geometric Origin

We consider the spectral behavior and noncommutative geometry of commutators$[P,f]$, where$P$is an operator of order 0 with geometric origin and$f$a multiplication operator by a function. When$f$is Hölder continuous, the spectral asymptotics is governed by singularities. We study precise spectral asymptotics through the computation of Dixmier traces; such computations have only been considered in less singular settings. Even though a Weyl law fails for these operators, and no pseudodifferential calculus is available, variations of Connes’ residue trace theorem and related integral formulas continue to hold. On the circle, a large class of nonmeasurable Hankel operators is obtained from Hölder continuous functions$f$, displaying a wide range of nonclassical spectral asymptotics beyond the Weyl law. The results extend from Riemannian manifolds to contact manifolds and noncommutative tori.

Embeddings of weighted Sobolev spaces into spaces of continuous functions

1992 ◽  
Vol 439 (1906) ◽  
pp. 279-296 ◽  
Keyword(s):  
Weighted Sobolev Spaces ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Hölder Continuous ◽  
Compact Embeddings ◽  
Spaces Of Continuous Functions ◽  
Hölder Continuous Functions ◽  
Continuous And Compact Embeddings ◽  
Necessary And Sufficient ◽  
Theoretical Results

We give sufficient conditions and necessary conditions (which in some cases are both necessary and sufficient) for continuous and compact embeddings of the weighted Sobolev space W 1,p ( Ω ;v 0 v 1 )into spaces of weighted continuous and Holder continuous functions. The theoretical results are illustrated by several examples.

On the transfer operator for rational functions on the Riemann sphere

1996 ◽  
Vol 16 (2) ◽  
pp. 255-266 ◽  
Author(s):  
Manfred Denker ◽  
Feliks Przytycki ◽  
Mariusz Urbański
Keyword(s):  
Equilibrium Measure ◽  
Riemann Sphere ◽  
Julia Set ◽  
Transfer Operator ◽  
Rational Functions ◽  
Continuous Functions ◽  
Hölder Continuous ◽  
Correlation Integral ◽  
Hölder Continuous Functions ◽  
Conformal Measure

AbstractLet T be a rational function of degree ≥ 2 on the Riemann sphere. Our results are based on the lemma that the diameter of a connected component of T−n(B(x, r)), centered at any point x in its Julia set J = J(T), does not exceed Lnrp for some constants L ≥ 1 and ρ > 0. Denote the transfer operator of a Hölder-continuous function φ on J satisfying P(T,φ) > supz∈Jφ(z). We study the behavior of {: n ≥ 1} for Hölder-continuous functions ψ and show that the sequence is (uniformly) normbounded in the space of Hölder-continuous functions for sufficiently small exponent. As a consequence we obtain that the density of the equilibrium measure μ for φ with respect to the -conformal measure is Höolder-continuous. We also prove that the rate of convergence of {ψ to this density in sup-norm is . From this we deduce the corresponding decay of the correlation integral and the central limit theorem for ψ.

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