Optimal factorization of Muckenhoupt weights

Peter Jones' theorem on the factorization of  weights is sharpened for weights with bounds near , allowing the factorization to be performed continuously near the limiting, unweighted case. When and is an  weight with bound , it is shown that there exist  weights  such that both the formula and the estimates hold. The square root in these estimates is also proven to be the correct asymptotic power as .

     

Muckenhoupt weights with singularities on closed lower dimensional sets in spaces of homogeneous type

We give sufficient conditions on a real number β and on a closed set F   in a general space of homogeneous type (X,d,μ)$(X,d,\mu )$ in such a way that μ(B(x,d(x,F)))^β$\mu {(B(x,d(x,F)))}^{\beta }$ becomes a Muckenhoupt weight. In order to prove our result, we modify the underlying space so that it becomes 1-Ahlfors regular.     

Paraexponentials, Muckenhoupt weights, and resolvents of paraproducts

We analyze the stability of Muckenhoupt's and classes of weights under a nonlinear operation, the -operation. We prove that the dyadic doubling reverse Hölder classes are not preserved under the -operation, but the dyadic doubling classes are preserved for . We give an application to the structure of resolvent sets of dyadic paraproduct operators.

     

Spectral analysis of biorthogonal expansions generated by Muckenhoupt weights

An arbitrary Muckenhoupt A₂-weight w² on the special contour ( 1/2) generates a function Y_,w (, t), which for =1, w²(z) 1 coincides with the exponential exp{it}. In the paper, with the aid of B. S. Pavlov's geometric approach, one obtains criteria for the unconditional basis property of families of functions of the form {y,w(_k,t):_k} in the space L₂(0, ). The analytic foundation of the constructions is a generalization of M. M. Dzhrbashyan's certain results (power weight) to the case of arbitrary Muckenhoupt A₂- weights.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 190, pp. 34–80, 1991.     

Fixed point sets through iteration schemes

We introduce the notion of a general fixed point iteration scheme to unify various fixed point iterations in the literatures, and extend the concept of virtual stability of a selfmap to an iteration scheme to obtain a connection, through an explicit retraction, between the convergence set of the scheme and its fixed point set. Moreover, we illustrate how to apply our results to obtain a new criterion for contractibility of the fixed point set of a certain quasi-nonexpansive selfmap.     

On the algebraic relation between principal components corresponding to two different sets of weights

In principal component analysis, it is sometimes useful to give weights to observations. When two different sets of weights are attached to the observations of a data table, we derive the algebraic relation between the corresponding two principal component matrices.     

Perturbations of strongly continuous operator semigroups,and matrix Muckenhoupt weights

Let A and A ₀ be linear continuously invertible operators on a Hilbert space ? such that A ^?1 ? A ₀ ^?1 has finite rank. Assuming that σ(A ₀) = ? and that the operator semigroup V ₊(t) = exp{iA ₀ t}, t ≥ 0, is of class C ₀, we state criteria under which the semigroups U _±(t) = exp{±iAt}, t ≥ 0, are of class C ₀ as well. The analysis in the paper is based on functional models for nonself-adjoint operators and techniques of matrix Muckenhoupt weights.     

Mixed modulus of smoothness with Muckenhoupt weights and approximation by angle

Mixed modulus of smoothness in weighted Lebesgue spaces with Muckenhoupt weights are investigated. Using mixed modulus of smoothness we obtain Potapov type direct and inverse estimates of angular trigonometric approximation of functions in these spaces. Also we obtain equivalences between mixed modulus of smoothness and K-functional and realization functional. Fractional order modulus of smoothness is considered as well.     

The Helmholtz decomposition of weightedLq-spaces for Muckenhoupt weights

We consider weights of Muckenhoupt classA _q, 1<q<∞. For a bounded Lipschitz domain Ω⊂ℝⁿ we prove a compact embedding and a Poincaré inequality in weighted Sobolev spaces. These technical tools allow us to solve the weak Neumann problem for the Laplace equation in weighted spaces on ℝⁿ, ℝⁿ ₊, on bounded and on exterior domains Ω with boundary of classC ¹, which will yield the Helmholtz decomposition ofL _ω ^q(Ω)ⁿ for general ω∈A _q. This is done by transferring the method of Simader and Sohr [4] to the weighted case. Our result generalizes a result of Farwig and Sohr [2] where the Helmholtz decomposition ofL _ω ^p(Ω)ⁿ is proved for an exterior domain and weights of Muckenhoupt class without singularities or degeneracies in a neighbourhood of ϖΩ.

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Sunto In questo lavoro consideriamo dei pesi della classe di MuckenhouptA _q, 1<q<∞. Per un dominio limitato lipschitziano Ω⊂ℝⁿ, dimostriamo una immersione compatta ed una disuguaglianza di Poincaré in spazi di Sobolev con peso. Questa tecnica ci consente di risolvere il problema debole di Neumann per l’equazione di Laplace in spazi pesati in ℝⁿ, ℝⁿ ₊ in domini limitati ed in domini esterni con frontiera di classeC ¹, che conduce alla decomposizione di Helmholtz diL _ω ^q(Ω)ⁿ per un qualsiasi ω∈A _q. Il risultato è ottenuto trasferendo il metodo di Simader e Sohr [4] al caso pesato. Quello qui presente estende un risultato di Farwig e Sohr [2] dove la decomposizione di Helmholtz diL _ω ^q(Ω)ⁿ è dimostrata per domini esterni e pesi della classe di Muckenhoupt privi di singolarità in un intorno di ϖΩ.

     

On determining the congruence of point sets in d dimensions

This paper considers the following problem: given two point sets A and B (|A| = |B| = n) in d dimensional Euclidean space, determine whether or not A is congruent to B. This paper presents an O(n^(d−1)/2 log n) time randomized algorithm. The birthday paradox, which is well-known in combinatorics, is used effectively in this algorithm. Although this algorithm is Monte-Carlo type (i.e., it may give a wrong result), this improves a previous O(n^d−2 log n) time deterministic algorithm considerably. This paper also shows that if d is not bounded, the problem is at least as hard as the graph isomorphism problem in the sense of the polynomiality. Several related results are described too.     

Fixed point theorems on closed sets through abstract cones

In this paper the authors consider the problem of the existence, and iteration to a fixed point or a zero, of an operator on a closed subset of an abstract space. The results generalize the construction mapping principle. A generalized or cone norm is used.     

The variational contact problem for elastic objects of different dimensions

We consider the variational free boundary problem describing the contact of an elastic plate with a thin elastic obstacle. The contact domain is unknown a priori and should be determined. The problem is described by a variational inequality for a fourth-order operator. The constraint on the displacement is given on a set of dimension less than that of the solution domain. We find the boundary conditions on the set of the possible contact and their exact statement. We justify the mixed statement of the problem and analyze the limit cases corresponding to the unbounded increase of the elasticity coefficients of the contacting bodies.     

Assouad dimensions of Moran sets and Cantor-like sets

We obtain the Assouad dimensions of Moran sets under suitable condition. Using the homogeneous set introduced in [J. Math. Anal. Appl., 2015, 432: 888–917], we also study the Assouad dimensions of Cantor-like sets.     

Instability of discrete point sets]{Instability of discrete point sets

Summary Let <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"10"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"11"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>X$ be a discrete subset of Euclidean $d$-space. We allow subsequently continuous movements of single elements, whenever the minimum distance to other elements does not decrease. We discuss the question, if it is possible to move all elements of $X$ in this way, for example after removing a finite subset $Y$ from $X$. Although it is not possible in general, we show the existence of such finite subsets $Y$ for many discrete sets $X$, including all lattices. We define the \textit{instability degree} of $X$ as the minimum cardinality of such a subset $Y$ and show that the maximum instability degree among lattices is attained by perfect lattices. Moreover, we discuss the $3$-dimensional case in detail.     

Assouad dimensions of Moran sets

We prove that the Assouad dimensions of a class of Moran sets coincide with their upper box dimensions and packing dimensions.     

Atomic decompositions of function spaces with Muckenhoupt weights,and some relation to fractal analysis

This paper deals with atomic decompositions in spaces of type B^s_p,q (?ⁿ , w), F^s_p,q (?ⁿ , w), 0 < p < ∞, 0 < q ≤ ∞, s ∈ ?, where the weight function w belongs to some Muckenhoupt class A_r. In particular, we consider the weight function w^Γ_κ (x) = dist(x, Γ)^κ, where Γ is some d ‐set, 0 < d < n, and κ > –(n – d). (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)