Space Lip Research Articles

Atomic decomposition of finite signed measures on compacts of R^n

Recently there has been interest in pairs of Banach spaces \((E_0,E)\) in an o-O relation and with \(E_0^{**}=E\). It is known that this can be done for Lipschitz spaces on suitable metric spaces. In this paper we consider the case of a compact subset \(K\) of \(\mathbf{R}^n\) with the Euclidean metric, which does not give an o-O structure, but we use part of the theory concerning these pairs to find an atomic decomposition of the predual of Lip\((K)\). In particular, since the space \(M(K)\) of finite signed measures on \(K\), when endowed with the Kantorovich-Rubinstein norm, has as dual space Lip\((K)\), we can give an atomic decomposition for this space.

A class of invariant valuations on Lip(Sn−1)

We provide an integral representation for continuous valuations defined on the space Lip(Sn−1) of Lipschitz continuous functions on the unit (n−1)-sphere, which are invariant under rotations and under the addition of linear functions restricted to the unit sphere (Λ-invariant).

Noncommutative dyadic martingales and Walsh–Fourier series

Let R be the hyperfinite II 1 -factor. For 0 < p < 1 , we show that the dual space of the Hardy space h p r ( R ) is the Lipschitz space Lip α c ( R ) , α = 1 p − 1 , which partially answers [Bekjan, Chen, Perrin and Yin, J. Funct. Anal. 258 (2010) 2483–2505, Problem 4]. If S is an operator of taking partial sum of the noncommutative Walsh–Fourier series of x ∈ L 1 ( R ) , then ∥ S ( x ) ∥ L 1 , ∞ ( R ) ⩽ c a b s ∥ x ∥ L 1 ( R ) . Furthermore, we show the closed subspace spanned by noncommutative Walsh system of multiplicity 2 is isomorphic to l 2 and is complemented in H 1 ( R ) . The latter result demonstrates a very substantial difference from its commutative counterpart due to Müller–Schechtman.

2-Local standard isometries on vector-valued Lipschitz function spaces

Under the right conditions on a compact metric space X and on a Banach space E, we give a description of the 2-local (standard) isometries on the Banach space Lip(X,E) of vector-valued Lipschitz functions from X to E in terms of a generalized composition operator, and we study when every 2-local (standard) isometry on Lip(X,E) is both linear and surjective.

Smoothness of the flow map for low-regularity solutions of the Camassa-Holm equations

It was recently proven by De Lellis, Kappeler, and Topalov that the periodic Cauchy problem for the Camassa-Holm equations is locally well-posed in the space Lip (T) endowed with the topology of H^1 (T). We prove here that the Lagrangian flow of these solutions are analytic with respect to time and smooth with respect to the initial data. These results can be adapted to the higher-order Camassa-Holm equations describing the exponential curves of the manifold of orientation preserving diffeomorphisms of T using the Riemannian structure induced by the Sobolev inner product H^l (T), for l in N, l > 1 (the classical Camassa-Holm equation corresponds to the case l=1): the periodic Cauchy problem is locally well-posed in the space W^{2l-1,infty} (T) endowed with the topology of H^{2l-1} (T) and the Lagrangian flows of these solutions are analytic with respect to time with values in W^{2l-1,infty} (T) and smooth with respect to the initial data. These results extend some earlier results which dealt with more regular solutions. In particular our results cover the case of peakons, up to the first collision.

Extreme points and isometries on vector-valued Lipschitz spaces

For a Banach space E and a compact metric space ( X , d ) , a function F : X → E is a Lipschitz function if there exists k > 0 such that ‖ F ( x ) − F ( y ) ‖ ⩽ k d ( x , y ) for all x , y ∈ X . The smallest such k is called the Lipschitz constant L ( F ) for F. The space Lip ( X , E ) of all Lipschitz functions from X to E is a Banach space under the norm defined by ‖ F ‖ = max { L ( F ) , ‖ F ‖ ∞ } , where ‖ F ‖ ∞ = sup { ‖ F ( x ) ‖ : x ∈ X } . Recent results characterizing isometries on these vector-valued Lipschitz spaces require the Banach space E to be strictly convex. We investigate the nature of the extreme points of the dual ball for Lip ( X , E ) and use the information to describe the surjective isometries on Lip ( X , E ) under certain conditions on E, where E is not assumed to be strictly convex. We make use of an embedding of Lip ( X , E ) into a space of continuous vector-valued functions on a certain compact set.

Sharp Ul’yanov-type inequalities using fractional smoothness

In this paper we obtain the following sharp ( p , q ) inequalities for moduli of smoothness of fractional order: ω k ( f , δ ) L q ( X ) ⩽ C ( ∫ 0 δ ( t − θ ω k + θ ( f , t ) L p ( X ) ) q d t t ) 1 q , 1 < p < q < ∞ , θ = d ( 1 / p − 1 / q ) , and X = T d or R d . Corresponding estimates for K -functionals and corresponding embedding theorems for the Lipschitz spaces Lip ( α , k , p ) = { ω k ( f , δ ) p ⩽ C δ α } are investigated.

A NOTE ON FUNCTIONS OF MEAN BLOCH TYPES

Abstract. A characterization of the holomorphic function spaces of me-an Bloch type on the unit disc is deduced in terms of the induced distance. 1. IntroductionWe introduce basic definitions, previous results, and the goal of this paperthat we will involve.Let D = {z ∈ C: |z| 0, let B α ( D ) be the space of holomorphic functions on D satisfyingsup z∈D i1 −|z| 2 ¢ α |f 0 ( z ) | 0 and 1 ≤ p < ∞ , let B pα ( D ) be the space holomorphic functionssatisfyingsup 0 <r< 1 (1 −r 2 ) α µZ T |f 0 ( rζ ) | p dσ ( ζ )¶ 1 p < ∞. The spaces B α ( D ) and B p ( D ) occurred in the literature in connection withthe Lipschitz space Lip α ( D ) and the mean Lipschitz space Lip p ( D ) which, for0 < α < 1 and 1 ≤ p < ∞ , are defined to consist of f holomorphic in D suchthat |f ( z ) −f ( w ) | ≤ C|z −w| α , z,w ∈ D, and of f ∈ H p ( D ) such thatµZ T |f

Heat kernel characterisation of Besov-Lipschitz spaces on metric measure spaces

We give a heat-kernel characterisation of the Besov-Lipschitz spaces Lip (α, p, q)(X) on domains which support a Markovian kernel with appropriate exponential bounds. This extends former results of Grigor’yan et al. (Trans Am Math Soc 355:2065–2095, 2008), Hu and Zahle (Studia Math 170:259–281, 2005), Pietruska-Paluba (Stoch Stoch Rep 67:267–285, 1999; 70:153–164, 2000), which were valid for \({\alpha = \frac{d_w}{2}, p = 2, q = \infty}\), where dw is the walk dimension of the space X.

Pointwise Lipschitz functions on metric spaces

For a metric space X, we study the space D ∞ ( X ) of bounded functions on X whose pointwise Lipschitz constant is uniformly bounded. D ∞ ( X ) is compared with the space LIP ∞ ( X ) of bounded Lipschitz functions on X, in terms of different properties regarding the geometry of X. We also obtain a Banach–Stone theorem in this context. In the case of a metric measure space, we also compare D ∞ ( X ) with the Newtonian–Sobolev space N 1 , ∞ ( X ) . In particular, if X supports a doubling measure and satisfies a local Poincaré inequality, we obtain that D ∞ ( X ) = N 1 , ∞ ( X ) .

Hölder seminorm preserving linear bijections and isometries

Let (X, d) be a compact metric and 0 < α < 1. The space Lip α (X) of Holder functions of order α is the Banach space of all functions ƒ from X into \( \mathbb{K} \) such that ∥ƒ∥ = max{∥ƒ∥∞, L(ƒ)} < ∞, where $$ L(f) = sup\{ \left| {f(x) - f(y)} \right|/d^\alpha (x,y):x,y \in X, x \ne y\} $$ is the Holder seminorm of ƒ. The closed subspace of functions ƒ such that $$ \mathop {\lim }\limits_{d(x,y) \to 0} \left| {f(x) - f(y)} \right|/d^\alpha (x,y) = 0 $$ is denoted by lip α (X). We determine the form of all bijective linear maps from lip α (X) onto lip α (Y) that preserve the Holder seminorm.

Compact embeddings of Brézis-Wainger type

Let \Omega be a bounded domain in \mathbb R^n and denote by id_\Omega the restriction operator from the Besov space B_{pq}^{1+n/p}(\mathbb R^n) into the generalized Lipschitz space Lip^{(1,-\alpha)}(\Omega) . We study the sequence of entropy numbers of this operator and prove that, up to logarithmic factors, it behaves asymptotically like e_k(id_\Omega) \sim k^{-1/p} if \alpha &gt; \max (1+2/p-1/q,1/p) . Our estimates improve previous results by Edmunds and Haroske.

Solutions in Sobolev spaces of vector refinement equations with a general dilation matrix

In this paper, we present a necessary and sufficient condition for the existence of solutions in a Sobolev space Wpk(ℝs) (1≤p≤∞) to a vector refinement equation with a general dilation matrix. The criterion is constructive and can be implemented. Rate of convergence of vector cascade algorithms in a Sobolev space Wpk(ℝs) will be investigated. When the dilation matrix is isotropic, a characterization will be given for the Lp (1≤p≤∞) critical smoothness exponent of a refinable function vector without the assumption of stability on the refinable function vector. As a consequence, we show that if a compactly supported function vector φ∈Lp(ℝs) (φ∈C(ℝs) when p=∞) satisfies a refinement equation with a finitely supported matrix mask, then all the components of φ must belong to a Lipschitz space Lip(ν,Lp(ℝs)) for some ν>0. This paper generalizes the results in R.Q. Jia, K.S. Lau and D.X. Zhou (J. Fourier Anal. Appl. 7 (2001) 143–167) in the univariate setting to the multivariate setting.

Lipschitz-free Banach spaces

We show that when a linear quotient map to a separable Banach space X has a Lipschitz right inverse, then it has a linear right inverse. If a separable space X embeds isometrically into a Banach space Y , then Y contains an isometric linear copy of X. This is false for every nonseparable weakly compactly generated Banach space X. Canonical examples of nonseparable Banach spaces which are Lipschitz isomorphic but not linearly isomorphic are constructed. If a Banach space X has the bounded approximation property and Y is Lipschitz isomorphic to X, then Y has the bounded approximation property. 1. Introduction. A (real) Banach space X is in particular a metric space, equipped with the distinguished point f0g, and to such a pointed metric space we can associate the space Lip 0 (X) of all real-valued Lipschitz functions which vanish at 0. We refer to (29) for basic facts on this space and some of its uses. It is clear that Lip 0 (X) is a Banach space when it is equipped with the Lipschitz norm, dened

Lipschitz — Orlicz Spaces and the Laplace Equation

Stein and Taibleson gave a characterization for f ϵ Lp(ℝn) to be in the spaces Lip (α, Lp) and Zyg(α, Lp) in terms of their Poisson integrals. In this paper we extend their results to Lipschitz-Orlicz spaces Lip (α, Lm) and Zygmund-Orlicz spaces Zyg (φ, Lm) and to the general function φ ϵ P instead of the power function φ(t)= tα. Such results describe the behavior of the Laplace equation in terms of the smoothness property of differences of f in Orlicz spaces Lm (IRn). More general spaces δk(φ,X, q) are also considered.

A Characterization of Smoothness in Terms of Approximation by Algebraic Polynomials in LP

We prove direct and inverse theorems for the classical modulus of smoothness and approximation by algebraic polynomials in Lp[−1, 1]. These theorems contain the well-known theorems of A. Timan, V. Dzyadyk, G. Freud, and Yu. Brudnyi as special cases when p = ∞. They also provide a characterization of the spaces Lip(α, p) (Lipschitz spaces in Lp) for 0 < α < ∞, 1 ≤ p ≤ ∞.

A Multiplier Problem for Fourier-Jacobi Expansions in a Banach Space

In the present paper we obtain a multiplier theorem for Fourier-Jacobi expansions in the space lip$(\gamma,p)$, $\gamma p>1$, which extends an earlier result of Trebels [11].

Some results about smoothing methods of fourier series

In this paper we compare the Fourier polynomials approximation, in C(Tn) or in Lp(Tn), with that one obtained by a class of smoothing methods, which naturally arise in solving ill posed problems. It is also given a sharp evaluation of the above approximation in the space Lip (α, C(T)), 0<α<1.

Lipschitz multipliers on compact O-dimensional groups

Let G be an infinite compact metrizable O-dimensional group. In this note we characterize the multipliers from one Lipschitz space Lip( α, p; G) to another Lipschitz space Lip( β, q; G) for 0 < α ⩽ β < ∞, 1 ⩽ p ⩽ ∞ and 1 ⩽ q ⩽ ∞. For Abelian G, similar results were obtained by the authors in an earlier paper ( J. Math. Anal. Appl. 86 (1982), 69–73).

Multipliers of Lipschitz spaces on a certain class of groups

Zygmund [lo], Mizuhara [7], and Anderson [l] have studied various multiplier problems of Lipschitz spaces defined on the circle group T. In [8,9], the authors have characterized the multipliers of Lipschitz spaces defined on a Vilenkin group G (i.e., G is an infinite compact O-dimensional metric Abelian group). In this paper we define a certain class S of metric locally compact Abelian groups, and study the multipliers from one Lipschitz space Lip(a,p; G) to another Lipschitz space Lip@, q; G) for G in .Y. The class .Y contains the classical groups T” and R”, as well as many other interesting groups. Our main result is a characterization of the multipliers from Lip(a,p; G) to Lip@‘, q; G), as restrictions of multipliers from Lp(G) to L9(G), for GE ,Y, 0 < a </3 < 1, 1 <p < co, and l<q<co.