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

     

Noncompactness and noncompleteness in isometries of Lipschitz spaces

We solve the following three questions concerning surjective linear isometries between spaces of Lipschitz functions Lip(X,E) and Lip(Y,F), for strictly convex normed spaces E and F and metric spaces X and Y:

(i)

Characterize those base spaces X and Y for which all isometries are weighted composition maps.

(ii)

Give a condition independent of base spaces under which all isometries are weighted composition maps.

(iii)

Provide the general form of an isometry, both when it is a weighted composition map and when it is not.

In particular, we prove that requirements of completeness on X and Y are not necessary when E and F are not complete, which is in sharp contrast with results known in the scalar context.     

Isometries of Lipschitz type function spaces

In this article, we describe isometries over the Lipschitz spaces under certain conditions. Indeed, we provide a unified proof for the main results of 3 and 5 in a more general setting. Finally, we extend our results for some other functions spaces like the space of vector‐valued little Lipschitz maps and pointwise Lipschitz maps.     

A proof of the trace theorem of Sobolev spaces on Lipschitz domains

A complete proof of the trace theorem of Sobolev spaces on Lipschitz domains has not appeared in the literature yet. The purpose of this paper is to give a complete proof of the trace theorem of Sobolev spaces on Lipschitz domains by taking advantage of the intrinsic norm on . It is proved that the trace operator is a linear bounded operator from to for .

     

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 N1,∞(X). In particular, if X supports a doubling measure and satisfies a local Poincaré inequality, we obtain that D^∞(X)=N1,∞(X).     

Algebraic reflexivity of the isometry group of some spaces of Lipschitz functions

We show that the isometry groups of Lip(X,d) and lip(X,dα) with α∈(0,1), for a compact metric space (X,d), are algebraically reflexive. We also prove that the sets of isometric reflections and generalized bi-circular projections on such spaces are algebraically reflexive. In order to achieve this, we characterize generalized bi-circular projections on these spaces.     

Convergence theorems of fixed points for Lipschitz pseudo-contractions in Hilbert spaces

Let C be a closed convex subset of a real Hilbert space H and assume that T is a κ-strict pseudo-contraction on C. Consider Mann's iteration algorithm given by

     

Projections and averages of isometries on Lipschitz spaces

We characterize projections on spaces of Lipschitz functions expressed as the average of two and three linear surjective isometries. Generalized bi-circular projections are the only projections on these spaces given as the convex combination of two surjective isometries.     

Some results on Gaussian Besov–Lipschitz spaces and Gaussian Triebel–Lizorkin spaces

In this paper we define Besov–Lipschitz and Triebel–Lizorkin spaces in the context of Gaussian harmonic analysis, the harmonic analysis of Hermite polynomial expansions. We study inclusion relations among them, some interpolation results and continuity results of some important operators (the Ornstein–Uhlenbeck and the Poisson–Hermite semigroups and the Bessel potentials) on them. We also prove that the Gaussian Sobolev spaces are contained in them. The proofs are general enough to allow extensions of these results to the case of Laguerre or Jacobi expansions and even further in the general framework of diffusion semigroups.     

Commutators on Lipschitz spaces and related function spaces

We characterize the symbol functions so that the associated commutators with symbol functions and the Hilbert transform are bounded on Lipschitz space Λ _α ^p , where 1 < p < ∞ and 0 < α < 1/p. Properties of such symbols are also discussed.      

Lipschitz spaces and Q_K type spaces

If f(z) =Σ∞ n=0 anzn and g(z) =Σ∞n=0bnzn for functions f, g are analytic in the unit disc, the Hadamard products of f and g is defined by f * g = ∞ n=0 a n b n z n . In this paper, the Lipschitz spaces Λ(s, α) and QK type spaces are studied in terms of the Hadamard products.     

New types of Lipschitz summing maps between metric spaces

Building upon the results of M. C. Matos and extending previous work of J. D. Farmer, W. B. Johnson and J. A. Chávez‐Domínguez we define a Lipschitz mixed summable sequence as the pointwise product of a strongly summable sequence and a weakly Lipschitz summable one. Then we introduce classes of Lipschitz maps satisfying inequalities between Lipschitz mixed summable sequence and strongly summable sequences analogously to the linear case. These classes generalize the classes of Lipschitz summable maps considered earlier in the literature. We use standard techniques to establish several basic properties, showing that these classes of maps are ideals and some relationships between them. We establish various composition and inclusion theorems between different classes of Lipschitz summing maps and several characterizations. Furthermore, we prove that the classes of Lipschitz p‐summing maps coincide and the nonlinear “Pietsch Domination Theorem” for the case . We also identify cases where all Lipschitz maps are in the aforementioned classes of Lipschitz maps and discuss a sufficient condition for a Lipschitz composition formula as in the linear case.     

On the numerical radius of Lipschitz operators in Banach spaces

We study the numerical radius of Lipschitz operators on Banach spaces via the Lipschitz numerical index, which is an analogue of the numerical index in Banach space theory. We give a characterization of the numerical radius and obtain a necessary and sufficient condition for Banach spaces to have Lipschitz numerical index 1. As an application, we show that real lush spaces and C  -rich subspaces have Lipschitz numerical index 1. Moreover, using the Gâteaux differentiability of Lipschitz operators, we characterize the Lipschitz numerical index of separable Banach spaces with the RNP. Finally, we prove that the Lipschitz numerical index has the stability properties for the _c0$c_0$-, _l1$l_1$-, and _l∞$l_{\infty }$-sums of spaces and vector-valued function spaces. From this, we show that the C(K)$C(K)$ spaces, _L1(μ)$L_1(\mu )$-spaces and _L∞(ν)$L_{\infty }(\nu )$-spaces have Lipschitz numerical index 1.     

Finite codimensional linear isometries on spaces of differentiable and Lipschitz functions

We characterize finite codimensional linear isometries on two spaces, C ⁽ⁿ⁾[0; 1] and Lip [0; 1], where C ⁽ⁿ⁾[0; 1] is the Banach space of n-times continuously differentiable functions on [0; 1] and Lip [0; 1] is the Banach space of Lipschitz continuous functions on [0; 1]. We will see they are exactly surjective isometries. Also, we show that C ⁽ⁿ⁾[0; 1] and Lip [0; 1] admit neither isometric shifts nor backward shifts.     

Surjective isometries on spaces of vector valued continuous and Lipschitz functions

For X a compact Hausdorff topological space and F a strictly convex and reflexive Banach space we characterize the surjective isometries of certain subspaces of C(X, F). It follows from this characterization that surjective isometries on spaces of vector valued Lipschitz functions equipped with a p-norm are also weighted composition operators.     

On the Lipschitz perturbation of monotonic functions

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Pseudo-differential operators on Sobolev and Lipschitz spaces

In this paper, by discovering a new fact that the Lebesgue boundedness of a class of pseudo- differential operators implies the Sobolev boundedness of another related class of pseudo-differential operators, the authors establish the boundedness of pseudo-differential operators with symbols in Sρ,δ^m on Sobolev spaces, where ∈ R, ρ≤ 1 and δ≤ 1. As its applications, the boundedness of commutators generated by pseudo-differential operators on Sobolev and Bessel potential spaces is deduced. Moreover, the boundedness of pseudo-differential operators on Lipschitz spaces is also obtained.