2023/06/22 by Ching-Jou Liao, Chih-Neng Liu, Liao, Ching-Jou +5
Computer Science · Mathematics · #26A16 #46B04 #51F30 #Approximation Theory and Sequence Spaces #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Matrix Theory and Algorithms
paper · pdf · doi:10.48550/arxiv.2306.12824
openalex publication_date 2023/06/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Lip(X), Lipb(X), Liploc(X) and Lippt(X) be the vector spaces of Lipschitz, bounded Lipschitz, locally Lipschitz and pointwise Lipschitz (real-valued) functions defined on a metric space (X, dX), respectively. We show that if a weighted composition operator Tf=h⋅ f∘ φ defines a bijection between such vector spaces preserving Lipschitz constants, local Lipschitz constants or pointwise Lipschitz constants, then h= ±1/α is a constant function for some scalar α>0 and φ is an α-dilation. Let U be open connected and V be open, or both U,V are convex bodies, in normed linear spaces E, F, respectively. Let Tf=h⋅ f∘φ be a bijective weighed composition operator between the vector spaces Lip(U) and Lip(V), Lipb(U) and Lipb(V), Liploc(U) and Liploc(V), or Lippt(U) and Lippt(V), preserving the Lipschitz, locally Lipschitz, or pointwise Lipschitz constants, respectively. We show that there is a linear isometry A: F→ E, an α>0 and a vector b∈ E such that φ(x)=αAx + b, and h is a constant function assuming value ± 1/α. More concrete results are obtained for the special cases when E=F=ℝn, or when U,V are n-dimensional flat manifolds.