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Lipschitz Free Spaces and Subsets of Finite-Dimensional Spaces
Bíma, Jan ; Cúth, Marek (advisor) ; Doucha, Michal (referee)
The present thesis is devoted to the geometry of Lipschitz free p-spaces Fp(M) over subsets of finite-dimensional vector spaces, where 0 < p ≤ 1. We solve an open problem and show that if M is an infinite subset of Rd endowed with the H¨older distorted metric | · |α , where 0 < α < 1, then Fp(M, | · |α ) ≃ ℓp for every 0 < p ≤ 1. Moreover, we tackle a question due to Albiac et al. and expound the role of p, d for the Lipschitz constant of a locally coordinatewise affine retraction from (K, | · |1), where K = ⋃︁ Q∈R Q is a union of a collection ∅ ̸= R ⊆ {Rw + R[0, 1]d : w ∈ Zd } of cubes in Rd with side length R > 0, into the Lipschitz free p-space Fp(V, | · |1) over their vertices. The last chapter is then dedicated to the Lipschitz extension problem Lip0(N, Z) → Lip0(M, Z), where N is a doubling subspace of a metric space M and Z is a p-Banach space. As it turns out, the problem can equivalently be stated in terms of a projective relation between the Lipschitz free p-spaces Fp(N) and Fp(M). 1
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