Lattice isomorphisms of the spaces W n 1 and quasiconformal mappings

Atomic representations in function spaces and applications to pointwise multipliers and diffeomorphisms, a new approach

“…There is nothing to prove for j = 0. Hence, we can use the moment conditions (9). Let L > 0, L = ⌊L⌋ + {L} as in Section 1.1 and let ψ ∈ C L (R n ).…”

“…For further sufficient results on diffeomorphisms including characterizations for classical Sobolev spaces W k p (R n ) we refer to Gol'dshtein, Reshetnyak, Romanov, Ukhlov and Vodop'yanov [8,9,10,35,34], [7,Chapter 4], Markina [13] as well as to Maz'ya and Shaposhnikova [14,15], while for Besov spaces B s p,q (R n ) with 0 < s < 1 we refer to Vodop'yanov, Bordaud and Sickel [35,1]. A special case of our result (Lipschitz diffeomorphisms) can be found in Triebel [30,Section 4].…”

Lattice homomorphisms between Sobolev spaces

“…Sergei Vodop'janov and Vladimir Gol'dšteȋn [24] showed in 1975 that an order isomorphism between the Sobolev spaces W 1,N (Ω 1 ) and W 1,N (Ω 2 ) for domains Ω 1 and Ω 2 in R N is a composition operator if it satisfies several additional quite natural order theoretic assumptions.…”

“…We can show for p > 2 and for p = 2 and N ≥ 2 that every isometry T from W 1,p (Ω 1 ) to W 1,p (Ω 2 ), where Ω 1 and Ω 2 are bounded open subsets of R N , is a weighted composition operator, whenever W 1,p 0 (Ω 2 ) ⊂ T W 1,p 0 (Ω 1 ) and T satisfies some mild additional order theoretical conditions. The choice of the Sobolev norm does not allow to reduce the question to the corresponding one for L p -spaces as in [12], and the order assumptions are by far not strong enough to reduce the proof to the one in [24]. Instead, we use quite different techniques, based on the first author's characterization of lattice homomorphisms between Sobolev spaces [7].…”

Spectral estimates of the p-Laplace Neumann operator and Brennan’s conjecture

“…for any f ∈ L 1 n ( Ω) [29]. Compositions of Sobolev functions of the spaces L 1 p (Ω ′ ), p = n, with quasiconformal mappings was studied also in [21].…”