Quasiconformal mappings and exponentially integrable functions

Fernando Farroni, and Raffaella Giova. "Quasiconformal mappings and exponentially integrable functions." Studia Mathematica 203.2 (2011): 195-203. <http://eudml.org/doc/285676>.

@article{FernandoFarroni2011,
abstract = {We prove that a K-quasiconformal mapping f:ℝ² → ℝ² which maps the unit disk onto itself preserves the space EXP() of exponentially integrable functions over , in the sense that u ∈ EXP() if and only if $u ∘ f^\{-1\} ∈ EXP()$. Moreover, if f is assumed to be conformal outside the unit disk and principal, we provide the estimate $1/(1 + K logK) ≤ (||u ∘ f^\{-1\}||_\{EXP()\})/(||u||_\{EXP()\}) ≤ 1 + K log K$ for every u ∈ EXP(). Similarly, we consider the distance from $L^\{∞\}$ in EXP and we prove that if f: Ω → Ω’ is a K-quasiconformal mapping and G ⊂ ⊂ Ω, then $1/K ≤ (dist_\{EXP(f(G))\}(u ∘ f^\{-1\},L^\{∞\}(f(G))))/(dist_\{EXP(f(G))\}(u,L^\{∞\}(G))) ≤ K$ for every u ∈ EXP(). We also prove that the last estimate is sharp, in the sense that there exist a quasiconformal mapping f: → , a domain G ⊂ ⊂ and a function u ∈ EXP(G) such that $dist_\{EXP(f(G))\}(u ∘ f^\{-1\},L^\{∞\}(f(G))) = K dist_\{EXP(f(G))\}(u,L^\{∞\}(G))$.},
author = {Fernando Farroni, Raffaella Giova},
journal = {Studia Mathematica},
keywords = {quasiconformal mapping; exponentially integrable function},
language = {eng},
number = {2},
pages = {195-203},
title = {Quasiconformal mappings and exponentially integrable functions},
url = {http://eudml.org/doc/285676},
volume = {203},
year = {2011},
}

TY - JOUR
AU - Fernando Farroni
AU - Raffaella Giova
TI - Quasiconformal mappings and exponentially integrable functions
JO - Studia Mathematica
PY - 2011
VL - 203
IS - 2
SP - 195
EP - 203
AB - We prove that a K-quasiconformal mapping f:ℝ² → ℝ² which maps the unit disk onto itself preserves the space EXP() of exponentially integrable functions over , in the sense that u ∈ EXP() if and only if $u ∘ f^{-1} ∈ EXP()$. Moreover, if f is assumed to be conformal outside the unit disk and principal, we provide the estimate $1/(1 + K logK) ≤ (||u ∘ f^{-1}||_{EXP()})/(||u||_{EXP()}) ≤ 1 + K log K$ for every u ∈ EXP(). Similarly, we consider the distance from $L^{∞}$ in EXP and we prove that if f: Ω → Ω’ is a K-quasiconformal mapping and G ⊂ ⊂ Ω, then $1/K ≤ (dist_{EXP(f(G))}(u ∘ f^{-1},L^{∞}(f(G))))/(dist_{EXP(f(G))}(u,L^{∞}(G))) ≤ K$ for every u ∈ EXP(). We also prove that the last estimate is sharp, in the sense that there exist a quasiconformal mapping f: → , a domain G ⊂ ⊂ and a function u ∈ EXP(G) such that $dist_{EXP(f(G))}(u ∘ f^{-1},L^{∞}(f(G))) = K dist_{EXP(f(G))}(u,L^{∞}(G))$.
LA - eng
KW - quasiconformal mapping; exponentially integrable function
UR - http://eudml.org/doc/285676
ER -