Mappings of finite distortion: Reverse inequalities for the Jacobian

Hencl, Stanislav; Koskela, Pekka; Zhong, Xiao

doi:10.1007/bf02930724

Cited by 11 publications

(8 citation statements)

References 30 publications

(17 reference statements)

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“…Proof By Martio , there exists

r > 1

for which

J_{f} \in L_{loc}^{r} false(M false)

; see also Meyers–Elcrat . In addition to this, there exists

ε > 0

for which

J_{f}^{- ε} \in L_{loc}^{1} false(M false)

; see for example, Hencl–Koskela–Zhong for a far‐reaching discussion. Let

U

be a normal neighborhood of

f

x

, that is, a precompact neighborhood

U

x

for which

\partial f U = f (\partial U)

.…”

Section: Conformal Cohomologymentioning

confidence: 99%

Uniform cohomological expansion of uniformly quasiregular mappings

Kangasniemi

Pankka

2018

Proc. London Math. Soc.

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Let f : M → M be a uniformly quasiregular self-map of a compact, connected, and oriented Riemannian n-manifold M without boundary, n 2. We show that, for k ∈ {0, . . . , n}, the induced homomorphism f * :is the kth singular cohomology of M , is complex diagonalizable and the eigenvalues of f * have absolute value (deg f ) k/n . As an application, we obtain a degree restriction for uniformly quasiregular self-maps of closed manifolds. In the proof of the main theorem, we use a Sobolev-de Rham cohomology based on conformally invariant differential forms and an induced push-forward operator.

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“…Proof By Martio , there exists

r > 1

for which

J_{f} \in L_{loc}^{r} false(M false)

; see also Meyers–Elcrat . In addition to this, there exists

ε > 0

for which

J_{f}^{- ε} \in L_{loc}^{1} false(M false)

; see for example, Hencl–Koskela–Zhong for a far‐reaching discussion. Let

U

be a normal neighborhood of

f

x

, that is, a precompact neighborhood

U

x

for which

\partial f U = f (\partial U)

.…”

Section: Conformal Cohomologymentioning

confidence: 99%

Uniform cohomological expansion of uniformly quasiregular mappings

Kangasniemi

Pankka

2018

Proc. London Math. Soc.

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“…This is a key fact in order to study the regularity of the Jacobian of quasiconformal mappings (see [13]). More generally, we refer for instance to [14,17] for the study of the self-improving property and the regularity of the Jacobian of a mapping of finite distortion.…”

Section: A P and G Q Classes We Recall Here The Definition Of The Mumentioning

confidence: 99%

Change of variables for A_\infty weights by means of quasiconformal mappings

Farroni¹,

Giova²

2013

Ann. Acad. Sci. Fenn. Math.

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Abstract. Let f : R n → R n be a quasiconformal mapping whose Jacobian is denoted by J f and let A ∞ be the Muckenhoupt class of weights w satisfyingfor every ball B ⊂ R n and for some positive constant A ≥ 1 independent of B. We consider two characteristic constantsÃ ∞ (w) andG 1 (w) which are simultaneously finite for every w ∈ A ∞ . We study the behaviour of theÃ ∞ -constant under the operator already considered by Johnson and Neugebauer [18] and establish the equivalence of the two constantsG. Our quantitative estimates are sharp.

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“…For detailed expositions, the interested reader may consult [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and the references therein.…”

Section: Introductionmentioning

confidence: 99%