Regularity of quasi-linear equations in the Heisenberg group

Capogna, Luca

doi:10.1002/(sici)1097-0312(199709)50:9<867::aid-cpa3>3.0.co;2-3

Cited by 110 publications

(129 citation statements)

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“…For further information on quasiconformal mappings on Carnot groups, see [48], [36], [34], [15], [16], [5] and [4], in addition to the references already mentioned.…”

Section: Examples Of Ahlfors Regular Loewner Spacesmentioning

confidence: 99%

Metric and geometric quasiconformality in Ahlfors regular Loewner spaces

Tyson

2001

Conform. Geom. Dyn.

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Abstract. Recent developments in geometry have highlighted the need for abstract formulations of the classical theory of quasiconformal mappings. We modify Pansu's generalized modulus to study quasiconformal geometry in spaces with metric and measure-theoretic properties sufficiently similar to Euclidean space. Our basic objects of study are locally compact metric spaces equipped with a Borel measure which is Ahlfors-David regular of dimension Q > 1, and satisfies the Loewner condition of Heinonen-Koskela. For homeomorphisms between open sets in two such spaces, we prove the equivalence of three conditions: a version of metric quasiconformality, local quasisymmetry and geometric quasiconformality.We derive from these results several corollaries. First, we show that the Loewner condition is a quasisymmetric invariant in locally compact Ahlfors regular spaces. Next, we show that a proper Q-regular Loewner space, Q > 1, is not quasiconformally equivalent to any subdomain. (In the Euclidean case, this result is due to Loewner.) Finally, we characterize products of snowflake curves up to quasisymmetric/bi-Lipschitz equivalence: two such products are bi-Lipschitz equivalent if and only if they are isometric and are quasisymmetrically equivalent if and only if they are conformally equivalent.

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“…For further information on quasiconformal mappings on Carnot groups, see [48], [36], [34], [15], [16], [5] and [4], in addition to the references already mentioned.…”

Section: Examples Of Ahlfors Regular Loewner Spacesmentioning

confidence: 99%

Metric and geometric quasiconformality in Ahlfors regular Loewner spaces

Tyson

2001

Conform. Geom. Dyn.

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“…[4,Proposition 2.3]. A more H n -specific feature of difference quotients is the way they connect the horizontal and vertical directions: As the next lemma shows, a horizontal derivative is comparable to "half" a vertical one.…”

Section: 1mentioning

confidence: 99%

“…In the Heisenberg group setting, up to now the conditions (1)(2)(3)(4) and (1)(2)(3)(4)(5) have been considered with q = p, i.e. in the standard p-growth case.…”

Section: Introductionmentioning

confidence: 99%

“…Hölder regularity results for the gradient of solutions of quasi-linear equations were given by Capogna [4,5], using a technique involving fractional pseudo-differential operators; this was also generalized to partial regularity for nonlinear systems by Capogna and Garofalo [7] (see also [17]). Regularity results focusing on the vertical derivative were developed by Marchi [39,40] and Domokos [8]: In particular, T u ∈ L p loc was established for solutions of equations with p-growth, under the condition that 2 ≤ p < 4, employing an iterative method based on fractional difference quotients.…”

Section: Introductionmentioning

confidence: 99%

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Regularity results for minimizers of (2, q)-growth functionals in the Heisenberg Group

Föglein

2010

manuscripta math.

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“…There are also proofs of Hölder continuity for some Carnot groups, see e.g. [2,3]. (5) Using the results and techniques of [30], it should be possible to prove continuity of extremal functions for a wide class of axiomatic Sobolev space (perhaps assuming that D is local and µ is doubling) …”

Section: The Case Of Condensersmentioning

confidence: 99%