On Geometric Properties of Functions With Generalized First Derivatives

Vodop’yanov, S. K.; Gol’dshtein, Vladimir; Reshetnyak, Yu. G.

doi:10.1070/rm1979v034n01abeh002871

Cited by 68 publications

(67 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…By Theorem 3.1 q = q ′ /(q ′ − 1) and q ≤ p. By elementary calculations [30]. Hence, using Theorem 2.2 we obtain…”

Section: Composition Operators and Brennan's Conjecturementioning

confidence: 77%

“…Then the function g = f • ϕ −1 is defined almost everywhere in D and belongs to the Sobolev space L 1 q (D) [30]. Hence, by the Sobolev embedding theorem g = f • ϕ −1 ∈ W 1,q (D) [24] and the classical Poincaré-Sobolev inequality,…”

Section: Poincaré-sobolev Inequalitiesmentioning

confidence: 99%

“…The property of the quasiconformal α-regularity implies the integrability of a Jacobian of quasiconformal mappings and therefore for any quasiconformal α-regular domain we have the embedding of weighted Lebesgue spaces L r (Ω, h) into nonweighted Lebesgue spaces L s (Ω) for s = α−2 α r. α r. Then using the change of variable formula for quasiconformal mappings [30], Hölder's inequality with exponents (r, rs/(r − s)) and equality…”

Section: Poincaré-sobolev Inequalitiesmentioning

confidence: 99%

“…Using the change of variable formula for quasiconformal mappings [30], the classical Poincaré-Sobolev inequality for the unit disc…”

Section: Poincaré-sobolev Inequalitiesmentioning

confidence: 99%

See 3 more Smart Citations

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

Gol’dshtein

Pchelintsev

Ukhlov

2017

Boll Unione Mat Ital

View full text Add to dashboard Cite

Abstract. In this paper we obtain estimates for the first nontrivial eigenvalue of the p-Laplace Neumann operator in bounded simply connected planar domains Ω ⊂ R 2 . This study is based on a quasiconformal version of the universal weighted Poincaré-Sobolev inequalities obtained in our previous papers for conformal weights. The suggested weights in the present paper are Jacobians of quasiconformal mappings. The main technical tool is the theory of composition operators in relation with the Brennan's Conjecture for (quasi)conformal mappings.

show abstract

“…By Theorem 3.1 q = q ′ /(q ′ − 1) and q ≤ p. By elementary calculations [30]. Hence, using Theorem 2.2 we obtain…”

Section: Composition Operators and Brennan's Conjecturementioning

confidence: 77%

Section: Poincaré-sobolev Inequalitiesmentioning

confidence: 99%

Section: Poincaré-sobolev Inequalitiesmentioning

confidence: 99%

“…Using the change of variable formula for quasiconformal mappings [30], the classical Poincaré-Sobolev inequality for the unit disc…”

Section: Poincaré-sobolev Inequalitiesmentioning

confidence: 99%

See 2 more Smart Citations

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

Gol’dshtein

Pchelintsev

Ukhlov

2017

Boll Unione Mat Ital

View full text Add to dashboard Cite

show abstract

“…G. Reshetnyak and his school (S. K. Vodop yanov, V. M. Goldstein and others), became classics of the mapping theory long ago, see, e.g., the monographs [146,27] and [60], and also [28], as well as the relatively recent papers [25,26]. G. Reshetnyak and his school (S. K. Vodop yanov, V. M. Goldstein and others), became classics of the mapping theory long ago, see, e.g., the monographs [146,27] and [60], and also [28], as well as the relatively recent papers [25,26].…”

Section: §1 Introductionmentioning

confidence: 99%

Toward the theory of Orlicz–Sobolev classes

Kovtonyuk

Ryazanov

Salimov

et al. 2014

St. Petersburg Math. J.

View full text Add to dashboard Cite

It is shown that, under a Calderón type condition on the function ϕ, the continuous open mappings that belong to the Orlicz-Sobolev classes W 1,ϕ loc have total differential almost everywhere; this generalizes the well-known theorems of Gehring-Lehto-Menchoff in the case of R 2 and of Väisälä in R n , n ≥ 3. Appropriate examples show that the Calderón type condition is not only sufficient but also necessary. Moreover, under the same condition on ϕ, it is also proved that the continuous mappings of class W 1,ϕ loc and, in particular, of class W 1,p loc for p > n−1 have Lusin's (N )-property on a.e. hyperplane. On that basis, it is shown that, under the same condition on ϕ, the homeomorphisms f with finite distortion of class W 1,ϕ loc and, in particular, those belonging to W 1,p loc for p > n−1, are what is called lower Q-homeomorphisms, where Q is equal to their outer dilatation K f ; also, they are so-called ring Q * -homeomorphisms with Q * = K n−1 f . The latter fact makes it possible to fully apply the theory of the boundary and local behavior of the ring and lower Q-homeomorphisms, as developed earlier by the authors, to the study of mappings in the Orlicz-Sobolev classes. Part 1. Differentiability and behavior of Hausdorff 's measures in the Orlicz-Sobolev classes2010 Mathematics Subject Classification. Primary 46E35. Key words and phrases. Moduli of families of curves and surfaces, mappings with bounded and finite distortion, differentiability, Lusin and Sard properties, Sobolev classes, Orlicz-Sobolev classes, boundary and local behavior.

show abstract

Whitney's Problem on Extendability of Functions and an Intrinsic Metric

Zobin

1998

Advances in Mathematics

View full text Add to dashboard Cite

We consider the space of functions with bounded (k+1) th derivatives in a general domain in R n . Is every such function extendible to a function of the same class defined on the whole R n ? H. Whitney showed that the equivalence of the intrinsic ( =geodesic) metric in this domain to the Euclidean one is sufficient for such extendability. There was an old conjecture (going back to H. Whitney) that this equivalence is also necessary for extendability. We disprove this conjecture and construct examples of domains in R 2 such that the above extendability holds but the analogous property for smaller k fails. Our study is based on a duality approach. 1998Academic Press

show abstract

On Geometric Properties of Functions With Generalized First Derivatives

Cited by 68 publications

References 27 publications

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

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

Toward the theory of Orlicz–Sobolev classes

Whitney's Problem on Extendability of Functions and an Intrinsic Metric

Contact Info

Product

Resources

About