Hölder Continuity of the Green Function and Markov Brothers’ Inequality

Advances in Mathematics

2015

Section: Ifmentioning

confidence: 97%

“…In [54], Pawłucki and Pleśniak introduced the notion of UPC condition (see also [7,8,18,19,55,56,58]). Definition 3.1.…”

Section: Spp Property Versus Upc Conditionmentioning

confidence: 99%

Remez-type inequality on sets with cusps

Advances in Mathematics

2015

“…Rafal.Pierzchala@im.uj.edu.pl 1 Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Łojasiewicza 6, 30-348 Kraków, Poland is called the pluricomplex Green function of K (with pole at infinity) or the Siciak-Zakharyuta extremal function; see for example [4,23,25,39,43,44] and the bibliography therein.…”

Section: B Rafał Pierzchałamentioning

confidence: 99%

Geometry of holomorphic mappings and Hölder continuity of the pluricomplex Green function

2020

Math. Ann.

We provide a solution to a long-standing open problem that lives in the interface of pluripotential theory and multivariate approximation theory. The problem is to characterize the holomorphic maps which preserve Hölder continuity of the pluricomplex Green function associated with a compact subset of C N . We also prove, under mild restrictions, that nondegenerate holomorphic maps preserve Markov's inequality for polynomials.Geometry of holomorphic mappings... Definition 1.2 (see [29]) We say that a compact set ∅ = K ⊂ C N has the HCP property if there exist , μ > 0 such that, for each z ∈ K (1) ,(1.4)In the above definition, and subsequently, we use the following notation: for each set ∅ = A ⊂ C N and each r > 0, we put A (r ) := {z ∈ C N : dist(z, A) ≤ r } .

“…There is now such extensive literature on Markov type inequalities that it is beyond the scope of this paper to give a complete bibliography. Let us mention only certain works which are most closely related to our paper (with emphasis on those dealing with generalizations of Markov's inequality on sets admitting cusps), for example [1][2][3][4][5][6][7]35,36,39,43,44,46,48,50,51,57,58]. We should stress here that the present paper owes a great debt particularly to Pawłucki and Pleśniak's work, because in [43] they laid the foundations for the theory of polynomial inequalities on "tame" (for example, semialgebraic) sets with cusps.…”

Section: Theorem 11 (Markov) If P Is a Polynomial Of One Variable Thenmentioning

confidence: 99%

“…Definition 1. 2 We say that a compact set ∅ = E ⊂ C N satisfies Markov's inequality (or: is a Markov set) if there exist ε, C > 0 such that, for each polynomial P ∈ C[z 1 , . .…”

Section: Theorem 11 (Markov) If P Is a Polynomial Of One Variable Thenmentioning

confidence: 99%

Markov’s inequality and polynomial mappings

2015

Math. Ann.

Markov's inequality is a certain estimate for the norm of the derivative of a polynomial in terms of the degree and the norm of this polynomial. It has many interesting applications in approximation theory, constructive function theory and in analysis (for instance, to Sobolev inequalities or Whitney-type extension problems). One of the purposes of this paper is to give a solution to an old problem, studied among others by Baran and Pleśniak, and concerning the invariance of Markov's inequality under polynomial mappings (polynomial images). We also address the issue of preserving Markov's inequality when taking polynomial preimages. Lastly, we give a sufficient condition for a subset of a Markov set to be a Markov set.