L-Regularity of Markov Sets and of m-Perfect Sets in the Complex Plane

Let V E be the pluricomplex Green function associated with a compact subset E of C N . The well-known Hölder continuity property of E means that there existThe main result of this paper says that this condition is equivalent to a Vladimir Markov-type inequality; i.e., D α P E ≤ M |α| (deg P ) m|α| (|α|!) 1−m P E , where m, M > 0 are independent of the polynomial P of N variables. We give some applications of this equivalence, e.g., for convex bodies in R N , for uniformly polynomially cuspidal sets and for some disconnect compact sets.

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“…and (8) is proved. We now proceed to show implication (9). For this purpose, observe that from (3), it is sufficient to prove…”

Section: Proofs Of the Main Resultsmentioning

confidence: 91%

“…In the case of r j < |α| Hence and from inequalities (18,19) we conclude that E ∈ V MI(2+µ, For the univariate case the answer to the second question is partially known because if E ⊂ R then it is L-regular (see [9]).…”

Section: Formula (3) Leads Us Tomentioning

confidence: 98%

Hölder Continuity of the Green Function and Markov Brothers’ Inequality

Baran

Białas-Cież

2013

Constr Approx

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“…Here we give an asymptotic for M n (K (α) ) which is new compared to the previous results about Markov's constants of Cantor-type sets (see [10,Example 7], [16], and [7]). …”

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confidence: 72%

On Smoothness of the Green Function for the Complement of a Rarefied Cantor-Type Set

Altun

Goncharov

2010

Constr Approx

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“…It should be noted though that Lemma 3.3 assumes L-regularity, which is guaranteed by HCP, but it is still not known whether all compact subsets of the complex plane admitting GMI are L-regular. In the real case this follows from the combination of [7] and [4].…”

Section: Proposition 32 For Any Compact Setmentioning

confidence: 99%

Jackson’s inequality in the complex plane and the Łojasiewicz–Siciak inequality of Green’s function

Białas-Cież

Eggink

2016

Journal of Approximation Theory

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Abstract. We prove a generalization of Dunham Jackson's famous approximation inequality to the case of compact sets in the complex plane admitting both upper and lower bounds for their Green's functions, i.e. the well known Hölder Continuity Property (HCP) and the less known but crucial Lojasiewicz-Siciak inequality ( LS). Moreover, we show that ( LS) is a necessary condition for our Jackson type inequality.

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L-Regularity of Markov Sets and of m-Perfect Sets in the Complex Plane

Cited by 8 publications

References 12 publications

Hölder Continuity of the Green Function and Markov Brothers’ Inequality

Hölder Continuity of the Green Function and Markov Brothers’ Inequality

On Smoothness of the Green Function for the Complement of a Rarefied Cantor-Type Set

Jackson’s inequality in the complex plane and the Łojasiewicz–Siciak inequality of Green’s function

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