Smoothness of the Green function for a special domain
Annales Polonici Mathematici
We consider a compact set K ⊂ R in the form of the union of a sequence of segments. By means of nearly Chebyshev polynomials for K, the modulus of continuity of the Green functions g C\K is estimated. Markov's constants of the corresponding set are evaluated. 2010 Mathematics Subject Classification: Primary 31A15; Secondary 41A10, 41A17. respect to a regular (in the sense of [ST]) measure. Following [G2], we construct a sequence of "nearly Chebyshev" polynomials for K and find the exact (up to
... d the exact (up to a constant) value of the modulus of continuity of g C\K . It should be noted that the general bound by V. Totik [T, Th. 2.2] of the Green functions, which is highly convenient to characterize optimal (that is, Lip 1/2) smoothness of g C\K for K ⊂ [0, 1], cannot be applied to our case. See Section 6 for more details. There are several applications of smoothness properties of the Green functions to solving different problems in analysis (see e.g. [T]). We are interested in applications to polynomial inequalities. In Section 7 we evaluate the Markov factors for our set K.