Markov-type inequalities on certain irrational arcs and domains

Abstract

Let P n d denote the set of real algebraic polynomials of d variables and of total degree at most n. For a compact set K ⊂ R d set ∥ P ∥ K = sup x ∈ K | P ( x ) | . Then the Markov factors on K are defined by M n ( K ) := max { ∥ D ω P ∥ K : P ∈ P n d , ∥ P ∥ K ⩽ 1 , ω ∈ S d - 1 } . (Here, as usual, S d - 1 stands for the Euclidean unit sphere in R d .) Furthermore, given a smooth curve Γ ⊂ R d , we denote by D T P the tangential derivative of P along Γ ( T is the unit tangent to Γ ). Correspondingly, consider the tangential Markov factor of Γ given by M n T ( Γ ) := max { ∥ D T P ∥ Γ : P ∈ P n d , ∥ P ∥ Γ ⩽ 1 } . Let Γ α := { ( x , x α ) : 0 ⩽ x ⩽ 1 } . We prove that for every irrational number α > 0 there are constants A , B > 1 depending only on α such that A n ⩽ M n T ( Γ α ) ⩽ B n for every sufficiently large n. Our second result presents some new bounds for M n ( Ω α ) , where Ω α := ( x , y ) ∈ R 2 : 0 ⩽ x ⩽ 1 ; 1 2 x α ⩽ y ⩽ 2 x α ( d = 2 , α > 1 ). We show that for every α > 1 there exists a constant c > 0 depending only on α such that M n ( Ω α ) ⩽ n c log n .