Markov-type inequalities on certain irrational arcs and domains

Let Pdn denote the set of real algebraic polynomials of d variables and of total degree at most n. For a compact set K ⊂ Rd set ||P||K = supx ∈ K|P(x)|. Then the Markov factors on K are defined by Mn(K): = max{||DωP||K : P ∈ Pdn, ||P||K ≤ 1, ω ∈ Sd-1}. (Here, as usual, Sd-1 stands for the Euclidean unit sphere in Rd) Furthermore, given a smooth curve Γ ⊂ Rd, we denote by DTP the tangential derivative of P along Γ (T is the unit tangent to Γ). Correspondingly, consider the tangential Markov factor of Γ given by MTn (Γ) : = max{||DTP||Γ : P ∈ Pdn ||P||Γ ≤ 1}. Let Γα: ={(x, xα) : 0 ≤ x ≤ 1}. We prove that for every irrational number α > 0 there are constants A, B > 1 depending only on α such that An ≤ MTn(Γα ≤ Bn for every sufficiently large n. Our second result presents some new bounds for Mn(Ωα), where Ωα: = {(x,y) ∈ R2 : 0 ≤ x ≤ 1; ½αx ≤ y ≤ 2xα} (d = 2, α > 1). We show that for every α > 1 there exists a constant c > 0 depending only on α such that Mn (Ωx ≤ nclog n.