Markov-type inequalities on certain irrational arcs and domains

AbstractLet Pnd 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∈Pnd,∥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 MnT(Γ):=max{∥DTP∥Γ:P∈Pnd,∥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⩽MnT(Γα)⩽Bnfor every sufficiently large n.Our second result presents some new bounds for Mn(Ωα), where Ωα:=(x,y)∈R2:0⩽x⩽1;12xα⩽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(Ωα)⩽nclogn