On Sharp Uniform Estimates of Intermediate Even-Order Derivatives in Sobolev Spaces

For functions from the Sobolev space Wⁿ_∞[0,1] and an arbitrary point a∈(0,1) , we study sharp estimating functions An,k,∞(a) in the inequality |f⁽ᵏ⁾(a)|≤ An,k,∞(a) · \|f⁽ⁿ⁾\|L_∞ [0,1] , 0 ≤ k < n . We establish a link between the functions An,k,∞ and best approximations of special splines by polynomials in L₁[0,1] . We introduce the Markov set An,k , 0 ≤ k < n , of values of the parameter a , on which we obtain the representation An, k, ∞(a)= 2⁻⁽ⁿ⁻ᵏ⁾|Vₙ⁽ᵏ⁾(2a-1)| of the function An,k,∞ in terms of the absolute value of the k th derivative of the Peano kernel Vₙ of order n . For arbitrary n and k , 0 ≤ k ≤ n-2 , we show that the embedding constant Λn,k,∞ of the Sobolev spaces Wⁿ_∞[0,1] Wᵏ_∞[0,1] is the maximum value of the function 2⁻⁽ⁿ⁻ᵏ⁾|Vₙ⁽ᵏ⁾(2a-1)| on the interval $$[0,1]$$ . For odd n and even k , 0 ≤ k < n , the embedding constant Λn,k,∞ is refined as Λn,k,∞=2⁻⁽ⁿ⁻ᵏ⁾|Vₙ⁽ᵏ⁾(0)| . We also express the embedding constants Λn,k,∞ for odd n and even k in terms of hypergeometric functions and study the asymptotic behavior of the embedding constants as n=2m+1 → ∞ with fixed k or $$n-k$$ .