Exact constants in generalized inequalities for intermediate derivatives
Anton A Lunev, Леонид Леонидович Оридорога · Mathematical Notes · 2009
Consider the Sobolev space W 2 (ℝ+) on the semiaxis with norm of general form defined by a quadratic polynomial in derivatives with nonnegative coefficients. We study the problem of exact constants A n,k in inequalities of Kolmogorov type for the values of intermediate derivatives |f (k)(0)| ≤ A n,k ‖f‖. In the general case, the expression for the constants A n,k is obtained as the ratio of two determinants. Using a general formula, we obtain an explicit expression for the constants A n,k in the case of the following norms: $$ \left\| f \right\|_1^2 = \left\| f \right\|_{L_2 }^2 + \left\| {f^{(n)} } \right\|_{L_2 }^2 and\left\| f \right\|_2^2 = \sum\limits_{l = 0}^n {\left\| {f^{(l)} } \right\|_{L_2 }^2 } . $$ In the case of the norm ‖ · ‖1, formulas for the constants A n,k were obtained earlier by another method due to Kalyabin. The asymptotic behavior of the constants A n,k is also studied in the case of the norm ‖ · ‖2. In addition, we prove a symmetry property of the constants A n,k in the general case.