Kolmogorov-type inequalities for mixed derivatives of functions of many variables
Let γ = (γ_1 ,..., γ_d ) be a vector with positive components and let D^γ be the corresponding mixed derivative (of order γ_j with respect to the j th variable). In the case where d > 1 and 0 < k < r are arbitrary, we prove that _x L^rγ_∞(T^d)D^rγx eq0 ||D^kγx||_L_∞(T^d) ||x||^1-k/r||D^rγ||^k/r_L_∞(T^d) = ∞ and ||D^kγx||_L_∞(T^d) ≤ K||x||^1 - k/r_L_∞(T^d)||D^rγx||_L_∞(T^d)^k/r (1 + ^+||D^rγx||_L_∞(T^d) ||x||_L_∞ (T^d) )^β for all x L^rγ_∞(T^d) Moreover, if \(β \) is the least possible value of the exponent β in this inequality, then ( d - 1 )( 1 - kr ) β ( d,γ ,k,r ) d - 1. , number=5, journal=Ukrains’kyi Matematychnyi Zhurnal, author=Babenko, V. F. and Korneichuk, N. P. and Pichugov, S. A., year=2004, month=May, pages=579–594