Vietnam J. Math. DOI 10.1007/s10013-014-0090-2 Some Extensions of the Kolmogorov–Stein Inequality Ha Huy Bang · Vu Nhat Huy Received: 2 September 2013 / Accepted: 14 March 2014 © Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2014 Abstract In this paper, we prove some extensions of the Kolmogorov–Stein inequality for derivatives in L p (R) norm to differential operators generated by a polynomial. Keywords L p spaces · Orlicz spaces · Kolmogorov inequality Mathematics Subject Classification (2010) 26A24 · 41A17 1 Introduction Let 1 ≤ p ≤∞, n ≥ 2, and 0 <k<n. In the space of all functions, f ∈ C n (R) such that f, f ′ ,...,f (n) belong to L p (R), consider the following inequality: ‖f (k) ‖ p ≤ A‖f ‖ p + B‖f (n) ‖ p . It is well known that this inequality is equivalent to the inequality (see [7]) ‖f (k) ‖ p ≤ A 1−k/n B k/n (‖f ‖ p +‖f (n) ‖ p ) and also equivalent to ‖f (k) ‖ p ≤ C n,k ‖f ‖ 1−k/n p ‖f (n) ‖ k/n p , where C n,k =  A 1 − (k/n)  1−(k/n)  B k/n  k/n . H. H. Bang () Institute of Mathematics, Vietnamese Academy of Science and Technology, 18 Hoang Quoc Viet Street, Cau Giay, Hanoi, Vietnam e-mail: hhbang@math.ac.vn V. N. Huy Department of Mathematics, College of Science, Vietnam National University, 334 Nguyen Trai Street, Thanh Xuan, Hanoi, Vietnam e-mail: nhat huy85@yahoo.com