Some Extensions of the Kolmogorov–Stein Inequality

Vietnam J Math DOI 10.1007/s10013-014-0090-2 Some Extensions of the Kolmogorov–Stein Inequality Ha Huy Bang · Vu Nhat Huy Received: 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 Lp (R) norm to differential operators generated by a polynomial Keywords Lp spaces · Orlicz spaces · Kolmogorov inequality Mathematics Subject Classification (2010) 26A24 · 41A17 Introduction Let ≤ p ≤ ∞, n ≥ 2, and < k < n In the space of all functions, f ∈ C n (R) such that f, f , , f (n) belong to Lp (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 ≤ A1−k/n B k/n ( f p + f (n) p) and also equivalent to f (k) p ≤ Cn,k f where Cn,k = A − (k/n) 1−k/n p 1−(k/n) f (n) k/n p , 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 H H Bang, V N Huy (1−k/n) For p = ∞, Kolmogorov proved in [15] that Cn,k = Kn−k /Kn Kj = π ∞ , where (−1)k /(2k + 1)j +1 k=0 for even j , while Kj = π ∞ 1/(2k + 1)j +1 k=0 for odd j Moreover the constants {Cn,k : n, k ∈ N, k < n} are sharp in the sense that these cannot be replaced by smaller ones This result of Kolmogorov has been extended by Stein to Lp (R) norm [20] and in [1] to any Orlicz norm The Kolmogorov–Stein inequality and its modifications are a problem of interest for many mathematicians and have various applications (see, for example, [1, 3, 5–12, 14, 20–22]) In this paper, we prove some extensions of the Kolmogorov– Stein inequality for derivatives in Lp (R) norm to differential operators generated by a polynomial Inequalities for Lp Spaces Denote by C k (R) the set of all functions f that are continuous in R together with all derivatives f (n) , n ≤ k, and Lp (R) (1 ≤ p ≤ ∞) the collection of all functions f specified on R for which the norm f p = [ R |f (x)| p dx]1/p , ess sup |f (x)|, ≤ p < ∞, p=∞ is finite The convolution f ∗ g of two functions f, g ∈ L1 (R) is defined as (f ∗ g)(x) = R f (x − y)g(y)dy (x ∈ R) (1) If f ∈ Lp (R), ≤ p ≤ ∞, and g ∈ L1 (R), then the integral (1) converges for almost all x ∈ R, and we have the following result Young inequality (see, e.g., [2]) Let ≤ p ≤ ∞, f ∈ Lp (R), g ∈ L1 (R) Then f ∗ g ∈ Lp (R) and f ∗ g p ≤ f p g Furthermore, S (R) stands for the Schwartz space on R and S (R) for the dual space of tempered distributions on R Recall that Lp (R) ⊂ S (R) for any ≤ p ≤ ∞ Let f ∈ L1 (R) and fˆ = F f be its Fourier transform fˆ(ξ ) = √ 2π +∞ −∞ e−ixξ f (x)dx The Fourier transform of a tempered generalized function f is defined via the formula F f, ϕ = f, F ϕ , ϕ ∈ S (R) n k Let n ∈ N and P (x) be a polynomial of degree n: P (x) := k=0 ak x We put n k k k (k) P1 (x) := and then the differential operator P (D) is k=0 |ak |x , D f = (−i) f Some Extensions of the Kolmogorov–Stein Inequality obtained from P (x) by substituting x → −i∂/∂x We define the function ρ ∈ C ∞ (R) as follows Ce |x|2 −1 if |x| < 1, if |x| ≥ 1, ρ(x) := where the constant C satisfies R ρ(x)dx = 1, and put (x) = F (1[−3/2,3/2] ∗ ρ1/4 )(x) ρ1/4 (x) = 4ρ(4x), Clearly, ∈ S (R) and ˆ (x) = ∀x ∈ [−1, 1], ˆ (x) = ∀x ∈ / (−2, 2) Put λ := We have the following theorem Theorem Let m, n ∈ N and f ∈ C m+n (R), ≤ p ≤ ∞, P (x) be a polynomial of degree n Assume f, D m (P (D)f ) ∈ Lp (R) Then P (D)f ∈ Lp (R) and P (D)f p ≤ λ(Km 2m + 1)P1 (2 ) f p + Km −m D m (P (D)f ) p ∀ > 0, where Km is the Favard constant To obtain Theorem 1, we need the following results Bohr-Favard inequality (see, e.g., [2, 4, 12]) Let ≤ p ≤ ∞, σ > 0, f ∈ C n (R) and supp fˆ ∩ (−σ, σ ) = ∅ Assume that f (n) ∈ Lp (R) Then f ∈ Lp (R) and f ≤ σ −n Kn f (n) p p, where the Favard constants Kn are best possible and have the following properties π < · · · < K3 ≤ K = π = K0 ≤ K2 < · · · < Bernstein inequality (see, e.g., [13, 18]) Assume ≤ p ≤ ∞, σ > 0, f ∈ Lp (R) and suppfˆ ⊂ [−σ, σ ] Then f (n) p ≤ σn f n = 1, 2, p, The constants are best possible Proof of Theorem for > we define (x) := ( x) x x Then, / = = λ and ˆ (x) = ˆ ( ) = ∀x ∈ [− , ], ˆ (x) = ˆ ( ) = ∀x ∈ (−2 , ) Therefore, supp ˆ ⊂ [−2 , ] Hence, by applying Bernstein inequality, we have k k k = 0, 1, , n (2) Dk ≤ (2 ) = λ(2 ) , We define the functions g, h as follows g=f ∗ h = f − g , Clearly, supp gˆ ⊂ supp ˆ ⊂ [−2 , ], supp hˆ ⊂ (−∞, − ] ∪ [ , +∞), f = g + h and D k g = f ∗ (D k ) It follows from Young inequality that g p ≤ f p =λ f p and then h p ≤ f p + g p ≤ (λ + 1) f p H H Bang, V N Huy Using Young inequality and D k g = f ∗ (D k Dk g ≤ f p p ), we obtain D k g ∈ Lp (R) and Dk k = 0, 1, , n 1, (3) From (2)–(3), we have the following estimate Dk g p ≤ λ(2 )k f k = 0, 1, , n p, (4) From (4), we get P (D)g ∈ Lp (R) and P (D)g p ≤ λP1 (2 ) f (5) p Similarly, we have D m (P (D)g) p ≤ (2 )m P (D)g p ≤ λ(2 )m P1 (2 ) f D m (P (D)h) p ≤ D m (P (D)g) p + D m (P (D)f ) p Therefore, ≤ λ(2 )m P1 (2 ) f p + D m (P (D)f ) p (6) p By Bohr–Favard inequality and P (D)h ⊂ (−∞, − ] ∪ [ , +∞), we obtain P (D)h −m ≤ Km p D m (P (D)h) p Then it follows from (6) that P (D)h p ≤ Km −m [λ(2 )m P1 (2 ) f p + D m (P (D)f ) p ] (7) From (5), (7) and f = g + h, we have P (D)f ∈ Lp (R) and P (D)f p ≤ P (D)g p + P (D)h (8) p By (5), (7) and (8), we obtain P (D)f p ≤ λ(Km 2m + 1)P1 (2 ) f p + Km −m D m (P (D)f ∀ > p The proof is complete By Theorem we obtain the following results: Corollary Let m, n ∈ N and f ∈ C m+n (R), ≤ p ≤ ∞, P (x) be a polynomial of degree n Assume f, D m (P (D)f ) ∈ Lp (R) Then P (D)f ∈ Lp (R) and π −m m D (P (D)f ) p ∀ > P (D)f p ≤ λ(π 2m−1 + 1)P1 (2 ) f p + Corollary Let m, n ∈ N Then, there exists a constant C < ∞ independent of f such that f + Dn f ≤C p f p + m+n f m p D m f + D m+n f p + π D m f + D m+n f m n p Proof Put H ( ) = λ(π 2m−1 + 1)(1 + (2 )n ) f By Theorem 1, f + Dn f p ≤ min{H ( ) : > 0} Therefore, since min{H ( ) : > 0} = H m+n A D m f + D m+n f f p p , p Some Extensions of the Kolmogorov–Stein Inequality where A = (π m)/(λn(π2m+n + 2n+1 )), we obtain that f + Dn f p ≤ H m+n A D m f + D m+n f f p = λ(π 2m−1 + 1) f p + C1 p m+n f m p D m f + D m+n f n, p where C1 is independent of f The proof is complete k Let Q(x) := m k=0 bk x be a polynomial of degree m ≥ with real coefficients, satm k isfying the condition |Q(x)| = ∀x ∈ R We define Q1 (x) := k=0 |bk |x , φ(x) = ˆ (1 − /4 (x))/Q(x) Since |Q(x)| = ∀x ∈ R, there exists a constant a > such that |Q(x)| ≥ a(1 + x ) and then φ(x) ∈ L1 (R) Hence φˆ is well defined on R and we can see that φˆ ∈ L1 (R) Put λ1 := √1 φˆ 2π Further, we have Theorem Let m, n ∈ N and f ∈ C m+n (R), ≤ p ≤ ∞, P (x) be a polynomial of degree n, Q(x) be a polynomial of degree m ≥ with real coefficients Assume |Q(x)| = ∀x ∈ R and f, (QP )(D)f ∈ Lp (R) Then P (D)f ∈ Lp (R) and P (D)f p ≤ λ1 (QP )(D)f p + λ(λ1 Q1 (2 ) + 1)P1 (2 ) f ∀ > p , g, h as in the proof of Theorem Arguing as in the Proof We define the functions proof of Theorem 1, we have (QP )(D)g p ≤ Q1 (2 ) P (D)g p ≤ λQ1 (2 )P1 (2 ) f p (9) Therefore, since h = f − g, we get (QP )(D)h p ≤ (QP )(D)g p + (QP )(D)f ≤ λQ1 (2 )P1 (2 ) f Further, put t (x) = − ˆ p p + (QP )(D)f p (10) /4 (x) Then t (x) = ∀x ∈ (−∞, − /2] ∪ [ /2, +∞) Hence, it follows from F ((QP )(D)h) = Q(x)F (P (D)h) and supp F ((QP )(D)h) ⊂ (−∞, − ] ∪ [ , +∞) that t (x)F ((QP )(D)h) = Q(x)F (P (D)h) Therefore F (P (D)h) = t (x) F ((QP )(D)h) Q(x) So t (x) P (D)h = √ ((QP )(D)h) ∗ F −1 Q(x) 2π Therefore, using Young inequality and (QP )(D)h ∈ Lp (R), we obtain P (D)h ∈ Lp (R) and t (x) (QP )(D)h p F −1 P (D)h p ≤ √ Q(x) 2π 1 t (x) (QP )(D)h p F = √ , Q(x) 2π H H Bang, V N Huy i.e., P (D)h p ≤ λ1 (QP )(D)h p From this and (10), we obtain P (D)h p ≤ λ1 (λQ1 (2 )P1 (2 ) f p + (QP )(D)f (11) p ) Since f = g + h, we have P (D)f ≤ P (D)g p p + P (D)h (12) p From (5), (11) and (12), we get P (D)f ∈ Lp (R) and P (D)f p ≤ λ1 (QP )(D)f p + λ(λ1 Q1 (2 ) + 1)P1 (2 ) f p ∀ > The proof is complete Note that Theorems 1, still hold for generalized derivatives which will be seen in the next section Inequalities for Orlicz Spaces Let : [0, +∞) → [0, +∞] be an arbitrary Young function, i.e., is convex Denote by ¯ (t) = sup{ts − (s)} (0) = 0, (t) ≡ and s≥0 and L (R) the space of all measurable functions u such the Young function conjugate to that | u, v | = R for all v with ρ(v, ¯ ) < ∞, where u(x)v(x)dx < ∞ ρ(v, ¯ ) = R ¯ (|v(x)|)dx Then L (R) is a Banach space with respect to the Orlicz norm u = sup R ρ(v, ¯ )≤1 u(x)v(x)dx , which is equivalent to the Luxemburg norm f ( ) = inf λ > : R (|f (x)|/λ)dx ≤ < ∞ Recall that · ( ) = · p where (t) = with ≤ p < ∞ and · ( ) = · ∞ when (t) = for ≤ t ≤ and (t) = ∞ for t > So, Lebesgue spaces Lp (R), ≤ p ≤ ∞ are special cases of Orlicz spaces (see [16, 17, 19]) Note that L (R) ⊂ S (R), where S (R) is the Schwartz space of tempered generalized functions Using Theorems 1, 2, and the method investigated in [1–3], we obtain the following theorems Theorem Let be an arbitrary Young function, m, n ∈ N and P (x) be a polynomial of degree n Assume f ∈ L (R) and D m (P (D)f ) ∈ L (R) in the S -sense Then P (D)f ∈ L (R) and P (D)f ( ) ≤ λ(Km 2m + 1)P1 (2 ) f ( ) + Km −m D m (P (D)f ( ) ∀ > Some Extensions of the Kolmogorov–Stein Inequality Theorem Let be an arbitrary Young function, m, n ∈ N, P (x) be a polynomial of degree n, Q(x) be a polynomial of degree m with real coefficients, and |Q(x)| = ∀x ∈ R Assume f ∈ L (R) and (QP )(D)f ∈ L (R) in the S -sense Then P (D)f ∈ L (R) and P (D)f ( ) ≤ λ1 (QP )(D)f ( ) + λ(λ1 Q1 (2 ) + 1)P1 (2 ) f (Φ) ∀ > Remark Theorems and still hold if we replace Luxemburg’s norm by Orlicz’s norm Acknowledgments This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2011.32 References 10 11 12 13 14 15 16 17 18 19 20 21 22 Bang, H.H.: A remark on the Kolmogorov–Stein inequality J Math Anal Appl 203, 861–867 (1996) Bang, H.H.: On an inequality of Bohr for Orlicz spaces Bull Pol Acad Sci 49, 383–389 (2001) Bang, H.H., Thu, M.T.: On a Landau–Kolmogorov inequality J Inequal Appl 7, 663–672 (2002) Bohr, H.: Ein allgemeiner Satz uă ber die Integration eines trigonometrischen Polynoms Prace Matem -Fiz 43, 273–288 (1935) Bojanov, B.D., Varma, A.K.: On a polynomial inequality of Kolmogorov type Proc Amer Math Soc 124, 491–496 (1996) Borwein, P., Erd´elyi, T.: Polynomials and Polynomial Inequalities Graduate Texts in Mathematics, vol 161 Springer-Verlag, New York (1995) Burenkov, V.I.: Exact constants in inequalities for norms of intermediate derivatives on a finite interval Proc Steklov Inst Math 156, 22–29 (1980) Certain, M.W., Kurtz, T.G.: Landau–Kolmogorov inequalities for semigroups and groups Proc Amer Math Soc 63, 226–230 (1977) Chernov, P.R.: Optimal Landau–Kolmogorov inequalities for dissipative operators in Hilbert and Banach spaces Adv Math 34, 137–144 (1979) Ditzian, Z.: Some remarks on inequalities of Landau and Kolmogorov Aequat Math 12, 145–151 (1975) Ditzian, Z.: A Kolmogorov-type inequality Math Proc Camb Philos Soc 136, 657–663 (2004) Favard, J.: Application de la formule sommatoire d’Euler ` la d´emonstration de quelques propri´et´es extr´emales des int´egrales des fonctions p´eriodiques et presque-p´eriodiques Mat Tidsskr B, 8194 (1936) Hăormander, L.: A new generalization of an inequality of Bohr Math Scand 2, 33–45 (1954) Kofanov, V.A.: On sharp Kolmogorov-type inequalities taking into account the number of sign changes of derivatives Ukr Math J 55, 548–565 (2008) Kolmogorov, A.N.: On inequalities between upper bounds of the successive derivatives of an arbitrary function on an infinite interval Amer Math Soc Transl Ser 2, 233–243 (1962) Krasnoselskii, M.A., Rutickii, Y.B.: Convex functions and orlicz spaces GITTL, Moscow (1958) Engl Transl Noordhoff (1961) Luxemburg, W.: Banach function spaces (Thesis) Technische Hogeschool te Delft., The Netherlands (1955) Nikolskii, S.M.: Approximation of functions of several variables and imbedding theorems Moscow, Nauka (1977) Rao, M.M., Ren, Z.D.: Theory of Orlicz spaces Marcel Dekker, New York (1991) Stein, E.M.: Functions of exponential type Ann Math 65, 582–592 (1957) Tikhomirov, V.M., Magaril-Il’jaev, G.G.: Inequalities for derivatives In: Kolmogorov, A.N Selected Papers, pp 387–390 Moscow, Nauka (1985) Trigub, R.M.: Comparison of linear differential operators Math Notes 82, 380–394 (2007) ... ∀ > p , g, h as in the proof of Theorem Arguing as in the Proof We define the functions proof of Theorem 1, we have (QP )(D)g p ≤ Q1 (2 ) P (D)g p ≤ λQ1 (2 )P1 (2 ) f p (9) Therefore, since h... (x) := and then the differential operator P (D) is k=0 |ak |x , D f = (−i) f Some Extensions of the Kolmogorov–Stein Inequality obtained from P (x) by substituting x → −i∂/∂x We define the function... (P (D)f ( ) ∀ > Some Extensions of the Kolmogorov–Stein Inequality Theorem Let be an arbitrary Young function, m, n ∈ N, P (x) be a polynomial of degree n, Q(x) be a polynomial of degree m with