A Remark on the Kolmogorov–Stein Inequality

JOUR NAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS AR TICLE NO . 203, 861]867 Ž1996. 0417 A R emark on the Kolmogorov]Stein Inequality* Ha Huy Bang† Institute of Mathematics, P.O. Box 631, Bo Ho, 10000 Hanoi, Vietnam Submitted by J. L. Brenner November 3, 1995 View metadata, citation and similar papersRateceived core.ac.uk In this paper, essentially developing the Stein method, we prove the Kolmogorov]Stein inequality for any Orlicz norm Žwith the same constants.. Q 1996 Academic Press, Inc. 1. INTR ODUCTION A. N. Kolmogorov has given the following result w1x: Let f 9Ž x ., . . . , f Ž n. Ž x . be continuous and bounded on R. Then f Ž x ., 5 f Ž k . 5 `n F Ck , n 5 f 5 `ny k 5 f Ž n. 5 `k , n where 0 - k - n, Ck, n s K nyk rK nŽ nyk ., Ki s 4 p ` Ý Ž y1. jr Ž 2 j q 1. iq1 js0 for even i, while Ki s 4 p ` Ý 1r Ž 2 j q 1 . iq1 js0 for odd i. Moreover the constants are best possible. This result has been extended by E. M. Stein to the L p-norm w2x. The Kolmogorov]Stein inequality and its variants are a problem of interest for *Supported by the National Basic R esearch Program in Natural Science and by the NCNST ‘‘Applied Mathematics.’’ † E-mail address: hhbang@thevinh.ac.vn. 861 0022-247Xr96 $18.00 Copyright Q 1996 by Academ ic P ress, Inc. All rights of reproduction in any form reserved. 862 HA HUY BANG many mathematicians and have various applications Žsee, for example, w3, 4x and their references .. In this paper, essentially developing the Stein method w2x, we prove this inequality for an arbitrary Orlicz norm 5 ? 5 F . The obtained result has been successfully applied to proving the corresponding imbedding theorems w5]7x for Sobolev]Orlicz spaces of infinite order and the result w8x for any Orlicz norm. 2. R ESULTS Let F Ž t .: w0, q`. ª w0, q`x be an arbitrary Young function w9]12x, i.e., Ž F 0. s 0, F Ž t . G 0, F Ž t . ' 0, and F Ž t . is convex. Denote by F Ž t . s sup  ts y F Ž s . 4 , sG0 which is the Young function conjugate to F Ž t . and LF ŽR., the space of measurable functions uŽ x . such that <² u, ¨ :< s H u Ž x . ¨ Ž x . dx -` for all ¨ Ž x . with r Ž ¨ , F . - `, where r Ž ¨ , F. s H F Ž < ¨ Ž x . <. dx. Then LF ŽR. is a Banach space with respect to the Orlicz norm 5 u5 F s sup r Ž ¨ , F .F1 H u Ž x . ¨ Ž x . dx , which is equivalent to the Luxembung norm ½ 5 f 5 ŽF . s inf l ) 0: H F Ž < f Ž x . <rl. dx F 1 5 - `. We have the following results w9x: L EMMA 1. Let u g LF ŽR. and ¨ g LF ŽR.. Then H < u Ž x . ¨ Ž x . < dx F 5 u 5 L EMMA 2. F 5 ¨ 5 ŽF . . Let u g LF ŽR. and ¨ g L1ŽR.. Then 5 u) ¨ 5 F F 5 u 5 F 5 ¨ 5 1 . R ecall that 5 ? 5 F s 5 ? 5 p when 1 F p - ` and F Ž t . s t p ; and 5 ? 5 F s 5 ? 5 ` when F Ž t . s 0 for 0 F t F 1 and F Ž t . s ` for t ) 1. 863 KOLMOGOR OV ] STEIN INEQUALITY T HEOR EM 1. Let F Ž t . be an arbitrary Young function, f Ž x . and its generalized deri¨ ati¨ e f Ž n. Ž x . be in LF ŽR.. Then f Ž k . Ž x . g LF ŽR. for all 0 - k - n and 5 f Ž k . 5 Fn F Ck , n 5 f 5 Fny k 5 f Ž n. 5 Fk . Ž 1. Proof. We begin to prove Ž1. with the assumption that f Ž k . Ž x . g LF ŽR., 0 F k F n. Fix 0 - k - n. It is known that r Ž ¨ , F . s 1 if and only if 5 ¨ 5 ŽF . s 1. Therefore, by the definition we get ` 5 f Žk. 5 F s Hy` f sup 5 ¨ 5 ŽF .F1 Žk. Ž x . ¨ Ž x . dx . Let e ) 0. We choose a function ¨e Ž x . g LF ŽR. such that 5 ¨e 5 ŽF . s 1 and ` Hy` f Žk. Ž x . ¨e Ž x . dx G 5 f Ž k . 5 F y e . Ž 2. Put ` Hy` f Ž x q y . ¨ Ž y . dy. Fe Ž x . s e Then Fe Ž x . g L`ŽR. by virtue of Lemma 1, and FeŽ r . Ž x . s ` Hy` f Žr. Ž x q y . ¨e Ž y . dy, Ž 3. 0 F r F n. Actually, for every function w Ž x . g C0`ŽR. it follows from the assumption and Lemma 1 that r ² FeŽ r . Ž x . , w Ž x . : s Ž y1 . ² Fe Ž x . , w Ž r . Ž x . : s Ž y1 . r s Ž y1 . r ž ` ž ` e ž ` ` s Hy` Hy` f s ¦H e ` y` So we have proved Ž3.. ` Hy` ¨ Ž y . Hy` f Ž x q y . w Hy` ¨ Ž y . Hy` f ž w Ž r . Ž x . dx Žr. Ž x . dx dy e s ` / ` Hy` Hy` f Ž x q y . ¨ Ž y . dy ` Žr. Žr. / / Ž x q y . w Ž x . dx dy / Ž x q y . ¨e Ž y . dy w Ž x . dx ; f Ž r . Ž x q y . ¨e Ž y . dy, w Ž x . . 864 HA HUY BANG For all x g R, clearly, < FeŽ r . Ž x . < F 5 f Ž r . Ž x q ? . 5 F 5 ¨e 5 Ž f . s 5 f Ž r . 5 F . Now we prove continuity of FeŽ r . Ž x . on R Ž0 F r F n.. We show this for r s 0 by contradiction: Assume that for some e ) 0, point x 0 , and subsequence < t k < ª 0 ` Hy` Ž f Ž x 0 q t k q y . y f Ž x 0 q y . . ¨e Ž y . dy G e , k G 1. Ž 4. Since f g LF we get easily f g L1, l o c ŽR.. Then for any n s 1, 2, . . . , f Ž t k q y . ª f Ž y . in L1Žyn, n.. Therefore, there exists a subsequence, denoted again by  t k 4 , such that f Ž t k q y . ª f Ž y . a.e. in Žyn, n.. Therefore, there exists a subsequence Žfor simplicity of notation we assume that it is coincident with  t k 4. such that f Ž x 0 q t k q y . ª f Ž x 0 q y . a.e. in Žy`, `.. On the other hand, without loss of generality we may assume that r Ž2 f, F . - `. Therefore by the Young inequality we get f Ž x 0 q tk q y . y f Ž x 0 q y . ¨e Ž y . F F Ž f Ž x 0 q tk q y . y f Ž x 0 q y . F 12 F Ž 2 f Ž x 0 q y . . q 12 . q F Ž < ¨e Ž y . < . F Ž 2 f Ž x 0 q t k q y . . q F Ž < ¨e Ž y . < . . The last expression belongs to L1ŽR., therefore by Lebesgue’s theorem we have ` H fŽ x kª` y` lim 0 q tk q y . y f Ž x 0 q y . ¨e Ž y . dy s 0, which contradicts Ž4.. The cases 1 F r F n are proved similarly. The continuity of FeŽ r . Ž x . has been proved. The functions FeŽ r . Ž x . are continuous and bounded on R. Therefore, it follows from the Kolmogorov inequality and Ž2. ] Ž3. that Ž 5 f Žk. 5 F y e . n F < FeŽ k . Ž 0 . < n F 5 FeŽ k . 5 `n F Ck , n 5 Fe 5 `ny k 5 FeŽ n. 5 `k . Ž 5. On the other hand, 5 Fe 5 ` F 5 f Ž x q y . 5 F 5 ¨e Ž y . 5 ŽF . s 5 f 5 F , Ž 6. 5 FeŽ n. 5 ` F 5 f Ž n. Ž x q y . 5 F 5 ¨e Ž y . 5 ŽF . s 5 f Ž n. 5 F . Ž 7. 865 KOLMOGOR OV ] STEIN INEQUALITY Combining Ž5. ] Ž7., we get Ž 5 f Žk. 5 F y e . n F Ck , n 5 f 5 Fnyk 5 f Ž n. 5 Fk . By letting e ª 0 we have Ž1.. To complete the proof, it remains to show that f Ž k . g LF ŽR., 0 k - n if f, f Ž n. g LF ŽR.. Let clŽ x . g C0`ŽR., clŽ x . G 0, clŽ x . s 0 for < x < G l and HclŽ x . dx s 1. We put fl s f ) cl . Then fl g C`ŽR. because of f g L1, l o c ŽR.. Therefore, flŽ k . s f ) clŽ k ., k G 0, and it is easy to check that flŽ n. s f Ž n. ) cl . On the other hand, it follows from Lemma 2 that flŽ k . s f ) clŽ k . g LF ŽR., k G 0. Therefore, by the fact proved above, we have 5 flŽ k . 5 Fn F Ck , n 5 fl 5 Fny k 5 flŽ n. 5 Fk , 0 - k - n. Therefore, since 5 fl 5 F F 5 f 5 F ? 5 cl 5 1 s 5 f 5 F , 5 flŽ n. 5 F F 5 f Ž n. 5 F 5 cl 5 1 s 5 f Ž n. 5 F we get that, for any 0 F k F n, the sequence flŽ k . is bounded in LF ŽR.. Now we prove that, for any 0 F k F n, there exists a subsequence, which is )-convergent to some g k g LF ŽR.. ŽWe say that hl is )-convergent to h, where hl, h g LF ŽR., if H hl¨ ª H h¨ for all ¨ g LF ŽR... We will show, for example, the fact that fl is )-convergent to f by contradiction: Assume that for some e 0 ) 0, ¨ g LF ŽR. and a subsequence l k ª 0, HŽf lk Ž x . y f Ž x . . ¨ Ž x . dx G e 0 , k G 1. Ž 8. Then, it is known that fl ª f, l ª 0 in L1, l o c ŽR.. Therefore, there exists a subsequence  k m 4 Žfor simplicity we assume that k m s m. such that flkŽ x . ª f Ž x . a.e. We may assume that r Ž2 f, F . - `. Then it follows from Young’s inequality that < fl Ž x . y f Ž x . < < ¨ Ž x . < F 12 F Ž 2 < fl Ž x . < . q 12 F Ž 2 < f Ž x . < . q F Ž < ¨ Ž x . < . , moreover, the right side of the last inequality belongs to L1ŽR.. Therefore, by virtue of Lebesgue’s theorem we get lim kª` H flkŽ x . y f Ž x . < ¨ Ž x . < dx s 0 because of flkŽ x . ª f Ž x . a.e., which contradicts Ž8.. 866 HA HUY BANG Finally, it follows from )-convergence fl ª f that for any w g C0`ŽR. ² flŽ k . Ž x . , w Ž x . : s Ž y1 . k ² fl Ž x . , w Ž k . Ž x . : k ª Ž y1 . ² f Ž x . , w Ž k . Ž x . : s ² f Ž k . Ž x . , w Ž x . : . Therefore, since the )-convergence of some subsequence of  flŽ k .4 to g k g LF ŽR., we get f Ž k . s g k g LF ŽR. Ž0 - k - n.. So we have proved the fact that f Ž k . g LF ŽR. for all 0 - k - n if f, f Ž n. g LF ŽR.. The proof is complete. Remark 1. To obtain Theorem 1 we have developed the Stein method because, for example, the property w g Ž x q h. y g Ž x .xrh ª g 9Ž x . in the L p mean Ž1 F p - `., which is used in w2x, holds for LF only if F Ž t . satisfies the D 2-condition Žsee w10x.. Remark 2. For periodic functions we have: T HEOR EM 2. Let F Ž t . be an arbitrary Young function, f Ž x . and its generalized deri¨ ati¨ e f Ž n. Ž x . be in LF ŽT.. Then f Ž k . Ž x . g LF ŽT. for all 0 - k - n and A f Ž k . A Fn F Ck , n A f A Fny k A f Ž n. A Fk , where T is the torus and A. A F is the corresponding norm. Remark 3. By the representation w11, p. 135x 5 u 5 ŽF . s sup 5 ¨ 5 FF1 H u Ž x . ¨ Ž x . dx , it is easy to see that the obtained results still hold for any Luxemburg norm. R EFER ENCES 1. A. N. Kolmogorov, On inequalities between upper bounds of the successive derivatives of an arbitrary function on an infinite interval, in ‘‘Amer. Math. Soc. Transl. Ser. 1,’’ Vol. 2, pp. 233]243, Amer. Math. Soc., Providence, 1962. 2. E. M. Stein, Functions of exponential type, Ann. Math. 65 Ž1957., 582]592. 3. M. W. Certain and T. G. Kurtz, Landau-Kolmogorov inequalities for semigroups and groups, Proc. Amer. Math. Soc. 63 Ž1977., 226]230. 4. V. M. Tikhomirov and G. G. Magaril-Il’jaev, Inequalities for derivatives, in ‘‘Kolmogorov A. N. Selected Papers,’’ pp. 387]390, Nauka, Moscow, 1985. 5. Ha Huy Bang, Some imbedding theorems for the spaces of infinite order of periodic functions, Math. Notes 43 Ž1988., 509]517. KOLMOGOR OV ] STEIN INEQUALITY 867 6. 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