On an inequality of Kolmogorov and Stein

BULL. AUSTRAL. MATH. SOC. VOL. 61 (2000) 26B35, 2 6 D 1 0 [153-159] ON AN INEQUALITY OF KOLMOGOROV AND STEIN HA HUY BANG AND HOANG MAI LE A.N. Kolmogorov showed that, if / , / ' , . . . , / ' " ' are bounded continuous functions on R, then H / ' * ' ^ s? Ck,n ||/||^"* /n ||/ ( n ) ||^" when 0 < k < n. This result was extended by E.M. Stein to Lebesgue lAspaces and by H.H. Bang to Orlicz spaces. In this paper, the inequality is extended to more general function spaces. 1. INTRODUCTION Kolmogorov [8] showed that, if / , / ' , . . . , / ( " ' are bounded continuous functions on R, then when 0 < k < n, where Ck>n = Kn_k / Kl~k'n, and 4 This is the best constant. Kolmogorov's result was extended to Lebesgue ZAspaces by Stein [10] and to Orlicz spaces by Bang [1]. In this paper, the methods of these authors are modified to prove the analogous result for other function spaces on R. For variants and applications of such results, see, for example, [4, 9, 11]. In particular, our results apply to amalgams of V and £9, as defined and studied in, for example, [2, 3, 5, 6, 7]. To formulate our result, we need several definitions. First, if / is a function on R, we denote by r(t)f its translate: r(t)f(s) = f(s +1). Next, let D(R) be a space of test functions, such as C£°(R) or <S(R). We require that translations act continuously in D(R). Received 1st April, 1999 Research supported by the National Basic Research Program in Natural Science. The authors would like to express their sincere thanks to the Editor, Professor Michael G. Cowling, who has written the new version of this paper. Copyright Clearance Centre, Inc. Serial-fee code: 0004-9727/00 SA2.00+0.00. 153 Downloaded from https://www.cambridge.org/core. IP address: 3.236.55.199, on 12 Jun 2020 at 15:06:17, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700022103 154 H.H. Bang and H.M. Le [2] Let X(M) be a Banach space of functions on R. We say that X(R) is r-stable provided that r(t)f is in X(R) whenever t is in R and / is in X(R) and further there is a constant Cx such that (1) V/6X(R) \\r(t)f\\x^Cx\\f\\x VtgR. Examples of r-stable spaces include Lebesgue spaces, Lorentz spaces, Orlicz spaces, and, more generally, rearrangement invariant function spaces, and spaces involving derivatives, such as the Sobolev spaces W'k(R). For all these spaces, Cx = 1. Other examples of r-stable spaces include amalgams and weighted Lebesgue spaces L^R, w) of functions / such that \ i \f(t)\pw(t)dt) / <oo, where the weight w is positive, bounded, and bounded away from 0. For these spaces, Cx > 1 in general. Let \\f\\x denote the norm of / in >f(R), and for g in D(R), let \\g\\x. denote the norm of duality with X(R), that is, g(t)f(t)dt : f € X(R), \\f\\ We say that X(R) is D(R)-full if the map from D{R) to X(R)* is continuous, so \\g\\x, < oo for all g in D(R), and, if / is in D(R)' (the dual space of D(R)) and g(t)f(t)dt :geD(R), \\g\\x. ^ 1 \ < oo, then / is in X(R) and | | / | | x is equal to the left hand side of the inequality above. For example, take D(R) to be Cf'(R). If X(R) is an amalgam of IP and (P where 1 < p,q < oo (in particular if X(R) = L"(R) for such p) then X(R) is £>(R)-full; however, if X(R) = Ll(R), then X(R) is not £>(R)-full: the problem is the measures. THEOREM 1 . Suppose that X(R) is a r-stable D(R)-M1 Banach space of functions on R. [The word "functions" here is intended to include generalised functions such as distributions.] Iff and its generalised derivative / ' " ' are in X(R), then /<*' is in X(R) when 0 < k < n and <? n n II fll 1 "*/" II /(")ll*/ n ^CxCk,n\\f\\x \\r '\\x • PROOF: Take a function h in D(R), and define F : R -*• C by the formula = f f(s + t)h(s)ds= JR ff(s)h(S-t)ds. JK Downloaded from https://www.cambridge.org/core. IP address: 3.236.55.199, on 12 Jun 2020 at 15:06:17, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700022103 [3] An inequality of Kolmogorov and Stein 155 Now \F(t) - F(H)\ = | j f / ( » ) [h(s - t ) - h(s - t')] ds as t -f if in R, since r(-t)h bounded, since -» r{-t!)h in D(R) and hence in X'(R). Furthermore, F is \\h\\x. < Cx \\f\\x \\h\\x.. \F(t)\ ^ Ht)f\\x Moreover, F (n) (*)= f f{n)(t + s)h(s)ds, JR and so, similarly, F ' " ' is continuous and bounded, and ll^'IL^^ii/wiui^.. Finally, since h^ is in D(R) and F'fc)(t) = (-1)* f f(s)hW(s - t)da, JR ( F *' is also bounded and continuous. By Kolmogorov's inequality applied t o F , Since X(R) is D(R)-full, by hypothesis, and for all h in D(R), it follows that / ( *' is in X(R), and C *.»I|/|IA- 11/ \\x ' as required. D This theorem implies, for instance, Stein's theorem, except for Z/'(R). Similarly, it does not give a result for amalgams involving L 1 . However, we have several extensions of this result which take care of these examples. Let Au(R) denote the closed subspace of X(R) of all functions / such that /L->0 as t->0. Downloaded from https://www.cambridge.org/core. IP address: 3.236.55.199, on 12 Jun 2020 at 15:06:17, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700022103 156 H.H. Bang and H.M. Le [4] COROLLARY 2 . Suppose that X(R) is a r-stable D(R)-full Banach space of functions on R. Iff is in XU(R) and its generalised derivative / ' " ' is in X(R), then /'*) is in XU(R) when 0 < k < n. P R O O F : By the theorem, - fw\\x < Cx Cktn\\r(t)f - fW1;"'" | as t —• 0, so /'*' is in XU(R), as required. D The second variant of the result involves another negative integer m, we say that a Banach space X(R) multiplication by C m (R) if, whenever / is in X(R) and continuous on R, the pointwise product <pf is in X(R) subspace of X(R). Given a nonof functions on R is stable under <p, <p', ... , f^ are bounded and and }=0 If X(R) is stable under multiplication by C m (R) for some m, then we denote by X0(R) the closed subspace of X(R) of all functions / for which lim \\<pef - f\\x = 0, where ip€(x) = e~ £I . COROLLARY 3 . Suppose that X(R) is a r-stable D(R)-M1 Banach space of functions on R, stable under multiplication by Cm(R) for some nonnegative integer m. If f is in XQ(R) and its generalised derivative / ' " ' is in X(R), then /'*) is in XQ(R) when 0 < k < n. PROOF: By the theorem, fW is in X(R), when 0 ^ j < n. By Leibniz's rule for the derivative of a product, f(k) - E ( j= 0 By the theorem, \\M- f)(k)\\x < cxck,n\\Vtf - f\\l~k/n\ Downloaded from https://www.cambridge.org/core. IP address: 3.236.55.199, on 12 Jun 2020 at 15:06:17, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700022103 [5] An inequality of Kolmogorov and Stein 157 as s -» 0, since | | y £ / — / | ^ —• 0 and | | ( V e / ~ / ) ^ n ' | | ^ i s bounded, by another application of Leibniz's rule. Further, when 0 ^ j < k, and it is easy t o check t h a t ||¥>e*~j)||c™ -* 0 as e - » 0 + . D A third variant of t h e result combines t h e themes of t h e two previous corollaries. Assume t h a t X(R) subspace of X(R) is stable under multiplication by C m ( R ) , and let -Xi(R) denote the of all functions / such t h a t (pef is in Xtt(R) COROLLARY 4 . Suppose that X(R) tions on R, stable under multiplication and its generalised derivative for all e in R + . is a T-stable D(R)-full Banach space of func- by C""(R) for some integer m. If f is in X i ( R ) / ' " ' is in X(R), then / ' * ' is in X i ( R ) when 0 < k < n. P R O O F : This proof combines the ingredients of t h e proofs of t h e last two corollaries. We need to show t h a t - iv.fw)\\x as t -> 0, which we do by induction. Suppose t h a t when 0 < j < k. Observe that - (<Pef(k)) j=t> Now cxck,n | {n) as t -> 0, since \\r(t)(<pj) - {<pj)\\x -*• 0 while ||(r(t)(^/) - (<pj)) \\x is bounded as t -» 0, by the arguments of the previous corollaries. Further, Downloaded from https://www.cambridge.org/core. IP address: 3.236.55.199, on 12 Jun 2020 at 15:06:17, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700022103 158 H.H. Bang and H.M. Le [6] as t —> 0, by straightforward estimates of <pe and its derivatives, and the inductive hypothesis. 0 EXAMPLES. The Lorentz spaces L ^ R ) (where 1 < p < oo and 1 ^ q ^ oo) are dual spaces except when 9 = 1, and are covered by the theorem. When q = 1, they are covered by Corollary 4. The amalgams P{IP) are covered by the theorem if p > 1. If q < oo and X = t?(M), where M denotes the space of bounded complex measures, then XQ = f (L1). If X = e°°(M), then Xi = ^{L1). REMARKS. It should be noted that the constant obtained in the theorem is, in general, not best possible. For example, it is easy to show that the best constant when X = I? is 1, by using the Fourier transform and Holder's inequality. The point of the theorem is that there is a constant which works for all Z?(R)-fulI, r-stable, Banach spaces for which the translation constant Cx of formula (1) is in a given range. The hypothesis of Corollary 2 can be varied a little without changing the conclusion: more precisely, we may assume that / is in X(R) and its generalised derivative / ' n ) is in XM(R). Similarly, the hypothesis of Corollary 3 can also be varied. If we are interested in proving additive inequalities, that is, those of the form (2) then more can be said. Indeed, the condition that X be r-stable can be replaced by the condition that translations act continuously on X (that is, the map (t,f) >-> r(t)f from R x X to X is continuous, which implies that | | T ( £ ) / | | X $ £)(t) | | / | | x for all f in X and t in E, where £l(t) grows at most exponentially as \t\ grows). By writing a function / as ip * f + (/ ~ ^ * /)> where ip is a suitable test function, one sees that The first term on the right hand side can be controlled by a weighted L'-norm of xji^ multiplied by | | / | | x , and the second, after some integrations by parts, by a weighted Ll norm of an (n — fc)-fold integral of S — ip multiplied by | | / ' n ' | | x , where S denotes the Dirac delta distribution. The conclusion at which one arrives is that the constant in the inequality (2) can be taken to depend only on k and n and the growth rate Q(t). This result applies to spaces such as i/(R, w), where the weight w does not vanish or grow too fast; the weights w(x) = (l + \x\)", where a is real, are examples of admissible weights. REFERENCES [1] H.H. Bang, 'A remark on the Kolmogorov-Stein inequality', J. Math. Anal. Appl. 203 (1996), 861-867. [2] J.P. Bertrandias and C. Dupuis, 'Transformation de Fourier sur les espaces lp(l/fy, Ann. Inst. Fourier Grenoble 29 (1979), 189-206. Downloaded from https://www.cambridge.org/core. IP address: 3.236.55.199, on 12 Jun 2020 at 15:06:17, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700022103 [7] An inequality of Kolmogorov and Stein 159 [3] W.R. Bloom, 'Estimates for the Fourier transform', Math. Scientist 10 (1985), 65-81. [4] M.W. Certain and T.G. Kurtz, 'Landau-Kolmogorov inequalities for semigroups and groups', PTOC. Amer. Math. Soc. 63 (1977), 226-230. [5] J.J.F. Fournier, 'On the Hausdorff-Young theorem for amalgams', Monatsh. Math. 95 [6] F. Holland, 'Harmonic analysis on amalgams of V and V, J. London Math. Soc. 10 (1975), 295-305. F. Holland, 'On the representation of functions as Fourier transforms of unbounded measures', Proc. London Math. Soc. 30 (1975), 347-365. A.N. Kolmogorov, 'On inequalities between upper bounds of the successive derivatives of an arbitrary function on an infinite interval', Uchen. Zap. Moskov. Gos. Uni. 30 (1939), 3-16. M.K. Kwong and A. Zettl, Norm inequalities for derivatives and differences, Lecture Notes in Math. 1536 (Springer-Verlag, Berlin, Heidelberg, New York, 1992). E.M. Stein, 'Functions of exponential type', Ann. Math. 65 (1957), 582-592. V.M. Tikhomirov and G.G. Magaril-Il'jaev, 'Inequalities for derivatives', in Kolmogorov, A.N., Selected Papers (Nauka, Moscow, 1985). (1983), 117-135. [7] [8] [9] [10] [11] Institute of Mathematics P.O. Box 61 10000 Bo Ho Hanoi Vietnam Thainguyen Pedagogic College Thainguyen Vietnam Downloaded from https://www.cambridge.org/core. IP address: 3.236.55.199, on 12 Jun 2020 at 15:06:17, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700022103