PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 28, Numer, Pages 3285 329 S 2-9939()564-9 Article electronically pulished on April 28, 2 ON SEQUENCES OF C MAPS WHICH CONVERGE IN THE UNIFORM C -NORM MOHAMED SAMI ELBIALY (Communicated y Michael Handel) Astract. We study maps f C (U, Y ) and give detailed estimates on D k f(x),x U, in terms of f and f. These estimates are used to prove a lemma y D. Henry for the case k 2. Here U X is an open suset and X and Y are Banach spaces.. Introduction Let X and Y e Banach spaces. Let U X e such that U is open. Let <δ,k =,, 2, and >. Define the following: C (U, Y )={u C (U, Y ) u }, u =max{ u, Du,, D k u,h δ (D k u)}, { D H δ (D k k u(x) D k u(x } ) u)=sup x x δ x x,x U, x U. Suppose we have a sequence of maps u n C (U, Y ),n, which converges to a map u : U Y in the uniform C -norm. That is, u u n asn. In this case one would say that u C (U, Y )andthatforx U, D k u(x) D k u n (x) asn. D. Henry showed that this is true for k =andthat D u(x) D u n (x) uniformly on any suset U o U which is uniformly ounded away from the oundary of U. See [3, page 5] and [, pages 35, 39]. This situation arises frequently when one is trying to otain a local C invariant manifold as a fixed point of a transformation Γ : C (U, Y ) C (U, Y )y showing that Γ is a contraction. It is ovious that it is much easier to show that Γ is a contraction on a ounded closed suset of C, (U, Y ) in the norm rather than working with the norm. We use these results in studying the dependence of invariant manifolds on parameters [2]. In this work (Lemma 2.) we give detailed estimates on D k f(x),x U, interms of f and f for any f C (U, Y ). These estimates are important in their own right. We point out that these estimates (H) and (Hk) are homogeneous in f and f in the sense that the sum of the exponents of f and f is. In addition, they lead to a proof of Henry s lemma for k 2. The case k =2does Received y the editors Decemer 8, 998. 2 Mathematics Suject Classification. Primary 37D. Key words and phrases. Invariant manifolds, linearization, Henry s lemma. 3285 c 2 American Mathematical Society License or copyright restrictions may apply to redistriution; see http://www.ams.org/journal-terms-of-use
3286 MOHAMED SAMI ELBIALY not follow y induction from the case k =. See Remark 2.3(3). In Proposition 2.5 we will show that in fact D k u D k u n k,β onu o for any <β<δ. 2. Main results 2.. Lemma (Estimate H). Let X and Y e Banach spaces. Let U X e such that U is open. Let k =, 2, 3, and <δ. Let U e the oundary of U. For x U, letd(x) :=min{,dist(x, U)}. For all f C (U, Y ) and all x U the following estimates are true: (H) d(x) Df(x) N f δ,δ, k =, (Hk) d(x) ak D k f(x) N k f βk f βk, k 2, where N =4,a =.Ifk =,β = δ/( + δ). Ifk 2,β =/2 and [ N j =2(j!) 2+ N + N 2! 2! + N ] j (j )! =4(j!)(3 j ), j =2,,k, j(j +) a j = a j + j =, 2 j =2,,k, β j = j + β j =, (j +)! j =2,,k, β k = δ k + δ β δ k = (k + δ)(k!). We prove this lemma in Section 3. f 2.2. Lemma (Henry s lemma). The following are true: () Any closed ounded suset C (U, Y ) is a closed and ounded suset of (C (U, Y ), ). That is, if we have a sequence {u n C (U, Y ) n< } and a map u : U Y such that u n u as n,thenu C (U, Y ). Moreover, D k u n (x) D k u(x) point-wise as n. (2) Let U o U e a suset which is uniformly ounded away from the oundary of U. ThenD k u n (x) D k u(x) uniformly on U o. Proof. () If k =, then for any x x, u(x) u(x ) u n (x) u n (x ) +2 u n u x x δ +2 u n u x x δ as n. When k, let f := u n u m. It follows from estimate (Hk) of Lemma 2. that d(x) a k D k u n (x) D k u m (x) N k u n u m β k u n u m β k (*). Thus u is C and D k u n (x) D k u(x) foreachx U. It also follows that u. Part (2) follows immediately from estimate (*) and the fact that U o U is uniformly ounded away from the oundary of U. 2.3. Remarks.. The proof of Henry s lemma 2.2 is ased on estimates (H) and (Hk) of Lemma 2.. License or copyright restrictions may apply to redistriution; see http://www.ams.org/journal-terms-of-use
CONVERGENCE IN THE UNIFORM C -NORM 3287 2. Estimate (H) and the proof of Henry s lemma for k = are given in [3, page 5] and [, page 39] without the explicit dependence on f,δ. 3. The k 2 case of Henry s lemma does not follow y induction from the C,δ case without estimate (Hk). This is ecause the convergence in the latter is point-wise and the uniform convergence holds only on any proper suset U o U which is uniformly ounded away from the oundary of U. 4. Moreover, estimates (H) and (Hk) of Lemma 2. are of interest y themselves. In particular, notice that in estimates (H) and (Hk) the sum of the exponents of the different norms of the function that appear on the right-hand side is always one. In other words, these inequalities are homogeneous so to speak. This is not the case with the estimates in [3, page 5] and [, page 39]. 5. In [4, page 82] Lanford gives a lemma similar to Henry s lemma. In Lanford s lemma, all limits, in the hypothesis and the conclusion, are pointwise in the weak topology. The following is a classical result which we will need later. We present a proof here for completeness. 2.4. Lemma. () Let g C,δ (U o,y) and <β<δ. Then (2.) g,β 2 (β/δ) g (β/δ) g β/δ,δ. (2) Let f n and f e C,δ (U o,y) and assume that f n f. Then f n f,β for all β<δ. Proof. Notice that part (2) follows from part () y setting g = f n f. So, we prove part (). (#) Notice that := g(x + h) g(x) h β { min 2 g } h β, h δ β g,δ. The two terms on the right-hand side of (#) are equal iff ( ) /δ 2 g h = := A. g,δ When h >A, we use the first term on the right-hand side of (#) to estimate. When h A we use the second term. In either case we otain 2 (β/δ) g (β/δ) g β/δ,δ which proves (). 2.5. Proposition. Let u n e a sequence in C (U, Y ) and u : U Y. Assume that u n u. LetU o U e an open suset which is uniformly ounded away from the oundary of U. Let <β<δ. Then u C (U, Y ). Moreover, on U o, u n u k,β as n. Proof. Applying Lemma 2.2, we can see that D j u n D j u onu o as n for j =,,,k. Notice that for f C (U, Y )wehaved k f C,δ (U, S(X k,y)) where S(X k,y) is the Banach space of symmetric k-multilinear maps from X k to Y. Thus we can apply assertion (2) of Lemma 2.4 to D k u n and D k u and otain D k u n D k u,β as n on U o. License or copyright restrictions may apply to redistriution; see http://www.ams.org/journal-terms-of-use
3288 MOHAMED SAMI ELBIALY 3. Proof of Estimate H Let x U e fixed ut aritrary. Let v X and v =andlet<t<d(x). Then Df(x)tv f(x) f(x + tv) + f(x + tv) f(x) Df(x)tv Thus 2 f + f,δ t 2 f + f,δ t. (*) Df(x) 2 f + f,δ t δ, t < d(x). t The two terms on the right-hand side of (*) are equal iff ( ) R (**) t = =: A, S R =2 f, S = f,δ. We have two cases. Case : d(x) A. Then for all <t<d(x) t Df(x) 2R 4 f 4 f δ Taking the supremum over <t<d(x), we otain (a) d(x) Df(x) 4 f δ f,δ. f,δ. Case 2: d(x) >A. In this case we evaluate the right-hand side of (*) at t = A. Thus () d(x) Df(x) Df(x) 2R δ δ S 4 f f It follows from (a) and () that estimate (H) holds. Now we consider the case k 2. In this case (H) holds with δ =. Assume that our estimate is true for i =, 2,,k. Since f i,δ f l,δ and β l <β i for i<l, it follows that (3.) f βi f βi i,δ f β l f β l l,δ. Let v (i) =(v,,v)withi components. For i =, 2,,k,let t i i! Di f(x)v (i) = F i F i (t), F i (t) :=f(x + tv) f(x) t! Df(x)v It follows that F k (t) = t s t2 2! D2 f(x)v (2) ti i! Di f(x)v (i), A i (t) := F i. s k,δ. [D k f(x + s k v) D k f(x)]v (k) ds k ds ds. License or copyright restrictions may apply to redistriution; see http://www.ams.org/journal-terms-of-use
CONVERGENCE IN THE UNIFORM C -NORM 3289 Thus (3.2) t A k (t) H δ (D k f) s s k s δ k ds k ds ds = tk+δ (k + δ)! H δ(d k f) tk+δ (k + δ)! f f t k+δ. Using (3.) we otain (3.3) A k (t) 2 f + tn f! f β β d(x) + t2 N 2 2! f β2 f β2 d(x) a2 + + tk N k (k )! f β k d(x) a k C k d(x) a f β k k, C k =2+ N! + N 2 2! + + N k (k )!. Using (3.2) and (3.3) we otain, for <t<d(x), (3.4) t k D k f(x) A k (t)+a k (t), k! D k f(x) R k! t k + Stδ, C k R := d(x) a f β k k S := f., The two quantities on the right-hand side of (3.4) are equal iff t = ( ) /(k+δ) R =: B. S As efore, there are two cases. Case : d(x) B. Then t<d(x) B and (ak) d(x) k Dk f(x) k! 2R 2C k d(x) a f β k k 2C k d(x) a k f β k f β k. License or copyright restrictions may apply to redistriution; see http://www.ams.org/journal-terms-of-use
329 MOHAMED SAMI ELBIALY Case 2: d(x) >B. In this case we evaluate (3.4) at t = B and otain D k f(x) (k) 2R δ/(k+δ) S k/(k+δ) k! 2C δ/(k+δ) k d(x) h [(k + δ)!] k/(k+δ) f r f s 2C k d(x) h f β k f β k, h = δa k k + δ <a k, r = δβ k k + δ = β k, s = δ( β k ) k + δ It follows from (ak) and (k) that + k k + δ = β k. d(x) k+a k D k f(x) 2(k!)C k f β k f β k which finishes the proof of the lemma. References [] S.-N. Chow and K. Lu, C k centre unstale manifolds, Proc. Roy. Soc. Edinurgh, 8A, 988, 33-32. MR 9a:5848 [2] M. S. ElBialy, Su-stale and weak-stale manifolds associated with finitely non-resonant spectral suspaces, Mathematische Zeitschrift, to appear. [3] D. Henry, Geometric theory of paraolic equations, Lecture Notes in Mathematics 84, Springer, New York, 98. MR 83j:3584 [4] O.E.Lanford,III,Bifurcation of periodic solutions into invariant tori: the work of Ruelle and Takens, innonlinear prolems in the Physical Sciences and Biology, Lecture Notes in Mathematics, vol. 322, Springer-Verlag, Berlin, Heidelerg, New York, 973. MR 5:7766 Department of Mathematics, University of Toledo, Toledo, Ohio 4366 E-mail address: melialy@math.utoledo.edu License or copyright restrictions may apply to redistriution; see http://www.ams.org/journal-terms-of-use