Nonlinear Maps Between Besov and Sobolev Spaces
ANNALES DE LA FACULTÉ DES SCIENCES
Mathématiques
PHILIP BRENNER,PETER KUMLIN Nonlinear Maps between Besov and Sobolev spaces
Tome XIX, no 1 (2010), p. 105-120.
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cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Annales de la Facult´e des Sciences de Toulouse Vol. XIX, n◦ 1, 2010 pp. 105–120
Nonlinear Maps between Besov and Sobolev spaces
Philip Brenner(1), Peter Kumlin(2)
RESUM´ E.´ — Notre r´esultat principal est que pour une grande famille d’applications non lin´eaires entre espaces de Besov et de Sobolev, l’interpo- lation est un ph´enom`ene propre aux petites dimensions. Ceci prolonge des r´esultats obtenus pr´ec´edemment par Kumlin [13] pour des applications analytiques au cas d’applications H¨older continues ou encore Lipschitz (Corollaires 1 and 2), et qui remontent aux id´ees de B.E.J. Dahlberg [8].
ABSTRACT. — Our main result shows that for a large class of nonlinear local mappings between Besov and Sobolev space, interpolation is an ex- ceptional low dimensional phenomenon. This extends previous results by Kumlin [13] from the case of analytic mappings to Lipschitz and H¨older continuous maps (Corollaries 1 and 2), and which go back to ideas of the late B.E.J. Dahlberg [8].
1. Main result
Our main result shows that for a large class of nonlinear local mappings between Besov and Sobolev space, interpolation is an exceptional low di- mensional phenomenon. We give results which are extensions of previous results by Kumlin [13] from the case of analytic mappings to Lipschitz and H¨older continuous maps (Corollaries 1 and 2), and which go back to ideas of the late B.E.J. Dahlberg [8].
(∗) Re¸cu le 30/09/08, accept´e le 22/06/09 (1) IT-university of G¨oteborg, University of Gothenburg and Chalmers University of Technology SE-412 96 G¨oteborg Sweden [email protected] (2) Mathematical Sciences, University of Gothenburg and Chalmers University of Tech- nology SE-412 96 G¨oteborg Sweden [email protected]
– 105 – Philip Brenner, Peter Kumlin
First an important definition in this context: In the formulation of the Main Theorem we use a notion of “a set of mappings F admitting interpo- lation on scales of Banach spaces” which is given by
Definition 1.1 A { } A { } .— Let = Aθ θ∈Θ=[s0,s1] and = Aθ θ ∈Θ =[s ,s ] ⊂ 0 1 be ordered scales of Banach spaces, i.e. Aθ1 Aθ2 whenever θ1 >θ2 and ⊂ F A A whenever θ1 >θ2 respectively. We say that a family admits θ1 θ2 interpolation on the ordered scales (A, A) if for every F ∈F
∈ ∈ 1. F (u) A for all u As0 , s0 ∈ ∈ 2. F (u) A for all u As1 , and s1 { ∈ ∈ 3. the increasing function sF (s) = sup t [s0,s1]:F (u) At all ∈ } u As is a mapping of (s0,s1) onto (s0,s1).
In the following we look at scales of interpolation spaces as the scales of Banach spaces and, in particular, we consider scales of Sobolev and Besov spaces. Here and in the following we assume that 1 p 2 p are dual 1 1 ∞ exponents, i.e. p + p = 1, and that 1 r . The Besov space (we s,q refer to section 3 for those unfamiliar with these spaces) Bp (Ω) will be s,q n written simply Bp in case Ω = R . The same convention will be used for s the Sobolev spaces H2 discussed below. Notice that the space dimension n will play an important role in the results. All functions u below are defined on Rn.
Definition 1.2. — Afamily F of mappings is said to admit interpola- s+1,r s ,r tion on the scale of Besov spaces (Bp ,Bp ), 0 s σ, 0 s σ if for every F ∈F
∈ ∈ 1,r 1. F (u) Lp for all u Bp ,
∈ σ ,r ∈ σ+1,r 2. F (u) Bp for all u Bp , and 3. there exists a realvalued function r = rF (s), rF (s) r, such that the the increasing function
{ ∈ ∈ t,r ∈ s+1,r} sF (s) = sup t [0,σ ]:F (u) Bp all u Bp is a mapping of (0,σ) onto (0,σ).
In the linear case, familiar interpolation methods give r = r and with s = θσ that s = θσ for 0 <θ<1. In case of nonlinear mappings, the
– 106 – NonlinearMaps between Besov- and Sobolev spaces
interpolation may result in sT (s) Theorem 1.3 (Main Theorem) ∈ [σ ]+1 1+2p .— Let f C where σ > p and 1 p 2 p with dual exponents p, p. Assume that the mapping F : u → f(u) s+1,r s ,r admits interpolation on the scale of Besov spaces (Bp ,Bp ), 0 s σ, 0 s σ. Moreover assume that there exists a β>0 such that sF (s) βs for 0 Notice that the result is independent of the interpolation method used. Remark 1.4. — The proof of the Main Theorem provides more detailed information:√ There is a strictly increasing function p(β)ofβ,1 [σ ]+1 Theorem 1.5 (Main Theorem L2 case).— Letf ∈ C , with σ > 5 2 , be a function such that the mapping F : u → f(u) s+1 s admits interpolation on the scale of Sobolev spaces (H2 ,H2 ), 0 s σ, 0 s σ . Assume that there is a β>0 such that sF , the function defined above, satisfies the inequality sF (s) βs for 0 – 107 – Philip Brenner, Peter Kumlin 10 − Then either the space dimension n β 1 or else f(z)=Dz for some constant D. Remark 1.6. — In the Main Theorem “L2 case”, the existence of a β>0 such that sF (s) βs for 0 For local mappings, the Main Theorem applies to the interpolation re- sults of Heintz and von Wahl for analytic mappings [11], those of Peetre [18] for Lipschitz mappings and of Maligranda [16] for H¨older continuous map- pings. In all cases, the Main theorem tells that for a large class of nonlinear mappings, interpolation is an exceptional, low dimensional phenomenon. We demonstrate this in two corollaries, which are consequences of the main theorem and Theorem 2.1 in section 2. 1 → ∈ We say that the mapping H2 u f(u) L2 is H¨older continuous of order α, Lipschitz continuous if α =1,if α 1 − 1 1 − ∈ f(u) f(v) L2 g( u H , v H ) u v 1 for u, v H2 , (1.1) 2 2 H2 2 where g(., .) is a locally bounded function on R+, increasing in each of its arguments. Corollary1.7 5 ∈ [σ ]+1 .— Let σ>0, σ > 2 and let f C . Assume that 1 → ∈ H2 u f(u) L2 is Lipschitz continuous and that ∈ σ+1 f(u) σ h( u H1 ) u σ+1 for u H2 , (1.2) H2 2 H2 where h(·) is a locally bounded increasing function on R+. Then either n< 10 or else f(z)=Dz for some constant D. We may weaken the assumptions on the mapping properties of f and still get results that are consequences of the Main Theorem, as we will prove below. Corollary1.8 5 ∈ [σ ]+1 .— Let σ>0, σ > 2 and let f C . Assume that 1 → ∈ H2 u f(u) L2 – 108 – Nonlinear Maps between Besov and Sobolev spaces is H¨older continuous of order α, 0 <α<1 and that the mapping σ+1 → ∈ σ H2 u f(u) H2 has at most power growth, i.e. µ ∈ σ+1 f(u) σ h( u 1 ) u σ+1 , for u H , (1.3) H H2 2 2 H2 where h(·) is a locally bounded increasing function on R+. Assume that µ α. Then there exists an integer n(α) such that either n In many cases, the growth condition (1.3) can be derived in low dimen- sions from (a possibly local version of) the inequality γ sup ∂x f(u) L2 |γ|=[s ] (l) (l−1) γ C max( f (u) Lr u L∞ :1 l [s ]) sup ∂x u L2 , (1.4) |γ|=[s ]+1 γ γ1 γ2 γn valid for r>n. Here we denote ∂x = ∂x1 ∂x2 ...∂xn with γ =(γ1,γ2,...,γn) | | n and γ =Σi=1γi. The inequality (1.4) follows from the Sobolev lemma and the Gagliardo-Nirenberg inequality (see [9], [17], and also e.g. H¨ormander [12], Corollary 6.4.5). Conditions under which u → f(u) is bounded as a mapping between Besov spaces (or between Lizorkin-Triebel spaces) are given e.g. in Bourdaud et al. [4] (see also Kumlin [13] and Dahlberg [8], and the references given in [4]). Examples of nonlinear, non-local mappings between Besov and Sobolev spaces generated by local nonlinear maps have been extensively studied in the context of initial value problems for nonlinear Klein-Gordon and Wave equations: 2 − 2 ∈ n ∂t u ∆xu + m u + f(u)=0,t>0,x R . In the case of not necessarily small initial data, f is usually assumed to have at most polynomial growth of the form f(u) |u|ρ−1u. In the subcritical 4 and critical case (i.e. ρ 1+ n−2 ), the solution operators are Lipschitz con- 1 tinuous mappings from H2 to L2 (cf. [10], [5] and [1]). Using constructions and ideas similar to those of Theorem 2.6 in the present paper, the authors proved in [6] that the solution operators for these nonlinear equations are 1 not Lipschitz continuous as mappings from H2 to L2 for supercritical expo- 4 nents ρ (ρ>1+ n−2 ). Related results for polynomial nonlinearities f have been obtained by Lebeau[14], [15]. – 109 – Philip Brenner, Peter Kumlin 2. Nonlinear maps and nonlinear interpolation In order to prove our results for α 1 we have to introduce and use s,q s ,q the Besov spaces Bp , Bp and real interpolation based on Peetre’s K- function (see [2], [3] and [7]), in fact mainly for p = p = 2. In general p 1 1 and p are assumed to be dual exponents, p + p =1, 1 p 2 and the standard inclusions (again see [3] pp. 142 and 152-153) between Besov and Sobolev spaces s,p ⊂ s ⊂ s,2 Bp Hp Bp s,p ⊃ s ⊃ s,2 Bp Hp Bp make the L2-results below to be consequences of the corresponding Besov space results. The following is a variation of results by Peetre [18] (for α = 1) and Maligranda [16]. Theorem 2.1.— Let 0 <α 1, and let f satisfy (1.1) and also the conditions of either Corollary 1.7 or Corollary 1.8, so that σ+1 n → ∈ σ n H2 (R ) u f(u) H2 (R ) with the estimate (1.2) for α =1, i.e. ∈ σ+1 f(u) σ h( u H1 ) u σ+1 for u H2 , H2 2 H2 or the estimate (1.3) in case 0 <α<1, i.e. µ ∈ σ+1 f(u) σ h( u 1 ) u σ+1 for u H , H H2 2 2 H2 with µ α. Here h(·) denotes a locally bounded increasing function on R+. For θ ∈ (0, 1) let s = σθ, s = σθ and let r 1. Then the inclusion α s , r s+1,r n → ∈ µ α n B2 (R ) u f(u) B2 (R ), where µ =1if α =1, holds. Corollary 2.2. — Under the assumptions of Theorem 3, with σ = σ, the inclusion s+1 n → ∈ s n H2 (R ) u f(u) H2 (R ) (2.1) n − α holds for 0 – 110 – Nonlinear Maps between Besov and Sobolev spaces The corollary follows as mentioned directly from Theorem 2.1 and the inclusions between Besov spaces above. Notice that in general, σ σ +1, and scaling gives the result in this slightly more general case. Remark 2.3. — The result of Theorem 2.1 is sufficient for our purposes, places in our context natural restrictions on f, and allows a simple proof. α The results of Maligranda will in our context replace µ s¯ withα ˜s¯ where s α α˜ = α(µ − (µ − α))−1 s0 µ again by assumption with µ =1ifα =1. α Correspondingly, in Corollary 2.2 s < µ s can be replaced by s < αs˜ . This can be used to give more detailed asymptotic estimates of n(α)as α → 0 in Corollary 1.8. The Lp-version of the following result due to Kumlin [13] is the main ingredient in the proof of the Main Theorem. Theorem 2.4 (Kumlin [13]). — Assume that n is a positive integer and s, s 0 satisfy n 1. 0 s−s +1 4. there exists an integer k 2 such that n>2s +2+2 k−1 . If under these assumptions f ∈ C[s ]+1 and s+1 n → ∈ s n H2 (R ) u f(u) H2 (R ) (2.2) then f is a polynomial of degree at most min([s ],k− 1) and f(0) = 0. Condition 4. in Theorem 2.4 is motivated by the following observation: Proposition 2.5 ∈ ∞ n .— Let Φ Co (R ), Φ(0) =0and define Hτ (x)= | |τ ∈ s¯ n x Φ(x). Then Hτ H2 if and only if s<τ¯ + 2 . We refer the reader to [7] (Proposition 4.2) for the straightforward proof of this proposition. Next we give a sketch of the proof of Theorem 2.4. For a complete proof we refer to the Appendix. – 111 – Philip Brenner, Peter Kumlin Proof. — (of Theorem 2.4): Assume that f(z) is a polynomial of degree k or higher in z. Since 4) holds for k¯ k if it holds for k, we may assume that the coefficient of zk is nonzero, and that f(z) is of degree k. Take ∈ s+1 k ∼| |τk k ∈ s z = Hτ H2 such that Hτ (x) x Φ(x) / H2 . By the proposition this is the case if n n τk + − s < 0 <τ+ − s − 1 2 2 which follows from 4) with a suitable choice of τ. It remains to prove that 1) through 3) imply that f is a polynomial of degree at most [s]. We construct a function v (to use as a counterexample) as follows. Compare Dahlberg [8]. j ∈ n ∈ ∞ {| | } Let y = (10j, 0,...,0) R and let u C0 with support in x 2 and such that u(x)=x1 in {|x| 1}. Define ∞ x − yj v(x)= A u( ),x∈ Rn, (2.3) j * j=1 j ↑∞ −λ where 0 2s −1 n−2s ∞ Aj *j = (2.5) j this contradicts the mapping property (2.2), i.e. s+1 n ⊂ s n f(H2 (R )) H2 (R ) and our Theorem will be proved. Assumptions 1) through 3) imply that it is possible to choose {Aj} and λ so that the properties (2.4) and (2.5) are satisfied. This completes the proof of Theorem 2.4. In the proof of Theorem 2.4 above, it is easy to see what happens if we replace the H2-spaces by Hp,Hp -spaces. Using the definition of the Besov – 112 – NonlinearMaps between Besov- and Sobolev spaces spaces in terms of moduli of continuity below (again, see [3],[2] and section 3), the construction and the proof via the Lp versions of (2.4) and (2.5) is essentially the same, although more technical (see Appendix or [13]), as that already given of Theorem 2.4. Theorem 2.6 (Kumlin [13]). — Assume that n is a positive integer, 1 1 1 p 2 p where p + p =1, q 1 and s, s 0 satisfy n 1. 0 1+p n 2. p − ss p −s−1 3. n>(p 1) 1+p , and s − p 1 − 1 1 − s 4. there exists an integer k 2 such that n( p k p ) >s+1 k . If under these assumptions f ∈ C[s ]+1 and s+1,q n → ∈ s ,q n Bp (R ) u f(u) Bp (R ) (2.6) then f is a polynomial of degree at most min([s],k− 1) and f(0) = 0. We are now in position to give the proof of the Main Theorem. Proof. — Since conditions in Theorem 2.6 are all strict inequalities, it is enough to prove the result with s replaced by βs, with 0 <β 1, in 1+2p 1+2p 1) through 4). Take s = p β so that s = p , which is allowed since by 1+2p assumption σ > p . Let us now refer to Theorem 2.6 conditions 1) through 4). With our choice of s, and with φ(p)=2(p−1)(1+2p) and k = 2, by elementary computations these are satisfied if ≡ φ(p) 1. n>φ1(p, β) 2β + p 2. n>φ2(p) ≡ 1+2p ≡ φ(p) − 3. n>φ3(p, β) β 1 ≡ φ(p) − p2+2p−1 4. n>φ4(p, β) (3−p)β +(p 3−p ) – 113 – Philip Brenner, Peter Kumlin Here 1),2) and 4) are exact reformulations of the corresponding inequalities φ(p) − 1 in Theorem 2.6, while in 3) the exact expression β p−1 has been re- φ(p) − placed by φ3(p, β)= β 1. This more restrictive choice will simplify the computations and results below. Notice that φ, and so φ1,φ2 and φ3 are strictly increasing functions of p. We first concentrate on the inequalities 1) through 3). Straightforward computations show that φ1 = φ2 = φ3 for p = p(β), 1 φ3 φ1 φ2 for p(β) p 2, φ3 φ1 φ2 for p(β) p 1. Thus 1) through 3) hold if n>max(φ3,φ2). We next show that φ4 max(φ3,φ2), and hence also 4) holds if n>max(φ3,φ2). If we use that φ(p) 2+2p for 1 p p(β), β φ(p) 2+2p for p(β) p 2, β straightforward computations show that φ(p) − p2+2p−1 φ4 = (3−p)β +(p 3−p ) 1+2p = φ2 for 1 p p(β), φ(p) − p2+2p−1 φ(p) − φ4 = (3−p)β +(p 3−p ) β 1=φ3 for p(β) p 2. Thus 1) through 4) hold for φ(p) − φ(p) − n>max(φ3,φ2) = max( β 1, 1+2p) = max( β , 2+2p) 1. Since we have assumed that the mapping u → f(u) admits interpolation, the condition (2.6) in Theorem 2.6 is also satisfied, and the main theorem is proved. For the convenience of the reader we will give a proof of Theorem 2.1 in next section and in that context also a very short introduction to the necessary concepts of real interpolation and Besov spaces. – 114 – NonlinearMaps between Besov- and Sobolev spaces 3. Besov spaces, real interpolation and the proof of Theorem 2.1 In this section we will shortly remind of the basic definitions and prop- erties of real interpolation and Besov spaces. The basic references are [2], [3] and [7], to which we refer the reader for additional information. Let C1 ⊂ C0 be a Banach space couple. Then the K-functional K(t, φ; C0,C1) is defined by { ∈ } K(t, φ; C0,C1) = inf φ0 C0 + t φ1 C1 : φ = φ0 + φ1,φi Ci , where φ ∈ C1 and t 0. Notice that K(t, φ; Co,C1) φ C0 for t 1. (3.1) We define Cθ,q =(C0,C1)θ,q as the completion of C1 in the norm ∞ −θ q dt 1/q φ Cθ,q =( (t K(t, φ; C0,C1)) ) . 0 t s1 s0 s,q If C1 = Hp ,C0 = Hp ,s1 >s0, then Cθ,q defines the Besov space Bp , where s =(1− θ)s0 + θs1. By definition, the family of Besov spaces have a number of natural convexity and inclusion properties, as mentioned earlier, for which we refer the reader to the already given basic references, and in particular to [3]. In this context let us remind of the following well-known interpolation result: This is important in the proof of Theorems 2.4 and 2.6, when we want to translate the effect of a lower bound of the derivatives of s,q f in terms of bounds on Besov space norms. Bp has the intrinsic norm (among many) ∞ 1 q −s+[s] (r) α q dt v s,q (t ω (t, ∂ v)) + v L , (3.2) Bp p t p |α|=[s] 0 (r) where [s] = sup{m ∈ Z : m s} and r 1 and we let ωp (t, v) denote the r-th modulus of continuity of v in Lp, i.e. r (r) − r−k · ωp (t, v) = sup rk( 1) v( + kh) Lp . (3.3) | | h t k=0 We may now begin the proof of Theorem 2.1. Proof. — In the following we let c denote a locally bounded function of 1 the H2 -norm of u, or a constant, where c may be different at each occurence. – 115 – Philip Brenner, Peter Kumlin Similarily we let C denote constants that may vary from line to line. Set ≡ 1 σ+1 ≡ σ ∈ K(t, v) K(t, v; H2 ,H2 ) and K (t, v) K(t, v; L2,H2 ). Choose e(t) σ+1 H2 such that − u e(t) H1 + t e(t) σ+1 2K(t, u) (3.4) 2 H2 In particular, by (3.4) we have e(t) 1 u − e(t) 1 + u 1 3 u 1 . (3.5) H2 H2 H2 H2 By (1.1) and (1.3) we get µ − µ K (t ,f(u)) f(u) f(e) L2 + t f(e) σ H2 − α µ µ g( u H1 , e(t) H1 ) u e 1 + h( e(t) H1 )t ( e σ+1 ) 2 2 H2 2 H2 and by the bound (3.5) and since by assumption, g is a locally bounded function increasing in both variables and h is a locally bounded increasing function g( u 1 , e(t) 1 ) c, h( e(t) 1 ) c, H2 H2 H2 where c, as remarked above, denotes a locally bounded function of u 1 . H2 Then by the choice of e(t), K(tµ,f(u)) cK(t, u)α + cK(t, u)µ Thus we get ∞ ∞ −θα µ r dt −θ r dt (t K (t ,f(u))) α c (t K(t, u)) 0 t 0 t ∞ −θ α rµ dt + c (t µ K(t, u)) α 0 t A change of variable in the first integral yields ∞ ∞ −θα µ r dt 1 −θ α r dt (t K (t ,f(u))) α = (t µ K (t, f(u))) α 0 t µ 0 t and hence we obtain ∞ ∞ −θ α r dt −θ r dt (t µ K (t, f(u))) α c (t K(t, u)) 0 t 0 t ∞ −θ α rµ dt + c (t µ K(t, u)) α 0 t – 116 – NonlinearMaps between Besov- and Sobolev spaces By the definition and the inclusions between the Besov spaces, noting that α µ 1 with equality only if α = µ = 1, this finally ends up as α µ f(u) s α , r c( u s+1,r ) + c( u α µ ) µ α B s +1,r B 2 µ α 2 B2 α µ c( u s+1,r ) + c( u s+1,r ) . B2 B2 This completes the proof of Theorem 2.1. Before we end, let us notice that we have wasted information in a number of places in the proof of Theorem 2.1, in order to avoid technical arguments involving advanced properties of real interpolation. For a more careful and complete discussion see Maligranda [16]. 4. Appendix In the appendix we supply the full proof of Theorems 2.4 and 2.6, where we use the formulation of the Besov norm given in (3.2). Theorem 2.4 is a special case of Theorem 2.6 so it is enough to prove the later one. Moreover we use the embeddings s,q ⊂ s,˜ q˜ Bp Bp for 1 p ∞, 1 q q˜ ∞ and s s˜. See [3]. Proof. — Since conditions 1)-4) in Theorem 2.6 only involve strict in- equalities we can without loss of generality assume that s, s ∈ R+ \ N due to the embeddings above. Set (as already mentioned in section 3) ∞ x − yj v(x)= A u( ),x∈ Rn, j * j=1 j ∈ ∞ {| | } {| | } where u C0 with support in x 2 such that u(x)=x1 in x 1 , j ∈ n ↑∞ −λ y = (10j, 0,...,0) R and with 0 – 117 – Philip Brenner, Peter Kumlin ∞ p n−(s+1)p ∞ ∈ s+1,p n Claim 1.— Σj=1Aj *j < implies that v Bp (R ). (1) α ≡ α · − α · Proof. — Consider ωp (t, ∂ v) sup|η|t ∂ v( + η) ∂ v( ) Lp where |α| =[s +1]=s +1− σ. For 0 − j − j ∂αu( x+η y )−∂αu( x y ) ∞ p | x j x j |p Σj=1Aj sup|η|t Rn tσ dx = α yj −η α yj − − ∂ u(x− )−∂ u(x− ) ∞ p n [s+1]p σp | x j x j |p =Σj=1Aj *j sup|η|t Rn ( t )σ dx j where α − yj −η − α − yj ∂x u(x % ) ∂x u(x % ) t t sup | j j |pdx C min(( )(1−σ)p, ( )−σp) t σ | | n ( ) *j *j η t R %j by the mean value theorem. This yields 1 −σ (1) α p dt 0 (t ωp (t, ∂ v)) t − Σ∞ Ap*n (s+1)p 1 min(( t )(1−σ)p, ( t )−σp) dt j=1 j j 0 %j %j t ∞ p n−(s+1)p CΣj=1Aj *j . For 1 t we have ∞ n −[s+1] 1 ∞ n −[s+1] (1) α p | α |p p p ωp (t, ∂ v) Σj=12Aj*j ( ∂x v(x) dx) CΣj=1Aj*j . Rn − n −[s+1] ∞ p n (s+1)p ∞ ∞ p ∞ But Σj=1Aj *j < implies that Σj=1Aj*j < assuming −λ −λj that *j = Aj =2 for some λ>0. Thus ∞ −σ (1) α p dt ∞ (t ωp (t, ∂ v)) < . 1 t This gives ∞ −σ (1) α p dt 1 v s+1,p v L +Σ| | ( (t ω (t, ∂ v)) ) p < ∞. Bp p α =[s+1] p 0 t – 118 – NonlinearMaps between Besov- and Sobolev spaces ∞ s p −1 n−s p ∞ Claim 2.— Σj=1Aj *j < implies that f is a polynomial of degree at most [s]. Proof. — Set s =[s]+σ. Assume that f is not a polynomial of degree at most [s]. Then there exists a and b, a d(a, b) ≡ inf |f ([s ]+1)(t)| > 0. t∈[a,b] Set { ∈ n | − j| Aj − j } Sj(η)= x R : x + η y <*j and a< (x1 + η1 y1) −σ (1) ([s ],0,0,...,0) p (t ωp (t, ∂ f(v))) = ([s ],0,0,...,0) ∞ x+η−yj ([s ],0,0,...,0) ∞ x−yj ∂ f(Σ Aj u( ))−∂ f(Σ Aj u( )) x1 j=1 j x1 j=1 j p = sup | | dx |η|t Rn tσ x +η −yk x −yk ([s ]) 1 1 1 ([s ]) 1 1 − f (Ak )−f (Ak ) j [s ]p [s ]p k k p sup| | Σ A * | | dx η t k=j0 k k Sk(0) Sk(η) tσ [s ]p +p −[s ]p −p − %n CΣj A * t(1 σ )p k . k=j0 k k Ak Thus we get 1 ∞ − (1) σ ([s ],0,0,...,0) p dt p f(v) s ,p ( (t ω (t, ∂ f(v))) ) B 0 p t p 1 ∞ tj −σ (1) ([s ],0,0,...,0) p dt C(Σ (t ω (t, ∂ f(v))) ) p j=1 tj+1 p t 1 ∞ j p ([s ]+1)−1 n−p ([s ]+1) % − % − C(Σ {(Σ A * )(( j )(1 σ )p − ( j+1 )(1 σ )p )}) p j=1 k=j0 k k Aj Aj+1 1 ∞ s p −1 n−s p C(Σ A * ) p = ∞. j=j0+1 j j This yields a contradiction by the assumption s+1,p → ∈ s ,p Bp v f(v) Bp in the theorem. 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