Nonlinear Maps between Besov and Sobolev spaces

ANNALES

DE LA FACULTÉ DES SCIENCES

Mathématiques

PHILIPBRENNER, PETERKUMLIN

Nonlinear Maps between Besov and Sobolev spaces

Tome XIX, n^o1 (2010), p. 105-120.

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Nonlinear Maps between Besov and Sobolev spaces

Philip Brenner⁽¹⁾, Peter Kumlin⁽²⁾

R^´ESUM´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]

First an important definition in this context: In the formulation of the Main Theorem we use a notion of “a set of mappingsF admitting interpo- lation on scales of Banach spaces” which is given by

Definition 1.1. — LetA={Aθ}_θ∈Θ=[s0,s1]andA={A_θ}θ∈Θ=[s₀,s₁]

be ordered scales of Banach spaces, i.e. Aθ1 ⊂ Aθ2 whenever θ1 > θ2 and A_θ

1 ⊂A_θ

2 whenever θ₁ > θ₂ respectively. We say that a family F admits interpolation on the ordered scales (A,A)if for everyF ∈ F

1. F(u)∈A_s

0 for allu∈As0, 2. F(u)∈A_s

1 for allu∈A_s1, and

3. the increasing function s_F(s) = sup{t ∈ [s₀, s₁] : F(u) ∈ A_t 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 1p2 p are dual exponents, i.e. ¹_p + _p¹ = 1, and that 1 r ∞. The Besov space (we refer to section 3 for those unfamiliar with these spaces) B^s,q_p (Ω) will be written simply B_p^s,q in case Ω =Rⁿ. The same convention will be used for the Sobolev spacesH₂^sdiscussed below.

Notice that the space dimensionnwill play an important role in the results. All functions ubelow are defined onRⁿ.

Definition 1.2. — AfamilyF of mappings is said to admit interpola- tion on the scale of Besov spaces (B_p^s+1,r, B_p^s,r), 0sσ,0 s σ if for everyF ∈ F

1. F(u)∈L_p for allu∈B^1,r_p ,

2. F(u)∈B_p^σ,r for allu∈B_p^σ+1,r, and

3. there exists a realvalued functionr=r_F (s),r_F (s)r, such that the the increasing function

s_F(s) = sup{t∈[0, σ] :F(u)∈B^t,r_p all u∈B^s+1,r_p } is a mapping of(0, σ)onto(0, σ).

In the linear case, familiar interpolation methods giver =r and with s = θσ that s = θσ for 0 < θ < 1. In case of nonlinear mappings, the

NonlinearMaps between Besov- and Sobolev spaces

interpolation may result in s_T(s) < s even when σ = σ as we will see in Theorem 2.1 below. We will (as in the references above) study the case when F = {F} is a singleton set. Here F is a local mapping consisting of the composition u → f(u) with a reasonably smooth function f. If as aboveF admits interpolation we say, for short, thatthe mappingF admits interpolation. Notice that in the case of compositions, necessarilyσ σ+1.

Here, and in the following [x] denotes the integer part ofx∈R.

Theorem 1.3 (Main Theorem). — Letf ∈ C^[σ]+1 where σ > ^1+2p_p

and1p2p with dual exponentsp, p. Assume that the mapping F : u→f(u)

admits interpolation on the scale of Besov spaces(B_p^s+1,r, B^s_p^,r),0sσ, 0sσ. Moreover assume that there exists a β >0 such that

s_F(s)βs for 0< s < σ,

wheres_F(s)is defined above. Set φ(p) = 2(p−1)(1 + 2p)and letn(p, β) = max(^φ(p)_β ,2 + 2p)−1. Then either the space dimension nn(p, β)or else f(z) =Dz for some constant D.

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 functionp(β) ofβ , 1< p(β)

1+√ 5

2 such that

n(p, β) = 1 + 2p for 1p < p(β) and

n(p, β) = φ(p)

β −1 for p(β)p2.

In additionp(β) = 1 +O(β) asβ→0.

In particular n(2, β) = ¹⁰_β −1 and we get the following important special case:

Theorem 1.5 (Main Theorem L₂ case). — Letf ∈C^[σ]+1, withσ>

5

2, be a function such that the mapping F : u→f(u)

admits interpolation on the scale of Sobolev spaces (H₂^s+1, H₂^s),0sσ, 0sσ. Assume that there is aβ >0such thats_F, the function defined above, satisfies the inequality

s_F(s)βs for 0< s < σ.

Then either the space dimension n ¹⁰_β −1 or else f(z) = Dz for some constant D.

Remark 1.6. — In the Main Theorem “L₂case”, the existence of aβ >0 such that

s_F(s)βs for 0< s < σ

follows for, say,f sufficiently smooth satisfying |f(x)|C|x|, |f(x)| C for allx∈Rⁿ. Here clearlys_F(s)_σ¹sholds for 0< s < σ.

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.

We say that the mappingH₂¹ u→f(u)∈L2 isH¨older continuous of orderα, Lipschitz continuous ifα= 1, if

f(u)−f(v)L2g(uH₂¹,vH₂¹)u−v^α_H¹

2 for u, v∈H₂¹, (1.1) where g(., .) is a locally bounded function onR²₊, increasing in each of its arguments.

Corollary1.7. — Letσ >0,σ> ⁵₂ and letf ∈C^[σ]+1. Assume that H₂¹u→f(u)∈L₂

is Lipschitz continuous and that f(u)_H^σ

2 h(uH¹₂)u_H^σ+1

2 for u∈H₂^σ+1, (1.2) whereh(·)is a locally bounded increasing function onR+. Then eithern <

10or else f(z) =Dz for some constantD.

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. — Letσ >0,σ> ⁵₂ and letf ∈C^[σ]+1. Assume that H₂¹u→f(u)∈L2

Nonlinear Maps between Besov and Sobolev spaces

is H¨older continuous of order α,0< α <1and that the mapping H₂^σ+1u→f(u)∈H₂^σ

has at most power growth, i.e.

f(u)_H^σ

2 h(uH₂¹)u^µ_Hσ+1 2

, for u∈H₂^σ+1, (1.3) where h(·) is a locally bounded increasing function on R₊. Assume that µα. Then there exists an integer n(α) such that eithern < n(α)or else f(z) =Dz for some constant D. Moreover n(α)O(_α¹)asα→0.

In many cases, thegrowth condition(1.3) can be derived in low dimen- sions from (a possibly local version of) the inequality

sup

|γ|=[s]

∂_x^γf(u)L2

Cmax(f^(l)(u)Lru^(l−1)_L∞ : 1l[s]) sup

|γ|=[s]+1

∂_x^γuL2, (1.4)

valid forr > n. Here we denote∂^γ_x=∂_x^γ1₁∂^γ_x2². . . ∂^γ_xnⁿwithγ= (γ1, γ2, . . . , γn) and|γ|= Σⁿ_i=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:

∂_t²u−∆_xu+m²u+f(u) = 0, t >0, x∈Rⁿ.

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 and critical case (i.e.ρ1 +_n−2⁴ ), the solution operators are Lipschitz con- tinuous mappings fromH₂¹ toL₂(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 not Lipschitz continuous as mappings fromH₂¹ toL₂for supercritical expo- nentsρ(ρ >1 +_n−⁴₂). Related results for polynomial nonlinearitiesf have been obtained by Lebeau[14], [15].

2. Nonlinear maps and nonlinear interpolation

In order to prove our results forα 1 we have to introduce and use the Besov spaces B_p^s,q, B_p^s,q and real interpolation based on Peetre’s K- function (see [2], [3] and [7]), in fact mainly for p= p = 2. In general p and p are assumed to be dual exponents, ¹_p +_p¹ = 1, 1 p2 and the standard inclusions (again see [3] pp. 142 and 152-153) between Besov and Sobolev spaces

B_p^s,p ⊂H_p^s ⊂B_p^s,2 B_p^s,p ⊃H_p^s ⊃B_p^s,2

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

H₂^σ+1(Rⁿ)u→f(u)∈H₂^σ(Rⁿ) with the estimate (1.2) forα= 1, i.e.

f(u)_H^σ

2 h(uH¹₂)u_H^σ+1

2 for u∈H₂^σ+1, or the estimate (1.3) in case 0< α <1, i.e.

f(u)_H^σ

2 h(uH¹₂)u^µ_Hσ+1 2

for u∈H₂^σ+1,

withµα. Hereh(·)denotes a locally bounded increasing function onR₊. Forθ∈(0,1)lets=σθ,s=σθ and letr1. Then the inclusion

B^s+1,r₂ (Rⁿ)u→f(u)∈B

α µs,_α^r 2 (Rⁿ), whereµ= 1 if α= 1, holds.

Corollary 2.2. — Under the assumptions of Theorem 3, withσ=σ, the inclusion

H₂^s+1(Rⁿ)u→f(u)∈H₂^s(Rⁿ) (2.1) holds for0< s < s0= ⁿ₂−1 and any s<^α_µsforα <1, and withs =sif α= 1.

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.

The results of Maligranda will in our context replace ^α_µs¯with ˜α¯swhere

˜

α=α(µ− s s0

(µ−α))⁻¹ α µ again by assumption withµ= 1 ifα= 1.

Correspondingly, in Corollary 2.2s < ^α_µs 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.

TheLp-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 ands, s0 satisfy

1. 0< s+ 1< ⁿ₂, 2. ³₂ < s <ⁿ₂, 3. n >^4ss_2s^−2s−2₋₃ , and

4. there exists an integer k2 such thatn >2s+ 2 + 2^s−s_k−1⁺¹. If under these assumptionsf ∈C^[s]+1 and

H₂^s+1(Rⁿ)u→f(u)∈H₂^s(Rⁿ) (2.2) then f is a polynomial of degree at most min([s], k−1)andf(0) = 0.

Condition 4. in Theorem 2.4 is motivated by the following observation:

Proposition 2.5. — LetΦ∈C_o^∞(Rⁿ), Φ(0)= 0 and defineH_τ(x) =

|x|^τΦ(x). ThenH_τ∈H₂^s¯if and only ifs < τ¯ +ⁿ₂.

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.

Proof. —(of Theorem 2.4): Assume thatf(z) is a polynomial of degree k or higher inz. Since 4) holds for ¯kkif it holds fork, we may assume that the coefficient of z^k is nonzero, and that f(z) is of degree k. Take z=H_τ ∈H₂^s+1 such that H_τ(x)^k ∼ |x|^{τ k}Φ(x)^k ∈/ H₂^s. By the proposition this is the case if

τ k+n

2 −s<0< τ+n

2 −s−1

which follows from 4) with a suitable choice ofτ. It remains to prove that 1) through 3) imply thatf is a polynomial of degree at most [s]. We construct a functionv(to use as a counterexample) as follows. Compare Dahlberg [8].

Lety^j = (10j,0, . . . ,0)∈Rⁿ and letu∈C₀^∞ with support in{|x|2}and such thatu(x) =x₁ in{|x|1}. Define

v(x) =

∞

j=1

A_ju(x−y^j

*j

), x∈Rⁿ, (2.3)

where 0 < Aj ↑ ∞ and*j =A^−λ_j withλ >0 to be chosen later. We note

thatv∈C^∞. If

j

A²_j*ⁿ_j^−2(s+1)<∞ (2.4) then a straightforward computation (at least for integerssand for fractional s see the Appendix) shows that v ∈ H₂^s+1(Rⁿ). If f is not a polynomial of degree at most [s], then there is an interval [a, b], a < b, such that

|f^([s]+1)(t)|>0 for atb. If we use the special form of our functionu, we find that

f(v)²_Hs

2 C

j

A^2s_j ⁻¹*^n−2s_j .

SeeClaim 2in the Appendix. If then

j

A^2s_j ⁻¹*^n−2s_j =∞ (2.5) this contradicts the mapping property (2.2), i.e.

f(H₂^s+1(Rⁿ))⊂H₂^s(Rⁿ)

and our Theorem will be proved. Assumptions 1) through 3) imply that it is possible to choose{A_j} andλ 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 byHp, Hp-spaces. Using the definition of the Besov

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 L_p 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, 1p2p where _p¹+_p¹ = 1,q1 ands, s 0satisfy

1. 0< s+ 1< ⁿ_p, 2. ^1+p_p < s<_pⁿ, 3. n >(p−1)^ssp−s−1

s−^1+p_p , and

4. there exists an integer k2 such thatn(¹_p−¹_kp¹)> s+ 1−^s_k.

If under these assumptionsf ∈C^[s]+1 and

B_p^s+1,q(Rⁿ)u→f(u)∈B_p^s,q(Rⁿ) (2.6) then f is a polynomial of degree at most min([s], k−1)andf(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) through 4). Takes= ^1+2p_pβ so thats = ^1+2p_p , which is allowed since by assumptionσ >^1+2p_p .

Let us now refer to Theorem 2.6 conditions 1) through 4). With our choice of s, and withφ(p) = 2(p−1)(1 + 2p) andk= 2, by elementary computations these are satisfied if

1. n > φ1(p, β)≡ ^φ(p)_2β +p 2. n > φ2(p)≡1 + 2p 3. n > φ₃(p, β)≡ ^φ(p)_β −1

4. n > φ4(p, β)≡ _(3−p)β^φ(p) + (p−^p2+2p−1_3−p )

Here 1),2) and 4) are exact reformulations of the corresponding inequalities in Theorem 2.6, while in 3) the exact expression ^φ(p)_β −_p−1¹ has been re- placed byφ3(p, β) =^φ(p)_β −1. This more restrictive choice will simplify the computations and results below.

Notice thatφ, and so φ₁, φ₂ andφ₃ 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 < p(β) < 2 the solution of

p²= (1 +p)1 +β 2

Differentiating, we see thatp(β) is strictly increasing, and so 1< p(β)p(1) = 1 +√

5 2 .

We also have p(β) = 1 +O(β) as β → 0. Since φ(p) is a second order polynomial in p with zeros−¹₂ and 1, we get

φ3φ1φ2forp(β)p2, φ₃φ₁φ₂forp(β)p1.

Thus 1) through 3) hold if n > max(φ3, φ2). We next show that φ4 max(φ3, φ2), and hence also 4) holds ifn >max(φ3, φ2). If we use that

φ(p)

β 2 + 2pfor 1pp(β), φ(p)

β 2 + 2pforp(β)p2, straightforward computations show that

φ4= _(3−p)β^φ(p) + (p−^p2+2p_3−p⁻¹)1 + 2p=φ2 for 1pp(β), φ₄= _(3−p)β^φ(p) + (p−^p2+2p−1_3−p )^φ(p)_β −1 =φ₃ forp(β)p2.

Thus 1) through 4) hold for

n >max(φ₃, φ₂) = max(^φ(p)_β −1,1 + 2p) = max(^φ(p)_β ,2 + 2p)−1.

Since we have assumed that the mappingu→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.

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.

LetC₁⊂C₀ be a Banach space couple. Then the K-functional K(t, φ;C₀, C₁) is defined by

K(t, φ;C₀, C₁) = inf{φ₀C0+tφ₁C1 :φ=φ₀+φ₁, φ_i∈C_i}, whereφ∈C₁ andt0. Notice that

K(t, φ;C_o, C₁)φC0 for t1. (3.1) We defineC_θ,q= (C₀, C₁)_θ,q as the completion ofC₁ in the norm

φCθ,q= (

_∞

0

(t^−θK(t, φ;C₀, C₁))^qdt t )^1/q.

IfC1=H_p^s1, C0=H_p^s0, s1> s0, then Cθ,q defines the Besov spaceB_p^s,q, wheres= (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 f in terms of bounds on Besov space norms. B_p^s,q has the intrinsic norm (among many)

vBp^s,q

|α|=[s]

_∞

0

(t^−s+[s]ω^(r)_p (t, ∂^αv))^qdt t

¹_q

+vLp, (3.2)

where [s] = sup{m∈Z:ms}andr1 and we letωp^(r)(t, v) denote the r-th modulus of continuity ofv inL_p, i.e.

ω^(r)_p (t, v) = sup

|h|t

r

k=0

rk(−1)^r−kv(·+kh)Lp. (3.3)

We may now begin the proof of Theorem 2.1.

Proof. — In the following we letcdenote a locally bounded function of theH₂¹-norm of u, or a constant, wherecmay be different at each occurence.

Similarily we let C denote constants that may vary from line to line. Set K(t, v)≡K(t, v;H₂¹, H₂^σ+1) and K(t, v)≡K(t, v;L₂, H₂^σ). Choosee(t)∈ H₂^σ+1 such that

u−e(t)H₂¹+te(t)_H^σ+1

2 2K(t, u) (3.4)

In particular, by (3.4) we have

e(t)H₂¹ u−e(t)H₂¹+uH₂¹ 3uH₂¹. (3.5) By (1.1) and (1.3) we get

K(t^µ, f(u)) f(u)−f(e)L2+t^µf(e)_H^σ

2

g(u_H2¹,e(t)_H2¹)u−e^α_H¹

2+h(e(t)_H2¹)t^µ(e_H^σ+1

2 )^µ and by the bound (3.5) and since by assumption, g is a locally bounded function increasing in both variables and his a locally bounded increasing function

g(uH₂¹,e(t)H¹₂)c, h(e(t)H₂¹)c,

where c, as remarked above, denotes a locally bounded function ofuH¹₂. Then by the choice ofe(t),

K(t^µ, f(u))cK(t, u)^α+cK(t, u)^µ Thus we get

_∞

0

(t^−θαK(t^µ, f(u)))^αrdt

t c

_∞

0

(t^−θK(t, u))^rdt t + c

_∞

0

(t^−θαµK(t, u))^rµα dt t A change of variable in the first integral yields

_∞

0

(t^−θαK(t^µ, f(u)))^rαdt t = 1

µ

_∞

0

(t^−θαµK(t, f(u)))^αr dt t and hence we obtain

_∞

0

(t^−θαµK(t, f(u)))^αrdt

t c

_∞

0

(t^−θK(t, u))^rdt t + c

_∞

0

(t^−θαµK(t, u))^rµα dt t

NonlinearMaps between Besov- and Sobolev spaces

By the definition and the inclusions between the Besov spaces, noting that

α

µ 1 with equality only ifα=µ= 1, this finally ends up as f(u)

B

sα µ, rα 2

c(u_B^s+1,r

2 )^α+c(u

B^{s αµ+1,r}

µ α 2

)^µ c(u_B^s+1,r

2 )^α+c(u_B^s+1,r

2 )^µ. 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

B_p^s,q⊂B_p^˜s,˜q for 1p∞,1qq˜∞ands˜s. See [3].

Proof. — Since conditions 1)-4) in Theorem 2.6 only involve strict in- equalities we can without loss of generality assume thats, s∈R₊\Ndue to the embeddings above.

Set (as already mentioned in section 3) v(x) =

∞

j=1

A_ju(x−y^j

*j

), x∈Rⁿ,

where u∈C₀^∞ with support in{|x|2}such thatu(x) =x1 in{|x|1}, y^j = (10j,0, . . . ,0)∈Rⁿ and with 0< Aj↑ ∞and*j =A^−λ_j ,λ >0, to be choosen later. We note thatv∈C^∞. It remains to prove that iff isnota polynomial of degree at most [s] there exists aλ >0 such that

v∈C^∞(Rⁿ)

B_p^s+1,p(Rⁿ) f(v)∈B_p^s,p(Rⁿ)

provided conditions 1)-3) in Theorem 2.6 are fullfilled. Note that we have used the first embedding result above here. The full statement of the theo- rem is then a direct consequence of Proposition 2.5.

Claim 1. — Σ^∞_j=1A^p_j*^n−(s+1)p_j <∞implies thatv∈B_p^s+1,p(Rⁿ).

Proof. — Considerω⁽¹⁾p (t, ∂^αv)≡sup_|η|t∂^αv(·+η)−∂^αv(·)Lp where

|α|= [s+ 1] =s+ 1−σ.

For 0< t <1 we have

(t^−σωp⁽¹⁾(t, ∂^αv))^p= sup_|η|t

Rⁿ|^{∂αxv(x+η)−∂}_tσ ^xαv(x)|^pdx Σ^∞_j=1A^p_jsup_|η|t

Rⁿ|^∂

α

xu(^x+η_j^−yj)−∂_x^αu(^x−_j^yj)

t^σ |^pdx=

= Σ^∞_j=1A^p_j*n−[s+1]p−σp

j sup_|η|t

Rⁿ|^∂

α

xu(x−^yj_j^−η)−∂^αxu(x−^yj_j) (_j^t)^σ |^pdx where

sup

|η|t

Rⁿ

|∂_x^αu(x−^yj_%^−η_j )−∂_x^αu(x−^y_%j^j) (_%^t

j)^σ |^pdxCmin((t

*j

)^(1−σ)p,(t

*j

)^−σp) by the mean value theorem. This yields

₁

0(t^−σω⁽¹⁾p (t, ∂^αv))^{p dt}_t Σ^∞_j=1A^p_j*ⁿ_j^−(s+1)p₁

0 min((_%^t

j)^(1−σ)p,(_%^t

j)^−σp)^dt_t CΣ^∞_j=1A^p_j*^n−(s+1)p_j .

For 1t we have ω⁽¹⁾_p (t, ∂^αv)Σ^∞_j=12A_j*

n p−[s+1]

j (

Rⁿ

|∂^α_xv(x)|^pdx)^p1 CΣ^∞_j=1A_j*

n p−[s+1]

j .

But Σ^∞_j=1A^p_j*ⁿ_j^−(s+1)p < ∞ implies that Σ^∞_j=1Aj*

n p−[s+1]

j < ∞ assuming that*_j=A^−λ_j = 2^−λj for some λ >0. Thus

_∞

1

(t^−σω_p⁽¹⁾(t, ∂^αv))^pdt t <∞.

This gives

v_B^s+1,p_p vLp+ Σ_|α|=[s+1](

_∞

0

(t^−σω_p⁽¹⁾(t, ∂^αv))^pdt

t )^1p <∞.

NonlinearMaps between Besov- and Sobolev spaces

Claim 2. — Σ^∞_j=1A^s_j^p−1*ⁿ_j^−sp < ∞implies that f is a polynomial of degree at most [s].

Proof. — Sets= [s] +σ. Assume thatf is not a polynomial of degree at most [s]. Then there existsaandb,a < b, such that

d(a, b)≡ inf

t∈[a,b]|f^([s]+1)(t)|>0.

Set

S_j(η) ={x∈Rⁿ:|x+η−y^j|< *_j anda <A_j

*j

(x₁+η₁−y^j₁)< b}. Since ^A_%^j

j ↑ ∞asj→ ∞it follows that the volume measure ofSj(0) Sj(η) is ¹_2A^%nj_j, where|η|t_j≡ _A^%j_j^b−₁₀^a, forj large enough, sayjj₀. Then we get fort_j+1tt_j,jj₀,

(t^−σω⁽¹⁾_p (t, ∂^([s],0,0,...,0)f(v)))^p =

= sup_|η|t

Rⁿ|^∂

([s],0,0,...,0)

x1 f(Σ^∞_j=1Aju(^x+η−yj_j ))−∂^([s],0,0,...,0)

x1 f(Σ^∞_j=1Aju(^x−yj_j ))

t^σ |^pdx

sup_|η|tΣ^j_k=j0A^[s_k^]p*⁻_k^[s]p

Sk(0)

Sk(η)|^f([s

])(Ak

x1 +η1−yk 1

k )−f^([s])(Ak x1−yk

1

k )

t^σ |^pdx

CΣ^j_k=j

0A^[s_k^]p+p*^−[s_k ^]p−pt^(1−σ)p_A^%nk

k. Thus we get

f(v)_B^s,p

p (_∞

0 (t^−σω_p⁽¹⁾ (t, ∂^([s],0,0,...,0)f(v)))^{p dt}_t)^p1 C(Σ^∞_j=1tj

tj+1(t^−σω_p⁽¹⁾ (t, ∂^([s],0,0,...,0)f(v)))^{p dt}_t)^p1 C(Σ^∞_j=1{(Σ^j_k=j0A^p_k^([s]+1)−1*^n−p_k ^([s]+1))((_A^%j

j)^(1−σ)p−(_A^%j+1

j+1)^(1−σ)p)})^p1 C(Σ^∞_j=j0+1A^s_j^p−1*^n−s_j ^p)^p1 =∞.

This yields a contradiction by the assumption B_p^s+1,pv→f(v)∈B_p^s,p

in the theorem. Hence d(a, b) = 0 for all a < b and f is a polynomial of degree at most [s].