LOCAL SOLVABILITY FOR SEMILINEAR PARTIAL DIFFERENTIAL EQUATIONS OF CONSTANT STRENGTH

F. Messina

LOCAL SOLVABILITY FOR SEMILINEAR PARTIAL DIFFERENTIAL EQUATIONS OF CONSTANT STRENGTH

Abstract. The main goal of the present paper is to study the local solv- ability of semilnear partial differential operators of the form

F(u)=P(D)u+ f(x,Q1(D)u, ...,QM(D)u),

where P(D),Q₁(D), ...,Q_M(D)are linear partial differntial operators of constant coefficients and f(x, v)is a C^∞function with respect to x and an entire function with respect tov.

Under suitable assumptions on the nonlinear function f and on P,Q1, ...,QM, we will solve locally near every point x⁰ ∈ Rⁿthe next equation

F(u)=g, g∈ B_p,k,

where Bp,kis a wieghted Sobolev space as in H¨ormander [13].

1. Introduction

During the last years the attention in the literature has been mainly addressed to the semilinear case:

(1) P(x,D)u+ f(x,D^αu)_|α|≤m−1=g(x)

where the nonlinear function f(x, v), x ∈ Rⁿ,v ∈ C^M, is in C^∞(Rⁿ,H(C^M))with H(C^M)the set of the holomorphic functions inC^M and where the local solvability of the linear term P(x,D)is assumed to be already known.

See Gramchev-Popivanov[10] and Dehman[4] where, exploiting the fact that the non- linear part of the equation (1) involves derivatives of order≤ m−1, one is reduced to applications of the classical contraction principle and Brower’s fixed point Theo- rem, provided the linear part is invertible in some sense. The general case of P(x,D) satisfying the(P)condition of Nirenberg and Tr`eves [21] has been settled in Hounie- Santiago[12], by combining the contraction principle with compactness arguments.

Corcerning the case of linear part with multiple characteristics, we mention the recent results of Gramchev-Rodino[11], Garello[6], Garello[5], Garello-Gramchev- Popivanov-Rodino[7], Garello-Rodino[8], Garello-Rodino[9], De Donno-Oliaro[3], Marcolongo[17], Marcolongo-Oliaro[18], Oliaro[22].

33

The main goal of the present paper is to study the local solvability of semilinear partial differential operators of the form

(2) F(u)=P(D)u+ f(x,Q1(D)u, ...,QM(D)u)

where P(D),Q₁(D), ...,Q_M(D)are linear partial differential operators with constant coefficients and f(x, v)is as before a C^∞ function with respect to x and an entire function with respect tov.

We will introduce suitable assumptions on the nonlinear function f and on P,Q₁, ...,Q_Min order to solve locally near a point x⁰∈Rⁿthe next equation

(3) F(u)=g, g∈Bp,k,

where Bp,kis a weighted Sobolev space as in H¨ormander [13], with 1≤ p≤ ∞and k temperate weight function. H¨ormander introduced these spaces exactly in connection with the problem of the solvability of linear partial differential operators with constant coefficients, namely one can find a fundamental solution T of P(D)belonging locally to B_∞,eP, see [13].

There are suitable assumptions on the temperate weight function k under which the space B_p,k forms an algebra, see [20], [19]. In [20] we have also proved, under the same conditions, invariance after composition with analytic functions. These results allow us to look for a solution u, in a related B_p,k, giving meaning to the nonlinear term of (2).

More precisely, from basic properties of these spaces (see [13], [24] and the next Sec- tion 2 for notations and results), we know that if u∈ B_p,keP then P(D)u ∈ B_p,k and Q_i(D)u∈B_p,keP/fQ

ifor i =1, ..,M. Assuming that P Q_i for every i , we get

(4) B_p,keP/fQ

i ,→ B_p,k,

then, one should require k satisfying the hypotheses which grant f(x,Q1(D)u, ...,QM(D)u) ∈ Bp,k. Under these conditions the equation (3) will be well defined in the classical sense, for g∈ Bp,kand u∈ B_p,keP.

In this paper we will prove two theorems about the local solvabilty of (3) under dif- ferent assumptions on the non-linear function f and the linear terms P,Q₁, ...,Q_M, completing the results of [20].

In the first theorem, Theorem 11, we assume ^fQ_e^{i(ξ )}

P(ξ ) → 0 when |ξ| → ∞ for i = 1, ...,M. From this hypothesis it follows that the inclusion B_p,keP/fQ

i ,→ Bp,k

is compact (see [13]).

However here we assume on the nonlinearity f(x,0) = 0, corresponding to the standard setting in literature; this is essentially weaker than the hypotheses in [20]

f(x0, v)=0 for everyv∈C^M.

In the proof, we will generalize a well-known property of the standard Sobolev spaces to the case of the weighted spaces B_p,k, namely, if k₁ and k₂ are temperate weight functions, k₂(ξ )≥N >0 for allξ ∈Rⁿ, x⁰is a point inRⁿand

(5) k₂(ξ )

k₁(ξ ) →0 for |ξ| → ∞

then

(6) kukp,k2 ≤C()kukp,k1 ∀u∈B_p,k1∩E⁰(B(x⁰)) where B(x⁰)= {x∈Rⁿ/kx−x⁰k< }and C()→0 for→0.

Applying the previous property and the Schauder Fixed Point Theorem, we will con- clude that the equation (3) is locally solvable at every x⁰∈Rⁿ, and the solution belongs to B_p,kPe.

In the second result, Theorem 12, the assumptions are weaker than those in the first theorem, but the functional frame is now limited to suitable spaces Hk:=B2,k. More precisely, we suppose g∈ Hkhaving the algebra property, and k of the partic- ular form k = ψ^r, with r ∈ R,ψ ∈ C^∞satisfying the ”slowly varying” estimates, stronger than the temperate condition. These estimates were introduced by Beals [1], cf. H¨ormander [14], in connection with the pseudodifferential calculus. Namely, as a particular case of the classes of Beals [1], we may consider the classes of symbols S_ψ^ν, whose definition is obtained by replacing(1+ |ξ|)in the classic S_1,0^ν bounds with the weightψ(ξ ). The corresponding pseudodifferential operators O P S_ψ^ν have a natural action on the weighted Sobolev spaces H_ψ^r. For most of the related properties and definitions recalled in the following let us refer to Rodino [23].

After having transformed the equation (3) into an equivalent fixed point problem, we will deduce, using the assumption Q_i ≺≺ P and applying the properties of the pseu- dodifferential calculus, that there exists C()such that C()→→00 and

kQ_i(D)ukHk ≤C()kP(D)ukHk

for i =1, ...,M and every u∈ H_keP∩E0(B(x⁰)), x⁰any point inRⁿ.

The Contraction Principle will allow us to conclude again that the equation (3) is locally sovable, and the solution belongs to H_keP, at every point ofRⁿ.

We end this introduction by giving a simple example of an operator with multiple characteristics to which our Theorems 11 and 12 apply. Namely consider

(7) D⁴_x

1u−L²u+ f(u, (D²_x

1−L)u, (D²_x

2+L)u)=g.

where L is a constant vector field inRⁿ. Our results will provide for (7) solutions in different scales of functional spaces B_p,k, cf. Section 2, exibiting novelty in com- parison with the papers mentioned at the beginning. We also point out that even for some semilinear partial differential equations with simple characteristics Theorem 11 and Theorem 12 imply new results for the local solvability in more general functional spaces.

2. Preliminary results

We begin with a short survey on Bp,kspaces.

DEFINITION 1. A positive function k defined inRⁿ is called a temperate weight function if there exist positive constants C and N such that

(8) k(ξ+η)≤(1+C|ξ|)^Nk(η); ξ, η∈Rⁿ.

The set of all such functions will be denoted by K.

EXAMPLE1. The basic example of a function in K, is the functioneP defined by

(9) Pe²(ξ )= X

|α|≥0

|P^(α)(ξ )|²

where P is a polynomial and so that the sum is finite. Here P^(α) =∂^αP. It follows immediately from Taylor’s formula that

e

P(ξ+η)≤(1+C|ξ|)^mP(η)e

where m is the degree of P and C is a constant depending only on m and the dimension n.

The proofs of the following results are omitted for shortness; let us refer for them to [13].

DEFINITION 2. If k ∈ K and 1 ≤ p ≤ +∞, we denote by Bp,k the set of all distributions u∈S0such that the Fourier Transformbu is a function and

(10) kukp,k =

(2π )⁻ⁿ

Z

|k(ξ )bu(ξ )|^pdξ 1/p

<+∞.

When p=+∞ we shall interpret kuk^p,k as ess.sup|k(ξ )bu(ξ )|. We shall also write Hk:=B2,k, endowed with the natural Hilbert structure.

EXAMPLE2. The usual Sobolev spaces H(s)correspond to the temperate weight function ks(ξ )=(1+ |ξ|²)^s/2, with p=2.

B_p,kis a Banach space with the norm (10). We have

(11) S⊂B_p,k ⊂S⁰

also in topological sense.

THEOREM1. If k1and k2belong to K and there exists C>0 such that k₂(ξ )≤Ck₁(ξ ), ξ∈Rⁿ,

it follows that B_p,k1 ,→B_p,k2.

REMARK1. Let k ∈ K , in view of estimate (8) withη =0 and Theorem 1, one can find s∈Rsuch that B_p,(1+|ξ|2)^s/2 ,→ B_p,k.

We shall now study how differential operators with constant coefficients act in the spaces Bp,k. Recall that if P(ξ )is a polynomial in n variablesξ1, ..., ξnwith complex coefficients, then a differential operator P(D) is defined by replacingξj by Dj =

−i∂/∂x_jand the functionP is defined by (9).˜

THEOREM2. If u∈ Bp,kit follows that P(D)u∈ B_p,k/P˜.

THEOREM3. If u₁∈B_p,k1∩E⁰and u₂∈ B_∞,k2, it follows that u₁∗u₂belongs to B_p,k1k2, and we have the estimate

(12) ku₁∗u₂k^p,k1k₂ ≤ ku₁k^p,k1ku₂k∞,k₂.

THEOREM4. If u∈ B_p,kandφ∈S, it follows that uφ∈ B_p,kand that (13) kφuk^p,k≤ kφk^1,Mkkuk^p,k,

where M_k ∈K is defined by M_k(ξ )=sup_η^k(ξ_k(η)^+η).

LEMMA1. Let k∈ K . For everyφ∈Sthere exists an equivalent weight function h (i.e. C⁻¹k(ξ )≤h(ξ )≤Ck(ξ )) such that

(14) kφukp,h ≤2kφk1,1kukp,h.

We stress that the weight h depends onφ. We will use Lemma 1 later.

THEOREM5. If k1and k2belong to K and H is a compact set inRⁿ, the inclusion mapping of Bp,k₁∩E0(H)into Bp,k₂ is compact if

(15) k2(ξ )

k1(ξ ) →0 for |ξ| → ∞.

Conversely, if the mapping is compact for one set H with interior points, it follows that (15) is valid.

The Fr`echet space B^loc_p,kis defined in the standard way and corresponding properties are valid for it, in particular we have the following variant of Theorem 3.

THEOREM6. Let u₁∈ B_p,k1 ∩E0and u₂∈ B_∞^loc_,k

2, it follows that u₁∗u₂belongs to B^loc_p,k

1k₂.

Under suitable conditions regarding a temperate weight function k the corresponding space Bp,kis an algebra; for the proof of the next theorems, let us refer to [20].

THEOREM 7. Let 1 < p < +∞, 1/p+1/q = 1, u, v ∈ Bp,k and K(ξ, η)=

k(ξ )

k(ξ−η)k(η) satisfy

(16) sup

ξ

Z

|K(ξ, η)|^qdη≤C₀<+∞.

then uv∈B_p,k andkuvkp,k≤Ckukp,kkvkp,k.

The previous theorem can be generalized to the invariance of the spaces Bp,kunder the composition with entire functions.

DEFINITION3. We write f(x, v)∈C^∞(Rⁿ

x,H(C^M)), whereH(C^M)is the set of the entire functions inC^M, if f(x, v) = P

|α|≥0cα(x)v^α where cα(x) ∈ C^∞(Rⁿ) and for every compact subset H⊂ Rⁿ, there exist Cβ(K), λα(K) > 0 such that sup_x∈K|∂^βcα(x)|<Cβλα,P

|α|≥0λαv^αbeing an entire function ofv∈C^M. THEOREM8. Let f(x, v)∈C^∞(Rⁿ

x,H(C^M)),u₁, ...,u_M∈ B_p,kwith k satisfying the hypotheses of Theorem 7, then f(x,u₁(x), ...,u_M(x))∈ B^loc_p,k.

Now we recall two particular definitions of comparison between differential polynomi- als and related theorems of characterization (see H¨ormander [13], Tr`eves [24] for the proofs).

Letbe an open subset ofRⁿ, P,Q two differential operators with C^∞ coeffi- cients in.

DEFINITION4. We say that P is stronger than Q inif to every relatively compact open subset⁰ of, there exists a constant C(P,Q, ⁰)such that, for all functions ψ ∈C₀^∞(⁰),

(17)

Z

|Q(x,D)ψ(x)|²d x≤C(P,Q, ⁰) Z

|P(x,D)ψ(x)|²d x.

DEFINITION5. Let x⁰be a point of. We say that P is infinitely stronger than Q at x⁰if to every > 0 there existsη > 0 such that, for all functionsψ ∈ C₀^∞() having their support in the open ball centered at x⁰, with radiusη,

(18)

Z

|Q(x,D)ψ(x)|²d x≤ Z

|P(x,D)ψ(x)|²d x.

In other words, P is infinitely stronger than Q at x⁰if estimate (17) holds for some open neighborhoord⁰of x⁰, and if we may choose the constant C(P,Q, ⁰)so that it converges to zero when⁰converges to the set{x⁰}. If P is stronger than Q and Q stronger than P in, we say that P and Q are equally strong or equivalent in.

If P and Q have constant coefficients, the validity of Definition 4 does not depend on

. In this situation, we simply say that P(D)is stronger than Q(D)and we shall write P(D) Q(D). Similarly, the translation invariance of P(D)and Q(D)implies that if P(D)is infinitely stronger than Q(D)at some point ofRⁿ, this is also true at any other point ofRⁿ. Thus we shall say that P(D)is infinitely stronger than Q(D), and write P(D)Q(D).

THEOREM9. Let P(D),Q(D)be two differential polynomials inRⁿ. The follow- ing properties are equivalent:

(a) P(D)is stronger than Q(D);

(b) the function ^{eQ(ξ )}_e

P(ξ ) is bounded inRⁿ; (c) the function^{|Q(ξ )}_e ^|

P(ξ ) is bounded inRⁿ.

REMARK 2. With reference to operator (2), let us observe the following. If we assume that Q_i ≺ P for every i =1, ...,M, in view of Theorem 9, we have ^fQ_e^{i(ξ )}

P(ξ ) ≤C;

therefore for every k∈ K

(19) k(ξ )≤Ck(ξ )Pe(ξ )

f Q_i(ξ ) . In view of Theorem 1 and (19) we obtain

(20) B

p,^k_f^Pe

Qi

,→ B_p,k, ∀i =1, ...,M.

We conclude this section by recalling the definition and the main properties of the pseu- dodifferential operators with symbols in the classes S_ψ, particular case of the classes of Beals [1].

We say that a positive continuous functionψ(ξ ) inRⁿ is a basic weight function if there are positive constants c,C such that

(1) c(1+ |ξ|)^c ≤ψ(ξ )≤C(1+ |ξ|),

(2) c≤ψ(ξ+θ )ψ(ξ )⁻¹≤C, if |θ|ψ(ξ )⁻¹≤c.

(21)

Forν ∈Rwe define S_ψ^ν to be the set of all a(x, ξ )∈C∞(Rⁿ×Rⁿ)which satisfy the estimates

(22) |D^α_xD^β_ξa(x, ξ )| ≤cαβψ(ξ )^ν−|β|, x∈Rⁿ, ξ∈Rⁿ. Let

(23) A f(x)=a(x,D)f(x)=(2π )⁻ⁿ Z

[i xξ]a(x, ξ )fˆ(ξ )dξ , f ∈C^∞

0 (Rⁿ), with a(x, ξ )∈S_ψ^ν. The standard rules of the calculus of the pseudo-differential opera- tors hold for operators of the form (23). Let us review shortly the properties which we shall use in the following.

Recall first that for every basic weight function ψ(ξ ) we may find a smooth ba- sic weight function ψ₀(ξ ), which is equivalent to ψ(ξ ) (i.e. ψ₀(ξ )ψ(ξ )⁻¹ and ψ₀(ξ )⁻¹ψ(ξ )are bounded inRⁿ), such that

(24) |D_ξ^βψ0(ξ )| ≤c_βψ0(ξ )ψ0(ξ )^−|β|, ξ ∈Rⁿ.

Note that, letψ be a smooth function satisfying (24) and (21), thenψ is a temperate weight function.

Equivalent basic weight functions define the same class of symbols; therefore we may assume in the following thatψ(ξ )satisfies (24).

Moreover from (24) it follows thatψ ∈S_ψ¹ and so, for every r∈R,ψ^r ∈S_ψ^r. The operator A in (23) maps continuouslyC∞

0 (Rⁿ)intoC∞(Rⁿ)and it extends to a linear continuous operator fromE0(Rⁿ)toD0(Rⁿ).

According to the previous notations, write Hψ^νfor the Hilbert space of the distributions f ∈S0(Rⁿ)which satisfy

(25) kfk²H_ψν =

Z

ψ(ξ )^2ν| ˆf(ξ )|²dξ <∞. If A has symbol in S_ψ^ν, then for every s∈R:

(26) A : H_ψν+s →H_ψs continuously.

A map A :C∞

0 (Rⁿ)→D0(Rⁿ)is said to be smoothing if it has a continuous extension mappingE0(Rⁿ)intoC∞(Rⁿ); for given operators A1,A2 : E0(Rⁿ) → D0(Rⁿ)we shall write A1∼A2if the difference A1−A2is smoothing.

If a(x, ξ )is in∩^νS_ψ^ν, then a(x,D)is smoothing.

THEOREM 10. Let a₁(x, ξ )be in S_ψ^ν1, let a₂(x, ξ ) be in S_ψ^ν2. Then the product a₁(x,D)a₂(x,D)is in S_ψ^ν1+ν2 with symbol

(27) a(x, ξ )∼X

α

(α!)⁻¹∂_ξ^αa₁(x, ξ )D^α_xa₂(x, ξ ).

3. Statement of the main results

We will study the following semilinear partial differential operator (2) where (1) P(D)is a linear partial differential operator with constant coefficients;

(2) f(x, v)∈C^∞(Rⁿ_x,H(C^M)) and f(x,0)=0 for every x∈Rⁿ;

(3) Q1(D), ...,QM(D)are linear partial differential operators with constant coeffi- cients such that Qi(D)≺ P(D)for i =1, ...,M.

We want to solve locally near every point x⁰∈Rⁿthe equation (3) under the following stronger assumptions on Q_i(D), cf. Introduction.

THEOREM11. Let g ∈ Bp,k, with k satisfying the assumptions of Theorem 7 and k(ξ )≥N >0,∀ξ ∈Rⁿ,1< p<∞. Consider the operator F defined by (2) where, for i =1, ...,M,

(28) fQ_i(ξ )

Pe(ξ ) →0, when|ξ| → ∞;

then for every x⁰∈Rⁿone can find a constant₀>0 and u⁰∈B_p,kP˜ such that

(29) F(u⁰)(x)=g(x), ∀x∈

where= {x∈Rⁿ/kx−x⁰k< 0}.

THEOREM12. Let g∈ Hk, k satisfying the assumptions of Theorem 7 with p=2 be such that k =ψ^r with r ∈R,ψ^r(ξ ) ≥ N >0 for everyξ ∈ Rⁿand assume that (21) holds. Consider the operator F defined by (2) where, for i=1, ...,M,

(30) Q_i ≺≺P;

then for every x⁰∈Rⁿone can find a constant0>0 and u⁰∈H_kePsuch that

(31) F(u⁰)(x)=g(x), ∀x∈

where= {x∈Rⁿ/kx−x⁰k< ₀}.

4. Proof of the results

To prove Theorem 11 we will apply a property of the Sobolev spaces true also for the weighted Sobolev spaces (see [16]).

THEOREM13. If k₁and k₂belong to K, k₂≥ N >0, x⁰is a point inRⁿand

(32) k2(ξ )

k1(ξ ) →0 for |ξ| → ∞ then

(33) kuk^p,k2 ≤C()kuk^p,k1 ∀u∈Bp,k₁∩E⁰(B(x⁰)) where B(x⁰)= {x∈Rⁿ/kx−x⁰k< }and C()→0 for→0.

Proof. First of all note that from the Theorem 5 it follows that the injection of B_p,k1∩ E0(B(x⁰))into B_p,k2is compact.

To prove the Theorem we have to verify that

(34) sup

u∈B_p,k1∩E0(B(x⁰))

kukp,k2

kuk^p,k1

=C() where C()→0 that is equivalent to prove that

(35) sup

u∈Bp,k1∩E0(B(x⁰)),kukp,k1=1

kukp,k2 =C() where C()→0.

We suppose, ab absurdo, that C()does not tend to 0 when tends to 0, then there exists a sequence{ν}ν∈N such thatν → 0 when ν → ∞and C(ν) 9 0 when ν → ∞. We can deduce that there exists a positive constant r and a subsequence {ν_j}j∈Nsuch that ν_j →0 when j → ∞and C(ν_j) >r for every j. Then we obtain that

(36) sup

u∈B_p,k1∩E0(B_ν

j(x⁰)),kukp,k1=1

kukp,k2 =C(_νj) >r for every j.

From the definition of supremum there exists a sequence uν_j ∈ Bp,k₁ with support contained in the ball of fixed center x⁰with radius_νj such that

(37) kuν_jk^p,k1 =1 and kuν_jk^p,k2 ≥r.

The sequence{uν_j}j∈N is bounded in Bp,k₁, then, according to the compactness of the injection of B_p,k1 ∩E0(B_ν

j(x⁰))into B_p,k2, we may assume that there exists a subsequence, still denoted by uν_j, and a distribution u∈ Bp,k₂such that

(38) ku_νj−uk^p,k2 →0 when j → ∞.

But from the properties of the topology of B_p,k2, see 11, we obtain that (39) kuν_j−uk^S0 →0 when j→ ∞.

Since u necessarly has support contained in{x⁰}, we have u=P

0≤|α|≤mc_αδ^(α)

x⁰. We have two possibilities:

1. u ≡ 0, which is absurd, indeed ku_νjkp,k2 → kukp,k2 when j → ∞ and ku_νjkp,k2 ≥r >0 for every j ∈N

2. u6≡0, which is absurd, indeed a nontrivial linear combination of derivatives of the distributionδ_x0 does not belong to B_p,kif k ≥N >0.

Proof of Theorem 11. Fix a point x⁰∈Rⁿ, choose

(40) ϕ∈C₀^∞(Rⁿ), suppϕ⊂B₁(0), ϕ≡1 in B_1/2(0) and define

(41) ϕ(x)=ϕ

x−x⁰

.

We also introduce the functionψ ∈C₀^∞(Rⁿ)defined asψ(x)=ϕ(x−x⁰). We observe that from the general theory of the linear partial differential operators with constant coefficients (see [13]) it follows that there exists a fundamental solution T ∈ B^loc

∞,P˜ of the linear part P(D)of the semilinear operator F.

In order to obtain our result we shall replace the weight function k by an equivalent function h∈ K ; obviously we have Bp,k = Bp,h. The function h will be determined later; Lemma 1 will play a crucial role in this connection (see H¨ormander [14] Vol. II Theorem 13.3.3 for a similar argument). Let us define

B_p,h;R= {u∈B_p,h/kuk^p,h≤ R}. We can consider the operator

(42) F˜: B_p,h;R →Bp,h

with R≥2kgk^p,hgiven by the following espression

(43) F˜[v]=g−ψf(x,Q₁(D)(ϕψLψv), ...,Q_M(D)(ϕψLψv)) where L :=T∗, T the fundamental solution of the operator P(D).

Note that, from the hypothesis (28),

(44) h(ξ )

h(ξ )P(ξ )˜

˜ Q_i(ξ )

→0 when |ξ| → ∞

and so, for every i=1, ..,M, the injection of B

p,^h_˜^P˜

Qi

∩E0(B(x⁰))into Bp,his compact.

In Theorem 11 we have also supposed that k(ξ )≥N >0, then we obtain that

(45) h(ξ )P(ξ )˜

˜

Qi(ξ ) ≥C>0 for everyξ.

From (44) and (45) it follows that the temperate weight functions^hP˜

˜ Qi

and h satisfy the hypothesis of the Theorem 13. Note that, ifv∈ B_p,h;Rthenψv∈B_p,h∩E⁰; therefore Lψv ∈ B^loc

p,hePandψLψv ∈ B_p,heP so we obtainF˜[v] ∈ B_p,hand, let x⁰be a fixed point inRⁿ, for every i =1, ..,M, there exists a function Ciof the variablesuch that Ci()→0 when → ∞and

(46) kuk_p,hP˜ Qi˜

≤Ci()kuk^p,h ∀u∈ Bp,h∩E⁰(B(x⁰)).

To prove Theorem 11 we will verify that there exists₀>0 such that the corresponding operator defined by

(47)

˜

F₀ : Bp,h;R→ Bp,h

˜

F₀[v]=g−ψf(x,Q1(D)(ϕ₀ψLψv), ...,QM(D)(ϕ₀ψLψv))

verifies the hypotheses of the Schauder Fixed Point Theorem, namely it is continuous, it is defined from Bp,h;Rto itself and it maps bounded sets into relatively compact sets.

First of all we will prove thatF˜(Bp,h;R)is relatively compact for every >0; to this end we will consider a sequence{vj}j∈Nof distributions in Bp,h;Rand we will verify that there exists a convergent subsequence in Bp,kof the sequence{ ˜F[vj]}j∈N. Set{uj}j∈N:= {ϕψLψvj}j∈N, thenkujkp,hP˜ ≤ R1(). Indeed, by Theorem 3 and Theorem 4,

ku_jk_p,hP˜ = kϕψ(T ∗ψv_j)k_p,hP˜=kϕψ(ψ˜T ∗ψv_j)k_p,hP˜≤Ck ˜ψT ∗ψv_jk_p,hP˜

≤C_1,k ˜ψTk_∞,P˜kψv_jk^p,h≤C_2,k ˜ψTk_∞,P˜kv_jk^p,h≤C_3,R (48)

whereψe∈C₀^∞andψe≡1 in K= {x∈Rⁿ/(x+suppψvj)∩suppϕψ 6= ∅}. Set

{E_iv_j}i=1,...,M={(E₁v_j, ...,E_Mv_j)}:={Q(D)u_j}={(Q₁(D)u_j, ...,Q_M(D)u_j)};

by Theorem 2, we also obtainkEivjk_p,^heP

fQi ≤ R2()for every i =1, ...,M and j∈N. The sequences{E_iv_j}j∈N are bounded in B

p,^h_f^eP

Qi

and belong to E0(B(x⁰)), indeed {uj} ∈E0(B(x⁰))and the differential operators Qi(D)do not increase the support.

Then, according to the compactness of the injection of B

p,^h_f^eP

Qi

∩E0(B(x⁰))into B_p,h, we may suppose that there exists a subsequence{v_jν}ν∈Nof{v_j}j∈Nand a distribution z₁such that

kE₁v_jν −z₁kp,h →0 when ν→ ∞.

The sequence{E₂v_jν}ν∈N is a subsequence of{E₂v_j}j∈N and so is still bounded in Bp,^h_g^eP

Q2

, then there exists{v_jν

l}l∈Nand a distribution z₂such that kE2vj_νl −z2k^p,h →0 when l → ∞.

Iterating this process for every i =1, ..,M we will find a subsequence{v_jm}m∈Nof {v_j}j∈Nand M distributions z₁, ..,z_Msuch that

kEivj_m −zik^p,h→0 when m→ ∞. From the continuity of the injection of B

p,^h_f^eP

Qi

∩E⁰(B(x⁰))into B_p,h, for every i , we may assume that the sequences{E_iv_jm}m∈Nare bounded in B_p,hand sokE_iv_jmkp,h≤ R^∗()for every 1≤i ≤M and m∈N.

To complete the proof we have to verify that{ψf(x,E₁v_jm, ..,E_Mv_jm)}m∈Nis conver- gent in B_p,h.

We will prove that the operator

(49) F : B_p,h;R∗×B_p,h;R∗×...×B_p,h;R∗→ B_p,h defined as

(50) F[w¹, .., w^M]=ψf(x, w¹, .., w^M) is sequentially continuous, then

keF[v_jn]+F[z₁, ..,z_M]−gkp,h=

= kg−F[E₁v_jn, ..,E_Mv_jn]+F[z₁, ..,z_M]−gkp,h

will tend to zero ifkF [E1vj_n, ..,EMvj_n]−F[z1, ..,zM]k^p,htends to zero.

Let{wn}n∈N:= {(w_n¹, ..., w_n^M)}n∈N∈B_p,h;R∗×...×B_p,h;R∗ and w:=(w¹, .., w^M)∈

B_p,h;R∗×...×B_p,h;R∗be such that(w_n¹, ...w_n^M)→(w¹, ..., w^M)for n→ ∞, we must show that

(51) F[w¹_n, ..., w_n^M]→ F[w¹, ..., w^M].

Reminding the Cavalieri Lagrange Formula, we have to estimate F[w_n¹, ...w_n^M]−F[w¹, ...w^M]k^p,h=

kψf(x, w_n¹, .., w_n^M)−ψf(x, w¹, .., w^M)k^p,h= (52)

kψ XM

i=1

(w_nⁱ −wⁱ) Z 1

0

∂_vif(x, w¹+t(w_n¹−w¹), .., w_n^M+t(w_n^M−w^M))dtk^p,h.

Set

(53) Gi(x,y,z):= Z 1

0

∂v_if(x,y¹+t(z¹−y¹), ..,y^M+t(z^M−y^M))dt

and note that G_i(x,y,z)∈C^∞(Rⁿ

x,H(C^2M)) for every i =1, ..,M.

Then G_i(x,y,z) = P

|α|≥0a_i,α(x)(y,z)^α where a_i,α(x) ∈ C^∞(Rⁿ) and for every compact subset H⊂⊂ Rⁿ, there exist C_β(H), λ_α(H) > 0 such that sup_x∈H|∂^βa_i,α(x)|<C_βλ_α, beingP

|α|≥0λ_α(y,z)^αan entire function.

Applying Theorem 7, we may further estimate (52) by

(54) C

XM

i=1

k(wⁱ_n−wⁱ)kp,hkψG_i(x, w_n, w)kp,h.

Set H :=suppψ, from the Fourier Transform properties we can deduce that there exist s∈Nand l∈Nsuch that

(55) kψai,αk^p,h≤ A(H)kψai,αk^l_C(s),∀α∈N.

Indeed, let us consider B_p,(1+|ξ|2)^s0/2 such that s⁰ ∈Nand B_p,(1+|ξ|2)^s0/2 ,→ B_p,h; set 1/p+1/q = 1, if p ≥ 2 then 1 ≤ q ≤ 2 and the Sobolev space W^q,s0 = {u ∈ S0/F−1((1+ |ξ|²)^s0/2bu)∈ L^q}is included with continuity in B_p,(1+|ξ|2)^s0/2. Then in this case

kψa_i,αkp,h≤ kψa_i,αkW^q,s0 = X

|β|≤s⁰

k∂^β(ψa_i,α)kL^q

≤ X

|β|≤s⁰

Z

K

(sup(|∂^β(ψai,α)|)^q)d x 1/q

=(measH)^1/q X

|β|≤s⁰

k∂^β(ψa_i,α)kC⁰ = Akψa_i,αk_C^(s0)

where A :=(measH)^1/q.

If 1< p<2 we apply the H¨older inequality with r =2/p:

kψa_i,αk^p_p,h = Z

|h(ξ )ψ[a_i,α(ξ )|^pdξ

= Z

((1+ |ξ|²)^s00|h(ξ )ψ[a_i,α(ξ )|)^p

1 (1+ |ξ|²)^s00

p

dξ

≤ Z

((1+ |ξ|²)^s00|h(ξ )ψ[a_i,α(ξ )|)²dξ

1/rZ 1 (1+ |ξ|²)^s00pr0

1/r⁰

≤ D Z

((1+ |ξ|²)^s∗|ψa[_i,α(ξ )|)²dξ 1/r

=Dkψa_i,αkH^p_(s∗),

where r⁰is such that 1/r+1/r⁰=1, s⁰⁰is such that s⁰⁰pr⁰>n/2 and s^∗=s⁰+s⁰⁰∈N. But we have

kψai,αk^H(s∗) ≤ A⁰kψai,αkC^(s∗), where A⁰:=√

measH, so in this case we have get (55) with l=p.

Therefore applying the algebra property of B_p,h, the estimate (55) and the algebra property of C^(s)we obtain

kψG_i(x, w_n, w)k^p,h≤ AX

α≥0

kψa_i,αk^l_C^(s)k(y,z)^αk^p,h

≤ A⁰kψk^l_C^(s)X

α≥0

ka_i,α|^Kk^l_C^(s)

C^|α|(kyk^p,h,kzk^p,h)^α

≤ A⁰⁰(K)X

α≥0

|maxβ|≤s{C^l_β}λα(C R^∗)^|α|

= A⁰⁰⁰(K)X

α≥0

λ_α(C R^∗)^|α| <D.

(56)

Replacing (56) in (54) we have

kF [w¹_n, ..., w_n^M]−F[w¹, ..., w^M]k^p,h≤ D XM

i=1

kwⁱ_n−wⁱk^p,h that tends to zero whenkw_nⁱ −wⁱkp,h→0 for every i =1, ..M.

As second step of the proof, now we will prove that there exists₀ >0 such that the corresponding operatorFe₀ is defined from B_p,h;Rto itself, namely letv∈ B_p,hsuch thatkvkp,h≤ R thenkFe₀[v]kp,h≤ R.

We have to estimate

(57) keF[v]k^p,h≤ kψf(x,Q₁ϕψLψv, ..,Q_MϕψLψv)k^p,h+ kgk^p,h.

We now determine our choice of the equivalent function h (see the beginning of the proof). Namely we take, as a new equivalent weight, h such that hP is equivalent to˜

kP, with k the original weight, and satisfies the estimate (14) in Lemma 1 with˜ φ=ϕ. We obtain

(58) kϕψLψvkp,hPe≤2kϕk^1,1kψLψvkp,heP≤2 Akϕk^1,1kvkp,heP.

We introduce the function,8 defined as8(x):=ϕ(x/)so from the Fourier Trans- form properties, we have thatkϕk^1,1= k8k^1,1, withϕ,ϕas in (40) and (41). Noting that

(59) 8c(ξ )=ⁿbϕ(ξ )

we obtain

kϕk^1,1= k8k^1,1= kb8kL¹ = Z

|ⁿbϕ(ξ )|dξ

= Z

|bϕ(z)|dz, (60)

so we have proved thatkϕk^1,1does not depend on.

Replacing (60) in (58), we get kϕψLψvkp,heP ≤ A⁰kvk^p,h = R3 and so kQ_iϕψLψvk_p,^heP

fQi ≤ R₄. Applying Theorem 13 we obtain

kQiϕψLψvk^p,h≤Ci()kQiϕψLψvk_p,^hPe

Qif ≤Ci()R4

with Ci()→0 when→0, for every i =1, ..,M.

Set QiϕψLψv := Ei,v and C() := maxi=1,..,M{Ci()}, then C() → 0 when →0 andkEi,vk^p,h≤C()R4. But f(x, v)∈C^∞(Rⁿ

x,H(C^M))with f(x,0)=0, then, by the same arguments used to obtain (56)

kψf(x,Q₁ϕψLψv, ..,Q_MϕψLψv)kp,h = kψf(x,E_1,v, ..,E_M,v)kp,h

= kψ X

|α|>0

c_α(x)(E_1,v, ..,E_M,v)^αkp,h

≤ X

|α|>0

C₁kψc_αkp,h(k(E_1,vkp,h, ..,kE_M,vkp,h)^αC^|α|

≤ X

|α|>0

C₂λ_α(R₄C(), ..,R₄C())^αC^|α|= X

|α|>0

C₂λ_α(eC())^|α|

=X

l>0

C₂µ_l(eC())^l (61)

where eC()→0 when →0 and P

l>0µlv^l <+∞for everyv∈C.

If we have chosensufficiently small, thenC() <e 1 and we may further estimate X

l>0

C₂µ_l(eC())^l =C₂C()e X

l≥0

µ_l+1(eC())^l

≤C₂C()e X

l≥0

µ_l+1=C₄C().e (62)