A Frobenius–Nirenberg theorem with parameter

Lemma A.1.

Let ri,si,ai,bi, where 1≤i≤m, be non-negative real numbers. Assume that (r1,…,rm,a1,…,am)∈N2⁢m or (s1,…,sm,b1,…,bm)∈N2⁢m. Let Di be a domain in Rni with the cone property and let fi={fiλ} be a family of functions on Di. Let

Assume that ∞>m≥2. Then

and

Here a=∑ai, b=∑bi, etc. Assume further that ri≥si and ai=bi=0 for all i. Then

Proof.

Let ki,ji∈ℕ. Let ∂xiki⁡∂λiji⁡fi denote a partial derivative of fλ⁢(x) of order ki in x and order ji in λ, evaluated at x=xi and λ=λi. Let k=∑ki and j=∑ji.

We will prove the inequality by estimating derivatives pointwise and Hölder ratios of derivatives. We apply (A.2) for domains Di to obtain

Estimating each term via pointwise derivatives, we can write

for some partial derivative ∂k′ with order k′≤k. While applying (A.2) for the domain I, we see that the last term is bounded by the right-hand side of (A.4). Thus, we have verified (A.4) when ri,sj are integers.

Let βi={ri} and ki=[ri], or βi=0 and ki≤[ri]. We estimate the product of Hölder ratios

We estimate each term in the last sum by considering pointwise derivatives and Hölder ratios. The former can be estimated by computations similar to (A.7)–(A.8). For the Hölder ratio, applying (A.2) for I, we can bound

by the right-hand side of (A.4).

To verify (A.5), it suffices to consider the case with (s1,…,sm,b1,…,bm)∈ℕ2⁢m. Furthermore, it is easy to reduce it to the case with si=0, which we now assume. Now let ki≤ri+ai and ji≤bi. We apply (A.2) to the domains Di to obtain

with a′=∑ℓ⁢|kℓ>⁢aℓ(kℓ-rℓ)≤a. Again we estimate each term via pointwise derivative. We can write

While applying (A.2) for the domain I, we see that the last term is bounded by the right-hand side of (A.5). For Hölder ratios on the right-hand side of (A.9), let βi={ai+r′} and ki=[ai+r′], or βi=0 and ki≤[ai+r′]; and for ℓ≠i, let βℓ={aℓ} or βℓ=0 and let kℓ≤[aℓ]. We estimate the product of Hölder ratios

We need to estimate each term in the last sum by considering pointwise derivatives and Hölder ratios. Applying (A.2) to Di, we can bound |∂λiji⁡fi|[r]⁢∏ℓ≠i|∂λℓjℓ⁡fℓ|0 by the right-hand side of (A.5). Let β={r} and k=[r]. Applying now (A.2) to the interval I, we can bound

by the right-hand side of (A.5). We have verified (A.5) and hence (A.6). ∎

Lemma A.2.

Let D be a domain in Rn with the cone property, and let m′,m be such that 1≤m′<m<∞. With all norms on D, we have

Proposition A.3.

Let r,s∈[0,∞). Let ϕi be Cr+s+1 functions in [ai,bi]. Let D be as in Lemma A.2. Let fi∈C*r,s⁢(D¯) and fiλ⁢(D)⊂[ai,bi]. Suppose that 0∈[ai,bi] and ϕi⁢(0)=0. Let 1≤m′<m. With all norms on D, we have

and

where Cr,s also depends on |ϕi|[ai,bi];[r]+[s]+1.

Proof.

Since ϕi∈𝒞1 and ϕi⁢(0)=0, we get |ϕi⁢(fi)|D;α,β≲|fi|α,β for α,β∈[0,1]. Applying the chain rule and Lemma A.1, we get

with all ri,sj are integers, except for possible one. We can verify (A.11) similarly by using Lemma A.1. ∎

Definition A.4.

Let k≥1 and k≥j≥0. Let Fλ be a family of mappings from D to D′. Let {uλ} be a family of functions on D and let {fiλ},…,{fmλ} be families of functions on D′. Define 𝒫k,j⁢({uλ∘Fλ};{fλ}) to be the linear space spanned by functions of the form

with

and

Analogously, define 𝒫k,j⁢({fλ}) to be the linear space spanned by

with |Kℓ|+jℓ≥1, ∑jℓ≤j, ∑(|Kℓ|+jℓ-1)<k.

Lemma A.5.

If 1≤|K|+j≤k, then

Proof.

Let y=Fλ⁢(x). Let ∂i be a derivative of order i in x. We use the chain rule. Applying ∂ or ∂λ one by one, we can verify that ∂k-j⁡∂λj⁡(uλ∘Gλ) is a linear combination of functions

Here ∑ℓ≥0jℓ=j, and j+∑ℓ≥1|Kℓ|=k. When |K0|=0, it is clear that (A.13)–(A.14) hold. When |K0|>0, we have

Thus (A.17) is in 𝒫k,j⁢({uλ∘Fλ};{Fλ}).∎

Lemma A.6.

Let r,s satisfy (2.3). Let D,D′ be bounded domains in Rm,Rn respectively, which have the cone property. Let Gλ:D→D′ be of class Cr,s⁢(D¯).

1. Assume that |G|D;1,0<2. Then

   (A.18)|{uλ∘Gλ}|D;α,0≤C⁢|u|D′;α,0,0≤α<1,

   (A.19)|{uλ∘Gλ}|D;r,0≤Cr⁢(|u|D′;r,0+|∂⁡u|D′;0,0⁢|∂⁡G|D;r-1,0),r≥1.

2. Assume that |G|D;1,0≤2 and s≥1. Then

   ∥{uλ∘Gλ}∥D;r,s≤Cr{∥u∥D′;r,s+Q~D′,D;r,s(∂u,∂~G)

   +|u|D′;1,0(|G|D;r,s+Q~D′,D′;r,s(∂~G,∂~G))}.

Proof.

(i) Inequality (A.18) is immediate and (A.19) is in [18] and [10].

(ii) Let α={r} and β={s}. Let r=k+α with k≥1. Let y=Gλ⁢(x). Let ∂i⁡uλ⁢(y) be a partial derivatives in y of order i. We know that

i.e. it is a linear combination of functions vλ⁢(x) of the form (A.17). We can express vλ as

where j=j0+…+jm′+m′, and by (A.14) we have j+m+∑ki≤k.

When m=0, it is immediate that by |G|1<2, we obtain

Suppose that m≥1. Computing the Hölder norms, we get

We use Lemma A.1 and obtain

as desired. ∎

Lemma A.7.

Let 1≤r<∞, let 0<ρ<∞, 0<θ<12 and ρi=(1-θ)i⁢ρ, and let Fλ=I+fλ be mappings from Dρ into RN. Assume that F∈C1,0⁢(Dρ¯) and

Then the following statements hold:

1. The mappings Fλ are injective in Dρ. There exist unique Gλ=I+gλ satisfying

   Gλ:Dρ1→Dρ,Fλ∘Gλ=I on Dρ1.

   Furthermore, Fλ:Dρ→D(1-θ)-1⁢ρ and Gλ∘Fλ=I in Dρ2.

2. Suppose that 14<ρ<2. Assume further that F∈𝒞r,0⁢(Dρ). Then {gλ}∈𝒞r,0⁢(Dρ1) and

   (A.20)|g|ρ1;r,0≤Cr⁢|f|ρ;r,0,|u∘G|ρ1;r,0≤Cr⁢(|u|ρ;r,0+|u|ρ;1,0⁢|f|ρ;r,0).

3. Suppose that 14<ρ<2. Assume further that r,s satisfy (2.3) with s≥1, |f|ρ;1<C, and f∈𝒞r,s⁢(Dρ). Then for u∈𝒞r,s⁢(Dρ),

   (A.21)∥g∥ρ1;r,s≤Cr⁢(∥f∥ρ;r,s+Q~ρ,ρ;r,s⁢(∂⁡f,∂~⁢f)+|f|ρ;1,0⁢Q~ρ,ρ;r,s⁢(∂~⁢f,∂~⁢f))

   and

   (A.22)∥uλ∘Gλ∥ρ1;r,s≤Cr⁢(∥u∥ρ;r,s+Q~ρ,ρ1;r,s⁢(∂⁡u,∂~⁢f)+|u|ρ;1,0⁢Q~ρ,ρ;r,s⁢(∂~⁢f,∂~⁢f)).