Products, Paraproducts and the Taylor formula

In this section we will understand how one can give an asymptotic expansion for products of conormal distributions. Note at first that the usual ansatz of a paramul-tiplication with the a priori Hölder-Zygmund estimates is insufficient, since even for type (1,0) symbols obeying infinite improved regularity the corresponding paradif-ferential operator yields a symbol of finite improved regularity.

This is due to the anisotropic regularity of our distributions. The regularity in the x⁰⁰-directions and in thex⁰-direction are to be separated, which implies a correspond-ing adaptation to paraproducts.

Furthermore we will introduce a Taylor expansion in the case of codimension 1.

There are several technical issues if one would want to extend this result to higher codimensions. Later on the Taylor expansion with remainder term will enable us to give an iterating scheme for deriving a symbolic solution to linearized problems.

3. PRODUCTS, PARAPRODUCTS AND THE TAYLOR FORMULA 27

3.1. Taylor formula. We start the Taylor expansion with a simple continua-tion result.

Proposition 2.15. Let v ∈ C∗^ρ(R^n−k) with ρ > 0¹ and let φ ∈ C_c^∞(R^k), with R φ(η⁰)đη⁰ = 1, be supported in an annulus, then

[e(v)](x⁰⁰, η⁰) =^X

ν

ψν(Dx⁰⁰)v(x⁰⁰)2^−νkφ(2^−νη⁰)∈S_1,1^−ρ−k(R^n−k×R^k) is a symbol yielding an extensionE(v)∈I_1,1^−ρ−k(Rⁿ,R^n−k) of v via

[E(v)](x) = Z

e^ix0η0[e(v)](x⁰⁰, η⁰)đη⁰.

Having further restrictions onφ we can deduce for α∈N^k ρ >|α|

Z

(iη⁰)^αφ(η⁰)đη⁰ = 0 ⇒ ∂_x^α0[E(v)](x⁰⁰,0) = 0.

(1)

Proof. Let k−, k₊ ∈ Z such that suppφ ⊆ B₂k+\B₂k−. Let η⁰ be given with

|η⁰| ≥2^k−−1, then we can restrict the sum to [e(v)](x⁰⁰, η⁰) =

dlog₂(η⁰)e+k++1

X

ν=blog₂(η⁰)c+k−−1

ψ_ν(D_x⁰⁰)v(x⁰⁰)2^−νkφ(2^−νη⁰)

and if|η⁰|<2^k−−1, then [e(v)](x⁰⁰, η⁰) = 0. Now taking partial derivatives, we obtain

|∂_x^α00∂_η^β0[e(v)](x⁰⁰, η⁰)|.

dlog₂(η⁰)e+k++1

X

ν=blog₂(η⁰)c+k−−1

2ν(|α|−ρ−|β|−k)

.η⁰−ρ−k+|α|−|β|

Thus [e(v)](x⁰⁰, η⁰) ∈ S_1,1^−ρ−k. The property for the restriction follows immediately from ^Rφ(η⁰)đη⁰ = 1 and v(x⁰⁰) =^P_νψν(Dx⁰⁰)v(x⁰⁰). Likewise the conclusion for the

derivative is immediate.

For the case k = 1 we will use this Proposition now extensively as a way to get a Taylor-like approximation of functions inI_1,1^m,ρwith (non-constant) Taylor coefficients having full symbols in S_1,1^m−ρ+j, to stay in G^m−ρ.

Further we will use this continuation to show, that we can put more restrains on the remainder part of a conormal distribution. Namely with the following definition.

Definition 2.16. We define the following subspace G^µ₀(Rⁿ,R^n−k)⊆G^µ(Rⁿ,R^n−k), with

u∈G^µ₀(Rⁿ,R^n−k)⇔u∈G^µ(Rⁿ,R^n−k)∧∂_x^α0u(x⁰⁰,0) = 0 ∀|α|<−µ−k Theorem2.17 (Taylor expansion). Let u∈I_1,1^m,ρ(Rⁿ,Rⁿ⁻¹)with` <−m−1≤`+ 1 for some `≥0. Then introducing

E_j(u) =E(∂_x^j0u(x⁰⁰,0))/j!∈I_1,1^m−ρ+j e_j(u) =e(∂_x^j0u(x⁰⁰,0))/j!∈S_1,1^m−ρ+j there is a symbol a^r<sub>`</sub> ∈S<sub>1,1</sub><sup>m+`,ρ</sup> with associated function u^r_{`</sub> obeying u<sup>r</sup><sub>`}(x⁰⁰,0) = 0 and a remainder function uG∈G^m−ρ₀ such that

u(x) =^X

j≤`

[E_j(u)](x)(x⁰)^j+u^r<sub>`</sub>(x)(x<sup>0</sup>)<sup>`</sup>+u_G(x).

1Indeed this can be dropped if instead of [e(v)](x⁰⁰,0) =v(x⁰⁰) we require lim→0[e(v)](x⁰⁰, ) =v(x⁰⁰) as forρ≤0 the trace is not well defined.

Proof. First assume some decomposition of the formu=u_C+˜u_G, withb(x⁰⁰, η⁰) as the full symbol foruC. We can use Proposition 2.15 with a choice forφsuch that (1) holds for all |α|<−m+ρ−1, to extend

∂_x^j0u_C(x⁰⁰,0)/j! = Z

(iη⁰)^jb(x⁰⁰, η⁰)đη⁰/j!∈C_∗^{−m+ρ−1−j}(Rⁿ⁻¹) for all j≤`. To understand the embedding observe

|ψ_ν(D_x⁰⁰)∂_x^j0u_C(x⁰⁰,0)|. We can thus reduce the symbolb via

b^r(x⁰⁰, η⁰) =b(x⁰⁰, η⁰)−^X

So to estimate symbol properties of b^r_`(x⁰⁰, η⁰) it is sufficient to study x⁰⁰-regularity behavior of b^r_j, since η⁰ derivatives follow by induction. Due to the alternative expression we may assume in our estimates, that η⁰ ≤ 0, else switch sign and the

3. PRODUCTS, PARAPRODUCTS AND THE TAYLOR FORMULA 29

expression to still obtain the integral as a tail integral. Thus we obtain for all m+s+j≤0 if 1 +|η⁰|>2^ν note thatu^r_C,2 a priori does not vanish of order`. Next, we can use Proposition 2.15 to obtain symbolsa^G_j ∈S^m−ρ+j_1,1 yielding an expansion of

∂_x^j0vG(x⁰⁰,0)∈C_∗^{−m+ρ−1−j}(Rⁿ⁻¹)

which fulfill the requirements by construction.

We also prove for codimension 1 thatG^{m,`</sup><sub>0</sub> = (x<sup>0</sup>)G<sup>m+1,`+1}₀ , which is handy in some computations.

Proposition 2.18. Let the codimension k = 1, let u ∈ G^m,` with m < −2, u(x⁰⁰,0) = 0, then

(x⁰)⁻¹u(x)∈G^m+1,`+1

Proof. As (x⁰)(x⁰)⁻¹u(x) =u(x)∈G^m,`, the we only thing we need to show is (x⁰)⁻¹u(x)∈G^m+1. Which is equivalent to

ψ_ν(D)(x⁰)⁻¹u(x)

L^p .2^{ν(m+2−1/p)}

for all 1< p≤ ∞with constants depending on p. Asu(x⁰⁰,0) = 0 we have (x⁰)⁻¹u(x) = (x⁰)⁻¹(u(x)−u(x⁰⁰,0)) =

Z 1 0

∂_x⁰u(x⁰⁰, sx⁰)ds Now let v(x) =∂_x⁰u(x), then

ψν(D)v(x⁰⁰, sx⁰) =ψν(D_x⁰⁰, sD_y⁰)v(x⁰⁰, y⁰)_y0=sx⁰

=



ψ_ν(D_x⁰⁰, sD_y⁰) ^X

µ≥ν−2

ψ_µ(D_x⁰⁰, D_y⁰)v(x⁰⁰, y⁰)





y⁰=sx⁰

Now we have from u∈G^m, that

kψ_µ(D)vk_Lp .2^{µ(m+2−1/p)}⇒ ^X

µ≥ν−2

kψ_µ(D)vk_Lp.2^{ν(m+2−1/p)} Now asψ_ν(D_x⁰⁰, sD_y⁰) is uniformly L^p continuous for 0≤s≤1, we obtain

ψ_ν(D_x⁰⁰, sD_y⁰)v(x⁰⁰, y⁰)_Lp .2^{ν(m+2−1/p)} And thus by scaling, we obtain

ψν(D)v(x⁰⁰, sx⁰)_Lp.s^−1/p2^{ν(m+2−1/p)} And we obtain

ψ_ν(D)(x⁰)⁻¹u(x)

L^p. Z ₁

0

s^−1/p2^{ν(m+2−1/p)}ds. 1

1−1/p2^{ν(m+2−1/p)}

Which provides the claim.

3.2. Products and Paraproducts. To approach the main theorem of this section introduce the cutoff functionχ(ζ, η)∈S⁰ with the following properties.

suppχ(ζ, η)⊆ {|ζ| ≤B(|η|+ 1),|η| ≤B(|ζ+η|+ 1)}

χ(ζ, η) = 1 ∀|η| ≥B(|ζ|+ 1) for someB >2. Then we can introduce

Φ(ζ, η) = 1−χ(ζ, η)−χ(η, ζ) with the properties

supp Φ⊆ {|ζ| ≥B(|η|+ 1),|η| ≥B(|η|+ 1)}

Φ(ζ, η) = 1 ∀B(|η+ζ|+ 1)≤min(|ζ|,|η|)

3. PRODUCTS, PARAPRODUCTS AND THE TAYLOR FORMULA 31

An example of such cutoff functions are for somed≥3 χ(ζ, η) = ^X

These cutoff functions are used to obtain a paraproduct. Observing that the Fourier transform can be split into these cutoff regions

uv(ξ) =c Z

u(ξˆ −η) (χ(ξ−η, η) +χ(η, ξ−η) + Φ(ξ−η, η)) ˆv(η)đη,

we can extract the formulas for the paradifferential operators given by the symbols u_χ(x, η) =χ(D, η)u(x) =

And thus we obtain the decomposition

u(x)v(x) =uχ(x, D)v(x) +vχ(x, D)u(x) +uΦ(x, D)v(x)

=uχ(x, D)v(x) +vχ(x, D)u(x) +vΦ(x, D)u(x).

In an abuse of notation, we also introduce for symbols aχ(x, ξ⁰) =

The rest of the section is devoted to provide subresults yielding the following main theorem.

Theorem 2.19. Let u ∈ I_1,1^m1,ρ1 and v ∈ I_1,1^m2,ρ2 with mi + 1 < 0, mi ∈/ Z and m1 ≥ m2, where u has full symbol a1 and b has full symbol a2. Let `i ∈ N0 be maximal with the propertym<sub>i</sub>+ 1 +`_i<0. We can give the following approximation for a product

In some of our following statements we are going to be a bit wasteful on the precise improved regularity of the component. This is due to the fact that these terms reach below the threshold of max(mi−ρi) in their remainder terms and those are neglectable in combination with other terms.

Firstly we are going to apply the standard paramultiplication following the descrip-tion in [Hö03b, Secdescrip-tion 10.2.]. We will establish, that any term in G^m−ρ₀ can be neglected within a multiplication regarding the product up toG^m−ρ₀ .

In the general paramultiplication scheme as described above, we obtain uχ(x, η) ∈ S_1,1^0,ρ and uΦ(x, η) ∈ S_1,1^−ρ if u ∈ C∗^ρ, and as an aside moreover we will obtain that for b(x, D) =u_χ(x, D)^∗ we have b(x, η) ∈ S_1,1^m,ρ, which we will not correspondingly get foruΦ(x, D)^∗. Again we refer to [Hö03b, Section 10.2.] for more details. Using this we prove the following Proposition.

Proposition 2.20. Let u∈I_1,1^m,ρ(Rⁿ,R^n−k) with m <−k andv∈G^m−ρ₀ , then u(x)v(x)∈G^m−ρ₀

Proof. Using paramultiplication, we obtain immediately withu∈C_∗^−m−k (uχ(x, D) +u_Φ(x, D))v∈G^m−ρ

Thus we need to investigate vχ(x, η) and as ∂_x^αv(x⁰⁰,0) = 0 for all |α| ≤ N with N ∈N such thatρ < N <−m+ρ−k, we can compute

∂_x^α∂_η^βv_χ(x⁰⁰,0, η)=∂_η^βχ(D, η)∂_x^αv(x⁰⁰,0)=−∂_η^β(1−χ(D, η))∂_x^αv(x⁰⁰,0) .hηi^{−|β|−N+|α|}

Note that as we a priori haveuχ ∈S_1,1^{0,−m+ρ−k} and thus for |α|> N we obtain

∂_x^α∂_η^βuχ(x, η).hηi^{−|β|+|α|−N}

We conclude that∂_x^α0uχ(x⁰⁰,0, η)∈S_1,1^−N+|α|(R^n−k×R^k). And thus by Theorem 2.14 we obtain symbol terms inS^−N_1,1 ^+m within the expansion of

vχ(x, η)#a(x⁰⁰, η⁰) = ^X

j<N

iD_y⁰⁰D_ξ⁰⁰−iD_x⁰D_ξ^0jvχ(x, ξ)a(y⁰⁰, ξ⁰)/j!_|

y00=x00 x0=ξ0=0

+rN(x⁰⁰, ξ⁰) Z

e^ix0ξ0rN(x⁰⁰, ξ⁰)đξ⁰ ∈I_1,1^m−N,ρ−N =I_1,1^m−ρ

⇒ v_χ(x, D)u∈G^m−ρ

Thus it remains to argue that u(x)v(x) vanishes at x⁰ = 0 of order M with M <

−m−k+ρ maximal. But as u ∈ C_∗^−m−k ⊆ C⁰ this follows immediately from v

vanishing of oder M there.

Thus all the relevant parts of multiplication takes place for symbols only. We are therefore able to alter the scheme of paramultiplication to take place only in η⁰ in order to sidestep some anisotropic improved regularity issues. Then we can analo-gously split

Z

a(x⁰⁰, ζ⁰)b(x⁰⁰, ξ⁰−ζ⁰)đζ⁰

= Z

a(x⁰⁰, ζ⁰) χ(ζ⁰, ξ⁰−ζ⁰) +χ(ξ⁰−ζ⁰, ζ⁰) + Φ(ζ⁰, ξ⁰−ζ⁰)b(x⁰⁰, ξ⁰−ζ⁰)đζ⁰ We first restate the symbol expansion formula for paradifferential operators acting on conormal distributions for this kind of operators. They evidently have anisotropic improved regularity but to have ax⁰-dependent symbol is mainly for consistency of notation, in most computations we will stick with the symbols.

3. PRODUCTS, PARAPRODUCTS AND THE TAYLOR FORMULA 33

Note that we can capture the behavior ofa_χ, which is the standard symbol behavior away fromm+k= 0 with a log(hη⁰i) term atm+k= 0 distorting the estimates in a neighborhood ofm+k= 0, with the following remark.

Remark 2.21. Leta(x⁰⁰, η⁰)∈S_1,1^m,ρ, then a_χ(x, η⁰)_|

x0=0 satisfies

|ψ_ν(D_x⁰⁰)∂_η^β0a_χ(x, η⁰)_|

x0=0|.2^−νρ











2^−νMhη⁰i^m+M+k hη⁰i ≤2^ν

2^ν(m+k)(1 +^(2−νhη_m+k^0i)m+k−1) hη⁰i ≥2^ν, m6=−k 1 + log(2^−νhη⁰i) hη⁰i ≥2^ν, m=−k But as we are not going to thoroughly study the neighborhood ofm+k= 0, we do not give a proof of this assertion. First we give the case m1 +k > 0, which yields only a qualitative statement and does not allow a symbol approximation.

Proposition 2.22. Let a(x⁰⁰, η⁰)∈ S_1,1^m1,ρ1 and b(x⁰⁰, η⁰) ∈S_1,1^m2,ρ2 with m₁+k > 0, then

c(x⁰⁰, ξ⁰) = Z

a(x⁰⁰, ζ⁰)χ(ζ⁰, ξ⁰−ζ⁰)b(x⁰⁰, ξ⁰−ζ⁰)đζ⁰ ∈S_1,1^{m1+m2+k,min(ρi)}

Proof. Due to the support properties ofχ, we can see that on the support of the integrand we havehξ⁰−ζ⁰i ∼ hξ⁰i, since we directly have|ξ⁰−ζ| ≤B(|ξ⁰|+1) and the converse is obtained if |ζ⁰| ≤ |ξ⁰|/2 by|ξ⁰−ζ⁰| ≥ |ξ⁰| − |ζ⁰|and if|ζ⁰| ≥ |ξ⁰|/2 by B(|ξ⁰−ζ⁰|+ 1)≥ |ζ⁰|. Thus we also have|ζ⁰|.hξ⁰i on the support of the integrand, making it compact. We can then estimate

|ψ_ν(D_x⁰⁰)∂_ξ^β0c(x⁰⁰, ξ⁰)|. Z

|ζ⁰|.hξ⁰i

2^−νρ1ζ^0m1ξ^0m2−|β|+ζ^0m12^−νρ2ξ^0m2−|β|đζ⁰ .2^{−νmin(ρi)}ξ^0m1+m2+k

And analogously for spacial derivatives ∂_x^α00 with|α|>min(ρi), we obtain

|∂_x^α00∂_ξ^β0c(x⁰⁰, ξ⁰)|. Z

|ζ⁰|.hξ⁰i

ζ^0m1ξ^0m2−|β|+|α|−min(ρ_i)

đζ⁰ .ξ^0m1+m2+k−|β|+|α|−min(ρ_i)

For the casem₁+k <0 we are able to give a symbol expansion.

Proposition 2.23. Let a(x⁰⁰, η⁰) ∈S_1,1^m1,ρ1 and b(x⁰⁰, η⁰) ∈ S^m_1,1^2,ρ2 with m₁+k < 0 and >0, then for N ≥0 define the approximation

b_N(x⁰⁰, ξ⁰) = ^X

|α|<N

(−∂_x⁰)^αD_ξ^α0a_χ(x, ξ⁰)b(x⁰⁰, ξ⁰)/α!_|

x0=0 ∈S_1,1^m2,ρ with² ρ= min(ρ₂, ρ₁−m₁−k−) and we obtain the estimate

r_N(x⁰⁰, ξ⁰) = Z

a(x⁰⁰, ζ⁰)χ(ζ⁰, ξ⁰−ζ⁰)b(x⁰⁰, ξ⁰−ζ⁰)đζ⁰−b_N(x⁰⁰, ξ⁰)∈S^m_1,1^{2−sN,ρ−sN} withs_N = min(N,−m₁−k)−_N, where _N = (N +m₁+k)^∆ .

2In fact droppingin the equation for ρ, we would only encounter a loghξ⁰i factor in the symbol estimates at the critical regularity level. Nevertheless this version is sufficient for the multiplication.

Proof. For computations one should first restate the approximationb_N in terms of symbols, id est

b_N(x⁰⁰, ξ⁰) = ^X

|α|<N

Z

(−iζ⁰)^α(−i∂_ξ⁰)^αa(x⁰⁰, ζ⁰)χ(ζ⁰, ξ⁰)b(x⁰⁰, ξ⁰)/α!đζ⁰

Ashζ⁰i.hξ⁰i on the support of χ, we can – for the first symbol estimate – restrict the case to |α|= 0. Then we obtain

2^νµ|ψ_ν(D_x⁰⁰)∂^β_ξ0b_N(x⁰⁰, ξ⁰)|

. Z

hζ⁰i.hξ⁰i

ζ^{0m1+(µ−ρ1)+}ξ^0m2−|β|+ζ^0m1ξ^{0m2+(µ−ρ2)+−|β|}đζ⁰ .ξ^{0m2+(µ+m1−ρ1+k+)+−|β|}+ξ^{0m2+(µ−ρ2)+−|β|}.ξ^{0m2+(µ−ρ)+−|β|}

Forr_N(x⁰⁰, ξ⁰) we can use the Taylor remainder formula with uniform bounds on the derivatives for|ζ⁰|<|η⁰|/2.

2^νµ|ψ_ν(D_x⁰⁰)∂^β_ξ0r_N(x⁰⁰, ξ⁰)|

. Z

|ζ⁰|.|ξ⁰|

ζ^{0m1+N+(µ−ρ1)+}ξ^{0m2−|β|−N} +ζ^0m1+Nξ^0m2−|β|−N+(µ−ρ2)⁺

đζ⁰ +

Z

|ζ⁰|h|ξ⁰|

X

|α|<N

ζ^{0m1+|α|+(µ−ρ1)+}ξ^{0m2−|β|−|α|}+ζ^0m1+|α|ξ^0m2−|β|−|α|+(µ−ρ₂)⁺

đζ⁰ +

Z

|ζ⁰|h|ξ⁰−ζ⁰|h|ξ⁰|

ζ^{0m1+(µ−ρ1)+}ξ⁰−ζ^0m2−|β|+ζ^0m1ξ⁰−ζ^{0m2−|β|+(µ−ρ2)+}đζ⁰ .ξ^{0(m1+k+N+(µ+−ρ1)+)++N+m2−|β|−N} +ξ^{0(m1+k+N)++N+m2}−|β|−N+(µ−ρ₂)⁺

+ξ^0m1+m2+k−|β|+(µ−min(ρ1,ρ2))⁺

.ξ^{0m2−sN+(µ−ρ+sN)+−|β|}

So having m ∈ Z results in a loss of approximation quality at the point where

∂^α_x0u(x⁰⁰, x⁰) is no longer continuous, e.g. the Heaviside function. This special case and some others might probably be overcome by other methods, which utilize their boundedness.

To give an expansion of the product with an error term in S_1,1^{m1+m2+k,min(ρi)} for the important case ofm_i ∈/Z and m_i+k <0 we are left with estimating the Φ term in the decomposition, which itself does not have a symbol expansion.

Proposition 2.24. Let a(x⁰⁰, η⁰)∈S_1,1^m1,ρ1, b(x⁰⁰, η⁰)∈S_1,1^m2,ρ2 with m₁+m₂+k <0 then we have

a_Φ(x, D_x⁰) Z

e^ix0ξ0b(x⁰⁰, ξ⁰)đξ⁰ ∈I_1,1;∞^{m1+m2+k,min(ρi)}

3. PRODUCTS, PARAPRODUCTS AND THE TAYLOR FORMULA 35

Combined with Proposition 2.7 and 2.8 this yields the claim.

We conclude the results so far in the following Corollary.

Corollary2.25. Letu∈I_1,1^m1,ρ1 andv∈I_1,1^m2,ρ2 withm₁+m₂+k <0, and without

To apply the Taylor formula to products, we are now going to briefly study [e_j(u)]_χ. Therefore recall that

So we have a decent understanding of the first approximation term for [e_j(u)]_χ. With the following Proposition we will learn that in the Taylor expansion this is fair enough.

and for m−ρ∈Z

For the estimates involving ∂^γ_x00 we use the same two ways to provide an estimate, taking the integralhζ⁰i.hξ⁰i ifm+|α|+|γ|+k >0 or thehζ⁰i&hξ⁰i ifm+|α|+

|γ|+k− |β|<0. The argument fails if and only if both equal 0, so indeed there is

only one other estimate that fails.

This finishes the proof of Theorem 2.19.

Im Dokument Paradifferential Operators and Conormal Distributions (Seite 26-36)