Symbolic Calculus: Composition Formula

2.3 SYMBOLIC CALCULUS: COMPOSITION FORMULA 41 By Fubini’s theorem and the fact that e^{2πiη¨px´yq}e^{2πiξ¨py´zq}“e2πipx´yq¨pη´ξqe^{2πipx´zq¨ξ} we have

pTaεT_b^εfqpxq “ ż

cεpx, ξqe^{2πipx´zq¨ξ}fpzqdz dξ, (2.41) withc_εpx, ξqgiven explicitly by

c_εpx, ξq “ ż

e2πipx´yq¨pη´ξqa_εpx, ηqb^εpy, ξqdy dη

“ ż

e^{2πix¨pη´ξq}aεpx, ηqpb^εpη´ξ, ξqdη

“ ż

e^2πix¨ηa_εpx, ξ`ηqpb^εpη, ξqdη,

(2.42)

wherepb^εp¨, ξq is the Fourier transform of b^εp¨, ξq as a function ofy PRⁿ. Note that, for each fixed 0ăεď1, this leads us to the following equality

pT_aεT_b^εfqpxq “T_cεfpxq @f PS. (2.43)

STEP 1.1 - Asymptotic Formula: By hypothesisbPS^m2 has compact support inx, and thus for each fixed ξ P Rⁿ, we have that b^εp¨, ξq P C_c⁸pRⁿq which implies that pb^εp¨, ξq P SpRⁿq. More explicitly, for each multiindex α and for eachξPRⁿ we have

p2πiηq^αpb^εpη, ξq “pB{_x^αb^εqpη, ξq “ ż

Rⁿ

e^´2πix¨ηB^α_xb^εpx, ξqdx.

Assupp_xB_x^αb^εpx, ξq Ăsupp_xb^εpx, ξq ĂK, we conclude that

p2πiηq^αpb^εpη, ξq

“

ż

Rⁿ

e^´2πix¨ηB_x^αb^εpx, ξqdx

ď ż

K

|B_x^αb^εpx, ξq|dx ďA_αp1`|ξ|q^m2mpKq.

This implies, in particular, that for every M PNthere exists a constant A_M such that

|pb^εpη, ξq|ďA_Mp1`|ξ|q<sup>m2</sup>p1`|η|q^´M. (2.44) We now proceed to make a careful analysis of the right-hand side of (2.42). Using Taylor’s formula we write

a_εpx, ξ`ηq “ ÿ

|α|ďN

1

α!B_ξ^αa_εpx, ξqη^α`R_N^ε px, ξ, ηq, (2.45) where

R^ε_Npx, ξ, ηq “ ÿ

|α|“N`1

η^α α!

ż₁

0

p1´θq^NB_ξ^αa_εpx, ξ`θηqdθ.

Inserting (2.45) into (2.42) we get c_εpx, ξq “

ż

e^2πix¨ηa_εpx, ξ`ηqpb^εpη, ξqdη

“ ÿ

|α|ďN

1 α!

ż

e^2πix¨ηB_ξ^αaεpx, ξqη^αpb^εpη, ξqdη` ż

e^2πix¨ηR^ε_Npx, ξ, ηqpb^εpη, ξqdη.

(2.46)

Now, each term in the sum of the above equality contributes with 1

α!

ż

B_ξ^αaεpx, ξqη^αpb^εpη, ξqe^2πix¨ηdη“ p2πiq^´|α|

α! B_ξ^αaεpx, ξq ż

p2πiq^|α|η^αpb^εpη, ξqe^2πix¨ηdη, which upon using the Fourier inversion formula onp2πiq^|α|η^αpb^εpη, ξq gives us

p2πiq^´|α|

α! B^α_ξaεpx, ξq ¨ B^α_xb^εpx, ξq,

and this is the term corresponding to each α in formula (2.39), except it carries 0ăεď1.

STEP 1.2 - Remainder Term: We will now show that the second term on the right-hand side of (2.46) belongs to S^m1`m2´N´1 uniformly in0 ăεď1. More precisely, we will show that given any multiindices γ, β there exists a constant A_γβ ą0 such that

B^γ_xB_ξ^β

ż

e^2πix¨ηR^ε_Npx, ξ, ηqpb^εpη, ξqdη

ďA_γβp1`|ξ|q<sup>m1`m2´N´1´|β|</sup> @x, ξPRⁿ, (2.47) uniformly in0ăεď1.

Note that differentiation under the integral sign gives us ż

B^γ_xB_ξ^βre^2πix¨ηR^ε_Npx, ξ, ηqpb^εpη, ξqsdη.

By Leibniz’s rule we get that

B_x^γB^β_ξre^2πix¨ηR^ε_Npx, ξ, ηqpb^εpη, ξqs Ăspante^2πix¨ηη^δB^σ_ξpb^εpη, ξq B^γ´δ_x B_ξ^β´σR^ε_Npx, ξ, ηq: δďγ, σďβu.

Since ż

e^2πix¨ηη^δB^σ_ξpb^εpη, ξq B^γ´δ_x B_ξ^β´σR^ε_Npx, ξ, ηqdη

ď

ż

|η|^|δ||B_ξ^σpb^εpη, ξq| |B_x^γ´δB^β´σ_ξ R^ε_Npx, ξ, ηq|dη, we may obtain (2.47) by showing that there exist constants Cγβδσ ą0such that

ż

|η|^|δ||B_ξ^σpb^εpη, ξq| |B_x^γ´δB_ξ^β´σR^ε_Npx, ξ, ηq|dηďC_γβδσp1`|ξ|q<sup>m1`m2´N´1´|β|</sup> (2.48) for allx, ξ PRⁿand uniformly in 0ăεď1. Let us prove the above inequality.

First, we observe that for any multiindices γ¹, β¹ we have B_x^γ1B^β

1

ξ R^ε_Npx, ξ, ηq “ ÿ

|α|“N`1

η^α α!

ż1 0

p1´θq^NB_x^γ1B^α`β

1

ξ aεpx, ξ`θηqdθ.

Thus we can find a constant C_{N γ}¹_β¹ ą0, which does not depend on0ăεď1, and such that

|B^γ_x¹B_ξ^β1R^ε_Npx, ξ, ηq|ď ÿ

|α|“N`1

|η|^N`1

α! maxt|B_x^γ1B_ξ<sup>α`β1</sup>aεpx, ζq|: ξ ďζ ďξ`ηu ďC_{N γ}¹_β¹|η|<sup>N`1</sup> maxt p1`|ζ|q^{m1´N´1´|β1|}: ξ ďζ ďξ`ηu, for allx, ξ, η PRⁿ. We now consider the following different cases for our estimates.

Case 1: When |η| ď |ξ|{2, we have |ξ|{2 ď |ζ| ď 3|ξ|{2, and therefore there exists a constant A_{N γ}¹_β¹ ą0such that

|B^γ_x¹B_ξ^β1R^ε_Npx, ξ, ηq|ďA_{N γ}¹_β¹|η|<sup>N`1</sup>p1`|ξ|q^{m1´N´1´|β1|} @xPRⁿ.

2.3 SYMBOLIC CALCULUS: COMPOSITION FORMULA 43 Case 2.1:When|η|ą|ξ|{2and m₁´N´1´|β¹|ą0, we use that |ζ|ď|ξ|`|η|ď3|η|to obtain a constantA_{N γ}¹_β¹ ą0 such that

|B^γ_x¹B^β_ξ¹R^ε_Npx, ξ, ηq|ďA_{N γ}¹_β¹|η|<sup>N`1</sup>p1`|η|q^{m1´N´1´|β1|} @xPRⁿ.

Case 2.2:When|η|ą|ξ|{2andm1´N´1´|β¹|ď0, we obtain a constant A_{N γ}¹_β¹ ą0 such that

|B_x^γ1B^β_ξ¹R^ε_Npx, ξ, ηq|ďA_{N γ}¹_β¹|η|^N`1 @xPRⁿ.

Now, in a similar fashion as in (2.44), we have that for every M PN and σ P Nⁿ0 there exists constantsA_{M σ} ą0 such that

|B_ξ^σpb^εpη, ξq|ďA_{M σ}p1`|ξ|q<sup>m2´|σ|</sup>p1`|η|q^´M. (2.49) Breaking the integral on the left-hand side of (2.48) according to cases 1 and 2 above we get

ż

|η|ď|ξ|{2

|η|^|δ||B_ξ^σpb^εpη, ξq| |B_x^γ´δB^β´σ_ξ R^ε_Npx, ξ, ηq|dη

` ż

|η|ą|ξ|{2

|η|^|δ||B^σ_ξpb^εpη, ξq| |B_x^γ´δB_ξ^β´σR^ε_Npx, ξ, ηq|dη (2.50) For the first integral we useCase 1together with (2.49) to obtain a constantAą0, which depends on γ, β, δ, σ, M and N, but does not depend on0ăεď1, and such that

ż

|η|ď|ξ|{2

|η|^|δ||B_ξ^σpb^εpη, ξq| |B_x^γ´δB^β´σ_ξ R^ε_Npx, ξ, ηq|dηďAp1`|ξ|q<sup>m1`m2´N´1´|β|</sup>

ż

Rⁿ

|η|^N`1`|δ|p1`|η|q^´Mdη.

It is enough to takeM ąn`N`1`|δ|to obtain our desired inequality.

We shall do only Case 2.2 for the estimate of the second integral in (2.50) since Case 2.1 follows by a similar calculation. Now, ifm₁´N´1´|β¹|ď0, it follows fromCase 2.2and (2.49) that we may obtain a constantB ą0, which depends onγ, β, δ, σ, M and N, but does not depend on 0ăεď1, and such that

ż

|η|ą|ξ|{2

|η|^|δ||B_ξ^σpb^εpη, ξq| |B_x^γ´δB^β´σ_ξ R^ε_Npx, ξ, ηq|dηďBp1`|ξ|q^m2´|σ|

ż

|η|ą|ξ|{2

|η|^N`1`|δ|p1`|η|q<sup>´M</sup>dη ďBp1`|ξ|q^m2´|σ|

ż

|η|ą|ξ|{2

p1`|η|q<sup>N`1`|δ|´M</sup>dη.

Note that ifM ąN `1`|δ|`n, then the last integral is absolutely convergent. Moreover, since the integrand is a radial function, we may use spherical coordinates to get

ż

|η|ą|ξ|{2

p1`|η|q<sup>N`1`|δ|´M</sup>dη“ω_n´1 ż₈

|ξ|{2

p1`|r|q<sup>N`|δ|`n´M</sup>dr,

whereω_n´1denotes the surface area of thepn´1q-sphere of radius 1. SettingM ąN`|δ|`n`k`1, wherekPNis still to be determined, we have that

ωn´1

ż₈

|ξ|{2

p1`|r|q<sup>N`|δ|`n´M</sup>drďωn´1

ż₈

|ξ|{2

p1`|r|q^´k´1dr“ ωn´1

k p1`|ξ|{2q^´k. Hence, there exists a constant Cą0, which does not depend on 0ăεď1 and such that

ż

|η|ą|ξ|{2

|η|^|δ||B_ξ^σpb^εpη, ξq| |B^γ´δ_x B_ξ^β´σR^ε_Npx, ξ, ηq|dηďCp1`|ξ|q^m2´|σ|´k.

AskPNcan be choosen arbitrarily large, we may take it so that the above expression is bounded by a constant timesp1`|ξ|q<sup>m1`m2´N´1´|β|</sup>.

Now that we have checked that (2.47) holds, we may proceed to the last step in the proof of the theorem for the scenario where bPS^m2 hasx-compact support.

STEP 1.3 - Symbol c P S^{m1`m2</sup>: Combining the results proved so far, we conclude that c<sub>ε</sub> P S<sup>m1`m2} uniformly in0ăεď1and that for every N PN we may write c_ε explicitly as

cεpx, ξq “ ż

e^2πix¨ηaεpx, ξ`ηqpb^εpη, ξqdη

“ ÿ

|α|ďN

p2πiq^´|α|

α! B_ξ^αaεpx, ξq ¨ B_x^αb^εpx, ξq ` ż

e^2πix¨ηR^ε_Npx, ξ, ηqpb^εpη, ξqdη.

By the Dominated Convergence Theorem and Proposition 2.6 we conclude that there exists c P S<sup>m1`m2</sup> such thatcεÑcpointwise and inS<sup>m1</sup> for allm<sup>1</sup> ąm1`m2. Moreover, we can writecpx, ξq explicitly as

cpx, ξq “ ż

e^2πix¨ηapx, ξ`ηqpbpη, ξqdη

“ ÿ

|α|ďN

p2πiq^´|α|

α! B_ξ^αapx, ξq ¨ B_x^αbpx, ξq ` ż

e^2πix¨ηRNpx, ξ, ηqpbpη, ξqdη.

Finally, taking the limit asεÑ0^`in (2.43) and using Theorem2.5leads us to the following equality pTaT_bfqpxq “ lim

εÑ0^`pTaεT_b^εfqpxq “ lim

εÑ0^`Tcεfpxq “Tcfpxq @f PS, and this finishes the proof for the case when bPS^m2 has compactx-support.

STEP 2 - General case:To consider the case when we do not assume thatbPS^m2 hasx-compact support we do as follows: Start by considering an arbitrary but fixed pointx0PRⁿ. TakeφPC_c⁸pRⁿq such that0ďφď1,suppφĂBpx0,2qandφ”1onBpx0,1qand writeb“φb` p1´φqb:“b<sup>1</sup>`b². It is clear thatb¹, b²PS^m2 and that

T_aT_bfpxq “T_aT_b¹fpxq `T_aT_b²fpxq @f PS.

Sincesuppb¹ĂBpx0,2q, it follows fromSTEP 1that there existsc¹ PS^m1`m2 such thatTaT_b¹ “T_c¹ and

c¹ ´ ÿ

|α|ďN

p2πiq^´|α|

α! pB_ξ^αaq ¨ pB_x^αb¹q P S^m1`m2´N´1 (2.51) for allN ě0.

We now show that there exists a symbol c² such that TaT_b² “ T_c² and such that for all x P Bpx₀,1{2q we have: For all γ, β multiindices there exists a constant C_{N γβ} such that

|B^γ_xB_ξ^βc²px, ξq|ďC_{N γβ}p1`|ξ|q<sup>m1`m2´N´1´|β|</sup> for allN ě0. (2.52) To see this, fix 0ăεď1 and consider the symbols a_ε, b²_ε. By (2.40)–(2.42) we have T_aεT_b²_ε “T_c²_ε with

c²_εpx, ξq “ ż

Rⁿ

ż

Bpx0,1q^c

e2πipx´yq¨pη´ξqa_εpx, ηqb²_εpy, ξqdy dη.

In the above integral we note that |x´y| ě |x₀´y|´1{2 ą 0 for all x P Bpx0,1{2q. We set

2.3 SYMBOLIC CALCULUS: COMPOSITION FORMULA 45 L_η “ p´4π²|x´y|²q^´1∆_η and thus

pL_ηqpe2πipx´yq¨pη´ξq

q “e2πipx´yq¨pη´ξq.

Inserting this operator N1 times on the above integral and integrating by parts 2N1 times in the η-variable we get

c²_εpx, ξq “ ż

Rⁿ

ż

Bpx0,1q^c

e2πipx´yq¨pη´ξq ∆^N_η¹a_εpx, ηq

p´4π²|x´y|²q^N1 b²_εpy, ξqdy dη.

Next, we setL_y “ p1`4π²|ξ´η|²q^´1p1´∆_yq, insert this operatorN₂ times on the above integral, and integrate by parts2N₂ times with respect to they-variable using the identity

L_ype2πipx´yq¨pη´ξq

q “e2πipx´yq¨pη´ξq. This calculation gives us the equality

c²_εpx, ξq “ ż

Rⁿ

ż

Bpx0,1q^c

e2πipx´yq¨pη´ξq ∆^N_η¹a_εpx, ηq

p1`4π²|ξ´η|²q^N2 p1´∆yq^N2

„ b²_εpy, ξq p´4π²|x´y|²q^N1

 dy dη.

(2.53) Since

∆^N_η¹aεpx, ηq

ďAN1p1`|η|q^m1´2N1,

p1´∆_yq^N2

„ b²_εpy, ξq p´4π²|x´y|²q^N1



ďA_N1N2p1`|ξ|q^m2

|x´y|^2N1 ,

(2.54)

and

1

1`4π<sup>2</sup>|ξ´η|<sup>2</sup> ď 1`4π²|η|² 1`2π²|ξ|², it follows that there exists a constant B_N1N2 ą0 such that

c²_εpx, ξq

ď

ż

Rⁿ

ż

Bpx0,1q^c

∆^N_η¹aεpx, ηq p1`4π²|ξ´η|²q^N2

p1´∆_yq^N2

„ b²_εpy, ξq p´4π²|x´y|²q^N1



dy dη

ďB_N1N2 ż

Rⁿ

ż

Bpx0,1q^c

p1`|η|q<sup>m1´2N1</sup> p1`4π²|η|²q^N2 p1`2π²|ξ|²q^N2

p1`|ξ|q^m2

|x´y|^2N1 dy dη.

As1`|ξ|<sup>2</sup> ď p1`|ξ|q² ď3p1`|ξ|²qfor everyξ PRⁿand|x´y|ě|x₀´y|´1{2for allxPBpx₀,1{2q, we have that there existsC_N1N2 ą0 such that

c²_εpx, ξq

ďC_N1N2p1`|ξ|q^m2´2N2

˜ż

Rⁿ

p1`|η|q<sup>m1`2N2´2N1</sup> dη

¸ ˜ż

Bp0,1q^c

1

p|z|´1{2q^2N1 dz

¸ .

From the above inequality it is clear that if we takeN2 sufficiently large and 2N1 ąmaxtn, m1` n`2N₂u we get (2.52) forc²_εpx, ξq when γ “β “0. The inequality for the general case involving derivatives follows from differentiation under the integral sign in (2.53), application of Leibniz’s rule, and similar inequalities as in (2.54). Then (2.52) follows from taking the limit of c²_εpx, ξq as εÑ0^` under the light of Theorem 2.5and Proposition 2.6.

Finally, take cx0 “c¹`c². Then Tcx0 “TaTb and, since b¹ “b for all xPBpx0,1{2q, it follows from (2.51) and (2.52) that

c_x0 ´ ÿ

|α|ďN

p2πiq^´|α|

α! pB^α_ξaq ¨ pB_x^αbq (2.55)

satisfies similar inequalities as of an element inS^{m1`m2´N´1</sup> for allN ě0, but uniformly only on x PBpx0,1{2q. Since x0 P R<sup>n</sup> is arbitrary and Theorem 2.9 guarantees that a pseudo-differential operator defines its symbol uniquely, it follows that there exists a unique symbolcPS<sup>m1`m2} such thatT_aT_b “T_c and such that (2.39) holds for all N ě0.

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