Estimates of the kernels

5.4 Kernels of pseudo-differential operators

5.4.1 Estimates of the kernels

Corollary 5.3.8 could be generalised by considering a finite family of positive Rockland operators which commute strongly between themselves (i.e. with com- muting spectral measures), with symbols possibly depending onxin a similar way to Corollary 5.3.7.

– if Q+m= 0,κ_x behaves like ln|y|: there existsC >0 anda, b, c∈N such that

∀(x, y)∈G×(G\{0}) |κx(y)| ≤Csup

π∈Gσ(x, π)S_ρ,δ^m,a,b,cln|y|; – if Q+m <0,κx is continuous onGand bounded:

sup

z∈G|κ_x(z)| ≤Csup

π∈G

σ(x, π)_Sρ,δ^m_,0,0,0.

Moreover, it is possible to replace the seminorm · S_ρ,δ^m,a,b,c in (i) and (ii) with a seminorm · _S^m,R

ρ,δ ,a,b given in (5.37).

Remark 5.4.2. Using Theorem 5.2.22 (i) Parts (3) and (2), and Corollary 5.2.25, we obtain similar properties forX_y^β1X˜_y^β2(X_x^βoq˜_α(y)κ_x(y)).

We start the proof of Theorem 5.4.1 with consequences of Proposition 5.2.16 as preliminary results on the right convolution kernels and then proceed to analy- sing the behaviour of these kernels both at zero and at infinity.

Proposition 5.2.16 has the following consequences:

Corollary 5.4.3. Let σ={σ(x, π)} be in S_ρ,δ^m with 1≥ρ≥δ≥0. Letκ_x denote its associated kernel.

1. If α, β1, β2, β_o∈Nⁿ0 are such that

m−ρ[α] + [β1] + [β2] +δ[βo]<−Q/2,

then the distribution X_z^β1X˜_z^β2(X_x^βoq˜_α(z)κ_x(z)) is square integrable and for every x∈G we have

G

X_z^β1X˜_z^β2(X_x^βoq˜_α(z)κ_x(z))²dz≤Csup

π∈G

σ(x, π)²_S^m

ρ,δ,a,b,c

wherea= [α],b= [β_o],c=ρ[α]+[β1]+[β2]+δ[β_o]andC=C_{m,α,β1,β2,βo} >0 is a constant independent of σ and x. If β1 = 0 then we may replace the seminorm · S^m_ρ,δ,a,b,cwith a seminorm · _S^m,R

ρ,δ ,a,b given in (5.37).

2. For any α, β1, β2, β_o∈Nⁿ0 satisfying

m−ρ[α] + [β1] + [β2] +δ[βo]<−Q,

the distribution z → X_z^β1X˜_z^β2X_x^βoq˜α(z)κx(z) is continuous on G for every x∈Gand we have

sup

z∈G

X_z^β1X˜_z^β2

X_x^βoq˜_α(z)κ_x(z)≤Csup

π∈G

σ(x, π)_Sρ,δ^m_,[α],[βo],[β2],

where C =C_{m,α,β1,β2,βo} >0 is a constant independent of σ and x. If β1 = 0 then we may replace the seminorm · _Sρ,δ^m_,[α],[βo],[β2] with the seminorm · _S^m,R

ρ,δ ,[α],[βo], see (5.37).

Consequently, if ρ > 0 then the map κ : (x, y) → κ_x(y) is smooth on G×(G\{0}).

Proof. Part (1) follows from Proposition 5.2.16 together with Theorem 5.2.22 (i) Parts (3) and (2), and Corollary 5.2.25 . Now by the Sobolev inequality in Theorem 4.4.25 (ii), if the right-hand side of the following inequality is finite:

sup

z∈G

X_z^β1X˜_z^β2

X_x^βoq˜_α(z)κ_x(z)≤C(I+R_z)^sνX_z^β1X˜_z^β2

X_x^βoq˜_α(z)κ_x(z)

L²(dz), fors > Q/2, then the distribution

z→X_z^β1X˜_z^β2

X_x^βoq˜_α(z)κ_x(z)

is continuous and the inequality of Part (2) holds. By Theorem 4.4.16, (I +Rz)^νsX_z^β1X˜_z^β2

X_x^βoq˜α(z)κx(z)

L²(dz)

≤C(I +R)^s+[βν1](I + ˜R)^[βν2]

X_x^βoq˜_α(z)κ_x(z)

L²(dz)

≤Cπ(I +R)^s+[βν1]X_x^βoΔ^ασ(x, π)π(I +R)^[βν2]

L²(G),

by the Plancherel formula (1.28). By Proposition 5.2.16 (together with Theorem 5.2.22 (ii)) as long as

m+s+ [β1]−ρ[α] +δ[βo] + [β2]<−Q/2, since

(I +R)^s+[βν1](I + ˜R)^[βν2]

X_x^βoq˜_α(z)κ_x(z) is the kernel of the symbol

π(I +R)^s+[βν1]X_x^βoΔ^ασ(x, π)π(I +R)^[βν2], we have

π(I +R)^s+[βν1]X_x^βoΔ^ασ(x, π)π(I +R)^[βν2]

L²(G)≤Cσ(x, π)_S^m_ρ,δ,[α],[βo],[β2], ifs+ [β1]≤ρ[α]−m−δ[βo]−[β2]. This shows Part (2).

Estimates at infinity

We will now prove better estimates for the kernel than the ones stated in Corollary 5.4.3. First let us show that the kernel has a Schwartz decay away from the origin.

Proposition 5.4.4. Let σ={σ(x, π)} be inS_ρ,δ^m with1≥ρ≥δ≥0. Letκ_xdenote its associated kernel.

We assume thatρ >0 and we fix a homogeneous quasi-norm| · |onG. Then for any M ∈ R and any α, β1, β2, βo ∈ Nⁿ0 there exist C > 0 and a, b, c ∈ N independent of σsuch that for all x∈Gandz∈Gsatisfying|z| ≥1, we have

X_z^β1X˜_z^β2(X_x^βoq˜_α(z)κ_x(z))≤Csup

π∈G

σ(x, π)_S^m_ρ,δ,a,b,c|z|^−M.

Furthermore, if β1 = 0 then we may replace the seminorm · _Sρ,δ^m_,a,b,c with a seminorm · _S^m,R

ρ,δ ,a,b given in (5.37).

Proof. We start by proving the stated result for α=β1 =β2 =β_o = 0 and for the homogeneous quasi-norm| · |_pgiven by (3.21). Herep >0 is a positive number to be chosen suitably. We also fix a numberbo >0 and a function ηo ∈ C^∞(R) valued in [0,1] withηo≡0 on (−∞,¹₂] andηo≡1 on [1,∞). We set

η(x) :=ηo(b^−p_o |x|^p_p).

Therefore, η is a smooth function on G such that η(z) = 1 if |z|_p ≥b_o. Conse- quently,

sup

|z|p≥bo

|z|^M_p κx(z)≤sup

z∈G

|z|^M_p κx(z)η(z)

≤C

[β]≤Q/2

X_z^β

|z|^M_p κx(z)η(z)

L²(G,dz)

(5.38)

by the Sobolev inequality in Theorem 4.4.25.

We study each term separately. We assume that p/2 is a positive integer divisible by all the weightsυ1, . . . , υn and we introduce the polynomial

|z|^p_p= n j=1

|zj|^υjp

and its inverse, so that X_z^β

|z|^M_p κ_x(z)η(z)

=X_z^β

|z|^M_p |z|^−p_p |z|^p_pκ_x(z)η(z)

=

[β₁]+[β₂]=[β]

Xz^β1

|z|^M_p |z|^−p_p η(z) Xz^β2

|z|^p_pκx(z) ,

where

means taking a linear combination, that is, a sum involving some con- stants. We observe that, using a polar change of coordinates,

Xz^β1

|z|^M_p |z|^−p_p η(z)

_L²(G,dz)<∞

as long as 2(M−p−[β1]) +Q−1<−1. We assume thatphas been chosen so that 2(M −p) +Q <0. Therefore, all theseL²-norms can be viewed as constants. By the Cauchy-Schwartz inequality and the properties of Sobolev spaces, we obtain

X_z^β

|z|^M_p κ_x(z)η(z)

_L²(G,dz) ≤ C

[β₂]≤[β]

Xz^β2

|z|^p_pκ_x(z)

_L²(G,dz)

≤ C

[β₂]≤[β]

[α]≤p

Xz^β2{q˜_ακ_x} 2,

since |z|^p_p = _n

j=1z_j^υjp is a polynomial of homogeneous degree p. Therefore, by Corollary 5.4.3 Part (1), we get

X_z^β

|z|^M_p κ_x(z)η(z)

_L²(G,dz)≤Csup

π∈G

σ(x, π)_S^m

ρ,δ,p,0,ρp+[β]

ifρp−m > Q/2 + [β]. We choosepaccordingly. Combining this with (5.38) yields sup

|z|p≥bo

|z|^M_p κx(z)≤Csup

π∈Gσ(x, π)S_ρ,δ^m,p,0,ρp+Q/2.

Therefore, we have obtained the result for the homogeneous norm| · |_p andα= β1=β2=β_o= 0.

The full result follows for any homogeneous norm and indices α, β1, β2, β_o from the equivalence of any two homogeneous norms and by Theorem 5.2.22 (i)

Parts (3) and (2), and Corollary 5.2.25.

Remark 5.4.5. 1. During the proof of Proposition 5.4.4, we have obtained the following statement which is quantitatively more precise. We keep the setting of Proposition 5.4.4. Then for any M ∈ R and bo > 0, there exists C = CM,bo,m >0 such that

sup

|z|p≥bo

|z|^M_p κ_x(z)≤Csup

π∈G

σ(x, π)_Sρ,δ^m_,p,0,ρp+Q/2,

where p∈N is the smallest positive integer such thatp/2 is divisible by all the weightsυ1, . . . , υn andp >max(Q/2 +M,¹_ρ(m+Q+ 1)).

2. Combining Part (1) above, Theorem 5.2.22 (i) Parts (3) and (2), and Corol- lary 5.2.25, it is possible (but not necessarily useful) to obtain a concrete expression for the numbersa, b, cappearing in Proposition 5.4.4, in terms of m, ρ, δ, α, β1, β2, βoand ofQ.

Furthermore, the same statement is true for |z| ≥bo for an arbitrary lower boundbo>0. However, the constantC may depend onbo.

Estimates at the origin

We now prove a singular estimate for the kernel near the origin which is (therefore) not covered by Corollary 5.4.3 (2).

Proposition 5.4.6. Let σ={σ(x, π)} be inS_ρ,δ^m with1≥ρ≥δ≥0. Letκ_xdenote its associated kernel.

We assume thatρ >0 and we fix a homogeneous quasi-norm| · |onG. Then for anyα, β1, β2, β_o∈Nⁿ0 with Q+m+δ[β_o]−ρ[α] + [β1] + [β2] ≥0 there exist a constant C >0 and computable integersa, b, c∈N0 independent of σsuch that for allx∈Gandz∈G\{0}, we have that if

Q+m+δ[β_o]−ρ[α] + [β1] + [β2]>0, thenX_z^β1X˜_z^β2(X_x^βoq˜α(z)κx(z))≤Csup

π∈Gσ(x, π)S^m_ρ,δ,a,b,c|z|⁻Q+m+δ[βo]−ρ[α]+[β1]+[β2]

ρ ,

and if

Q+m+δ[β_o]−ρ[α] + [β1] + [β2] = 0, then X_z^β1X˜_z^β2(X_x^βoq˜α(z)κx(z))≤Csup

π∈G

σ(x, π)S_ρ,δ^m,a,b,cln|z|.

In both estimates, ifβ1= 0 then we may replace the seminorm · S^m_ρ,δ,a,b,c with a seminorm · _S^m,R

ρ,δ ,a,b given in (5.37).

During the proof of Proposition 5.4.6, we will need the following technical lemma which is of interest on its own.

Lemma 5.4.7. Letσ={σ(x, π)} be inS_ρ,δ^m with 1≥ρ≥δ≥0. Let η∈ D(R)and co>0. We also fix a positive Rockland operatorR of homogeneous degreeν with corresponding seminorms for the symbol classesS_ρ,δ^m.

Then for any ∈N0, the symbols given by

σL,(x, π) :=η(2^−coπ(R))σ(x, π) and σR,(x, π) :=σ(x, π)η(2^−coπ(R)), are in S^−∞. Moreover, for any m1 ∈R and a, b, c ∈N0, there exists a constant C=Cm,m1,ρ,δ,a,b,c,η,co >0 such that for any∈N0 we have

σL,(x, π)_S^m1

ρ,δ,a,b,c≤Csup

π∈G

σ(x, π)S^m_ρ,δ,a,b,c2^coν(m−m1).

The same holds forσR,(x, π), but with a possibly different seminorm on the right hand side.

Only forσR,(x, π), we also have for the seminorm·_S^m,R

ρ,δ ,a,bgiven in (5.37), the estimate

σR,(x, π)_S^m1,R

ρ,δ ,a,b≤Csup

π∈Gσ(x, π)_S^m,R

ρ,δ ,a,b2^coν(m−m1).

Proof of Lemma 5.4.7. For each ∈N0, the symbolη(2^−coπ(R)) is in S^−∞ by Proposition 5.3.4. Therefore, by Theorem 5.2.22 (ii) and the inclusions (5.31),σ_L, andσ_R,are inS^−∞.

Let us fixαo, βo∈Nⁿ0 andγ∈R. By the Leibniz formula (see (5.28)), π(I +R)^{ρ[αo]−m1ν−δ[βo]+γ}X_x^βoΔ^αoσ_L,π(I +R)^−γν

=π(I +R)^{ρ[αo]−m1ν−δ[βo]+γ}X_x^βoΔ^αo

η(2^−coπ(R))σ(x, π)

π(I +R)^−γν

=

[α1]+[α2]=[αo]

c_α1,α2π(I +R)^{ρ[αo]−m1ν−δ[βo]+γ}Δ^α1η(2^−coπ(R)) X_x^βoΔ^α2σ(x, π)π(I +R)^−γν. Therefore, taking the operator norm, we obtain

π(I +R)^{ρ[αo]−m1ν−δ[βo]+γ}X_x^βoΔ^αoσ_L,π(I +R)^−γν

≤C

[α1]+[α2]=[αo]

π(I +R)^{ρ[αo]−m1ν−δ[βo]+γ}Δ^α1η(2^−coπ(R))π(I +R)^{−ρ[α2]−m−δ[βo]+γν} π(I +R)^{ρ[α2]−m−δ[βo]+γν} X_x^βoΔ^α2σ(x, π)π(I +R)^−γν

≤Cσ(x, π)_Sρ,δ^m_,[αo],[βo],|γ|

[α1]+[α2]=[αo]

π(I +R)^{ρ[αo]−m1ν−δ[βo]+γ}Δ^α1η(2^−coπ(R))π(I +R)^{−ρ[α2]−m−δ[βo]+γν} .

By Proposition 5.3.4,

π(I +R)^{ρ[αo]−m1ν−δ[βo]+γ}Δ^α1η(2^−coπ(R))π(I +R)^{−ρ[α2]−m−δ[βo]+γν}

≤Cη(2^−co·)_Mm2 ν ,k, for somek, wherem2 is such that

[α1]−m2=ρ[α_o]−m1−δ[β_o] +γ−(ρ[α2]−m−δ[β_o] +γ), that is,

m2=m1−m+ [α1](1−ρ). Now, we can estimate

η(2^−co·)_Mm2

ν ,k = sup

λ>0, k=0,...,k(1 +λ)^k−mν2∂_λ^k(η(2^−coλ))

= sup

λ>0, k=0,...,k(1 +λ)^k−mν22^−cok(∂^kη)(2^−coλ)

≤ C2^−comν2.

Therefore,

[α1]+[α2]=[αo]

π(I +R)^{ρ[α]−mν 1+γ}Δ^α1η(2^−coπ(L))π(I +R)^{−ρ[α2]−m−δ[βo]+γν}

≤C

[α1]+[α2]=[αo]

2^{−com1−m+[αν1](1−ρ)} ≤C2^{−com1ν−m}, and we have shown that

π(I +R)^{ρ[αo]−m1ν−δ[βo]+γ}X_x^βoΔ^αoσL,π(I +R)^−γν

≤Cαoσ(x, π)_S^m_ρ,δ,[αo],[βo],|γ|2^{−com1ν−m}.

The desired property forσL,follows easily. The property forσR,may be obtained by similar methods and its proof is left to the reader.

Proof of Proposition 5.4.6. By Theorem 5.2.22 (i) Parts (3) and (2), and Corollary 5.2.25, it suffices to show the statement forα=β1=β2=β_o= 0. By equivalence of homogeneous quasi-norms (Proposition 3.1.35), we may assume that the homo- geneous quasi-norm is | · |_p given by (3.21) where p > 0 is such that p/2 is the smallest positive integer divisible by all the weights υ1, . . . , υ_n. Since κ_x decays faster than any polynomial away from the origin (more precisely see Proposition 5.4.4), it suffices to prove the result for|z|_p<1.

So letσ∈S_ρ,δ^m withQ+m≥0. By Lemma 5.4.11 (to be shown in Section 5.4.2) we may assume that the kernelκ: (x, y)→κx(y) ofσis smooth onG×G and compactly supported inx. By Proposition 5.4.4 it is also Schwartz iny.

We fix a positive Rockland operatorRof homogeneous degreeνand a dyadic decomposition of its spectrum: we choose two functionsη0, η1 ∈ D(R) supported in [−1,1] and [1/2,2], respectively, both valued in [0,1] and satisfying

∀λ >0

∞ =0

η(λ) = 1, where for∈Nwe set

η(λ) :=η1(2^−(−1)νλ).

For each∈N0, the symbolη(π(R)) is inS^−∞by Proposition 5.3.4 and its kernel η(R)δ0 is Schwartz by Corollary 4.5.2. Furthermore, by the functional calculus, _N

=0η(R) converges in the strong operator topology ofL(L²(G)) to the identity operator I asN → ∞, and thus_N

=0η(R)δ0converges inK(G) and inS(G) to the Dirac measureδ0 at the origin asN→ ∞.

By Theorem 5.2.22 (ii), the symbolσ given by

σ(x, π) :=σ(x, π)η(π(R)), (x, π)∈G×G,

is in S^−∞. The kernel associated withσ isκgiven by κ(x, y) =κ_,x(y) = (η(R)δ0)∗κ_x(y). For eachx, we haveκ,x∈ S(G). The sum_N

=0κ,x converges inS(G) toκxas N → ∞since

N =0

Op(σ(x,·)) = Op(σ(x,·)) N =0

η(R)

converges to Op(σ(x,·)) in the strong operator topology of L(L²(G), L²_−m(G)).

This convergence is in fact stronger. Indeed, by Lemma 5.4.7, σ_S^m1

ρ,δ,a,b,c≤Csup

π∈G

σ_Sρ,δ^m_,a,b,c2^(m−m1),

thus

∈N

σ_S^m1

ρ,δ,a,b,c<∞ ifm1> m. Consequently, the sum

σ is convergent in S_ρ,δ^m1 and, fixingx∈G, the sum

sup_z∈S|κ_,x(z)|is convergent whereSis any compact subset ofG\{0} by Proposition 5.4.4 or more precisely the first part in Remark 5.4.5. Necessarily, the limit of

σ is σ and the limit of

κ_,x for the uniform convergence on any compact subset ofG\{0} isκ_x with

|κ_x(z)| ≤^∞

=0

|κ_,x(z)|, z∈G\{0}.

By Corollary 5.4.3 (2), for anym1<−Qandr∈N0, we have sup

z∈G|z|^pr_p |κ_,x(z)| ≤ C

[α]=pr

sup

π∈G

Δ^ασ(x, π)_S^m1 ρ,δ,0,0,0

≤ Cc_σ,r2^{(m−m1−ρpr)} (5.39) by Lemma 5.4.7 and its proof, withc_σ,r:= sup_π∈Gσ(x, π)_Sρ,δ^m_,pr,0,0.

We write |z|_p ∼2^−o in the sense that_o∈N0 is the only integer satisfying

|z|_p∈(2^−(o+1),2^−o].

Let us assume that Q+m >0. We use (5.39) withr= 0 andm1 such that m−m1= (Q+m)/ρ. In particular,

m1=m(1−1 ρ)−Q

ρ <−Q.

The sum over= 0, . . . , o−1, can be estimated as

o−1 =0

|κ,x(z)| ≤

o−1 =0

Ccσ,02^(m−m1)≤cσ,02^o(m−m1)

≤ Cc_σ,0|z|⁻p^Q+mρ .

We now chooser∈Nandm1<−Qsuch that

m−m1−ρpr <0 and pr(1−ρ) +m−m1= Q+m ρ .

More precisely, we setr:=(m+Q)/(ρp) , that is,ris the largest integer strictly greater than (m+Q)/(ρp), while m1 is defined by the equality just above; in particular,

m−m1> Q+m

ρ −(1−ρ)Q+m

ρ thus m1<−Q.

We may use (5.39) and sum over=_o, _o+ 1. . ., to get ∞

=o

|z|^pr_p |κ_,x(z)| ≤Cc_σ,r^∞

=o

2^{(m−m1−ρpr)}≤Cc_σ,r2^{o(m−m1−ρpr)}. Therefore, we obtain

∞ =o

|κ,x(z)| ≤ Ccσ,r2^{o(m−m1−ρpr)}|z|^−pr_p

≤ Ccσ,r|z|^−pr−_p ^{(m−m1−ρpr)}=Ccσ,r|z|⁻p^Q+mρ . This yields the desired estimate forκx whenQ+m <0.

Let us assume that Q+m= 0. Using (5.39) with r= 0 andm1 =−m, we obtain

o−1 =0

|κ,x(z)| ≤

o−1 =0

Ccσ,02^(m−m1)≤cσ,0o

≤ Cc_σ,0ln|z|_p.

Proceeding as above for the sum over ≥ o, we obtain that _∞

=o|κ,x(z)| is bounded. This yields the desired estimate forκ_x in the caseQ+m= 0.

Remark 5.4.8. It is possible to obtain a concrete expression for the numbersa, b, c appearing in Proposition 5.4.6, in terms ofm, ρ, δ, α, β1, β2, βoand ofQ.