Proof of the case S 0,0 0

5.7 Calder´ on-Vaillancourt theorem

5.7.2 Proof of the case S 0,0 0

This section is devoted to the proof of the following result which is a particular case of Theorem 5.7.1. We also give an explicit estimate on the number of derivatives and differences of the symbol needed for theL²-boundedness.

Proposition 5.7.7. Let T ∈Ψ⁰_0,0. ThenT extends to a bounded operator onL²(G).

Furthermore, if we fix a positive Rockland operatorR(in order to define the semi- norms onΨ^m_ρ,δ) then

∀φ∈ S(G) T φ_L²(G)≤CTΨ⁰_0,0,a,b,cφ_L²(G),

where C > 0 and a, b, c ∈ N0 are independent of T. In particular, this estimate holds with a=rpo,b=rν+^Q₂ ,c=rν, whereν is the degree ofR,po/2 is the smallest positive integer divisible byυ1, . . . , υn, and r∈N0 is the smallest integer such that rp_o> Q+ 1.

Throughout Section 5.7.2, we fix the homogeneous norm | · | =| · |_po given by (3.21), where p_o/2 is the smallest positive integer divisible by υ1, . . . , υ_n. We fix a maximal family{B(x_i,_2C¹

o)}^∞_i=1of disjoint balls and a sequence of functions (χ_i)^∞_i=1 so that the properties of Lemma 5.7.5 hold. We also fix ψ0, ψ1 ∈ D(R) supported in [−1,1] and [1/2,2], respectively, such that 0≤ψ0, ψ1≤1 and

∀λ≥0

∞ j=0

ψ_j(λ) = 1 with ψ_j(λ) :=ψ1(2^−(j−1)λ), j∈N.

Let us start the proof of Proposition 5.7.7. Letσ∈S0⁰,0. For each I= (i, j)∈N×N0, we define

σI(x, π) :=χi(x)σ(x, π)ψj(π(R)). We denote byTI andκI the corresponding operator and kernel.

Roughly speaking, the parametersi andj correspond to localising in space and frequency, respectively. The localisation in space corresponds to the covering ofGby the balls centred at thexi’s, while the localisation in frequency is deter- mined by the spectral projection ofRto theL²(G)-eigenspaces corresponding to eigenvalues close to each 2^j.

It is not difficult to see that eachTI is bounded onL²(G):

Lemma 5.7.8. Each operatorT_I is bounded onL²(G).

Since σ_I is localised both in space and in frequency, we may use one of the two localisations.

Proof of Lemma 5.7.8 using frequency localisation. Letα, β∈Nⁿ0. By the Leibniz formulae for difference operators (see Proposition 5.2.10) and for vector fields, we have

X_x^βΔ^ασ_I(x, π) =

[β1]+[β2]=[β] [α1]+[α2]=[α]

X_x^β1χ_i(x)X_x^β2Δ^α1σ(x, π) Δ^α2ψ_j(π(R)).

Therefore,

π(I +R)^[α]+γν X_x^βΔ^ασ_I(x, π)π(I +R)^−γν_L(Hπ)

≤C

[β2]≤[β] [α1]+[α2]=[α]

π(I +R)^[α]+γν X_x^β2Δ^α1σ(x, π) Δ^α2ψ_j(π(R))π(I +R)^−γν_L(Hπ)

≤C

[β2]≤[β] [α1]+[α2]=[α]

π(I +R)^[α]+γν X_x^β2Δ^α1σ(x, π)π(I +R)^{−[α2]+ν γ}_L(Hπ)

π(I +R)^{[α2]+ν γ}Δ^α2ψ_j(π(R))π(I +R)^−γν_L(Hπ).

Therefore, by Lemma 5.4.7, we obtain

σI_S1,0⁰ _,a,b,c≤ σ_S0,0⁰ _,a,b,c+a2^ja/ν.

This shows that the operator TI is in Ψ⁰ and is therefore bounded on L²(G) by

Theorem 5.4.17.

Proof of Lemma 5.7.8 using space localisation. Another proof is to apply the fol- lowing lemma since the symbolσ_I(x, π) has compact support inx. Lemma 5.7.9. Letσ(x, π)be a symbol (in the sense of Definition 5.1.33) supported inx∈S, and assume thatSis compact. Then the operator norm of the associated operator onL²(G)is

Op(σ)_L(L²(G))≤C|S|^1/2 sup

x∈G [β]≤^Q₂

X_x^βσ(x, π)_L∞(G).

Proof of Lemma 5.7.9. Let T = Op(σ) and let κx be the associated kernel. We have by the Sobolev inequality in Theorem 4.4.25,

|T φ(x)|² = |φ∗κ_x(x)|²≤ sup

xo∈G|φ∗κ_xo(x)|²

≤ C

[β]≤^Q₂

φ∗X_x^β_oκ_xo(x)²

L²(dxo).

Hence

T φ²_L2(G) ≤ C

[β]≤^Q₂

G

G|φ∗X_x^β_oκ_xo(x)|²dx_odx

≤ C

[β]≤^Q₂

Gφ∗X_x^β_oκ_xo²_L2(dx)dx_o

≤ C|S| sup

xo∈G,[β]≤^Q₂

φ∗X_x^β_oκ_xo(x)²

L²(dx).

Now by Plancherel’s Theorem, φ∗X_x^β_oκxo(x)

L²(dx)≤ φ_L2(dx)X_x^β_oσ(xo, π)_L∞(G). This implies that theL²-operator norm ofT is

≤C|S|^1/2 sup

xo∈G,[β]≤^Q₂X_x^β_oσ(xo, π)_L∞(G),

and concludes the proof of Lemma 5.7.9.

Let us go back to the proof of Proposition 5.7.7. The approach is to apply the following version of Cotlar’s lemma:

Lemma 5.7.10(Cotlar’s lemma here). Suppose thatr∈N0is such thatrp_o> Q+1 and that there exists A_r>0satisfying for all (I, I)∈N×N0:

max

T_IT_I^∗_L(L²(G)),T_I^∗T_IL(L²(G))

≤A_r2^−|j−j|r(1 +|x⁻_i¹x_i|)^−rpo.

ThenT = Op(σ)isL²-bounded with operator norm≤C√ A_r.

Lemma 5.7.10 can be easily shown, adapting for instance the proof given in [Ste93, ch. VII §2] using Part (4) of Lemma 5.7.5. Indeed, the numbering of the sequence of operators to which the Cotlar-Stein lemma (see Theorem A.5.2) is applied is not important, and the condition rp_o> Q+ 1 is motivated by Lemma 5.7.5, Part (4). This is left to the reader.

Lemma 5.7.11 which follows gives the operator norm for TIT_I^∗ and T_I^∗TI. Combining Lemmata 5.7.10 and 5.7.11 gives the proof of Proposition 5.7.7.

Lemma 5.7.11. 1. For any r∈N0, the operator norm ofTIT_I^∗ onL²(G)is TIT_I^∗_L(L²(G))≤Cr1_|j−j|≤1(1 +|x⁻_i¹xi|)^−rpoσ²_S0

0,0,rpo,^Q₂,0. 2. For any r∈N0, the operator norm ofT_I^∗TI on L²(G)is

T_I^∗T_IL(L²(G))≤C_r1_|x−1

i xi|≤4Co2^−|j−j|rσ²_S0

0,0,0,rν+^Q₂,rν. In the proof of Lemma 5.7.11, we will also use the symbols σi, i∈N, given by

σi(x, π) :=χi(x)σ(x, π),

and the corresponding operatorsTi= Op(σi) and kernelsκi. We observe thatσiis compactly supported inx, therefore by Lemma 5.7.9, the operator Ti is bounded onL²(G).

Proof of Lemma 5.7.11 Part (1). We have (see the end of Lemma 5.5.4) T_I = Op(σ_I) =T_i ψ_j(R),

thus

TIT_I^∗ =Tiψj(R)ψj(R)T_i^∗.

Sinceψj(R)ψj(R) = (ψjψj)(R), this is 0 if|j−j|>1. Let us assume|j−j| ≤1.

We set

T_ijj:=T_i◦(ψ_jψ_j)(R) = Op (σ_i◦(ψ_jψ_j) (π(R))),

see again the end of Lemma 5.5.4. ThereforeT_IT_I^∗ =T_iT_i^∗jj, and we have by the Sobolev inequality in Theorem 4.4.25,

|T_IT_I^∗φ(x)| =

GT_i^∗jjφ(z)κ_ix(z⁻¹x)dz

≤ sup

xo

GT_i^∗jjφ(z)κixo(z⁻¹x)dz 1_x∈B(x

i,2)

≤ C

[β]≤^Q₂

X_x^β_o

GT_i^∗jjφ(z)κ_ixo(z⁻¹x)dz L²(dxo)

1_x∈B(xi,2).

Hence,

T_IT_I^∗φ_L² ≤ C

[β]≤^Q₂

GT_i^∗jjφ(z)X_x^β_oκ_ixo(z⁻¹x)dz 1_x∈B(xi,2)

L²(dxodx)

.

The idea of the proof is to use a quantity which will help the space localisa- tion; so we introduce this quantity 1 +|z⁻¹x|^rpo and its inverse, where the integer r∈Nis to be chosen suitably. Notice that for the inverse we have

(1 +|z⁻¹x|^rpo)⁻¹≤C_r(1 +|z⁻¹x|)^−rpo ≤C_r(1 +|x⁻_i¹x_i|)^−rpo, for anyz∈suppχi andx∈B(xi,2). Therefore, we obtain

GT_i^∗_jjφ(z)X_x^β_oκixo(z⁻¹x)dz 1_x∈B(x

i,2)

L²(dxodx)

=

GT_i^∗jjφ(z)1 +|z⁻¹x|^rpo

1 +|z⁻¹x|^rpoX_x^β_oκ_ixo(z⁻¹x)dz1_x∈B(xi,2)

L²(dxo,dx)

≤C(1 +|x⁻_i¹x_i|)^−rpoT_i^∗jjφ(z1)

L²(dz1)

(1 +|z⁻2¹x|^rpo)X_x^β_oκixo(z⁻2¹x) 1_x∈B(x

i,2)

L²(dz2,dxo,dx)

by the observation just above and the Cauchy-Schwartz inequality. The last term can be estimated as

(1 +|z2⁻¹x|^rpo)X_x^β_oκ_ixo(z2⁻¹x) 1_x∈B(xi,2)

L²(dz2,dxo,dx)

≤ |B(xi,2)| sup

xo∈G

(1 +|z|^rpo)X_x^β_oκixo(z)

L²(dz)

≤C sup

xo∈G rpo

[α]=0

X_x^β_oΔ^ασ_i(x_o, π)

L^∞(G)

by the Plancherel theorem and Theorem 5.2.22, since |z|^rpo can be written as a linear combination of ˜q_α(z), [α] =rp_o. Combining the estimates above, we have

obtained

T_IT_I^∗φ_L² ≤C(1 +|x⁻_i¹x_i|)^−rpoT_i^∗jjφ

L² sup

xo∈G [β]≤^Q₂,[α]≤rpo

Δ^αX_x^β_oσ(x_o, π)

L^∞(G).

The supremum is equal toσ_S0

0,0,rpo,^Q₂,0. So we now want to study the operator norm of T_i^∗jj, which is equal to the operator norm ofTijj. Since the symbol of Tijj is localised in space we may apply Lemma 5.7.9 and obtain

T_i^∗_jjL(L²(G))=Tijj_L(L²(G)) =Op (σi(ψjψj) (π(R)))_L(L²(G))

≤C|B(x_i,2)|^1/2 sup

[β]≤Q/x∈G2

X_x^β{χ_i(x)σ(x, π) (ψ_jψ_j) (π(R))} _L∞(G)

≤C sup

x∈G, π∈G [β]≤Q/2

[β1]+[β2]=[β]

|X^β1χi(x)| X_x^β2σ(x, π)_L(Hπ)(ψjψj) (π(R))_L(Hπ)

≤C sup

x∈G, π∈G [β2]≤Q/2

X_x^β2σ(x, π)_L(Hπ)=Cσ_S⁰

0,0,0,Q/2,0,

since theX^β2χi’s are uniformly bounded onGand overi. Thus, we have obtained

T_IT_I^∗φ_L² ≤C(1 +|x⁻_i¹x_i|)^−rpoσ_S⁰

0,0,0,Q/2,0φ_L2σ_S0

0,0,rpo,^Q₂,0, and this concludes the proof of the first part of Lemma 5.7.11.

Proof of Lemma 5.7.11 Part (2). Recall that each κIx(y) is supported, with re- spect tox, in the ballB(x_i,2). We compute easily that the kernel ofT_I^∗T_I is

κ_I∗I(x, w) =

Gκ_Ixz⁻¹(wz⁻¹)κ^∗_Ixz−1(z)dz.

Therefore,κI∗Iis identically 0 if there is nozsuch thatxz⁻¹∈B(xi,2)∩B(xi,2).

So if|x⁻_i¹x_i|>4C_o(which impliesB(x_i,2)∩B(x_i,2) =∅) thenT_I^∗T_I = 0. So we may assume|x⁻_i¹x_i| ≤4C_o.

The idea of the proof is to use a quantity which will help the frequency localisation; so we introduce this quantity (I +R)^r and its inverse, where the integer r∈Nis to be chosen suitably. We can write

T_I^∗T_I =T_I^∗T_iψ_j(R) =T_I^∗T_i(I +R)^r (I +R)^−rψ_j(R). By the functional calculus (see Corollary 4.1.16),

(I +R)^−rψj(R)_L(L²(G))= sup

λ≥0

(1 +λ)^−rψj(λ)≤Cr2^−jr.

Thus we need to studyT_I^∗T_i(I +R)^r. We see that its kernel is κx(w) =

G

(I + ˜R)^rκ_ixz⁻¹(wz⁻¹)κ^∗_Ixz−1(z)dz

=

G

(I + ˜R)^rκ_ixw⁻¹_z(z)κ^∗_Ixw−1z(z⁻¹w)dz.

We introduce (I +R)^r(I +R)^−r on the first term of the integrand acting on the variable ofκ_ixw⁻¹_z, and then integrate by parts to obtain

κ_x(w) =

[β1]+[β2]+[β3]=rν

GX_z^β₁¹=z(I +R)^−r(I + ˜R)^rκ_ixw⁻¹_z1(z) X_z^β₂²=zX_z^β₃³=zκ^∗_Ixw−1z2(z⁻3¹w)dz

=

[β1]+[β2]+[β3]=rν

GX_z^β1

1=xw⁻¹z(I +R)^−r(I + ˜R)^rκ_iz1(z) X_z^β2

2=xw⁻¹z(X^β3κ_Iz2)^∗(z⁻¹w)dz.

Re-interpreting this in terms of operators, we obtain T_I^∗T_i(I +R)^r=

[β1]+[β2]+[β3]=rν

Op

π(X^β3)X_x^β2σ_I(x, π)_∗ Op

π(I +R)^−rX_x^β1σ_i(x, π)π(I +R)^r . By Lemma 5.7.9,

Op

π(I +R)^−rX_x^β1σ_i(x, π)π(I +R)^r

_L(L²(G))

≤C sup

x∈G [β]≤^Q₂

π(I +R)^−rX_x^βX_x^β1σi(x, π)π(I +R)^r_L∞(G)

≤ σ_S0

0,0,0,[β1]+^Q₂,rν, and

Op

π(X^β3)X_x^β2σ_I(x, π)

_L(L²(G))

≤ sup

[β]≤^Q₂π(X^β3)X_x^βX_x^β2σi(x, π)ψj(π(R))_L∞(G)

≤ sup

[β]≤^Q₂π(X^β3)π(I +R)^−[βν3]_L∞(G)×

×π(I +R)^[βν3]X_x^β+β2σ_i(x, π)π(I +R)^−[βν3]_L∞(G)×

×π(I +R)^[βν3]ψ_j(π(R))_L∞(G)

≤C_β2^j[βν3]σ_S0

0,0,0,[β2]+^Q₂,[β3],

by Lemma 5.4.7. Hence we have obtained T_I^∗T_IL(L²(G)) ≤ C_r2^−jr

[β1]+[β2]+[β3]=rν

σ²_S0

0,0,0,rν+^Q₂,rν2^j[βν3]

≤ Cr2^(j−j)rσ²_S0

0,0,0,rν+^Q₂,rν.

This shows Part 2 of Lemma 5.7.11 up to the fact that we should have −|j−j| instead of (j−j) but this can be deduced easily by reversing the rˆole ofIandI, and usingT_L(L²(G))=T^∗_L(L²(G)). This concludes the proof of Lemma 5.7.11. Therefore, by Lemma 5.7.10, Proposition 5.7.7 is also proved.