On the L p -boundedness of pseudo-differential operators with non-regular symbols

c 2009 by Institut Mittag-Leffler. All rights reserved

On the L ^p -boundedness of pseudo-differential operators with non-regular symbols

Naohito Tomita

Abstract. In this paper, we consider the continuity property of pseudo-differential oper- ators with symbols whose Fourier transforms have compact support. As applications, we obtain theL^p-boundedness for symbols in Besov spaces and in modulation spaces.

1. Introduction

The continuity property of pseudo-differential operators has been well inves- tigated. Among many works, Calder´on–Vaillancourt [3] first treated the bounded- ness for the classS_0,0⁰ , which means that the boundedness of all the derivatives of a symbol assures theL²-boundedness of the corresponding operator. It should be mentioned that the boundedness of all the derivatives of a symbol is not necessary in their proof.

Authors such as Coifman–Meyer [4], Cordes [5], Miyachi [11] and Muramatu [12], improved the result of [3] along this line, that is, investigated the minimal assumption on the regularity of a symbol. In fact, they proved that the boundedness of the derivatives of a symbol up to a certain order, which exceedsn/2, assures the boundedness onL²(Rⁿ) of the corresponding operator. In particular, Sugimoto [16]

showed that symbols σ(x, ξ) in the Besov spaceB^∞_(n/2,n/2)^,1 (Rⁿ×Rⁿ) imply the L²- boundedness. This assumption means thatσ(x, ξ) belongs toB_n/2^∞,1(Rⁿ) with respect to bothxandξ. [16, Theorem 2.1.2] also stated that the classB_(n/2,n/2)^∞,1 (Rⁿ×Rⁿ) can be replaced by a wider oneB(1/2,...,1/2)^∞,1 (R×...×R), which means thatσ(x, ξ)=

σ(x₁, ..., x_n, ξ₁, ..., ξ_n) belongs to B_1/2^∞,1(R) with respect to x_j and ξ_k for all j, k=

1, ..., n. This is equivalent to the result of Boulkhemair [1, Corollary 6].

Theorem 1.1. There exists a constant C >0 such that σ(X, D)fL²≤C(R1...R2n)^1/2σ(x, ξ)L^∞_x,ξfL²

for allσ(x, ξ) with supp ˆσ⊂2n

j=1[−Rj, Rj], Rj≥1,j=1, ...,2n,and all f∈L²(Rⁿ).

Here σ=ˆ F1,2σ andC >0 is independent of Rj≥1, j=1, ...,2n.

The objective of this paper is to induce a similar estimate for theL^p-bounded- ness, p=2. In this direction, Miyachi [11] proved that symbols σ(x, ξ) in the weighted Besov space B^∞_(s^,∞

1,s₂),m(p)(Rⁿ×Rⁿ), s₁>min{n/p, n/2}, s₂>max{n/p, n/2}, m(p)=n|1/p−1/2| imply the L^p-boundedness, 1<p<∞. This assumption means that fractional derivatives ofσ(x, ξ) inx(resp.ξ) higher than min{n/p, n/2} (resp. max{n/p, n/2}) are bounded byC ξ^−m(p). Sugimoto [17] treated the critical cases₁=n/2 ands₂=n/pas in the argument for theL²-boundedness, and showed that symbolsσ(x, ξ) in the weighted Besov spaceB(n/2,n/p),m(p)^∞,1 (Rⁿ×Rⁿ) imply the L^p-boundedness for 1<p≤2. The following main theorem of this paper extends this result to the casep>2. Furthermore, in the casep<2, it replaces the symbol class B(n/2,n/p),m(p)^∞,1 (Rⁿ×Rⁿ) by the wider oneB^∞(1/2,...,1/2,1/p,...,1/p),m(p)^,1 (R×...×R) as in the case ofL²-boundedness.

Theorem 1.2. Let 0<p<∞. Then there exists a constant C >0 such that σ(X, D)fL^p≤C(R1...Rn)^min{1/p,1/2}(Rn+1...R2n)^max{1/p,1/2}

×σ(x, ξ) ξ^n|1/p−1/2|L^∞

x,ξfH^p

for allσ(x, ξ)with supp ˆσ⊂2n

j=1[−Rj, Rj], Rj≥1,j=1, ...,2n,and allf∈H^p(Rⁿ).

Here H^p(Rⁿ)is the Hardy space and C >0 is independent of Rj≥1, j=1, ...,2n.

The statement for symbols in the class B(1/2,...,1/2,1/p,...,1/p),m(p)^∞,1 with p<2 (resp. B^∞(1/p,...,1/p,1/2,...,1/2),m(p)^,1 withp>2) can be obtained from Theorem1.2as a corollary (see Corollary5.2(2)). We remark that symbols in the weighted modula- tion spaceM_m(p)^∞,1 also induce theL^p-boundedness. Such a result is just a corollary of Theorem 1.2 again (see Corollary 5.2(3)). The L²-boundedness with symbols in the modulation space M₀^∞,1 was studied by Sj¨ostrand [13], Boulkhemair [1], Gr¨ochenig–Heil [8] and Toft [18].

Finally, we explain the organization of this paper. Section 2 is for the pre- liminaries, including the notation and definitions of function spaces. Sections 3

and 4 are devoted to the proofs of Theorem 1.2 with p≤2 and p≥2, respectively.

In Section5, we state several results induced from Theorem 1.2.

2. Preliminaries

LetS(Rⁿ) andS(Rⁿ) be the Schwartz spaces of all rapidly decreasing smooth functions and tempered distributions, respectively. We define theFourier transform Ff and theinverse Fourier transform F⁻¹f off∈S(Rⁿ) by

Ff(ξ) = ˆf(ξ) =

Rⁿ

e^−ix·ξf(x)dx and F⁻¹f(x) = 1 (2π)ⁿ

Rⁿ

e^ix·ξf(ξ)dξ.

Let σ(x, ξ)∈S(R²ⁿ), where x, ξ∈Rⁿ. We denote by F1σ(y, ξ) and F2σ(x, η) the partial Fourier transforms ofσwith respect to the first variable and the second vari- able, respectively. That is, F1σ(y, ξ)=F[σ(·, ξ)](y) and F2σ(x, η)=F[σ(x,·)](η).

We also denote byF1⁻¹σandF2⁻¹σthe partial inverse Fourier transforms ofσwith respect to the first variable and the second variable, respectively. We writeF1,2= F1F2andF1,2⁻¹=F1⁻¹F2⁻¹, and note thatF1,2andF1,2⁻¹ are the usual Fourier trans- form and inverse Fourier transform of distributions on Rⁿ×Rⁿ. For σ∈S(R²ⁿ), the pseudo-differential operatorσ(X, D) is defined by

σ(X, D)f(x) = 1 (2π)ⁿ

Rⁿ

e^ix·ξσ(x, ξ) ˆf(ξ)dξ forf∈ S(Rⁿ).

We use the notation ξ=(1+|ξ|²)^1/2, whereξ∈Rⁿ. For 1≤p≤∞,pis the conjugate exponent ofp(that is, 1/p+1/p=1).

We introduce Hardy spaces based on Stein [14, Chapter 3]. Let 0<p<∞, and let Φ∈S(Rⁿ) be such that

RⁿΦ(x)dx=0. Then theHardy space H^p(Rⁿ) consists of allf∈S(Rⁿ) such that

fH^p= sup

0<t<∞|Φt∗f|

L^p<∞,

where Φt(x)=t⁻ⁿΦ(x/t). It is known thatH^p(Rⁿ)=L^p(Rⁿ) if 1<p<∞([14, Chap- ter 3, Section 1.2.1]), and the definition ofH^p(Rⁿ) is independent of the choice of Φ∈S(Rⁿ) with

RⁿΦ(x)dx=0 ([14, Chapter 3, Theorem 1]). Let 0<p≤1. We say thatais anH^p-atom ifasatisfies

(2.1) suppa⊂B, aL^∞≤ |B|^−1/p and

Rⁿ

x^βa(x)dx= 0

for all|β|≤[n/p−n], whereB is an open ball and [n/p−n] stands for the largest integer ≤n/p−n. It is also known that everyf∈H^p(Rⁿ) can be written as a sum ofH^p-atoms:

f= ∞ j=1

λjaj inS(Rⁿ),

where {aj}j≥1 is a collection of H^p-atoms and {λj}j≥1 is a sequence of complex numbers with_∞

j=1|λj|^p<∞. Moreover, 1

Cinf _∞

j=1

|λj|^p

1/p

≤ fH^p≤Cinf _∞

j=1

|λj|^p

1/p

,

where the infimum is taken over all representations of f ([14, Chapter 3, Theo- rem 2]). By virtue of atomic decomposition of H^p, if an L²-bounded operator T satisfiesT aL^p≤Cfor allH^p-atomsa, thenT is bounded fromH^p(Rⁿ) toL^p(Rⁿ), where 0<p≤1.

We recall the definitions of Besov and modulation spaces. Assume that 0<q≤∞andρ∈R. Letψ0, ψ∈S(R^N) be such that

suppψ0⊂ {ξ∈R^N:|ξ| ≤2}, suppψ⊂ {ξ∈R^N: 2⁻¹≤ |ξ| ≤2}, (2.2)

ψ0(ξ)+

∞ j=1

ψ(2^−jξ) = 1 for allξ∈R^N, and setψ_j(ξ)=ψ(2^−jξ) if j≥1. Theweighted Besov space B_(s^∞,q

1,s2),ρ(Rⁿ×Rⁿ) con- sists of all σ∈S(R²ⁿ) such that

σB^∞,q

(s1,s2 )

= _∞

j=0

∞ k=0

(2^js12^ks2ψ_j(D_x)ψ_k(D_ξ)σ(x, ξ) ξ^ρL^∞_x,ξ)^q

1/q

<∞,

wheres1, s2∈R,{ψj}j≥0,{ψk}k≥0 are as in (2.2) withN=n, ψ_j(D_x)ψ_k(D_ξ)σ(x, ξ) =F_(y,η)⁻¹ _→(x,ξ)[ψ_j(y)ψ_k(η)ˆσ(y, η)]

andx, y, ξ, η∈Rⁿ. Similarly, B_(s^∞,q

1,...,s2n),ρ(R×...×R) is defined by the norm σB^∞,q

(s1,...,s2n),ρ=

_∞

j1,...,jn=0

∞ k₁,...,k_n=0

(2^{j1s1+...+jnsn}2^{k1sn+1+...+kns2n}

×ψj₁(Dx₁)...ψj_n(Dx_n)ψk₁(Dξ₁)...ψk_n(Dξ_n)σ(x, ξ) ξ^ρL^∞_x,ξ)^q

1/q

,

wheres1, ..., s2n∈R,{ψj1}j1≥0, ...,{ψkn}kn≥0 are as in (2.2) withN=1, ψj₁(Dx₁)...ψj_n(Dx_n)ψk₁(Dξ₁)...ψk_n(Dξ_n)σ(x, ξ)

=F_(y,η)⁻¹ _→(x,ξ)[ψj₁(y1)...ψj_n(yn)ψk₁(η1)...ψk_n(ηn)ˆσ(y, η)], y=(y1, ..., yn)∈Rⁿ andη=(η1, ..., ηn)∈Rⁿ. Letϕ∈S(Rⁿ) be such that

(2.3) suppϕ⊂[−1,1]ⁿ and

k∈Zⁿ

ϕ(ξ−k) = 1 for allξ∈Rⁿ.

Then theweighted modulation spaceM_ρ^∞,q(Rⁿ×Rⁿ) consists of allσ∈S(R²ⁿ) such that

σM_ρ^∞,q=

k∈Zⁿ

l∈Zⁿ

ϕ(D_x−k)ϕ(D_ξ−l)σ(x, ξ)ξ^ρq_L∞ x,ξ

1/q

<∞,

where

ϕ(D_x−k)ϕ(D_ξ−l)σ(x, ξ) =F_(y,η)⁻¹ _→(x,ξ)[ϕ(y−k)ϕ(η−l)ˆσ(y, η)].

3. Proof of Theorem1.2 withp≤2

In this section we prove Theorem1.2 withp≤2.

Lemma 3.1. Let s∈R. Then there exists a constant C >0 such that σ(X, D)D^sfL²≤C(R1...R2n)^1/2σ(x, ξ)ξ^sL^∞

x,ξfL²

for allσ(x, ξ)with supp ˆσ⊂2n

k=1[−R_k, R_k], R_k≥1,k=1, ...,2n,and allf∈L²(Rⁿ).

Here C >0 is independent ofRk≥1,k=1, ...,2n.

Proof. Letωs(ξ)= ξ^s, and let{ψj}j≥0 be as in (2.2) withN=n. Then (3.1) σ(x, ξ)ξ^s=

∞ j=0

σ(x, ξ)ψ_j(D)ω_s(ξ) = ∞ j=0

σ_j(x, ξ).

Since supp ˆσ⊂2n

k=1[−R_k, R_k] and suppψ_j(D)ω_s⊂{ξ∈R:|ξ|≤2^j+1}, we have suppσ_j⊂

n k=1

[−R_k, R_k] × 2n

k=n+1

[−(R_k+2^j+1), R_k+2^j+1] for allj≥0.

Hence, by Theorem1.1, σj(X, D)fL²

(3.2)

≤C(R1...Rn)^1/2((Rn+1+2^j)...(R2n+2^j))^1/2σj(x, ξ)L^∞_x,ξfL²

for allf∈L²(Rⁿ) andj≥0. If j=0 then

|σ₀(x, ξ)|= σ(x, ξ)

Rⁿ

Ψ₀(η) ξ−η^sdη

≤C|σ(x, ξ)|

Rⁿ|Ψ₀(η)| ξ^s η^|s|dη=C|σ(x, ξ)| ξ^s, where Ψ₀=F⁻¹ψ₀, and consequently

(3.3) σ0(x, ξ)L^∞_x,ξ≤Cσ(x, ξ) ξ^sL^∞_x,ξ. We consider the case j≥1, and note that

Rⁿη^βΨ(η)dη=i^|β|∂^βψ(0)=0 for allβ, where Ψ=F⁻¹ψ. LetN=[n/2]. Since

ψj(D)ωs(ξ) =

Rⁿ

2^jnΨ(2^j(ξ−η))

ωs(η)−

|β|≤N

(η−ξ)^β

β! ∂^βωs(ξ) dη

=

Rⁿ

2^jnΨ(2^j(ξ−η))

×

(N+1)

|β|=N+1

(η−ξ)^β β!

1 0

(1−t)^N∂^βωs(ξ+t(η−ξ))dt dη

andωs∈S^s_1,0 (that is,|∂^βωs(ξ)|≤Cβ ξ^s−|β|), we see that

|ψj(D)ωs(ξ)| ≤C

Rⁿ|Ψ(η)| |2^−jη|^N+1 1

0

ξ−2^−jtη^s−(N+1)dt dη

≤C

Rⁿ|Ψ(η)| |2^−jη|^N+1 1

0

ξ^s−(N+1) 2^−jtη^|s−(N+1)|dt dη

≤C2^−j(N+1) ξ^s−(N+1)

Rⁿ|Ψ(η)| |η|^N+1 η^|s−(N+1)|dη for allj≥1. This gives

(3.4) σ_j(x, ξ)L^∞_x,ξ=σ(x, ξ)ψ_j(D)ω_s(ξ)L^∞_x,ξ≤C2^−j(N+1)σ(x, ξ) ξ^sL^∞_x,ξ

for allj≥1. Note that ifa, b≥1 thena+b≤2ab. Therefore, since 2^j≥1,j≥0,Rk≥1, 1≤k≤2n, andN+1>n/2, we have by (3.1)–(3.4),

σ(X, D) D^sfL²≤ ∞ j=0

σj(X, D)fL²

≤C ∞ j=0

(R1...Rn)^1/2((Rn+1+2^j)...(R2n+2^j))^1/2

×σj(x, ξ)L^∞_x,ξfL²

≤C ∞ j=0

(R1...Rn)^1/2((2^jRn+1)...(2^jR2n))^1/2

×σ_j(x, ξ)L^∞_x,ξfL²

≤C(R₁...R_2n)^1/2 ∞ j=0

2^{−j((N+1)−n/2)}σ(x, ξ) ξ^sL^∞_x,ξfL²

=C(R1...R2n)^1/2σ(x, ξ)ξ^sL^∞_x,ξfL². The proof is complete.

We are now ready to prove Theorem1.2withp≤2.

Proof of Theorem 1.2 withp≤2. Letσ(x, ξ) be such that supp ˆσ⊂

2n j=1

[−Rj, Rj], Rj≥1, j= 1, ...,2n.

We first consider the case 0<p<1. By virtue of atomic decomposition of H^p, it is enough to prove that

(3.5) σ(X, D)fL^p≤C(R1...Rn)^1/2(Rn+1...R2n)^1/pσ(x, ξ)ξ^n(1/p−1/2)L^∞

x,ξ

for allf∈H^p(Rⁿ) satisfying

(3.6) suppf⊂ {x∈Rⁿ:|x| ≤r}, fL^∞≤r^−n/p and

Rⁿ

x^βf(x)dx= 0 for all |β|≤[n/p−n], where C >0 is independent of r. In fact, since · L^p and · H^p are translation invariant,σ(X, D)(f(· −x₀))(x)=σ(X+x₀, D)f(x−x₀) and suppF(x,ξ)→(y,η)[σ(x+x0, ξ)]⊂2n

j=1[−Rj, Rj], if (3.5) holds for allf satisfying (3.6)

then (3.5) holds for allf satisfying (2.1) (see Remark 5.1). Note that iff satisfies (3.6) thenfL²≤Cr^{−n(1/p−1/2)}.

Letf be as in (3.6) withr>1, and splitσ(X, D)f^p_Lp as σ(X, D)^pL^p

=

n

j=1[−2rRn+j,2rRn+j]

+

cn

j=1[−2rRn+j,2rRn+j] |σ(X, D)f(x)|^pdx.

Since suppF2⁻¹σ(x,·)⊂n

j=1[−R_n+j, R_n+j] for all x∈Rⁿ, suppf⊂{x∈Rⁿ:|x|≤r} and

(3.7) σ(X, D)f(x) =

Rⁿ

1 (2π)ⁿ

Rⁿ

e^i(x−y)·ξσ(x, ξ)dξ f(y)dy

=

RⁿF2⁻¹σ(x, x−y)f(y)dy, we see that

suppσ(X, D)f⊂ n j=1

[−R_n+j, R_n+j]+{x∈Rⁿ:|x| ≤r} ⊂ n j=1

[−(R_n+j+r), R_n+j+r].

By H¨older’s inequality and Theorem1.1,

n

j=1[−2rR_n+j,2rR_n+j]

|σ(X, D)f(x)|^pdx

1/p

(3.8)

≤ n

j=1

4rR_n+j

1/p−1/2

σ(X, D)fL²

≤Cr^n(1/p−1/2)(Rn+1...R2n)^1/p−1/2(R1...R2n)^1/2σ(x, ξ)L^∞_x,ξfL²

≤C(R1...Rn)^1/2(Rn+1...R2n)^1/pσ(x, ξ) ξ^n(1/p−1/2)L^∞

x,ξ, where we have used the fact that 1/p−1/2>0. On the other hand, (3.9)

cn

j=1[−2rRn+j,2rRn+j]|σ(X, D)f(x)|^pdx= 0.

In fact, sincer>1 andR_n+j≥1, 1≤j≤n, suppσ(X, D)f⊂

n j=1

[−(Rn+j+r), Rn+j+r]⊂ n j=1

[−2rRn+j,2rRn+j].

Hence, (3.8) and (3.9) give (3.5) forf satisfying (3.6) withr>1.

Letf be as in (3.6) withr≤1. In this case, our proof is similar to that of [17], but we give it for the reader’s convenience. Since

suppσ(X, D)f⊂ n j=1

[−(Rn+j+r), Rn+j+r]

⊂ n j=1

[−(R_n+j+1), R_n+j+1]⊂ n j=1

[−2R_n+j,2R_n+j], we have by H¨older’s inequality

(3.10) σ(X, D)fL^p≤C(Rn+1...R2n)^1/p−1/2σ(X, D)fL².

Letφ∈S(Rⁿ) be such thatφ=1 on{ξ∈Rⁿ:|ξ|≤1}and suppφ⊂{ξ∈Rⁿ:|ξ|≤2}, and splitσ(x, ξ) as

σ(x, ξ) =σ(x, ξ)φ(rξ)+σ(x, ξ)(1−φ(rξ)) =σ⁽¹⁾(x, ξ)+σ⁽²⁾(x, ξ).

SetK^(l)(x, y)=F2⁻¹σ^(l)(x, y), l=1,2, andN=[n(1/p−1)]. By (3.6) and (3.7), σ⁽¹⁾(X, D)f(x) =

|y|≤r

K⁽¹⁾(x, x−y)−

|β|≤N

(−y)^β

β! ∂₂^βK⁽¹⁾(x, x) f(y)dy

=

|y|≤r

(N+1)

|β|=N+1

(−y)^β β!

× 1

0

(1−t)^N∂₂^βK⁽¹⁾(x, x−ty)dt f(y)dy and

σ⁽²⁾(X, D)f(x) =

|y|≤r

K⁽²⁾(x, x−y)f(y)dy,

where∂₂^βK⁽¹⁾(x, y)=∂^β_yK⁽¹⁾(x, y). Then, by Minkowski’s inequality for integrals,

(3.11)

σ⁽¹⁾(X, D)fL²≤Cr^{N+1−n(1/p−1)} sup

0≤t≤1

|y|≤r

|β|=N+1

∂^β₂K⁽¹⁾(x, x−ty)L²_x,

σ⁽²⁾(X, D)fL²≤Cr^{−n(1/p−1)} sup

|y|≤r

K⁽²⁾(x, x−y)L²_x.

SettingFg⁽¹⁾_t,y,β,r(ξ)=φ(rξ)(iξ)^βe^−ity·ξ ξ^{−n(1/p−1/2)}, we obtain

∂₂^βK⁽¹⁾(x, x−ty) = 1 (2π)ⁿ

Rⁿ

e^{i(x−ty)·ξ}σ(x, ξ)φ(rξ)(iξ)^βdξ

= 1

(2π)ⁿ

Rⁿ

e^ix·ξσ(x, ξ)ξ^n(1/p−1/2)

×φ(rξ)(iξ)^βe^−ity·ξ ξ^{−n(1/p−1/2)}dξ

=σ(X, D)D^n(1/p−1/2)g_t,y,β,r⁽¹⁾ (x).

Similarly,

K⁽²⁾(x, x−y) =σ(X, D) D^n(1/p−1/2)g⁽²⁾_y,r(x), whereFg⁽²⁾y,r(ξ)=(1−φ(rξ))e^−iy·ξ ξ^{−n(1/p−1/2)}. Then, by Lemma3.1,

∂^β₂K⁽¹⁾(x, x−ty)L²_x=σ(X, D)D^n(1/p−1/2)g⁽¹⁾_t,y,β,rL²

(3.12)

≤C(R1...R2n)^1/2σ(x, ξ)ξ^n(1/p−1/2)L^∞_x,ξg⁽¹⁾_t,y,β,rL²

and

K⁽²⁾(x, x−y)L²_x=σ(X, D) D^n(1/p−1/2)g⁽²⁾_y,rL²

(3.13)

≤C(R1...R2n)^1/2σ(x, ξ)ξ^n(1/p−1/2)L^∞_x,ξg⁽²⁾_y,rL². Since N+1−n(1/p−1/2)>−n/2 and −n(1/p−1/2)<−n/2, by Plancherel’s theo- rem,

sup

0≤t≤1

|y|≤r

|β|=N+1

g⁽¹⁾_t,y,β,rL²= sup

0≤t≤1

|y|≤r

|β|=N+1

(2π)^−n/2Fg_t,y,β,r⁽¹⁾ L²

(3.14)

≤(2π)^−n/2φ(rξ)|ξ|^{N+1−n(1/p−1/2)}L²_ξ

=Cr^{−(N+1−n(1/p−1))} and

(3.15) sup

|y|≤r

g_y,r⁽²⁾L²_x≤(2π)^−n/2(1−φ(rξ))|ξ|^{−n(1/p−1/2)}L²_ξ=Cr^n(1/p−1). Combining (3.10)–(3.15), we have (3.5) forf satisfying (3.6) with r≤1. Thus, we obtain Theorem1.2with 0<p<1.

We next consider the case 1≤p<2 (since we already have the case p=2 as Theorem1.1). We use the interpolation theorem for analytic families of operators (see Stein–Weiss [15] and Calder´on–Torchinsky [2]). Since our proof is very similar to that of Sugimoto [17, Section 5] (or Miyachi [11, p. 150]), we shall only indicate the necessary modifications.

Let 0<p0<1 and S={z∈C:0≤Rez≤1}. For 1≤p<2, we take 0<θ<1 such that 1/p=(1−θ)/p0+θ/2. Set

σz(x, ξ) =e^(z−θ)2σ(x, ξ) ξ^{n(1/p0−1/2)(z−θ)} forz∈S, and note that

(3.16) σθ(X, D) =σ(X, D).

Letω_{n(1/p0−1/2)}(ξ)= ξ^{n(1/p0−1/2)}, and let{ψj}j≥0be as in (2.2) withN=n. Then σz(x, ξ) =

∞ j=0

e^(z−θ)2σ(x, ξ)ψj(D)ω^z_n(1/p^−θ

0−1/2)(ξ) = ∞ j=0

σz,j(x, ξ).

In the same way as in the proof of Lemma3.1, we can prove that

|ψ_j(D)ω_n(1/p^it−θ

0−1/2)(ξ)| ≤Cmax{1, t}^N+12^−j(N+1) ξ^{−nθ(1/p0−1/2)} for allt∈Rand j≥0, whereN=[n/p0]. Hence, since

supp ˆσit,j⊂ n

k=1

[−Rk, Rk] × 2n

k=n+1

[−(Rk+2^j+1), Rk+2^j+1]

for all t∈Rand j≥0 (see the proof of Lemma 3.1), we have by Theorem1.2 with 0<p₀<1,

σ_it(X, D)f^p_L⁰p0≤^∞

j=0

σ_it,j(X, D)f^p_L⁰p0

≤C ∞ j=0

(R1...Rn)^p0/2((Rn+1+2^j)...(R2n+2^j))^p0/p0

×σ_it,j(x, ξ)ξ^{n(1/p0−1/2)p}_L⁰∞

x,ξf^p_H^0p0

≤C(R1...Rn)^p0/2(Rn+1...R2n)^p0/p0

× ∞ j=0

(e^−t2+θ2max{1, t}^N+12^{−j(N+1−n/p0)})^p0

×σ(x, ξ)ξ^{−nθ(1/p0−1/2)} ξ^{n(1/p0−1/2)p}_L⁰∞

x,ξf^p_H^0p0

≤C(R₁...R_n)^p0/2(R_n+1...R_2n)^p0/p0

×σ(x, ξ)ξ^{n(1/p−1/2)p}L^0∞_x,ξf^pH^0p0,

that is,

σit(X, D)fL^p0≤C(R1...Rn)^1/2(Rn+1...R2n)^1/p0 (3.17)

×σ(x, ξ) ξ^n(1/p−1/2)L^∞_x,ξfH^p0

for allf∈S(Rⁿ)∩H^p0(Rⁿ) andt∈R. Similarly,

σ1+it(X, D)fL²≤C(R1...Rn)^1/2(Rn+1...R2n)^1/2 (3.18)

×σ(x, ξ) ξ^n(1/p−1/2)L^∞_x,ξfL²

for all f∈S(Rⁿ) and t∈R. Therefore, by the interpolation theorem for analytic families of operators with (3.16)–(3.18), we obtain Theorem1.2with 1≤p<2.

4. Proof of Theorem 1.2with p≥2

Hwang–Lee [9] essentially proved the following lemma, and obtained Miyachi’s result (see the introduction) as a corollary. In order to prove Theorem 1.2 with p≥2, we modify their estimate as follows:

Lemma 4.1. Let 2≤p<∞and m=−n(1/2−1/p). Forϕ1, ..., ϕn, ψ1, ..., ψn∈ S(R)andf, g∈S(Rⁿ), set

ϕ(y) = [ϕ₁⊗...⊗ϕ_n](y), ψ(η) = [ψ1⊗...⊗ψn](η), V(f, g)(y, η) =

Rⁿ

e^iy·tf(η+t)g(t)dt, W(ϕ, ψ, f, g)(x, ξ) =

R²ⁿ

e^{−i(x·y+ξ·η)}ϕ(y)ψ(η)V(f, g)(y, η)dy dη,

where [ϕ1⊗...⊗ϕn](y)=ϕ1(y1)...ϕn(yn) and y=(y1, ..., yn)∈Rⁿ. Then there exists a constant C >0 such that

R²ⁿ

ξ^m|W(ϕ, ψ, f, g)(x, ξ)|dx dξ

≤C n

j=1

ϕjL^p+ϕ⁽¹⁾_j

L^p+ϕ⁽²⁾_j

L^p

× n

k=1

ψˆkL²+ψ⁽¹⁾_k

L¹

+(ψ_k⁽¹⁾)⁽¹⁾

L¹ fL^pgL^p, where ϕ⁽¹⁾_j (y_j)=dϕ_j(y_j)/dy_j, ϕ⁽²⁾_j (y_j)=d²ϕ_j(y_j)/dy²_j and C >0 is independent of ϕ₁, ..., ϕn, ψ1, ..., ψn∈S(R)and f, g∈S(Rⁿ).

To prove Lemma4.1, we use the following result.

Lemma 4.2.[9, Lemma 2.4] Let 2≤p<∞ andp≤q≤p. Then there exists a constantC >0 such that

Rⁿ|fˆ(ξ)|^q|ξ|^{−n(1−q/p)}dξ

1/q

≤CfL^p.

Proof of Lemma 4.1. By the Fourier inversion formula, we have W(ϕ, ψ, f, g)(x, ξ) =

R²ⁿ

e^{−i(x·y+ξ·η)}ϕ(y)ψ(η)V(f, g)(y, η)dy dη

=

R²ⁿ

e^{−i(x·y+ξ·η)}ϕ(y)ψ(η)

× Rⁿ

e^iy·tf(η+t) 1

(2π)ⁿ

Rⁿ

e^it·ζˆg(ζ)dζ dt dy dη

= 1

(2π)ⁿ

R³ⁿ

e^{−i(x·y+ξ·η)}ϕ(y)ψ(η)ˆg(ζ)

× Rⁿ

e^{i(y−ζ)·(t−η)}f(t)dt dy dη dζ

= 1

(2π)ⁿ

R³ⁿ

e^−ix·ye^i(y−ζ)·tϕ(y)f(t)ˆg(ζ)

× Rⁿ

e^{−i(y+ξ−ζ)·η}ψ(η)dη dt dy dζ

= 1

(2π)ⁿ

R²ⁿe^−iζ·tf(t)ˆg(ζ)

× Rⁿ

e^i(t−x)·yϕ(y) ˆψ(y+ξ−ζ)dy dt dζ

= 1

(2π)ⁿ

R²ⁿ

e^{ix·(ξ−ζ)}e^−it·ξ

× Rⁿ

e^i(t−x)·yϕ(y+ζ−ξ) ˆψ(y)dy f(t)ˆg(ζ)dt dζ.

For the sake of simplicity, we only consider the case n=2 (the case n=2 can be proved in the same way). This means that ϕ(y)=ϕ₁(y1)ϕ2(y2) and ψ(η)=