dedicated to the memory of cora sadoskyvnaibo/herbertnaibo_2015.pdf · 2018. 10. 16. · dedicated...

BESOV SPACES, SYMBOLIC CALCULUS AND BOUNDEDNESSOF BILINEAR PSEUDODIFFERENTIAL OPERATORS.

JODI HERBERT AND VIRGINIA NAIBO

Dedicated to the memory of Cora Sadosky

Abstract. Mapping properties of bilinear pseudodifferential operators with sym-bols of limited smoothness in terms of Besov norms are proved in the context ofLebesgue spaces. Techniques used include the development of a symbolic calculusfor certain classes of symbols considered.

1. Introduction and main results

In this article we present results concerning boundedness properties in the settingof Lebesgue spaces of bilinear pseudodifferential operators associated to symbols oflimited smoothness measured in terms of Besov norms. Closely connected to suchsymbols are the bilinear Hörmander classes BSm0,0, m ∈ R, whose elements are infin-itely differentiable complex-valued functions σ = σ(x, ξ, η), x, ξ, η ∈ Rn, that satisfy(1.1) sup

|α|≤N|β|,|γ|≤M

supx,ξ,η∈Rn

〈ξ, η〉−m|∂αx∂βξ ∂γησ(x, ξ, η)|

2 JODI HERBERT AND VIRGINIA NAIBO

smallest numbers N and M for which condition (1.1) is sufficient for boundedness ofthe associated operator.

We next present the results in this article, which continue the work originated in[7] and [8]. See Section 2 for notation.

Theorem 1.1. Consider 1 ≤ p1, p2 ≤ ∞ and p = 1 such that 1p1 +1p2

= 1p

or,

2 ≤ p1, p2 ≤ ∞ and p satisfying 1p1 +1p2

= 1p. If m < m(p1, p2), s(p) is as in Definition

2.1 and s is a vector of the same dimension as s(p), the following statements holdtrue:

(a) If 0 < q ≤ 1 and s ≥ s(p) component-wise, then‖Tσ(f, g)‖Lp . ‖σ‖Bs,m∞,q ‖f‖Lp1 ‖g‖Lp2 ,

for all f, g ∈ S(Rn) and all x-independent symbols σ in Bs,m∞,q(R3n).(b) If 1 < q ≤ ∞ and s > s(p) component-wise, then

‖Tσ(f, g)‖Lp . ‖σ‖Bs,m∞,q ‖f‖Lp1 ‖g‖Lp2 ,

for all f, g ∈ S(Rn) and all x-independent symbols σ in Bs,m∞,q(R3n).If 2 ≤ p1, p2, p ≤ ∞ with 1p1 +

1p2

= 1p, the above statements hold for any σ ∈

Bs,m∞,q(R3n).

Theorem 1.2. Consider 1 ≤ p1, p2, p ≤ ∞ such that 1p1 +1p2

= 1p

and m < m(p1, p2).

(a) If 0 < q ≤ 1, s = (s1, s2, n2 ) ∈ R3 or s = (s1, . . . , s2n, 12 , . . . , 12) ∈ R3n andσ ∈ Bs,m∞,q(R3n) is such that

σ̂(y, a, b) =

∫R3n

σ(x, ξ, η) e−2πi(y·x+a·ξ+b·η) dx dξ dη, y, a, b ∈ Rn

has compact support in y and a uniformly in b, then Tσ is bounded from Lp1(Rn)×

Lp2(Rn) into Lp(Rn), where 2 ≤ p1, p2, p ≤ ∞ or, 1 ≤ p, p1 ≤ 2 and 2 ≤ p2 ≤ ∞.The same conclusion is true for q > 1 and s = (s1, s2, s3) with s3 >

n2

or

s = (s1, . . . , s3n) with sj >12

for j = 2n+ 1 . . . , 3n.

(b) If 0 < q ≤ 1, s = (s1, n2 , s3) ∈ R3 or s = (s1, . . . , sn, 12 , . . . , 12 , s2n+1, . . . s3n) ∈ R3nand σ ∈ Bs,m∞,q(R3n) is such that

σ̂(y, a, b) =

∫R3n


has compact support in y and b uniformly in a, then Tσ is bounded from Lp1(Rn)×

Lp2(Rn) into Lp(Rn), where 2 ≤ p1, p2, p ≤ ∞ or, 1 ≤ p, p2 ≤ 2 and 2 ≤ p1 ≤ ∞.The same conclusion is true for q > 1 and s = (s1, s2, s3) with s2 >

n2

or

s = (s1, . . . , s3n) with sj >12

for j = n+ 1 . . . , 2n.

Figure 1 shows the value of m(p1, p2) according to the region in the square [0, 1]×[0, 1] to which the point ( 1

p1, 1p2

) belongs. Theorem 1.1 states, in particular, that

operators with symbols in Bs(p),m∞,1 (R3n) are bounded from Lp1(Rn) × Lp2(Rn) into

BESOV SPACES, SYMBOLIC CALCULUS AND BOUNDEDNESS OF OPERATORS. 3

Lp(Rn) if m < m(p1, p2) and ( 1p1 ,1p2

) belongs to region II, and that operators with x-

independent symbols in Bs(p),m∞,1 (R3n) are bounded from Lp1(Rn)×Lp2(Rn) into Lp(Rn)

if m < m(p1, p2) and (1p1, 1p2

) belongs to region I or to the line segment joining (1, 0)

and (0, 1). The mapping properties of Theorem 1.2 correspond to regions II and IIIfor item (a) and to regions II and IV for item (b).

(0, 1)

(0, 12)

(0, 0) (1, 0)

(1, 1)

(12, 0)

1p1

1p2

− np2

− np1

−n2

−n(1− 1p) I

II

III

IV

Figure 1. Visualization of m(p1, p2),1p1

+ 1p2

= 1p.

Corollary 1.3. Consider 1 ≤ p1, p2 ≤ ∞ and p = 1 such that 1p1 +1p2

= 1p

or,

2 ≤ p1, p2 ≤ ∞ and p satisfying 1p1 +1p2

= 1p. If m < m(p1, p2) then Tσ is bounded from

Lp1(Rn)× Lp2(Rn) into Lp(Rn) for any σ satisfying one of the following conditions:(a) If 2 ≤ p ≤ ∞,(1.2) sup

|α|≤[np

]+1

|β|,|γ|≤[n2

]+1

supx,ξ,η∈Rn



Corollary 1.4. Consider 1 ≤ p1, p2, p ≤ ∞ such that 1p1 +1p2

= 1p

and m < m(p1, p2).

(a) If σ = σ(x, ξ, η), x, ξ, η ∈ Rn, is such that

σ̂(y, a, b) =

∫R3n


has compact support in y and a uniformly in b, and

(1.6) sup|α|≤1,|β|≤1|γ|≤[n

2]+1

supx,ξ,η∈Rn



The statements of Theorem 1.1 and Corollary 1.3 for the case 1 ≤ p < 2 correspondto operators associated to x-independent symbols, i.e. bilinear multipliers. Addi-tional results concerning minimal smoothness conditions for the symbols of relatedbilinear multipliers that are sufficient for boundedness can be found in [6, 12, 13] andreferences therein. In particular, a consequence of the results in [13] is that bilinearmultipliers with symbols σ = σ(ξ, η), ξ, η ∈ Rn, satisfying

sup|β+γ|≤[−m(p1,p2)]+1

supξ,η∈Rn

〈ξ, η〉−m(p1,p2)|∂βξ ∂γησ(ξ, η)|


{wj}j∈N0 , satisfying the following conditions:w0 ∈ S(RN), supp(w0) ⊂ {ξ ∈ RN : |ξ| ≤ 2},w ∈ S(RN), supp(w) ⊂ {ξ ∈ RN : 1

2≤ |ξ| ≤ 2},(2.11)

wj(ξ) := w(2−jξ) for ξ ∈ Rn and j ∈ N,

∞∑j=0

wj(ξ) = 1 ∀ξ ∈ RN .

Given such a Littlewood-Paley partition of unity we set:

w̃(ξ) :=2∑

l=−2

w(2−lξ) for ξ ∈ RN ,

w̃j(ξ) := w̃(2−jξ) =

j+2∑l=j−2

wl(ξ) for j ≥ 2 and ξ ∈ RN ,(2.12)

w̃j(ξ) :=

j+2∑l=0

wl(ξ) for j = 0, 1 and ξ ∈ RN .

Note that w̃j(ξ) ≡ 1 for 2j−2 ≤ |ξ| ≤ 2j+2 and j ≥ 2, and that w̃j(ξ) ≡ 1 for |ξ| ≤ 2j+2and j = 0, 1; therefore wjw̃j = wj for all j ∈ N0.Besov spaces of product type. For m ∈ R, 0 < r, q ≤ ∞, and s ∈ R3 or s ∈ R3n,the Besov spaces Bs,mr,q (R3n) are defined as follows:• Given s = (s1, s2, s3) ∈ R3 and {wj}j∈N0 satisfying (2.11) with N = n, Bs,mr,q (R3n)

denotes the space of complex-valued functions σ(x, ξ, η), x, ξ, η ∈ Rn, such that

‖σ‖Bs,mr,q :=

∑k∈N30

(2s·k

∥∥〈ξ, η〉−mF−1(wkσ̂)∥∥Lr)q 1q


contains S(R3n), and that S(R3n) is dense if 0 < r, q < ∞. We refer the reader to[17], where a variety of Besov spaces of product type are defined and many of theirproperties are presented. See also Proposition 2.1 below.

The classes Csm(R3n). We next define classes of symbols that are closely connectedwith both Bs,mr,q (R3n) and the Hörmander classes BSm0,0. Given s ∈ N30 or s ∈ N3n0 andm ∈ R, a complex-valued functions σ(x, ξ, η), x, ξ, η ∈ Rn, belongs to Csm(R3n) if itsatisfies the following conditions.

• If s = (s1, s2, s3) ∈ N30 :∂αx∂

βξ ∂

γησ ∈ C(R3n) for α, β, γ ∈ Nn0 , |α| ≤ s1, |β| ≤ s2, |γ| ≤ s3, and

‖σ‖Csm := sup|α|≤s1|β|≤s2,|γ|≤s3

supx,ξ,η∈Rn



(b) If 1 ≤ r, r̃ ≤ ∞, 0 < q, q̃ ≤ ∞, s, s̃ are vectors of the same dimension (dimension3 or 3n) with positive components, and m ∈ R, then Bs,mr,q (R3n) = B s̃,mr̃,q̃ (R3n) ifand only if r = r̃, q = q̃ and s = s̃.

(c) Let 1 ≤ r ≤ ∞, 0 < q ≤ ∞, m ∈ R, s = (s1, . . . , s3n) ∈ R3n with sk > 0 fork = 1, . . . , 3n, and s̃ = (s1 + . . .+ sn, sn+1 + . . .+ s2n, s2n+1 + . . .+ s3n). Then thefollowing continuous inclusion holds:

B s̃,mr,q (R3n) $ Bs,mr,q (R3n).

We end this section with the following definitions.

Definition 2.1. Given 1 ≤ p ≤ ∞, s(p) will denote the 3-dimensional vector givenby (min(n

2, np),max(n

2, np),max(n

2, np)) or the 3n-dimensional vector (s1, . . . , s3n) where

s1 = · · · = sn = min(12 , 1p) and sn+1 = · · · = s3n = max(12 , 1p). This is,s(p) = (n

2, np, np) or s(p) = (1

2, . . . , 1

2︸︷︷︸n

, 1p, . . . , 1

p︸︷︷︸2n

), 1 ≤ p < 2,

s(p) = (np, n

2, n

2) or s(p) = (1

p, . . . , 1

p︸︷︷︸n

, 12, . . . , 1

2︸︷︷︸2n

), 2 ≤ p ≤ ∞.

It will be clear from the context which of these definitions of s(p) is being used ineach case.

Definition 2.2. If s = (s1, s2, s3) ∈ R3+ then s∗1 := (s1, s2 − s3, s3) and s∗2 :=(s1, s2, s3−s2). If s = (s1, . . . , s3n) ∈ R3n+ , then s∗1 := (s1, . . . , sn, sn+1−s2n+1, . . . , s2n−s3n, s2n+1, . . . , s3n) and s

∗2 := (s1, . . . , sn, sn+1, . . . , s2n, s2n+1 − sn+1, . . . , s3n − s2n).

3. Symbolic calculus: Proof of Theorem 1.5

We start with some preliminaries in relation to the symbols of the transposes ofTσ for σ ∈ Bs,m∞,1(R3n), where m ∈ R and s is a vector in R3+ or R3n+ . For such σ letΣ ∈ S ′(R3n) be given by

Σ := F−1(σ̂(y,−a, b− a)e2πia·y),where F−1 andˆare taken in R3n and the tempered distribution σ̂(y,−a, b− a)e2πia·yis understood in the usual sense, this is,

〈σ̂(y,−a, b− a)e2πia·y, F 〉 = 〈σ̂, F (y,−a, b− a)e−2πia·y〉, F ∈ S(R3n),where 〈T, ϕ〉 denotes the action of a tempered distribution T on a Schwartz functionϕ. Assume that Σ ∈ B s̃,m∞,1(R3n) for some s̃ with positive components; let us see thatσ∗1 = Σ. Indeed, define

V (f, g, h)(y, a, b) :=

∫R3n

e2πi(y·x+a·ξ+b·η)f̂(ξ)ĝ(η)e2πix·(ξ+η)h̄(x) dxdξdη

=

∫Rne2πiy·xh̄(x)f(x+ a)g(x+ b) dx,


which belongs to S(R3n) for any f, g, h ∈ S(Rn). SetṼ (f, g, h̄)(y, a, b) := e−2πia·yV (f, g, h̄)(y,−a, b− a) = V (h, g, f̄)(y, a, b)

and note that since σ ∈ Bs,m∞,1(R3n) ⊂ C[s]m (R3n) the action of σ as a tempered dis-tribution in R3n is given through absolutely convergent integrals in R3n. We thenhave ∫

RnTσ(h, g)(x)f(x) dx =

∫R3n

σ(x, ξ, η)ĥ(ξ)ĝ(η)e2πix·(ξ+η)f(x) dxdξdη

= 〈σ̂, V (h, g, f̄)〉 = 〈σ̂, Ṽ (f, g, h̄)〉 = 〈Σ̂, V (f, g, h̄)〉

=

∫R3n

Σ(x, ξ, η)f̂(ξ)ĝ(η)e2πix·(ξ+η)h(x) dxdξdη

=

∫RnTΣ(f, g)(x)h(x) dx,

which shows that σ∗1 = Σ. In the second to the last equality we have used the fact

that Σ ∈ B s̃,m∞,1(R3n) ⊂ C[s̃]m (R3n) and therefore the action of Σ over the function inS(R3n) given by f̂(ξ)ĝ(η)e2πix·(ξ+η)h(x) is through integration.

An analogous explanation leads to σ∗2 = F−1(σ̂(y, a−b,−b)e2πib·y). We also observethat in the particular case when σ is x-independent, then σ∗1(ξ, η) = σ(−ξ − η, η)and σ∗2(ξ, η) = σ(ξ,−η − ξ).

We next make a remark about the norm of x-independent symbols belonging to theBesov spaces Bs,m∞,q(R3n). Let {wj}j∈N0 be as in (2.11) with N = n, s = (s1, s2, s3) ∈R3, m ∈ R and 0 < q ≤ ∞. If σ = σ(ξ, η), ξ, η ∈ Rn, is an x-independent symbol inBs,m∞,q(R3n) it easily follows that

(3.14) ‖σ‖Bs,m∞,q ∼

∑k∈N20

(2s̄·k

∥∥〈ξ, η〉−mF−1(wkσ̂)∥∥L∞)q 1q ,

where s̄ = (s2, s3), wk(a, b) = wk1(a)wk2(b) for k = (k1, k2) ∈ N20, F−1 andˆdenoteinverse Fourier transform and Fourier transform in R2n, respectively, and the L∞norm is taken in R2n. This is, an x-independent symbols σ = σ(ξ, η), ξ, η ∈ Rn,belongs to Bs,m∞,1(R3n) if an only if σ ∈ B s̄,m∞,1(R2n), where B s̄,m∞,1(R2n) is defined as theclass of symbols τ = τ(ξ, η), ξ, η ∈ Rn, such that the righthand side of (3.14), with τreplacing σ, is finite. An analogous remark corresponds to the case s ∈ R3n, with s̄being the vector in R2n whose components are the last 2n components of s.

Lemma 3.1 below will be useful in the proof of Theorem 1.5. We state it inR2n and R3n because it is convenient for our settings, but more general versionsalso hold (compare with [17] or [18, p. 25-28]). If h is a function defined inRN and d = (d1, . . . , dN) ∈ RN , denote Sd(h)(y1, . . . , yN) := h(d1y1, . . . , dNyN)and Sd−1(h)(y1, . . . , yN) := h(d

−11 y1, . . . , d

−1N yN). In particular, if h(ξ, η) = 〈ξ, η〉

for ξ, η ∈ Rn and d ∈ R2n, we write 〈ξ, η〉d−1 instead of Sd−1(h) and we have〈ξ, η〉d−1 := 〈(d−11 ξ1, . . . , d−1n ξn), (d−1n+1η1, . . . , d−12n ηn)〉. In Lemma 3.1, integration is


in R2n in part (a) and in R3n in part (b); additionally, the inverse Fourier transformand the Fourier transform are taken in R2n in part (a) and in R3n in part (b).Lemma 3.1. Let 1 ≤ r ≤ ∞ and t ∈ R.(a) For every continuous function g(ξ, η) defined for ξ, η ∈ Rn such that ‖〈ξ, η〉tg‖Lr <∞ and every M ∈ S(R2n),

(3.15)∥∥〈ξ, η〉td−1F−1Mĝ∥∥Lr ≤ ∥∥∥〈ξ, η〉|t|d−1F−1M∥∥∥L1 ∥∥〈ξ, η〉td−1g∥∥Lr

for any d ∈ R2n. In particular, if d = (d1, . . . , d2n) and di ≥ 1 for i = 1, . . . , 2n,(3.16)

∥∥〈ξ, η〉tF−1Mĝ∥∥Lr≤∥∥〈ξ, η〉|t|F−1Sd(M)∥∥L1 ∥∥〈ξ, η〉tg∥∥Lr .

(b) For every continuous function g(x, ξ, η) defined for x, ξ, η ∈ Rn such that ‖〈ξ, η〉tg‖Lr <∞ and every M ∈ S(R3n),

(3.17)∥∥〈ξ, η〉td̄−1F−1Mĝ∥∥Lr ≤ ∥∥∥〈ξ, η〉|t|d̄−1F−1M∥∥∥L1 ∥∥〈ξ, η〉td̄−1g∥∥Lr

for any d ∈ R3n and where d̄ is the vector in R2n whose components are thelast 2n components of d. In particular, if d = (d1, . . . , d3n) and di ≥ 1 for i =n+ 1, . . . , 3n,

(3.18)∥∥〈ξ, η〉tF−1Mĝ∥∥

Lr≤∥∥〈ξ, η〉|t|F−1Sd(M)∥∥L1 ∥∥〈ξ, η〉tg∥∥Lr .

Proof of Lemma 3.1. We prove item (a), with part (b) following analogously. Wehave 〈u+ y, v + z〉t . 〈u, v〉|t|〈y, z〉t for all u, v, y, z ∈ Rn. Then

|〈ξ, η〉td−1F−1(Mĝ)(ξ, η)| .∫R2n〈a, b〉|t|d−1|M̌(a, b)|〈ξ − a, η − b〉td−1|g(ξ − a, η − b)| da db,

from where (3.15) follows by Minkowski’s integral inequality. For (3.16), apply (3.15)

with M replaced by Sd(M) and g replaced by Sd−1(g) and note that 〈ξ, η〉|t|d−1 ≤ 〈ξ, η〉|t|since di ≥ 1 for i = 1, . . . , 2n.

�

Proof of part (a) of Theorem 1.5. We will prove the result for σ∗1, with the result forσ∗2 following in an analogous way. We will also assume q = 1; the case 0 < q < 1can be deduced in the same way. Fix m ∈ R and let σ be an x-independent symbolin Bs,m∞,1(R3n). We have σ∗1(ξ, η) = σ(−ξ − η, η) for ξ, η ∈ Rn.

Consider first the case when s = (s1, s2, s3) ∈ R3+, then s∗1 = (s1, s2 − s3, s3).As already observed, since σ is x-independen, s1 does not play any role here. Letw and w0 be radial functions that satisfy (2.11) for N = n and let {wj}j∈N0 be thecorresponding Littlewood-Paley partition of unity. In view of (3.14) we have to provethat(3.19)∑

k∈N20

2(s2−s3,s3)·k∥∥∥〈ξ, η〉−mF−1(wkσ̂∗1)∥∥∥

L∞.∑k∈N20

2(s2,s3)·k∥∥〈ξ, η〉−mF−1(wkσ̂)∥∥L∞

where wk(a, b) = wk1(a)wk2(b) for k = (k1, k2) ∈ N20, F−1 andˆdenote inverse Fouriertransform and Fourier transform in R2n, respectively, and the L∞ norm is taken inR2n.


Given k = (k1, k2) ∈ N20 and noting that σ̂∗1(a, b) = σ̂(−a, b − a), a change ofvariables gives

F−1(wkσ̂∗1)(ξ, η) = F−1(wk1(a)wk2(b− a)σ̂(a, b))(−η − ξ, η).

Since 〈ξ, η〉 ∼ 〈ξ + η, η〉, it then follows that

(3.20)∥∥∥〈ξ, η〉−mF−1(wkσ̂∗1)∥∥∥

L∞∼∥∥〈ξ, η〉−mF−1(wk1(a)wk2(b− a)σ̂(a, b))∥∥L∞ .

We will split the summation in (k1, k2) ∈ N20 according to the following regions:

R = N2, R1 = {(k1, 0) : k1 ≥ 3}, R2 = {(0, k2) : k2 ≥ 3},R3 = {(0, 0), (0, 1), (0, 2), (1, 0), (2, 0)}.

Let {w̃j}j∈N0 be as in (2.12). Recall that w̃j(a) ≡ 1 for 2j−2 ≤ |a| ≤ 2j+2 andj ≥ 2, and that w̃j(a) ≡ 1 for |a| ≤ 2j+2 and j = 0, 1; in particular, wjw̃j = wj forall j ∈ N0. For (k1, k2) ∈ N20 and a, b ∈ Rn set h(k1,k2)(a, b) := wk1(a)w̃k2(b).

Summation in region R: Consider the following subregions

RA = {(k1, k2) ∈ N2 : k1 − k2 > 2}, RB = {(k1, k2) ∈ N2 : k1 − k2 < −2},RC = {(k1, k2) ∈ N2 : −2 ≤ k1 − k2 ≤ 2}.

We first estimate the summation in region RA. If (k1, k2) ∈ RA,

supp(wk1(a)wk2(b− a)) ⊂ {(a, b) : 2k1−1 ≤ |a| ≤ 2k1+1 and 12 2k1−1 ≤ |b| ≤ 98 2k1+1}

and therefore wk1(a)wk2(b− a) = w̃k1(a)wk2(b− a)wk1(a)w̃k1(b). Then (3.20) implies∥∥∥〈ξ, η〉−mF−1(wkσ̂∗1)∥∥∥L∞∼∥∥〈ξ, η〉−mF−1[w̃k1(a)wk2(b− a)F(F−1(h(k1,k1)σ̂))(a, b)]∥∥L∞ .

By (3.16) in Lemma 3.1 it follows that,∥∥〈ξ, η〉−mF−1[w̃k1(a)wk2(b− a)F(F−1(h(k1,k1)σ̂))(a, b)]∥∥L∞.∥∥〈ξ, η〉|m|F−1[w̃k1(2k2a)wk2(2k2(b− a))]∥∥L1 ∥∥〈ξ, η〉−mF−1(h(k1,k1)σ̂)∥∥L∞ .

An elementary computation shows that for 1 ≤ k2 ≤ k1 − 3,∥∥〈ξ, η〉|m|F−1[w̃k1(2k2a)wk2(2k2(b− a))]∥∥L1 . 1,therefore ∑

(k1,k2)∈RA

2(s2−s3,s3)·(k1,k2)∥∥∥〈ξ, η〉−mF−1(wkσ̂∗1)∥∥∥

L∞

.∞∑k1=4

k1−3∑k2=1

2(s2−s3,s3)·(k1,k2)∥∥〈ξ, η〉−mF−1(h(k1,k1)σ̂)∥∥L∞ .


We have then obtained∑k∈RA

2(s2−s3,s3)·k∥∥∥〈ξ, η〉−mF−1(wkσ̂∗1)∥∥∥

L∞

.∞∑k1=4

2(s2−s3,s3)·(k1,k1)∥∥〈ξ, η〉−mF−1(h(k1,k1)σ̂)∥∥L∞ .(3.21)

We now look at the summation in the region RB. Note that if (k1, k2) ∈ RB, thensupp(wk1(a)wk2(b− a)) ⊂ {(a, b) : 2k1−1 ≤ |a| ≤ 2k1+1 and 12 2k2−1≤|b| ≤ 98 2k2+1}.

For 1 ≤ k1 ≤ k2 − 3 we have wk1(a)wk2(b − a) = wk2(b − a)w̃k2(b)wk1(a)w̃k2(b), and(3.20) implies∥∥∥〈ξ, η〉−mF−1(wkσ̂∗1)∥∥∥

L∞∼∥∥〈ξ, η〉−mF−1[wk2(b− a)w̃k2(b)F(F−1(h(k1,k2)σ̂))(a, b)]∥∥L∞ .

By (3.16) in Lemma 3.1 and noting that∥∥〈ξ, η〉|m|F−1(wk2(2k2(b− a))w̃k2(2k2b))∥∥L1 . 1we obtain ∑

k∈RB

2(s2−s3,s3)·k∥∥∥〈ξ, η〉−mF−1(wkσ̂∗1)∥∥∥

L∞

.∞∑k2=4

k2−3∑k1=1

2(s2−s3,s3)·(k1,k2)∥∥〈ξ, η〉−mF−1(h(k1,k2)σ̂)∥∥L∞ .(3.22)

For (k1, k2) ∈ RC it follows thatsupp(wk1(a)wk2(b− a)) ⊂ {(a, b) : 2k1−1 ≤ |a| ≤ 2k1+1 and |b| ≤ 10 · 2k1}.

Set χk1 :=∑k1+4

j=0 wj for k1 ∈ N; since χk1(b) = 1 for all b in the set {b : |b| ≤ 10 · 2k1}then

wk1(a)wk2(b− a) =k1+4∑j=0

wk2(b− a)χk1(b)wk1(a)wj(b).

From this and (3.20) it follows that for k1 ∈ N,k1+2∑

k2=max(0,k1−2)

2(s2−s3,s3)·(k1,k2)∥∥∥〈ξ, η〉−mF−1(w(k1,k2)σ̂∗1)∥∥∥

L∞

.k1+4∑j=0

k1+2∑k2=max(0,k1−2)

2(s2−s3,s3)·(k1,k2)∥∥〈ξ, η〉−mF−1[wk2(b− a)χk1(b)F(F−1(w(k1,j)σ̂))(a, b)]∥∥L∞ .

By (3.16) in Lemma 3.1 and since

k1+2∑k2=max(0,k1−2)

2(s2−s3,s3)·(k1,k2)∥∥〈ξ, η〉|m|F−1(wk2(2k1(b− a))χk1(2k1b))∥∥L1 . 2(s2−s3,s3)·(k1,k1),


it follows that(3.23)∑k∈RC

2(s2−s3,s3)·k∥∥∥〈ξ, η〉−mF−1(wkσ̂∗1)∥∥∥

L∞.

∞∑k1=1

k1+4∑j=0

2(s2,0)·(k1,j)∥∥〈ξ, η〉−mF−1(w(k1,j)σ̂)∥∥L∞ .

Summation in region R1: In view of (3.20), we have to estimate∞∑k1=3

2(s2−s3)k1∥∥〈ξ, η〉−mF−1(wk1(a)w0(b− a)σ̂(a, b))∥∥L∞ .

For k1 ≥ 3 it holds thatsupp(wk1(a)w0(b− a)) ⊂ {(a, b) : 2k1−1 ≤ |a| ≤ 2k1+1 and 2k1−2≤|b| ≤ 2k1+2}

and therefore

wk1(a)w0(b− a) = w0(b− a)w̃k1(b)wk1(a)w̃k1(b).It easily follows that ∥∥〈ξ, η〉|m|F−1[w0(b− a)w̃k1(b)]∥∥L1 . 1,and reasoning as above we obtain(3.24)∑k∈R1

2(s2−s3,s3)·k∥∥∥〈ξ, η〉−mF−1(wkσ̂∗1)∥∥∥

L∞.

∞∑k1=3

2(s2−s3)k1∥∥〈ξ, η〉−mF−1(h(k1,k1)σ̂)∥∥L∞ .

Summation in region R2: In this case we have to estimate, again by (3.20),∞∑k2=3

2s3k2∥∥〈ξ, η〉−mF−1(w0(a)wk2(b− a)σ̂(a, b))∥∥L∞ .

For k2 ≥ 3 it holds thatsupp(w0(a)wk2(b− a)) ⊂ {(a, b) : |a| ≤ 2 and 2k2−2≤|b| ≤ 2k2+2}

and therefore

w0(a)wk2(b− a) = wk2(b− a)w̃k2(b)w0(a)w̃k2(b).Since ∥∥〈ξ, η〉|m|F−1[wk2(2k2(b− a))w̃k2(2k2b)]∥∥L1 . 1,by (3.16) in Lemma 3.1, we get(3.25)∑

k∈R2

2(s2−s3,s3)·k∥∥∥〈ξ, η〉−mF−1(wkσ̂∗1)∥∥∥

L∞.

∞∑k2=3

2s3k2∥∥〈ξ, η〉−mF−1(h(0,k2)σ̂)∥∥L∞ .

Summation in region R3: We first observe that for (k1, k2) ∈ R3supp(wk1(a)wk2(b− a)) ⊂ {(a, b) : |a| ≤ 8 and |b| ≤ 16}.


Therefore, for (k1, k2) in region R3 it follows that∑k∈R3

2(s2−s3,s3)·k∥∥∥〈ξ, η〉−mF−1(wkσ̂∗1)∥∥∥

L∞

.3∑

k1=0

4∑k2=0

2(s2−s3,s3)·(k1,k2)∥∥〈ξ, η〉−mF−1(w(k1,k2)σ̂)∥∥L∞ .(3.26)

Inequalities (3.21), (3.22), (3.23), (3.24), (3.25) and (3.26) then lead to the desiredestimate (3.19).

We now briefly describe the proof corresponding to σ∗1 when s = (s1, . . . , s3n) ∈R3n+ in which case s∗1 := (s1, . . . , sn, sn+1 − s2n+1, . . . , s2n − s3n, s2n+1, . . . , s3n). Letw and w0 be radial functions that satisfy (2.11) for N = 1 and {wj}j∈N0 be thecorresponding Littlewood-Paley partition of unity. It must be proved that∑

k∈N2n0

2s∗1·k∥∥∥〈ξ, η〉−mF−1(wkσ̂∗1)∥∥∥

L∞.∑k∈N2n0

2s·k∥∥〈ξ, η〉−mF−1(wkσ̂)∥∥L∞ ,

where for k = (k1, . . . , k2n), a = (a1, . . . , an), b = (b1, . . . , bn) we have wk(a, b) =

wk1(a1) · · ·wkn(an)wkn+1(b1) · · ·wk2n(bn), s and s∗1 are the vectors in R2n made ofthe last 2n components of s and s∗1, respectively, F−1 andˆdenote inverse Fouriertransform and Fourier transform in R2n, respectively, and the L∞ norm is taken inR2n. For such k, we have∥∥∥〈ξ, η〉−mF−1(wkσ̂∗1)∥∥∥

L∞∼∥∥〈ξ, η〉−mF−1(wK1(a)wK2(b− a)σ̂(a, b))∥∥L∞ ,

where wK1(a) = wk1(a1) · · ·wkn(an) and wK2(x) = wkn+1(b1) · · ·wk2n(bn). The processis now similar to the case previously treated but much heavier in notation and theresult follows by splitting the summation in (k1, . . . , k2n) ∈ N2n0 based on the regionsR, R1, R2 and R3 for each pair (kj, kn+j), j = 1, . . . , n. �

Proof of part (b) of Theorem 1.5. We prove item (i), with item (ii) following in ananalogous way. Without loss of generality, we assume q = 1.

Let m ∈ R, s = (s1, s2, s3) ∈ R3+, {wj}j∈N0 as in (2.11) for N = n, {w̃j}j∈N0 as in(2.12) forN = n, wk(x, ξ, η) = wk1(x)wk2(ξ)wk3(η) and hk(x, ξ, η) := wk1(x)wk2(ξ)w̃k3(η)for k = (k1, k2, k3) ∈ N0, and RA, RB, RC , R1, R2, R3 as in the proof of part (a) ofTheorem 1.5. Consider σ ∈ Bs,m∞,1(R3n); recall that σ̂∗1(y, a, b) = σ̂(y,−a, b− a)e2πia·yand define Γ ∈ S ′(R3n) by Γ̂(y, a, b) = σ̂(y, a, b)e−2πia·y. It then follows that∥∥∥〈ξ, η〉−mF−1(wkΣ̂)∥∥∥

L∞∼∥∥∥〈ξ, η〉−mF−1(wk1(y)wk2(a)wk3(b− a)Γ̂(y, a, b))∥∥∥

L∞,


where k = (k1, k2, k3) ∈ N30. Similar calculations to those in the proof of part (a) ofTheorem 1.5 (using item (b) of Lemma 3.1) give

∞∑k1=0

∑(k2,k3)∈RA

2(s1,s2,s3)·(k1,k2,k3)∥∥∥〈ξ, η〉−mF−1(w(k1,k2,k3)σ̂∗1)∥∥∥

L∞

.∞∑k1=0

∞∑k2=4

2(s1,s2,s3)·(k1,k2,k2)∥∥∥〈ξ, η〉−mF−1(h(k1,k2,k2)Γ̂)∥∥∥

L∞,

∞∑k1=0

∑(k2,k3)∈RB


L∞

.∞∑k1=0

∞∑k3=4

k3−3∑k2=1

2(s1,s2,s3)·(k1,k2,k3)∥∥∥〈ξ, η〉−mF−1(h(k1,k2,k3)Γ̂)∥∥∥

L∞,

∞∑k1=0

∑(k2,k3)∈RC


L∞

.∞∑k1=0

∞∑k2=1

k2+4∑j=0

2(s1,s2+s3,0)·(k1,k2,j)∥∥∥〈ξ, η〉−mF−1(w(k1,k2,j)Γ̂)∥∥∥

L∞,

∞∑k1=0

∑(k2,k3)∈R1


L∞

.∞∑k1=0

∞∑k2=3

2(s1,s2,0)·(k1,k2,k2)∥∥∥〈ξ, η〉−mF−1(h(k1,k2,k2)Γ̂)∥∥∥

L∞,

∞∑k1=0

∑(k2,k3)∈R2


L∞

.∞∑k1=0

∞∑k3=3

2(s1,s2,s3)·(k1,0,k3)∥∥∥〈ξ, η〉−mF−1(h(k1,0,k3)Γ̂)∥∥∥

L∞,

∞∑k1=0

∑(k2,k3)∈R3


L∞

.∞∑k1=0

3∑k2=0

4∑k3=0

2(s1,s2,s3)·(k1,k2,k3)∥∥∥〈ξ, η〉−mF−1(w(k1,k2,k3)Γ̂)∥∥∥

L∞.

The inequalities above imply that

(3.27)∥∥σ∗1∥∥

Bs,m∞,1. ‖Γ‖

B(s1,s2+s3,s3),m∞,1

.


We next estimate∥∥∥〈ξ, η〉−mF−1(wkΓ̂)∥∥∥

L∞for k = (k1, k2, k2) ∈ N0 (the same com-

putations apply to the factors with h instead of w.) Set w̃k(x, ξ, η) := w̃k1(x)w̃k2(ξ)w̃k3(η)and use part (b) of Lemma 3.1 to get∥∥∥〈ξ, η〉−mF−1(wkΓ̂)∥∥∥

L∞

≤∥∥∥〈2−k2ξ, 2−k3η〉|m|F−1(w(y)w(a)w(b)e−2πi2k1+k2a·y)∥∥∥

L1

∥∥〈ξ, η〉−mF−1(w̃kσ̂)∥∥L∞ ,(3.28)

where w should be replaced by w0 in the corresponding cases when k1 = 0 or k2 = 0or k3 = 0. With λ := 2

−(k1+k2), it follows that

F−1(w(y)w(a)w(b)e−2πiλ−1a·y)(x, ξ, η) = w̌(η)λnVŵ(w(λ·))(x,−λξ),where Vŵ(h)(x, ξ) :=

∫Rn h(a)ŵ(a − x)e−2πiξ·a da is the short-time Fourier transform

of h using the window ŵ. An estimate of the L1 norm in (3.28) is based on a boundfor the L1 norm with respect to x and ξ of (1 + |2−k2ξ|)|m|λnVŵ(w(λ·))(x,−λξ). Aftera change of variables and using that 2k1 ≥ 1 and 0 < λ < 1, we obtain∫

R2n(1 + |2−k2ξ|)|m|λn|Vŵ(w(λ·))(x,−λξ)| dxdξ

≤ 2k1|m|∫R2n

(1 + |ξ|)|m||Vŵ(w(λ·))(x, ξ)| dxdξ . 2k1|m|λ−n = 2k1(|m|+n)2k2n.

The last inequality follows from the scaling properties of the modulation spacesM1,1τ,0 (Rn) with τ ∈ R (see [5, Theorem 3.2]).

From (3.27) and the above computations we then have that∥∥σ∗1∥∥Bs,m∞,1

. ‖Γ‖B

(s1,s2+s3,s3),m∞,1

. ‖σ‖B

(s1+|m|+n,s2+s3+n,s3),m∞,1

.

If we now assume the hypothesis that σ̂(y, a, b) has compact support in y and a uni-formly in b then the summations in k1 and k2 appearing in ‖σ‖B(s1+|m|+n,s2+s3+n,s3),m∞,1 arefinite and therefore ‖σ‖

B(s1+|m|+n,s2+s3+n,s3),m∞,1

. Cσ ‖σ‖B(s1,s2,s3),m∞,1 , where Cσ dependson the size of the support of σ in the variables y and a.

For the case s ∈ R3n we can proceed in a similar way by splitting the summationin (k1, . . . , k3n) ∈ N3n0 based on the regions RA, RB, RC , R1, R2, R3 for each pair(kj, kn+j), j = n+ 1, . . . , 2n.

�

4. Proof of Theorems 1.1 and 1.2 and their corollaries

In this section, we start by explaining how Corollaries 1.3 and 1.4 follow fromTheorems 1.1 and 1.2, respectively. We then continue with the proofs of Theorems 1.1and 1.2. Finally, we state some improvements of Theorem 1.1 corresponding to thecase 1 ≤ p < 2.

Corollary 1.3 follows directly from Theorem 1.1 in view of the definition of s(p)

(see Definition 2.1) and the fact that C[s(p)]+1m (R3n) is continuously contained in


Bs(p),m∞,1 (R3n) (see (2.13)). As for Corollary 1.4, the assumption (1.6) and (2.13) imply

that σ ∈ C(1,1,[n2

]+1)m (R3n) ⊂ B(s1,s2,

n2

)

∞,1 (R3n) for any 0 < s1, s2 < 1; similarly, (1.7) and(2.13) give that σ ∈ C(1,...,1)m (R3n) ⊂ Bs,m∞,1(R3n) for any s ∈ R3n with components inthe interval (0, 1). Then part (a) of Corollary 1.4 follows from part (a) of Theorem 1.2.An analogous reasoning shows that part (b) of Corollary 1.4 follows from part (b) ofTheorem 1.2.

We next proceed to prove Theorem 1.1. Recall that if s = (s1, s2, s3) ∈ R3 or s =(s1, . . . , s2n) ∈ R3n, s denotes the vector (s2, s3) ∈ R2 or the vector (sn+1, . . . , s3n) ∈R2n, respectively. Moreover, an x-independent symbol σ belongs to Bs,m∞,q(R3n) if andonly if σ ∈ Bs,m∞,q(R2n) (see (3.14) and the corresponding discussion).Proof of Theorem 1.1. The boundedness results from Lp1(Rn)×Lp2(Rn) into Lp(Rn)for 2 ≤ p1, p2 ≤ ∞ and 2 ≤ p


previously treated cases, T satisfies the following boundedness properties:

T is bounded from Bs(2),m∞,1 (R2n)× L2(Rn)× L∞(Rn) into L2(Rn) for m < −n2 ,

T is bounded from Bs(2),m∞,1 (R2n)× L∞(Rn)× L2(Rn) into L2(Rn) for m < −n2 ,

T is bounded from Bs(1),m∞,1 (R2n)× L2(Rn)× L2(Rn) into L1(Rn) for m < −n2 .

From the facts shown in Section 5, it follows that

(Bs(2),m∞,1 (R2n), B

s(1),m∞,1 (R2n))[θ] = B

s(p),m∞,1 (R2n), 1p =

1−θ2

+ θ, m ∈ R.Using trilinear complex interpolation (see [4, Theorem 4.4.1]), we conclude that T is

bounded from Bs(p),m∞,1 (R2n)×L2(Rn)×Lp2(Rn) into Lp(Rn) for m < −n2 and p2, p such

that 12

+ 1p2

= 1p

and (12, 1p2

) is on the line segment joining (12, 0) and (1

2, 1

2) in Figure 2.

Similarly, we obtain the desired mapping properties for the indices corresponding tothe segment joining (0, 1

2) and (1

2, 1

2) in Figure 2. If ( 1

p1, 1p2

) is a point in the interior

of region I in Figure 2, the boundedness result for operators with x-independent

symbols in Bs(p),m∞,1 (R3n), m < m(p1, p2) = −n2 , follows by complex interpolation and

the boundedness corresponding to the indices given by the pairs ( 1u, 1

2) and (1

2, 1v) that

satisfy 1u

+ 12

= 12

+ 1v

= 1p1

+ 1p2

= 1p.

�

(0, 1)

(0, 12)

(0, 0) (1, 0)

(1, 1)

(12, 0)

III

(1p, 0)

(0, 1p)

( 1p1 ,1p2)

(1u,12)

(12,1v)

Figure 2. Interpolation argument in the proof of Theorem 1.1

Proof of Theorem 1.2. In view of Proposition 2.1 it is enough to consider q = 1. Also,s ∈ R3 will be assumed; a similar argument applies to the case s ∈ R3n.

We first observe that if σ ∈ Bs,m∞,1(R3n) is a symbol as in part (a) of Theorem 1.2,the assumption on the support of σ̂ allows to conclude that σ ∈ B s̃,m∞,1(R3n) for anys̃ = (s̃1, s̃2,

n2) ∈ R3. By taking s̃ = (n

p, n

2, n

2) the boundedness corresponding to


2 ≤ p1, p2, p ≤ ∞ is a particular case of Theorem 1.1. The same reasoning is validfor symbols as in part (b) of Theorem 1.2.

Fix now 1 ≤ p1, p ≤ 2 and 2 ≤ p2 ≤ ∞ such that 1p1 +1p2

= 1p, s = (s1, s2,

n2) ∈ R3,

m < m(p1, p2) and let σ ∈ Bs,m∞,1(R3n) be a symbol as in part (a) of Theorem 1.2. Fromthe assumptions on the support of σ̂ it follows that σ ∈ B s̃,m∞,1(R3n) for s̃ = ( np′1 ,

n2, n

2).

By item (i) in part (b) of Theorem 1.5 we have that σ∗1 ∈ B s̃,m∞,1(R3n); applyingthe mapping properties corresponding to region II in Figure 2, it follows that Tσ∗1is bounded from Lp

′(Rn) × Lp2(Rn) into Lp′1(Rn). Duality then implies that Tσ is

bounded from Lp1(Rn)× Lp2(Rn) into Lp(Rn) as desired.Part (b) of Theorem 1.2 follows analogously through the use of σ∗2. �

4.1. An improved version of Theorem 1.1 for 1 ≤ p < 2. We present animprovement of Theorem 1.1 in the sense that boundedness into Lp(Rn), 1 ≤ p < 2,holds for operators with x-independent symbols belonging to certain Besov spacesthat have regularity index below s(p).

If s is a vector in R2 or R2n and σ = σ(ξ, η) is in Bs,m∞,1(R2n) for some m ∈ R, wedefine ‖σ‖Bs,m∞,1 := ‖σ‖BS,m∞,1 , where S is a vector in R

3 or R3n, respectively, such thatS = s (see (3.14) and the corresponding discussion).

For 1 ≤ p1, p2 ≤ ∞ such that 1p1 +1p2

= 1 define s(p1, p2) as the vector in R2 givenby

s(p1, p2) =

{(n, n

2) if 1 ≤ p1 ≤ 2,

(n2, n) if 2 ≤ p1 ≤ ∞,

or as the vector in R2n given by

s(p1, p2) =

(1, . . . , 1︸︷︷︸

n

, 12, . . . , 1

2︸︷︷︸n

) if 1 ≤ p1 ≤ 2,

(12, . . . , 1

2︸︷︷︸n

, 1, . . . , 1︸︷︷︸n

) if 2 ≤ p1 ≤ ∞.

Note that s(2, 2) receives two possible values in each dimension considered. The proofof Theorem 1.1 and part (c) of Proposition 2.1 give that

‖Tσ(f, g)‖L1 . ‖σ‖Bs(p1,p2),m∞,1 ‖f‖Lp1 ‖g‖Lp2 ,

for all f, g ∈ S(Rn) and all symbols σ ∈ Bs(p1,p2),m∞,1 (R2n) with m < m(p1, p2) andwhere s(p1, p2) is in R2 or in R2n. This an improvement on the corresponding resultin Theorem 1.1 since B

s(1),m∞,1 (R2n) $ B

s(p1,p2),m∞,1 (R2n). Recall that s(1) = (n, n) or

s(1) = (1, . . . , 1) ∈ R2n, then the above states that boundedness still holds even ifthe symbol has n

2fewer derivatives (in the sense of Besov spaces) with respect to one

of its frequency variables.For 2 ≤ p1, p2 ≤ ∞ such that 1p1 +

1p2

= 1p

and p1 = 2 or p2 = 2 (segments joining

points (12, 1

2) and (1

2, 0) and points (1

2, 1

2) and (0, 1

2), respectively, in Figure 2) define


s(p1, p2) as the vector in R2 given by

s(p1, p2) =

{(n

2, np) if p1 = 2,

(np, n

2) if p2 = 2,

or as the vector in R2n given by

s(p1, p2) =

(1

2, . . . , 1

2︸︷︷︸n

, 1p, . . . , 1

p︸︷︷︸n

) if p1 = 2,

(1p, . . . , 1

p︸︷︷︸n

, 12, . . . , 1

2︸︷︷︸n

) if p2 = 2.

In this case we also have Bs(p),m∞,1 (R2n) $ B

s(p1,p2),m∞,1 (R2n), if (p1, p2) 6= (2,∞) and

(p1, p2) 6= (∞, 2). Symbols in the latter class are allowed to have np − n2 > 0 fewerderivatives (in the sense of Besov spaces) than those in B

s(p),m∞,1 (R2n).

Finally, if p1, p2 are such that1p1

+ 1p2

= 1p

and ( 1p1, 1p2

) is in the interior of region

I of Figure 2, define s(p1, p2) := (1 − θ)s(u, 2) + θs(2, v) where θ ∈ (0, 1) is suchthat ( 1

p1, 1p2

) = (1 − θ)( 1u, 1

2) + θ(1

2, 1v) and u and v are as in Figure 2. Once more,

Bs(p),m∞,1 (R2n) $ B

s(p1,p2),m∞,1 (R2n).

An analogous proof to that of Theorem 1.1 gives then the following:

Theorem 4.1. Consider 1 ≤ p1, p2 ≤ ∞ and p = 1 such that 1p1 +1p2

= 1p

or,

2 ≤ p1, p2 ≤ ∞ and 1 ≤ p < 2 satisfying 1p1 +1p2

= 1p. If m < m(p1, p2), s(p1, p2) is

as defined above and s is a vector of the same dimension as s(p1, p2), the followingstatements hold true:

(a) If 0 < q ≤ 1 and s ≥ s(p1, p2) component-wise, then‖Tσ(f, g)‖Lp . ‖σ‖Bs,m∞,q ‖f‖Lp1 ‖g‖Lp2 ,

for all f, g ∈ S(Rn) and all symbols σ in Bs,m∞,q(R2n).(b) If 1 < q ≤ ∞ and s > s(p1, p2) component-wise, then

‖Tσ(f, g)‖Lp . ‖σ‖Bs,m∞,q ‖f‖Lp1 ‖g‖Lp2 ,

for all f, g ∈ S(Rn) and all symbols σ in Bs,m∞,q(R2n).

Corresponding improvements of (1.4) and (1.5) in Corollary 1.3 follow from The-orem 4.1.

5. Complex interpolation of Besov spaces of product type

We refer the reader to [4, Chapter 4] regarding the method of complex interpolationand some of the notation used in this section. The following theorem is a particularcase of [17, Theorem 1.3.3] (see also [9, Theorem 4]).


Theorem 5.A. Let 1 ≤ r0, r1, q0, q1 ≤ ∞ (excluding the cases r0 = r1 = ∞ andq0 = q1 =∞), m0,m1 ∈ R, s0, s1 both in R3 or both in R3n. If 0 < θ < 1,

(Bs0,m0r0,q0 (R3n), Bs1,m1r1,q1 (R

3n))[θ] = Bs,mr,q (R3n),(5.29)

(Bs0,m0r0,q0 (R2n), Bs1,m1r1,q1 (R

2n))[θ] = Bs,mr,q (R2n),(5.30)

where 1r

= 1−θr0

+ θr1, 1q

= 1−θq0

+ θq1, m = (1− θ)m0 + θm1 and s = (1− θ)s0 + θs1.

In this section we will address the case r0 = r1 =∞ and m0 = m1 and show that(5.29) and (5.30) also hold in such situation. This is, with the parameters as in thestatement of Theorem 5.A and m0 = m1 = m,

(Bs0,m∞,q0(R3n), Bs1,m∞,q1(R

3n))[θ] = Bs,m∞,q(R3n),(5.31)

(Bs0,m∞,q0(R2n), Bs1,m∞,q1(R

2n))[θ] = Bs,m∞,q(R2n).(5.32)

For the sake of notation, only (5.31) will be treated; a completely analogous reasoningleads to (5.32).

We start by introducing some definitions and notation. Given a Banach spaceX, 1 ≤ q ≤ ∞, and s ∈ RL, denote by lqs(X) the Banach space of all sequencesx = {xk}k∈NL0 in X such that

‖x‖lqs(X) :=

∑k∈NL0

(2s·k ‖xk‖X)q 1q


Define P : lqs(C0m(R3n)) → Bs,m∞,q(R3n) as P({xk}k∈NL0 ) :=∑

k∈NL0F−1(w̃kx̂k). The

mapping P is bounded. Indeed, for k = (k1, . . . , kL) ∈ NL0 we haveF−1(wkF(P({xν}ν∈NL0 )) =

∑j=(j1,...,jL)∈ZL|j`|≤2, k`+j`≥0

F−1(wk+jx̂k+j).

Therefore, by part (b) of Lemma 3.1,∥∥∥F−1(wkF(P({xν}ν∈NL0 ))∥∥∥C0m = ∑j=(j1,...,jL)∈NL0|j`|≤2, k`+j`≥0

∥∥F−1(wk+jx̂k+j)∥∥C0m. c(w,w0)

∑j=(j1,...,jL)∈NL0|j`|≤2, k`+j`≥0

‖xk+j‖C0m ,

which implies that∥∥∥P({xk}k∈NL0 )∥∥∥Bs,m∞,1 .

∥∥∥{xk}k∈NL0 ∥∥∥lqs(C0m(R3n)) . Moreover, P is lin-ear and we clearly have P ◦I = Id.

�

Lemma 5.2. Let s0, s1 ∈ RL, 1 ≤ q0, q1 ≤ ∞ and 0 < θ < 1. If X0 and X1 areBanach spaces then, with equal norms,

(lq0s0(X0), lq1s1

(X1))[θ] = lqs(X),

where s = (1− θ)s0 + θs1, 1q = 1−θq0 +θq1, and X = (X0, X1)[θ].

Proof. This is a consequence, for instance, of [4, Theorem 5.6.3], where the casecorresponding to L = 1 is proved. �

Proof of (5.31). Note that I and P in the proof of Lemma 5.1 are independent ofthe parameters of the spaces involved. In particular I and P are linear mappingsfromBs0,m∞,q0(R

3n)+Bs1,m∞,q1(R3n) into lq0s0(C0m(R3n))+lq1s1(C0m(R3n)) and from lq0s0(C0m(R3n))+

lq1s1(C0m(R3n)) into Bs0,m∞,q0(R3n) + Bs1,m∞,q1(R3n), respectively, such that I is linear andbounded from Bsk,m∞,qk(R

3n) into lqksk(C0m(R3n)) for k = 0, 1, P is linear and boundedfrom lqksk(C0m(R3n)) into Bsk,m∞,qk(R3n) for k = 0, 1, and P(I (σ)) = σ for all σ ∈Bs0,m∞,q0(R

3n) +Bs1,m∞,q1(R3n). By complex interpolation we then conclude that the space

(Bs0,m∞,q0(R3n), Bs1,m∞,q1(R

3n))[θ] is a retract of (lq0s0

(C0m(R3n)), lq1s1(C0m(R3n)))[θ] with the map-pings I and P. This implies that

P((lq0s0(C0m(R3n)), lq1s1(C0m(R3n)))[θ]) = (Bs0,m∞,q0(R3n), Bs1,m∞,q1(R3n))[θ].By Lemma 5.2 and the fact that (C0m(R3n), C0m(R3n))[θ] = C0m(R3n), we have

P(lqs(C0m(R3n))[θ])) = (Bs0,m∞,q0(R3n), Bs1,m∞,q1(R3n))[θ].Since P(lqs(C0m(R3n))) = Bs,m∞,q(R3n), (5.31) follows. The equality of the norms fol-lows from the fact that I is an isometry and from the equality in norms given inLemma 5.2. �


Remark 5.1. The above reasoning in the proof of (5.31) breaks down when m0 6= m1.Indeed, assume without lost of generality thatm0 < m1; then (C0m0(R3n), C0m1(R3n))[θ] 6=C0m(R3n) for m = (1 − θ)m0 + θm1 since C0m0(R3n) ∩ C0m1(R3n) = C0m0(R3n) is dense in(C0m0(R3n), C0m1(R3n))[θ] while C0m0(R3n) is not dense in C0m(R3n).

6. Boundedness from L∞ × L∞ into L∞ for Bs(∞),m∞,1 (R3n),m < m(∞,∞) = −n

Boundedness from Lp1(Rn)×Lp2(Rn) into Lp(Rn) for operators associated to sym-bols in B

s(p),m∞,1 (R3n) with m < m(p1, p2), 2 ≤ p1, p2 ≤ ∞, 2 ≤ p


Proof of (6.33). Defining

Jk := {~j = (j1, . . . , jn) ∈ {1, . . . , n}n : jl 6= jl̃ if l 6= l̃, j1 < · · · < jk, jk+1 < · · · < jn}for k = 0, . . . , n, and denoting W (f, g, h, ϕ, ψ, θ) by W, the proof in [8, Theorem 4]shows that

W (x, ξ, η) =∑

k=0,...,n

~j∈Jk

ck,~jWk,~j(x, ξ, η),

where for k = 0, . . . , n and ~j = (j1, . . . , jn) ∈ Jk,

Wk,~j(x, ξ, η) :=

∫R3n

e−2πi(x·(−ξ−η+τ)+ξ·a+η·b)¯̂h(τ)f(a)g(b)Sk,~j(x, τ, ξ, η, a, b) dτdadb,

Sk,~j(x, τ, ξ, η, a, b) :=

∫Rne−2πix·yϕk,~j(τ − ξ − η)Φk,~j(y, τ, ξ, η)Âa,b(y) dy,

Aa,b(t) := ψ(a+ t)θ(b+ t),

ϕk,~j(τ − ξ − η) :=k∏l=1

ϕjl(τjl − ξjl − ηjl), (ϕ0,~j(τ − ξ − η) := 1),

Φk,~j(y, τ, ξ, η) :=n∏

l=k+1

yjl

∫ 10

ϕ(1)jl

(sjlyjl + τjl − ξjl − ηjl) dsjl

= yjk+1 · · · yjn∫

[0,1]n−k

n∏l=k+1

ϕ(1)jl

(sjlyjl + τjl − ξjl − ηjl) dsjk+1 . . . dsjn ,

with Φn,~j := 1 and ϕ(1)jl

denoting the first derivative of ϕjl . It is then enough to provethe inequality (6.33) for each Wk,~j.

For the case k = n, define Fx(a) := f(a)ψ(a − x), Gx(b) := g(b)θ(b − x), andHx(τ) := h̄(τ)ϕ̂(x− τ), and obtain as in [8, Theorem 4] that∫R3n〈ξ, η〉m|Wn,~j(x, ξ, η)| dxdξdη .

∫Rn‖Fx‖L2 ‖Gx‖L2

(∫R2n〈ξ, η〉2m|Ȟx(ξ + η)|2dξdη

) 12

dx.

Then∫R3n〈ξ, η〉m|Wn,~j(x, ξ, η)| dxdξdη

. supx∈Rn

(‖Fx‖L2 ‖Gx‖L2)∥∥∥∥∥(∫

R2n〈ξ, η〉2m|Ȟx(ξ + η)|2dξdη

) 12

∥∥∥∥∥L1

. ‖f‖L∞ ‖g‖L∞ ‖ψ̂‖L2‖θ̂‖L2∥∥∥∥∥(∫


) 12

∥∥∥∥∥L1

.

Since m(∞,∞) = −n and m < m(∞,∞), we have m = −n − ε for some ε > 0.Set m1 = m2 := −n2 − ε2 . The change of variable η → η − ξ and the fact that


〈ξ, η − ξ〉2m ≤ (1 + |ξ|)2m1(1 + |η|)2m2 imply(∫R2n〈ξ, η〉2m|Ȟx(ξ + η)|2dξdη

) 12

≤(∫

Rn(1 + |ξ|)2m1 dξ

) 12(∫

Rn(1 + |η|)2m2|Ȟx(η)|2 dη

) 12

.

The integral in ξ is finite and(∫Rn

(1 + |η|)2m2|Ȟx(η)|2 dη) 1

2

=

(∫Rn

(1 + |η|)−n−ε|Ȟx(η)|2 dη) 1

2

.∥∥Ȟx∥∥L∞ . ‖Hx‖L1 = ∥∥h̄(·)ϕ̂(x− ·)∥∥L1 ,

which implies ∥∥∥∥∥(∫


) 12

∥∥∥∥∥L1

. ‖h‖L1 ‖ϕ̂‖L1 .

We then obtain∫R3n〈ξ, η〉m|Wn,~j(x, ξ, η)| dxdξdη . ‖ψ̂‖L2‖θ̂‖L2 ‖ϕ̂‖L1 ‖f‖L∞ ‖g‖L∞ ‖h‖L1 .

For the case k ∈ {0, . . . , n−1}, set Ft,α1,γ1(a) := f(a)(∂α1+γ1ψ)(a+t), Gt,α2,γ2(b) :=g(b)(∂α2+γ2θ)(b+ t) and

Hx,s̄k,ȳk(τ) := h̄(τ + x)F−1(ϕk,~j(·)

(n∏

l=k+1

ϕ(jl)l (·l)sjl−1l

))(τ) e−2πi

∑nl=k+1 slylτl ,

where s̄k := (sk+1, . . . , sn) ∈ [0, 1]n−k, ȳk := (yk+1, . . . , yn) ∈ Rn−k, x, τ ∈ Rn andϕl

(jl) denotes the jl derivative of ϕl with jl ≤ 3. Proceeding as in [8, Theorem 4], itis enough to analyze

Wk,~j,2(x, ξ, η) =

∫R2n

∫[0,1]n−k

e2πix·(ξ+η)e−2πi(x+t)·yH(y)∏nj=1(1 + i(tj + xj))

2

× F̂t,α1,γ1(ξ)Ĝt,α2,γ2(η)F−1 (Hx,s̄k,ȳk) (ξ + η) dsk+1 . . . dsn dydt,

where H is an appropriate fixed function on L1(Rn) and αi, γi, i = 1, 2, are multi-indices with components at most equal to 3. We have∫

R3n〈ξ, η〉m|Wk,~j,2(x, ξ, η)| dξdηdx .

∫R3n|H(y)|

∫[0,1]n−k

‖Ft,α1,γ1‖L2 ‖Gt,α2,γ2‖L2∏nj=1(1 + |tj + xj|2)

×(∫

R2n〈ξ, η〉2m|F−1 (Hx,s̄k,ȳk) (ξ + η)|2 dξdη

) 12

dsk+1 . . . dsn dy dt dx,


and therefore

∫R3n〈ξ, η〉m|Wk,~j,2(x, ξ, η)| dξdηdx . sup

t‖Ft,α1,γ1‖L2 ‖Gt,α2,γ2‖L2

∫Rn|H(y)|

(6.34)

×∫

[0,1]n−k

(∫Rn

(∫R2n〈ξ, η〉2m|F−1 (Hx,s̄k,ȳk) (ξ + η)|2 dξdη

) 12

dx

)dsk+1 . . . dsn dy.

The first factor in the right hand side above satisfies

(6.35) supt‖Ft,α1,γ1‖L2 ‖Gt,α2,γ2‖L2 . ‖ ̂∂α1+γ1ψ‖L2‖∂̂α2+γ2θ‖L2 ‖f‖L∞ ‖g‖L∞ .

For the other factor, we proceed as in the case k = n and recalling that sj ∈ [0, 1], itfollows that(∫

Rn

(∫R2n〈ξ, η〉2m|F−1 (Hx,s̄k,ȳk) (ξ + η)|2 dξdη

) 12

dx

)(6.36)

.

(∫R2n|Hx,s̄k,ȳk(τ)| dτdx

). ‖h‖L1

∥∥∥∥∥ϕ̂k,~jn∏

j=k+1

ϕ̂(jl)l

∥∥∥∥∥L1

.

Putting (6.34), (6.35), and (6.36) together and using that H ∈ L1(Rn), we get∫R3n〈ξ, η〉m|Wk,~j,2(x, ξ, η)| dξdηdx

. ‖ ̂∂α1+γ1ψ‖L2‖∂̂α2+γ2θ‖L2∥∥∥∥∥ϕ̂k,~j

n∏j=k+1

ϕ̂(jl)l

∥∥∥∥∥L1

‖f‖L∞ ‖g‖L∞ ‖h‖L1 .

�

7. Acknowledgements

The authors thank Stefania Marcantognini, Cristina Pereyra, Alex Stokolos andWilfredo Urbina for the invitation to contribute to this volume in honor of CoraSadosky. The second author is also grateful to Laura De Carli, Alex Stokolos, andWilfredo Urbina for inviting her to speak in the special session “Harmonic Analysisand Operator Theory, in memory of Cora Sadosky,” held at the University of NewMexico (Albuquerque, NM) during the Spring 2014 Meeting of the Western Sectionof the American Mathematical Society.

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Jodi Herbert, Maranatha Baptist University, 745 W Main St., Watertown, WI53094, USA

E-mail address: [email protected]

Virginia Naibo, Department of Mathematics, 138 Cardwell Hall, Kansas StateUniversity, Manhattan, KS 66506, USA.

E-mail address: [email protected]

1. Introduction and main results2. preliminariesBesov spaces of product typeThe classes Csm(R3n)Connections between the spaces.

3. Symbolic calculus: Proof of Theorem 1.54. Proof of Theorems 1.1 and 1.2 and their corollaries4.1. An improved version of Theorem 1.1 for 1p

dedicated to the memory of cora sadoskyvnaibo/herbertnaibo_2015.pdf · 2018. 10. 16. · dedicated...

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