Bilinear Pseudo-Differential Operators with Exotic Class Symbols of Limited Smoothness

Abstract

We consider the bilinear pseudo-differential operators with symbols in the bilinear Hörmander class $BS_{\rho , \rho }^m ({\mathbb {R}}^n)$ for $m \in {\mathbb {R}}$ and $0 \le \rho < 1$. The aim of this paper is to discuss smoothness conditions for symbols to assure the boundedness from $h^p \times h^q$ to $h^r$ for $1 \le r \le 2 \le p,q \le \infty$ satisfying $1/p+1/q=1/r$, where $h^\infty$ is replaced by the space bmo.

Appendices

Appendix A: Existence of Decomposition

In this appendix, we determine functions used to decompose symbols in Sect. 5. Let $\phi \in {\mathcal {S}}({\mathbb {R}}^n)$ satisfy that $\mathrm {supp}\,\phi \subset \{ \xi \in {\mathbb {R}}^n : | \xi | \le 2 \}$ and $\phi = 1$ for $| \xi | \le 1$. For $k \in {\mathbb {Z}}$, we write $\phi _k = \phi (\cdot / 2^k )$. Then, we see that $\mathrm {supp}\,\phi _k \subset \{ \xi \in {\mathbb {R}}^n : | \xi | \le 2^{k+1} \}$ and $\mathrm {supp}\,(1-\phi _k) \subset \{ \xi \in {\mathbb {R}}^n : | \xi | \ge 2^{k} \}$ for $k \in {\mathbb {Z}}$, since $\phi _k = 1$ for $| \xi | \le 2^{k}$.

Let $\{ \Psi _{j} \}_{j\in {\mathbb {N}}_0}$ be a Littlewood–Paley partition on $({\mathbb {R}}^n)^2$. Then, for $j \ge 1$, $\Psi _j$ can be expressed into the following form:

$$\begin{aligned} \Psi _j ( \xi , \eta )&= \Psi _j ( \xi , \eta ) \phi _{j+1} (\xi ) \phi _{j+1} (\eta ) \\&\quad \times \left\{ \phi _{j-6} (\xi ) + (1- \phi _{j-6} (\xi ) ) \right\} \left\{ \phi _{j-6} (\eta ) + (1- \phi _{j-6} (\eta ) ) \right\} . \end{aligned}$$

Since $\phi _{j} \, \phi _{j'} = \phi _{j'}$ if $j>j'$ and $\phi _{j-6} (\xi ) \phi _{j-6} (\eta )$ vanishes on $\mathrm {supp}\,\Psi _j$, $j \ge 1$, we have

$$\begin{aligned} \Psi _j ( \xi , \eta )&= \Psi _j ( \xi , \eta ) \phi _{j-6} (\xi ) \, \phi _{j+1} (\eta ) (1- \phi _{j-6} (\eta ) ) \\&\quad + \Psi _j ( \xi , \eta ) \phi _{j+1} (\xi ) (1- \phi _{j-6} (\xi ) ) \, \phi _{j-6} (\eta ) \\&\quad + \Psi _j ( \xi , \eta ) \phi _{j+1} (\xi ) (1- \phi _{j-6} (\xi ) ) \, \phi _{j+1} (\eta ) (1- \phi _{j-6} (\eta ) ) . \end{aligned}$$

We further decompose the first factor above as follows:

$$\begin{aligned} \Psi _j ( \xi , \eta ) \phi _{j-6} (\xi ) \, \phi _{j+1} (\eta ) (1- \phi _{j-6} (\eta ) ) \left\{ \phi _{j-4} (\eta ) + (1- \phi _{j-4} (\eta ) ) \right\} . \end{aligned}$$

Then, this is equal to $\Psi _j ( \xi , \eta ) \phi _{j-6} (\xi ) \, \phi _{j+1} (\eta ) (1- \phi _{j-4} (\eta ) )$, since $\phi _{j-6} (\xi ) \phi _{j-4} (\eta )$ vanishes on $\mathrm {supp}\,\Psi _j$, $j \ge 1$, and $(1-\phi _{j}) (1-\phi _{j'}) = (1-\phi _{j})$ if $j>j'$. The second factor can be expressed similarly because of symmetry. Therefore, we have

$$\begin{aligned} \Psi _j ( \xi , \eta )&= \Psi _j ( \xi , \eta ) \phi _{j-6} (\xi ) \, \phi _{j+1} (\eta ) \big (1- \phi _{j-4} (\eta ) \big ) \\&\quad + \Psi _j ( \xi , \eta ) \phi _{j+1} (\xi )\big (1- \phi _{j-4} (\xi ) \big ) \, \phi _{j-6} (\eta ) \\&\quad + \Psi _j ( \xi , \eta ) \phi _{j+1} (\xi )\big (1- \phi _{j-6} (\xi ) \big ) \, \phi _{j+1} (\eta ) \big (1- \phi _{j-6} (\eta ) \big ) \\&=: \Psi _j ( \xi , \eta ) \, \phi _{j}' (\xi ) \psi _{j}' (\eta ) + \Psi _j ( \xi , \eta ) \, \psi _{j}' (\xi ) \phi _{j}' (\eta ) + \Psi _j ( \xi , \eta ) \, \psi _{j}'' (\xi ) \psi _{j}'' (\eta ) \end{aligned}$$

with $\phi _{j}' = \phi ' (\cdot /2^{j})$, $\psi _{j}' = \psi ' (\cdot /2^{j})$, and $\psi _{j}'' = \psi '' (\cdot /2^{j})$, and then we realize that

$$\begin{aligned} \mathrm {supp}\,\phi '&\subset \left\{ \zeta \in {\mathbb {R}}^n : |\zeta | \le 2^{-5}\right\} , \quad \\ \mathrm {supp}\,\psi '&\subset \left\{ \zeta \in {\mathbb {R}}^n : 2^{-4} \le |\zeta | \le 2^{2}\right\} , \\ \mathrm {supp}\,\psi ''&\subset \left\{ \zeta \in {\mathbb {R}}^n : 2^{-6} \le |\zeta | \le 2^{2}\right\} . \end{aligned}$$

Hence, we obtain the information (5.2), (5.3), and (5.4) given in Sect. 5.

Appendix B: Boundedness from $L^2 \times L^2$ to $L^1$

In this appendix, we shall prove the following boundedness stated in Remark 3.4.

Theorem B.1

Let $0 \le \rho <1$, $m=-(1-\rho )n/2$, and ${\varvec{s}}=(s_0,s_1,s_2) \in [0,\infty )^3$ satisfy $s_{0}, s_{1}, s_{2} \ge n/2$. Then, if $\sigma \in BS_{\rho ,\rho }^m ({\varvec{s}};{\mathbb {R}}^{n})$, the bilinear pseudo-differential operator $T_\sigma$ is bounded from $L^2 ({\mathbb {R}}^n) \times L^2 ({\mathbb {R}}^n)$ to $L^1 ({\mathbb {R}}^n)$.

The proof is much simpler than that of the boundedness to $h^1$ done in Sect. 6.

Proof

As in Sect. 5 and Appendix A, we decompose a Littlewood–Paley partition $\{ \Psi _{j} \}_{j \in {\mathbb {N}}_0}$ on $({\mathbb {R}}^n)^2$ into the following form: For $j \ge 1$,

$$\begin{aligned} \Psi _j ( \xi , \eta ) = \Psi _j ( \xi , \eta ) \phi '_{j} (\xi ) \psi '_{j} (\eta ) + \Psi _j ( \xi , \eta ) \psi '_{j} (\xi ) \phi '_{j} (\eta ), \end{aligned}$$

where $\phi '_{j} = \phi ' (\cdot /2^{j})$, $\psi '_{j} = \psi ' (\cdot /2^{j})$, $\mathrm {supp}\,\phi ' \subset \{ \zeta \in {\mathbb {R}}^n : |\zeta | \le 2^{-2} \}$, and $\mathrm {supp}\,\psi ' \subset \{ \zeta \in {\mathbb {R}}^n : 2^{-3} \le |\zeta | \le 2^{2} \}$. Then, repeating the same lines as in Sect. 5, the dual form of $T_{\sigma }(f,g)$ can be expressed by the sum of the forms $I_{0}$, $I_{1}$, and $I_{2}$ as follows. The form $I_{0}$ is the same as in (5.9). The form $I_{1}$ is the following:

$$\begin{aligned} I_{1} = \sum _{ j\gg 1 } \sum _{{\varvec{k}} \in ({\mathbb {N}}_0)^3} \sum _{\varvec{\nu } \in ({\mathbb {Z}}^{n})^2} 2^{-j\rho n} \int _{{\mathbb {R}}^n} T_{ \sigma _{ j, {\varvec{k}}, \varvec{\nu }}^\rho } (\square _{\nu _1} f_{j}', \square _{\nu _2} g_{j}' ) (x) \, {h_j(x)} \, dx, \end{aligned}$$

where $f_{j}' = \phi '_{j}(D) f (2^{-j\rho } \cdot )$, $g_{j}' = \psi '_{j}(D) g (2^{-j\rho } \cdot )$, and $h_j = h(2^{-j\rho } \cdot )$. Also, we have

$$\begin{aligned} \mathrm {supp}\,\widehat{f_{j}'}&\subset \left\{ \xi \in {\mathbb {R}}^n : |\xi | \le 2^{j(1-\rho )-2}\right\} , \\ \mathrm {supp}\,\widehat{g_{j}'}&\subset \left\{ \eta \in {\mathbb {R}}^n : 2^{j(1-\rho )-3} \le |\eta | \le 2^{j(1-\rho )+2}\right\} . \end{aligned}$$

The form $I_{2}$ is in a symmetrical position with $I_{1}$, and thus we omit stating it.

We shall consider the three forms above. However, we only consider $I_{1}$, since the proof for $I_{0}$ is exactly the same as in Sect. 6.4 and the proof for $I_{2}$ is similar to that for $I_{1}$ because of symmetry. We take a Littlewood–Paley partition $\{ \psi _{\ell } \}$ on ${\mathbb {R}}^n$ and decompose the factor of f as

$$\begin{aligned} I_{1} = \sum _{ j\gg 1 } \sum _{{\varvec{k}} \in ({\mathbb {N}}_0)^3} \sum _{\ell \in {\mathbb {N}}_0} \sum _{\varvec{\nu } \in ({\mathbb {Z}}^{n})^2} 2^{-j\rho n} \int _{{\mathbb {R}}^n} T_{ \sigma _{ j, {\varvec{k}}, \varvec{\nu }}^\rho } (\square _{\nu _1} \Delta _{\ell } f_{j}', \square _{\nu _2} g_{j}' ) (x) \, {h_j(x)} \, dx. \end{aligned}$$

Then, the sums over $\varvec{\nu }$ and $\ell$ are restricted to $\nu _1 \in \{ \nu _1 \in {\mathbb {Z}}^n : |\nu _1| \lesssim 2^{\ell } \}$, $\nu _2 \in \{ \nu _2 \in {\mathbb {Z}}^n : |\nu _2| \lesssim 2^{j(1-\rho )} \}$, and $\ell \le j(1-\rho )$ (see Sect. 6.1). Applying Lemma 4.4 (1) with $p=q=2$ and $r=\infty$ to the restricted sums, we have

$$\begin{aligned} \left| I_{1} \right| \lesssim \sum _{ j, \, {\varvec{k}}} \sum _{\ell :\, \ell \le j(1\!-\!\rho )} 2^{ \ell n/2 } \, 2^{ (k_0 \!+\! k_1 \!+\! k_2 ) n/2} \Vert \Delta _{{\varvec{k}}} [ \sigma _j^\rho ] \Vert _{ L^{2}_{ul} } \Vert \Delta _{\ell +j\rho } f \Vert _{L^2} \Vert \psi '_{j}(D)g \Vert _{L^2} \Vert h \Vert _{L^\infty } \end{aligned}$$

where we used the calculation as in (6.8) to the factors of f and g. Since we are not dividing the sum over j, the right hand side above is simply bounded by

$$\begin{aligned} \Vert \sigma \Vert _{ BS_{\rho ,\rho }^{m} ({\varvec{s}};{\mathbb {R}}^{n}) } \Vert h \Vert _{L^\infty } \sum _{ j \gg 1 } \sum _{\ell :\, \ell \le j(1-\rho )} 2^{ \ell n/2 } \, 2^{-j(1-\rho ) n/2} \Vert \Delta _{\ell +j\rho } f \Vert _{L^2} \Vert \psi '_{j}(D)g \Vert _{L^2}, \end{aligned}$$

where $m=-(1-\rho )n/2$ and ${\varvec{s}} = (n/2,n/2,n/2)$. The sums over j and $\ell$ are bounded by a constant times $\Vert f \Vert _{L^2} \Vert g \Vert _{L^2}$, recalling the proof of (6.12). Hence, we obtain

$$\begin{aligned} \left| I_{1} \right| \lesssim \Vert \sigma \Vert _{ BS_{\rho ,\rho }^{m} ({\varvec{s}};{\mathbb {R}}^{n}) } \Vert f \Vert _{L^2} \Vert g \Vert _{L^2} \Vert h \Vert _{L^\infty }, \end{aligned}$$

where $m=-(1-\rho )n/2$ and ${\varvec{s}} = (n/2,n/2,n/2)$. This completes the proof. $\square$