Bilinear Pseudodifferential Operators and the Hörmander Classes

Introduction

As suggested by their name, bilinear pseudodifferential operators go beyond the class of differential operators. Their study is motivated by applications to analysis and partial differential equations as well as by the corresponding linear theory. This article is devoted to bilinear pseudodifferential operators associated to the so-called Hörmander classes. Different aspects of the theory will be described, with an emphasis on their boundedness properties in a variety of function spaces.

In order to motivate the definition of a bilinear pseudodifferential operator, consider

$$\begin{equation*} L(f,g)(x)=\sum _{\left\vert \mu \right\vert \le m_1,\left\vert \nu \right\vert \le m_2}a_{\mu , \nu }(x)\,\partial ^\mu f(x)\,\partial ^\nu g(x), \end{equation*}$$

where $m_1,m_2\in \mathbb{N}_0,$ $\mu , \nu \in \mathbb{N}_0^n$ are multi-indices, $a_{\mu ,\nu },$ $f,$ and $g$ are functions defined on $\mathbb{R}^n,$ $|\mu |=\mu _1+\cdots +\mu _n$ for $\mu =(\mu _1,\dots ,\mu _n),$ $\partial ^\mu =\partial _1^{\mu _1}\cdots \partial _n^{\mu _n},$ and similarly for $|\nu |$ and $\partial ^\nu .$ Recalling that the Fourier transform is defined for an integrable function $h$ on $\mathbb{R}^n$ as $\widehat{h}(\xi )=\int _{\mathbb{R}^n}h(x)e^{-2\pi i x\cdot \xi }\,dx$ for $\xi \in \mathbb{R}^n,$ basic properties give that the partial differential operator $L$ can be written as

$$\begin{equation} L(f,g)(x) = \int _{\mathbb{R}^{2n}} P(x,\xi ,\eta ) \widehat{f}(\xi ) \widehat{g}(\eta ) e^{2\pi i x \cdot (\xi + \eta )} \, d\xi \, d\eta , \tag{1}{} \end{equation}$$

where $x \cdot (\xi + \eta )$ stands for the dot product of $x$ with $\xi +\eta$ and $P(x,\xi ,\eta )$ is the characteristic polynomial of $L,$ that is,

$$\begin{equation*} P(x,\xi ,\eta )=\sum _{\left\vert \mu \right\vert \le m_1,\left\vert \nu \right\vert \le m_2}a_{\mu , \nu }(x)(2\pi i \xi )^\mu (2\pi i \eta )^\nu , \end{equation*}$$

with $\xi ^\mu =\xi _1^{\mu _1}\cdots \xi _n^{\mu _n}$ for $\xi =(\xi _1,\dots ,\xi _n)$ and similarly for $\eta ^\nu .$

We also note that if the functions $a_{\mu , \nu }$ are infinitely differentiable with bounded derivatives of all orders, then the characteristic polynomial $P$ satisfies the estimates

$$\begin{equation} |\partial _x^\alpha \partial _\xi ^\beta \partial _\eta ^\gamma P(x,\xi , \eta )|\leq C_{\alpha \beta \gamma } (1+|\xi |+|\eta |)^{m - (|\beta |+|\gamma |)} \tag{2}{} \end{equation}$$

for all $x,\xi , \eta \in {\mathbb{R}}^n,$ for all multi-indices $\alpha$, $\beta$, $\gamma \in \mathbb{N}_0^n,$ for some constants $C_{\alpha \beta \gamma }>0,$ and with $m=m_1+m_2.$ Thus, when taking a derivative of $P$ with respect to a component of the frequency variables $\xi$ or $\eta ,$ the power on the right-hand side of 2 decreases by one; on the other hand, when taking a derivative of $P$ with respect to a component of the space variable $x,$ the power on the right-hand side of 2 stays unchanged.

The expression on the right-hand side of 1 is an example of a bilinear pseudodifferential operator with symbol $P(x,\xi ,\eta ).$ More generally, a bilinear pseudodifferential operator is of the form

$$\begin{equation} T_\sigma (f,g)(x) = \int _{\mathbb{R}^{2n}} \sigma (x,\xi ,\eta ) \widehat{f}(\xi ) \widehat{g}(\eta ) e^{2\pi i x \cdot (\xi + \eta )} \, d\xi \, d\eta , \tag{3}{} \end{equation}$$

where $\sigma (x,\xi ,\eta )$ is a complex-valued function defined for $x,\xi ,\eta \in \mathbb{R}^n$ called the symbol of the operator. If $\sigma$ depends only on the frequency variables $\xi$ and $\eta ,$ $T_\sigma$ is usually referred to as a bilinear multiplier operator and $\sigma$ as a bilinear multiplier. For example, if $L$ is a partial differential operator with constant coefficients, then $L$ is a bilinear multiplier operator.

We next present a number of examples of bilinear pseudodifferential operators with symbols that satisfy conditions in the spirit of 2.

Motivated by the study of certain commutator operators, R. Coifman and Y. Meyer investigated boundedness properties in Lebesgue spaces of bilinear pseudodifferential operators. Conditions satisfied by the symbols included the following types: for $x$-independent $\sigma ,$

$$\begin{equation} |\partial _\xi ^\beta \partial _\eta ^\gamma \sigma (\xi , \eta )|\leq C_{ \beta , \gamma } (|\xi |+|\eta |)^{ - (|\beta |+|\gamma |)} \tag{4}{} \end{equation}$$

or, for $x$-dependent $\sigma ,$

$$\begin{equation} |\partial _x^\alpha \partial _\xi ^\beta \partial _\eta ^\gamma \sigma (x, \xi , \eta )|\leq C_{\alpha , \beta , \gamma } (1+|\xi |+|\eta |)^{ - (|\beta |+|\gamma |)} \tag{5}{} \end{equation}$$

or, for $x$-dependent $\sigma$ and some $0\le \rho \le 1,$

$$\begin{equation} |\partial _x^\alpha \partial _\xi ^\beta \partial _\eta ^\gamma \sigma (x, \xi , \eta )|\leq C_{\alpha , \beta , \gamma } (1+|\xi |+|\eta |)^{ \rho |\alpha |- \rho (|\beta |+|\gamma |)} \tag{6}{} \end{equation}$$

for every $x,\xi , \eta \in {\mathbb{R}}^n$ and $\alpha ,\beta ,\gamma \in \mathbb{N}_0^n.$ Thus, comparing with 2, when taking a derivative of $\sigma$ with respect to a component of the frequency variables $\xi$ or $\eta ,$ the powers on the right-hand sides of 4 and 5 decrease by one while the power on the right-hand side of 6 decreases by $\rho ;$ on the other hand, when taking a derivative of $\sigma$ with respect to a component of the space variable $x,$ the power on the right-hand side of 5 stays unchanged while the power on the right-hand side of 6 increases by $\rho .$

The building blocks of operators with symbols satisfying 4 are the so-called paraproducts, whose use emerged with the works of J. M. Bony and of R. Coifman and Y. Meyer on paradifferential operators. The paraproduct of $f$ and $g$ is defined as

$$\begin{equation} \Pi (f,g)=\sum _{j=-\infty }^\infty \Delta _jf S_{j-3} g, \tag{7}{} \end{equation}$$

where $S_j$ and $\Delta _j$ are Littlewood–Paley operators, with $S_j$ capturing frequency components of its argument up to the order of $2^j$ and $\Delta _j$ capturing frequency components of its argument of the order of $2^j.$ More precisely,

$$\begin{align*} &\widehat{S_j(f)}(\xi )=\varphi (2^{-j}\xi ) \widehat{f}(\xi ),\\ &\widehat{\Delta _j(f)}(\xi )=\psi (2^{-j}\xi ) \widehat{f}(\xi ), \end{align*}$$

where $\varphi$ is an infinitely differentiable function on $\mathbb{R}^n$ such that $\varphi (\xi )=1$ if $|\xi |\le 1,$ $\varphi (\xi )=0$ if $|\xi |\ge 2,$ and $\psi (\xi )=\varphi (\xi )-\varphi (2\xi );$ see Figure 1. It follows that the product of two functions $f$ and $g$ defined on $\mathbb{R}^n$ can be decomposed as

$$\begin{align*} fg=\Pi (f,g)+\Pi (g,f)+\sum _{j=-\infty }^\infty \Delta _j f\sum _{\ell =-2}^2 \Delta _{j+\ell }g. \end{align*}$$

Figure 1.

The functions $\varphi ,$ $\psi ,$ and $\psi (2^{-j}x)$.

Note that the $j$th term in the paraproduct $\Pi$ given in 7 only contributes with frequency components of the order of $2^j,$ since the support of its Fourier transform is contained in the set $\{\xi :2^{j-2}\le |\xi |\le 2^{j+2}\};$ simple computations show that $\Pi$ is an operator of the type 3 with symbol $\sum _{j=-\infty }^\infty \psi (2^{-j}\xi )\varphi (2^{-j+3}\eta ),$ which satisfies 4. Moreover, it can be proved that any operator 3 with symbol satisfying 4 can be written as a superposition of paraproduct-type operators; more precisely, it is the sum of an operator of the form

$$\begin{equation*} \int _{\mathbb{R}^{2n}}\sum _{j\in \mathbb{Z}}m_j(u,v)\,\Delta ^u_jf(x)\,S_j^vg(x)\frac{du\,dv}{(1+|u|^2+|v|^2)^N} \end{equation*}$$

and a similar one with the roles of $f$ and $g$ interchanged, where $\Delta ^u_j$ and $S_j^v$ are appropriate Littlewood–Paley operators and $\sup _{j\in \mathbb{Z},u,v\in \mathbb{R}^n}|m_j(u,v)|<\infty .$

Several different flavors of paraproduct operators, adapted to a variety of applications, have appeared since their introduction; the reader is referred for more details to the survey “What is …a paraproduct?” (Notices Amer. Math. Soc. 57 (2010), no. 7, 858–860) by A. Bényi, D. Maldonado, and the author.

Operators with symbols satisfying estimates of the type 4 or 5 are also essential in the study of bilinear estimates known as fractional Leibniz rules. In turn, these stand as important tools for results on local and global well-posedness of nonlinear partial differential equations such as Euler, Navier–Stokes (see T. Kato and G. Ponce KP88) and Korteweg–de Vries, as well as in the study of smoothing properties of Schrödinger semigroups.

Fractional Leibniz rules in the setting of Lebesgue spaces for functions in $\mathbb{R}^n$ state that for $1 \le p_1, p_2 \leq \infty$ and $1/2\le p\le \infty$ satisfying $1/p=1/{p_1}+1/{p_2}$ and for $s>\max \{0,n({1}/{p}-1)\}$ or $s\in 2\mathbb{N},$ it holds that

$$\begin{align} \left\|D^s(f g)\right\|_{L^{p}}& \lesssim \left\|D^sf\right\|_{L^{p_1}} \left\|g\right\|_{L^{p_2}} + \left\|f\right\|_{L^{p_1}} \left\|D^sg\right\|_{L^{p_2}},\tag{8}{}\\ \left\|J^s(f g)\right\|_{L^{p}} &\lesssim \left\|J^sf\right\|_{L^{p_1}} \left\|g\right\|_{L^{p_2}} + \left\|f\right\|_{L^{p_1}} \left\|J^sg\right\|_{L^{p_2}}, \tag{9}{} \end{align}$$

where $\|h\|_{L^p}=\left(\int _{\mathbb{R}^n}|h(x)|^p\,dx\right)^{1/p}$ and the operators $D^s$ and $J^s$ are the homogeneous and inhomogeneous fractional differentiation operators of order $s\ge 0,$ respectively, given by

$$\begin{align*} &\widehat{D^s f}(\xi )= |\xi |^s \widehat{f}(\xi ), \quad \xi \in {\mathbb{R}}^n,\\ &\widehat{J^s f}(\xi )= (1+|\xi |^2)^{\frac{s}{2}} \widehat{f}(\xi ), \quad \xi \in {\mathbb{R}}^n. \end{align*}$$

$D^s$ can be thought of as an operator that takes derivatives of order $s$ of its argument while $J^s$ can be interpreted as an operator that takes derivatives up to order $s$ of its argument; therefore, the estimates 8 and 9 are reminiscent of the Leibniz rule for the derivative of a product of functions taught in the calculus courses.

As it turns out, operators with symbols satisfying 4 or 5 constitute building blocks for the bilinear operators $(f,g)\mapsto D^s(fg)$ and $(f,g)\mapsto J^s(fg),$ respectively. More precisely, a decomposition in terms of frequency variables leads to a representation for $J^s(fg)$ given by

$$J^s(fg)=T_{\sigma _1}(J^sf,g)+T_{\sigma _2}(f,J^sg)+ T_{\sigma _3}(J^sf,g),$$

where $\sigma _1$ and $\sigma _2$ satisfy 5 while $\sigma _3$ verifies 5 for $s$ sufficiently large. Consequently, boundedness properties in Lebesgue spaces of the form

$$\begin{equation*} \left\|T_{\sigma _j}(f,g)\right\|_{L^p}\lesssim \left\|f\right\|_{L^{p_1}}\left\|g\right\|_{L^{p_2}},\quad j=1,2,3, \end{equation*}$$

lead to the right-hand side of 9. An analogous argument, now involving the condition 4, applies to $D^s(fg).$

Numerous works have recently emerged showcasing the connections between operators with symbols satisfying 4 or 5 and related fractional Leibniz-type rules in various function spaces; see Á. Bényi and R. H. Torres BT03, L. Grafakos and S. Oh GO14, A. Thomson and the author NT19b, R. H. Torres Tor20, and the references therein.

Operators with symbols satisfying estimates of the type 5 or 6 also play a role in the proofs of boundedness properties of commutators. More precisely, given a linear pseudodifferential operator $T$ and a function $A,$ the commutator of $T$ and the operator $M_A$ corresponding to pointwise multiplication by $A$ is defined by $[T,M_A](f)=T(Af)-AT(f).$ Assuming that $A$ is a Lipschitz function in $\mathbb{R}^n$ (that is, $|A(x)-A(y)|\lesssim |x-y|$ for all $x,y\in \mathbb{R}^n$) and the symbol of $T$ satisfies appropriate conditions, R. Coifman and Y. Meyer CM78 proved $L^p$ estimates for $[T,M_A].$ Depending on the conditions assumed on the symbol of $T,$ the proof of these results involves the decomposition of the commutator into pieces that include bilinear pseudodifferential operators with symbols satisfying 5 or 6.

After the foundational work of R. Coifman and Y. Meyer, the field of bilinear (and multilinear) Fourier analysis experienced a rapid growth with highlights that include the study of boundedness properties of the bilinear Hilbert transform in Lebesgue spaces by M. Lacey and C. Thiele, the further development of the bilinear Calderón–Zygmund theory by M. Christ and J.-L. Journé, C. Kenig and E. Stein, and L. Grafakos and R. H. Torres GT02, as well as the first works regarding bilinear Hörmander classes by Á. Bényi and R. H. Torres BT03BT04.

Spurred by BT03BT04, significant efforts have been dedicated in recent years to the study of bilinear pseudodifferential operators with symbols in classes that unify conditions 2, 5, and 6. Namely, given $m \in \mathbb{R}$ and $0 \leq \delta \leq \rho \leq 1,$ the bilinear Hörmander class $BS^m_{\rho , \delta }$ consists of symbols $\sigma (x,\xi ,\eta )$ satisfying

$$\begin{equation} |\partial _x^\alpha \partial _\xi ^\beta \partial _\eta ^\gamma \sigma (x,\xi , \eta )|\leq C_{\alpha \beta \gamma } (1+|\xi |+|\eta |)^{m + \delta |\alpha | - \rho (|\beta |+|\gamma |)} \tag{10}{} \end{equation}$$

for all $x,\xi , \eta \in {\mathbb{R}}^n,$ for all multi-indices $\alpha$, $\beta$, $\gamma \in \mathbb{N}_0^n$, and for some constants $C_{\alpha \beta \gamma }>0.$ The index $m$ is referred to as the order of (the symbols in) the class $BS^m_{\rho ,\delta }.$ Hence, we have that $BS^m_{1,0}$ corresponds to symbols satisfying 2, $BS^{0}_{1,0}$ corresponds to symbols satisfying 5, and $BS^0_{\rho ,\rho }$ corresponds to symbols satisfying 6.

Two fundamental aspects of the study of the bilinear Hörmander classes are the symbolic calculus for transposes of operators and mapping properties in a variety of settings that include Lebesgue and Hardy spaces, the space BMO of functions of bounded mean oscillations, and Besov and Triebel–Lizorkin spaces. The corresponding literature is vast and includes works by Á. Bényi and R. H. Torres BT03BT04, Á. Bényi, D. Maldonado, R. H. Torres, and the author BMNT10, Á. Bényi, F. Bernicot, D. Maldonado, R. H. Torres, and the author BBM$^{+}$13, A. Miyachi and N. Tomita MT13, N. Michalowski, D. Rule, and W. Staubach MRS14, the author Nai15aNai15b, K. Koezuka and N. Tomita KT18, A. Miyachi and N. Tomita MT20MT19, A. Thomson and the author NT19a, and the references therein.

The classes $BS^m_{\rho ,\delta }$ stand as the bilinear counterparts of the Hörmander classes $S^m_{\rho ,\delta }$ associated to linear pseudodifferential operators. As we will see, while the bilinear and linear theories have some features in common, they require the use and development of different tools and techniques in several aspects.

A Brief on the Linear Setting

With earlier works by S. Mihlin, by A. Calderón and A. Zygmund, by A. Unterberger and J. Bokobza, and by R. Seeley, the theory of linear pseudodifferential operators was formalized in the 1960s by J. Kohn and L. Nirenberg and by L. Hörmander. These operators, in particular those with symbols in the Hörmander classes $S^m_{\rho ,\delta }$ defined below, have played a central role in the analysis of partial differential equations.

In the linear setting, a pseudodifferential operator is associated to a symbol $\sigma (x,\xi )$ defined for $x,\xi \in \mathbb{R}^n$ and is given by

$$\begin{equation*} T_\sigma f(x) = \int _{{\mathbb{R}}^n} \sigma (x,\xi ) \widehat{f}(\xi ) e^{2\pi i \xi \cdot x} \, d\xi . \end{equation*}$$

For $m\in \mathbb{R}$ and $0\le \delta \le \rho \le 1,$ the Hörmander class $S^m_{\rho ,\delta }$ is defined as the family of symbols $\sigma :\mathbb{R}^n\times \mathbb{R}^n\to \mathbb{C}$ such that

$$\begin{equation} |\partial _x^\alpha \partial _\xi ^\beta \sigma (x,\xi )|\le C_{\alpha \beta }\, (1+|\xi |)^{m+\delta |\alpha |-\rho |\beta |} \tag{11}{} \end{equation}$$

for all $x,\xi \in \mathbb{R}^n$ and all multi-indices $\alpha ,\beta \in \mathbb{N}_0^n.$

The study of boundedness properties of pseudodifferential operators with symbols in $S^m_{\rho ,\delta }$ in the setting of Lebesgue spaces goes back to the 1960s and 1970s with the works of I. Hirschman, S. Wainger, L. Hörmander, A. Calderón and R. Vaillancourt, and C. Fefferman and E. Stein.

A seminal result in this direction is due to A. Calderón and R. Vaillancourt CV72, who established that symbols in the class $S^0_{\rho ,\rho }$ for $0\le \rho <1$ (that is, symbols of order zero) give rise to pseudodifferential operators that are bounded on $L^2(\mathbb{R}^n)$. Due to the nested properties of the Hörmander classes, this implies boundedness on $L^2(\mathbb{R}^n)$ for the operators with symbols in the classes $S^0_{\rho ,\delta }$ with $0\le \delta < \rho <1,$ previously proved by L. Hörmander Hör67. In turn, the boundedness of operators with symbols of order zero plays a significant role in the proof of endpoint boundedness properties for pseudodifferential operators with symbols in other Hörmander classes (C. Fefferman Fef73). More precisely, the following result holds.

Theorem 1.

Let $0\le \delta \le \rho \le 1,$ $0\le \delta <1,$ and $1\le p\le \infty ;$ define $m_\rho (p)=-n(1-\rho )|\frac{1}{p}-\frac{1}{2}|.$

(1)

If $1<p<\infty$ and $m\le m_\rho (p),$ then $T_\sigma$ is bounded from $L^p(\mathbb{R}^n)$ into $L^p(\mathbb{R}^n)$ for all $\sigma \in S^m_{\rho ,\delta }.$

(2)

If $p=1$ and $m\le m_\rho (1)=-n(1-\rho )/2,$ then $T_\sigma$ is bounded from $H^1(\mathbb{R}^n)$ into $L^1(\mathbb{R}^n)$ for all $\sigma \in S^m_{\rho ,\delta }.$

(3)

If $p=\infty$ and $m\le m_\rho (\infty )=-n(1-\rho )/2,$ then $T_\sigma$ is bounded from $L^\infty (\mathbb{R}^n)$ into $BMO(\mathbb{R}^n)$ for all $\sigma \in S^m_{\rho ,\delta }.$

(4)

If $1<p<\infty ,$ $0<\rho <1,$ and $m> m_\rho (p),$ there exists $\sigma \in S^m_{\rho ,\delta }$ such that $T_\sigma$ fails to be bounded from $L^p(\mathbb{R}^n)$ into $L^p(\mathbb{R}^n).$

The space $H^1(\mathbb{R}^n)$ is the Hardy space (defined on page 1), which is contained in $L^1(\mathbb{R}^n),$ and the space $BMO(\mathbb{R}^n)$ is the space of functions of bounded mean oscillation (modulo constants), which contains $L^\infty (\mathbb{R}^n).$ Both $H^1(\mathbb{R}^n)$ and $BMO(\mathbb{R}^n)$ are natural substitutions for $L^1(\mathbb{R}^n)$ as a domain and $L^\infty (\mathbb{R}^n)$ as a codomain, respectively, when studying endpoint boundedness properties corresponding to $p=1$ and $p=\infty .$ In view of parts 1 and 4 of Theorem 1, $m_\rho (p)$ is called a critical order.

Homogeneous $x$-independent versions of the symbols in the class $S^0_{1,0}$ are given by multipliers satisfying Mihlin’s condition, a version of 4 in $\mathbb{R}^n$; that is, multipliers