$L^p$-Multipliers Sensitive to the Group Structure on Nilpotent Lie Groups

Abstract

We propose new sufficient conditions for $L^p$-multipliers on homogeneous nilpotent groups. The multipliers generalise the flag multipliers of Nagel–Ricci–Stein–Wainger, but the approach and the techniques applied are entirely different. Our multipliers are better adapted to the specific commutation rules on the Lie algebra of the given group. The proofs are based on a new symbolic calculus fashioned after Hörmander. We also take advantage of the Cotlar–Stein lemma, and the Littlewood–Paley theory in the spirit of Duoandikoetxea–Rubio de Francia.

Appendix: Convolution of Distributions

Let X be an N-dimensional vector space as described in Sect. 2.

10.1

(Sobolev inequality) We have

$$\begin{aligned} \Vert f\Vert _{A(X)}\lesssim \max _{|\alpha |\le {N/2+1}} \Vert D^{\alpha }f\Vert _{L^2(X)}, \quad f\in \mathcal {S}(X), \end{aligned}$$

where $\Vert f\Vert _{A(X)}=\int _{\mathfrak {g}^{\star }}|\widehat{f}(\xi )|d\xi$. (Proposition 3.5.14 of Narasimhan [14].)

The following is a direct consequence of (10.1).

10.2

Let F be a measurable function on an open subset of X. If, for every $\alpha \in \mathcal {A}_{\varvec{N}}$, $D^{\alpha }F$ is a locally bounded function, then F is smooth.

Let $\mathfrak {g}$ be a nilpotent Lie group as described in Sect. 5. The following definition of the general convolution is due to Chevalley. See Chevalley [1], Sect. 8.

Definition 10.3

We say that distributions $S,T\in \mathcal {S}'(\mathfrak {g})$ are convolvable if

$$\begin{aligned} \int _{\mathfrak {g}}\left| \Big (\widetilde{S}\star f\Big )(x)\Big (T\star \widetilde{g}\Big )(x)\right| dx<\infty , \quad f,g\in \mathcal {S}(\mathfrak {g}). \end{aligned}$$

Proposition 10.4

If S,T are convolvable, then there exists a unique distribution $S\star T$ such that

$$\begin{aligned} \langle S\star T,f\star g\rangle =\int _{\mathfrak {g}}\Big (\widetilde{S}\star f\Big )(x)\Big (T\star \widetilde{g}\Big )(x)dx. \quad f,g\in \mathcal {S}(\mathfrak {g}). \end{aligned}$$

(10.5)

10.6

If $S,T\in \mathcal {S}'(\mathfrak {g})$ are convolvable, then

$$\begin{aligned} (S\star T)\star \varphi =S\star (T\star \varphi ), \quad \varphi \star (S\star T)=(\varphi \star S)\star T, \end{aligned}$$

(10.7)

for $\varphi \in \mathcal {S}(\mathfrak {g})$. Moreover, $\widetilde{T}$ and $\widetilde{S}$ are also convolvable and

$$\begin{aligned} (S\star T)^{\sim }=\widetilde{T}\star \widetilde{S}. \end{aligned}$$

10.8

A distribution $R\in \mathcal {S}'(\mathfrak {g})$ is said to be an $\mathcal {S}$-convolver if $R\star f,f\star R\in \mathcal {S}(\mathfrak {g})$, for every $f\in \mathcal {S}(\mathfrak {g})$.

10.9

If $K\in \mathcal {S}'(\mathfrak {g})$ and R is an $\mathcal {S}$-convolver, then K,R and R,K are convolvable and

$$\begin{aligned} \langle K\star R,f\rangle =\langle K,f\star \widetilde{R}\rangle , \quad \langle R\star K,f\rangle =\langle K,\widetilde{R}\star f\rangle , \quad f\in \mathcal {S}(\mathfrak {g}). \end{aligned}$$

10.10

A distribution $R\in \mathcal {S}'(\mathfrak {g})$ is central if $R\star f=f\star R$, for every $f\in \mathcal {S}(\mathfrak {g})$. If R is a central $\mathcal {S}$-convolver, then $R\star K=K\star R$, for every $K\in \mathcal {S}'(\mathfrak {g})$.

10.11

If S and T are convolvable and R is an $\mathcal {S}$-convolver, then S, $T\star R$ and $R\star S$, T are convolvable, and

$$\begin{aligned} (S\star T)\star R=S\star (T\star R), \quad R\star (S\star T)=(R\star S)\star T. \end{aligned}$$

If, moreover, R is central, then

$$\begin{aligned} (S\star R)\star T=S\star (R\star T). \end{aligned}$$