Spectral Multipliers on 2-Step Stratified Groups, I

Abstract

Given a 2-step stratified group which does not satisfy a slight strengthening of the Moore–Wolf condition, a sub-Laplacian ${\mathcal {L}}$ and a family ${\mathcal {T}}$ of elements of the derived algebra, we study the convolution kernels associated with the operators of the form $m({\mathcal {L}}, -\,i {\mathcal {T}})$. Under suitable conditions, we prove that: (i) if the convolution kernel of the operator $m({\mathcal {L}},-\,i {\mathcal {T}})$ belongs to $L^1$, then m equals almost everywhere a continuous function vanishing at $\infty$ (‘Riemann–Lebesgue lemma’); (ii) if the convolution kernel of the operator $m({\mathcal {L}},-\,i{\mathcal {T}})$ is a Schwartz function, then m equals almost everywhere a Schwartz function.

Appendix: Composite Functions

We collect in this appendix a number of technical results used throughout the paper to establish properties (RL) and (S) of ‘image families.’

1.1 Continuous Functions

In this subsection, we consider the following problem: given three Polish spaces X, Y, Z, a positive measure $\mu$ on X, a $\mu$-measurable mapping $\pi :X\rightarrow Y$, and a function $m:Y\rightarrow Z$ such that $m\circ \pi$ equals $\mu$-almost everywhere a continuous function, does m equal $\pi _*(\mu )$-almost everywhere a continuous function?

To this end, we introduce the following definition.

Definition 10.1

Let X be a Polish space, Y a set, $\mu$ a positive Radon measure on X, and $\pi$ a mapping from X into Y. We say that two points $x,x'$ of $\text {Supp}\left( \mu \right)$ are $(\mu ,\pi )$-connected if $\pi (x)=\pi (x')$ and there are $x=x_1,\dots , x_k=x'\in \pi ^{-1}(\pi (x))\cap \text {Supp}\left( \mu \right)$ such that, for every $j=1,\dots , k$, for every neighbourhood $U_j$ of $x_j$ in $\text {Supp}\left( \mu \right)$, and for every neighbourhood $U_{j+1}$ of $x_{j+1}$ in $\text {Supp}\left( \mu \right)$, the set $\pi ^{-1}(\pi (U_j)\cap \pi (U_{j+1}))$ is not $\mu$-negligible. We say that $\mu$ is $\pi$-connected if every pair of elements of $\text {Supp}\left( \mu \right)$ having the same image under $\pi$ are $(\mu ,\pi )$-connected.

Observe that $(\mu ,\pi )$-connectedness actually depends only on the equivalence class of $\mu$ and the equivalence relation induced by $\pi$ on X. In addition, notice that, if Y is a topological space and $\pi$ is open at some point of each fibre (in the support of $\mu$), then $\mu$ is $\pi$-connected.

We emphasize that, in the definition of $(\mu ,\pi )$-connectedness, the points $x_1,\dots ,x_k$ are fixed before considering their neighbourhoods. In other words, if for every neighbourhood U of x in $\text {Supp}\left( \mu \right)$ and for every neighbourhood $U'$ of $x'$ in $\text {Supp}\left( \mu \right)$ we found $x=x_1,\dots , x_k=x'$ and neighbourhoods $U_j$ of $x_j$ in $\text {Supp}\left( \mu \right)$ so that $U=U_1$, $U'=U_k$ and, for every $j=1,\dots , k$, the set $\pi ^{-1}(\pi (U_j)\cap \pi (U_{j+1}))$ were not $\mu$-negligible, then we would not be able to conclude that x and $x'$ are $(\mu ,\pi )$-connected.

Now we can prove our main result. Notice that, even though its hypotheses are quite restrictive, it still gives rise to important consequences.

Proposition 10.2

Let X, Y, Z be three Polish spaces, $\pi :X\rightarrow Y$ a $\mu$-measurable mapping, and $\mu$ a $\pi$-connected positive Radon measure on X. Assume that $\pi$ is $\mu$-proper and that there is a disintegration $(\lambda _y)_{y\in Y}$ of $\mu$ relative to $\pi$ such that $\text {Supp}\left( \lambda _y\right) \supseteq \text {Supp}\left( \mu \right) \cap \pi ^{-1}(y)$ for $\pi _*(\mu )$-almost every $y\in Y$.

Take a continuous map $m_0:X\rightarrow Z$ and assume that there is map $m_1:Y\rightarrow Z$ such that $m_0(x)= (m_1\circ \pi )(x)$ for $\mu$-almost every $x\in X$. Then, there is a $\pi _*(\mu )$-measurable mapping $m_2:Y\rightarrow Z$ such that $m_0=m_2\circ \pi$pointwise on $\text {Supp}\left( \mu \right)$.

Notice that, if $\pi$ is also proper, then $m_2$ is actually continuous on $\pi (\text {Supp}\left( \mu \right) )$.

Proof

Observe first that there is a $\pi _*(\mu )$-negligible subset N of Y such that $m_1\circ \pi =m_0$$\lambda _y$-almost everywhere for every $y\in Y\setminus N$. Notice that we may assume that $\text {Supp}\left( \mu \right) =X$ and that, if $y\in Y\setminus N$, then the support of $\lambda _y$ contains $\pi ^{-1}(y)$. Since $m_0$ is continuous and since $m_1\circ \pi$ is constant on the support of $\lambda _y$, it follows that $m_0$ is constant on $\pi ^{-1}(y)$ for every $y\in Y\setminus N$.

Now, take $y\in \pi (X)\cap N$ and $x_1,x_2\in \pi ^{-1}(y)$. Let ${\mathfrak {U}}(x_1)$ and ${\mathfrak {U}}(x_2)$ be the filters of neighbourhoods of $x_1$ and $x_2$, respectively. Assume first that $\pi (U_1)\cap \pi (U_2)$ is not $\pi _*(\mu )$-negligible for every $U_1\in {\mathfrak {U}}(x_1)$ and for every $U_2\in {\mathfrak {U}}(x_2)$. Take $U_1\in {\mathfrak {U}}(x_1)$ and $U_2\in {\mathfrak {U}}(x_2)$. Then, there is $y_{U_1,U_2}\in \pi (U_1)\cap \pi (U_2)\setminus N$, and then $x_{h,U_1,U_2}\in U_h\cap \pi ^{-1}(y_{U_1,U_2})$ for $h=1,2$. Now, $m_0(x_{1,U_1,U_2})=m_0(x_{2,U_1,U_2})$ for every $U_1\in {\mathfrak {U}}(x_1)$ and for every $U_2\in {\mathfrak {U}}(x_2)$. In addition, $x_{h,U_1,U_2}\rightarrow x_h$ in X along the product filter of ${\mathfrak {U}}(x_1)$ and ${\mathfrak {U}}(x_2)$. Since $m_0$ is continuous, passing to the limit we see that $m_0(x_1)=m_0(x_2)$. Since $\mu$ is $\pi$-connected, this implies that $m_0$ is constant on $P^{-1}(y)$ for every$y\in \pi (X)$. The assertion follows. $\square$

In the following proposition we give sufficient conditions in order that a measure be connected.

Proposition 10.3

Let $E_1,E_2$ be two finite-dimensional vector spaces, $L:E_1\rightarrow E_2$ a linear mapping, C a closed convex subset of $E_1$ and $\mu$ a positive Radon measure on $E_1$ with support C. Take a Polish subspace X of $E_1$ such that $\mu (E_1\setminus X)=0$, and assume that either $X=C$ or C is a convex cone. Then, $\mu _X$, that is, the measure induced by $\mu$ on X, is -connected.

Actually, there is no need that X be a Polish space, but we did not consider Radon measures on more general Hausdorff spaces.

Proof

We may assume that C has non-empty interior. Then, we may find a non-empty bounded convex open subset U of C and an convex open neighbourhood V of 0 in $\ker L$ such that $U+V\subseteq C$. Take $r\in ]0,1]$ and $x,y\in C\cap X$ such that $y-x\in V$; take $R_x>0$ so that $U\subseteq B(x,R_x)$. Then, for every $u\in U$ we have $y+ r(u-x)\in B(y,r R_x)\cap [y,y-x+u]\subseteq B(y,r R_x)\cap C$; analogously, $x+r(U-x)\subseteq B(x,r R_x)\cap C$. Since $L(x)=L(y)$, we infer that

$$\begin{aligned} L^{-1}(L( B(x, r R_x)\cap C \cap X)\cap L(B(y,r R_x)\cap C\cap X))\supseteq [x+r(U-x)]\cap X. \end{aligned}$$

Now, $x+r(U-x)$ is a non-empty open subset of $C=\text {Supp}\left( \mu \right)$, so that $\mu _X([x+r(U-x)]\cap X)=\mu (x+r(U-x))>0$. The arbitrariness of r then implies that x and y are $(\mu ,L)$-connected. The assertion then follows easily if $X=C$.

Finally, assume that $X\ne C$, so that C is a convex cone; we may assume that C has vertex 0. Then, given $x,y\in C\cap X$ such that $L(x)=L(y)$, we may find $r_{x,y}>0$ such that, with the above notation, $y-x\in r_{x,y} V$. Then, $r_{x,y} U+r_{x,y} V\subseteq r_{x,y} V\subseteq C$, so that the above argument shows that x and y are $(\mu ,L)$-connected. The arbitrariness of x and y then implies that $\mu _X$ is -connected. $\square$

Now we present a result on the disintegration of Hausdorff measures, which is particularly useful to check the assumptions of Proposition 10.2, and is a straightforward consequence of the general coarea formula (cf. [28, Theorem 3.2.22]). Recall that a subset of $\mathbb {R}^n$ is said to be countably ${\mathcal {H}}^k$-rectifiable if it is the union of an ${\mathcal {H}}^k$-negligible set and a countable family of Lipschitz images of bounded subsets of $\mathbb {R}^k$. For example, any countable union of k-dimensional submanifolds (of class $C^1$) of $\mathbb {R}^n$ is ${\mathcal {H}}^k$-measurable and countably ${\mathcal {H}}^k$-rectifiable, as well as any countable union of images of $C^\infty$ functions (defined on $C^\infty$ manifolds with a countable base) of maximum rank k, thanks to [63, Theorem 1]. We refer the reader to [28, Theorem 3.2.22] for the definition of the approximate k-dimensional Jacobian $\text {ap}\,J_{k} P$ of a Lipschitz mapping P defined on an ${\mathcal {H}}^{k}$-measurable and countably ${\mathcal {H}}^{k}$-rectifiable subset E of $\mathbb {R}^{n}$ and taking values in $\mathbb {R}^m$. Notice that, if E is a submanifold (of class $C^1$) of $\mathbb {R}^{n}$ and P is of class $C^1$, then $\text {ap}\,J_{k}P(x)$ is simply $\Vert \bigwedge ^{k} T_x (P) \Vert$ for every $x\in E$, where $T_x (P)$ denotes the differential of P.

Proposition 10.4

Let, for $j=1,2$, $E_j$ be an ${\mathcal {H}}^{k_j}$-measurable and countably ${\mathcal {H}}^{k_j}$-rectifiable subset of $\mathbb {R}^{n_j}$. Assume that $k_2\leqslant k_1$, and let P be a locally Lipschitz mapping of $E_1$ into $E_2$. Take a positive function $f\in L^1_\text {loc}({\mathcal {H}}^{k_1})$ which vanishes on the complement of $E_1$, and assume that $f(x){ ap }J_{k_2} P(x)\ne 0$ for ${\mathcal {H}}^{k_1}$-almost every $x\in E_1$, and that P is $(f\cdot {\mathcal {H}}^{k_1})$-proper.¹⁴

Then, the following hold:

• 1. the mapping

  $$\begin{aligned} g:\mathbb {R}^{n_2}\ni y \mapsto \int _{P^{-1}(y)} \frac{f}{{ap } J_{k_2} P}\,\text {d}{\mathcal {H}}^{k_1-k_2} \end{aligned}$$

  is well-defined ${\mathcal {H}}^{k_2}$-almost everywhere and measurable; in addition,

  $$\begin{aligned} P_*(f\cdot {\mathcal {H}}^{k_1})= g\cdot {\mathcal {H}}^{k_2}; \end{aligned}$$

• 2. the measure

  $$\begin{aligned} \beta _y{:}{=}\frac{1}{g(y)}\frac{f}{{ap } J_{k_2} P} \chi _{P^{-1}(y)}\cdot {\mathcal {H}}^{k_1-k_2} \end{aligned}$$

  is well-defined and Radon for $P_*(f\cdot {\mathcal {H}}^{k_1})$-almost every $y\in \mathbb {R}^{n_2}$; in addition, $(\beta _y)$ is a disintegration of $f\cdot {\mathcal {H}}^{k_1}$ relative to P;

• 3. $\beta _y$ is equivalent to $\chi _{P^{-1}(y)}\cdot {\mathcal {H}}^{k_1-k_2}$ for $P_*(f\cdot {\mathcal {H}}^{k_1})$-almost every $y\in E_2$.

1.2 Schwartz Functions

In this subsection we shall extend some results on composite differentiable functions by Bierstone, Milman and Schwarz in [9, 11] to the case of Schwartz functions by means of the techniques developed by Astengo, Di Blasio and Ricci in [5].

We shall take advantage of the remarkable works of Bierstone, Milman and Schwarz about the composition of smooth functions on analytic manifolds. Recall that, if M is a (real, finite-dimensional) analytic manifold, then a subset A of M is called analytic if every $x\in M$ has a neighbourhood U such that $U\cap A$ is the zero locus of a (real) analytic function on U. The set A is semianalytic if every $x\in M$ has a neighbourhood U such that $U\cap A$ belongs to the algebra of subsets of U generated by the sets of the form $f^{-1}(]0,\infty [)$, where f is a (real) analytic function on U. The set A is subanalytic if every $x\in M$ has a neighbourhood U such that $U\cap A=\text {pr}_1(B)$ for some analytic manifold N and some relatively compact semianalytic subset B of $M\times N$. The set A is Nash subanalytic of pure dimension k if it is closed, subanalytic, and, for every $x\in A$, $\dim _x Y=\dim Z_x=k$, where $Z_x$ denotes the smallest germ of an analytic set at x containing the germ of A at x (cf. [11, 1.5]). A closed subanalytic set is Nash subanalytic if it is the locally finite union of Nash subanalytic sets of pure dimension. We refer the reader to [9,10,11] for an account of the main properties of semianalytic and (Nash) subanalytic sets. As a matter of fact, in the applications we shall only need to know that any closed convex subanalytic set is automatically Nash subanalytic, since it is contained in an affine space of the same dimension, and that closed semianalytic sets are Nash subanalytic (cf. [9, Proposition 2.3]).

Our starting point is the following result (cf. [9, Theorem 0.2] and [11, Theorem 0.2.1]). If C is a closed subset of $\mathbb {R}^n$, then we denote by ${\mathcal {E}}(C)$ is the quotient of ${\mathcal {E}}(\mathbb {R}^n)$ by the space of functions of class $C^\infty$ which vanish on C.

Theorem 10.5

Let C be a closed subanalytic subset of $\mathbb {R}^n$ and let $P:\mathbb {R}^n\rightarrow \mathbb {R}^m$ be an analytic mapping. Assume that P is proper on C and that P(C) is Nash subanalytic. Then, the canonical mapping

$$\begin{aligned} \Phi :{\mathcal {E}}(\mathbb {R}^m)\ni \varphi \mapsto \varphi \circ P\in {\mathcal {E}}(C) \end{aligned}$$

has a closed range, and admits a continuous linear section defined on $\Phi ({\mathcal {E}}(\mathbb {R}^n))$.

In addition, $\psi \in {\mathcal {E}}(C)$ belongs to the image of $\Phi$ if and only if for every $y\in P(C)$ there is $\varphi _y\in {\mathcal {E}}(\mathbb {R}^m)$ such that, for every $x\in C$ such that $P(x)=y$, the Taylor series of $\varphi _y\circ P$ and $\psi$ at x differ by the Taylor series of a function of class $C^\infty$ which vanishes on C.

In order to simplify the notation, we shall simply say that $\psi$ is a formal composite of P if the second condition of the statement holds.

We shall now describe how Theorem 10.5 can be extended to the case of Schwartz functions. The strategy developed in [5] is the following: first, decompose dyadically a given Schwartz function in a sum of dilates of a family of test functions with a suitable decay; then, apply the section given by Theorem 10.5, truncate the resulting functions (so that they are still test functions), and finally sum their dilates. In order to do that, however, we need homogeneity.

Theorem 10.6

Let $P:\mathbb {R}^n\rightarrow \mathbb {R}^m$ be a polynomial mapping, and assume that $\mathbb {R}^n$ and $\mathbb {R}^m$ are endowed with dilations such that $P(r\cdot x)=r\cdot P(x)$ for every $r>0$ and for every $x\in \mathbb {R}^n$. Let C be a dilation-invariant subanalytic closed subset of $\mathbb {R}^n$, and assume that P is proper on C and that P(C) is Nash subanalytic. Then, the canonical mapping

$$\begin{aligned} \Phi :{\mathcal {S}}(\mathbb {R}^m)\ni \varphi \mapsto \varphi \circ P\in {\mathcal {S}}(C) \end{aligned}$$

has a closed range and admits a continuous linear section defined on $\Phi ({\mathcal {S}}(\mathbb {R}^m))$. In addition, $\psi \in {\mathcal {S}}(C)$ belongs to the image of $\Phi$ if and only if it is a formal composite of P.

As a matter of fact, in our applications C is (a subset of) $\sigma ({\mathcal {L}}_A)$. Then, Theorem 10.6 gives sufficient conditions in order that some $f\in {\mathcal {S}}_{P({\mathcal {L}}_A)}(G)$ which also belongs to ${\mathcal {K}}_{{\mathcal {L}}_A}({\mathcal {S}}(\sigma ({\mathcal {L}}_A)))$ should belong to ${\mathcal {K}}_{P({\mathcal {L}}_A)}({\mathcal {S}}(\sigma (P({\mathcal {L}}_A))))$ (cf. Sect. 5).

Notice, however, that sometimes it is convenient to take C so as to be a subset of $\sigma ({\mathcal {L}}_A)$ such that $P(C)=\sigma (P({\mathcal {L}}_A))$, since $\sigma ({\mathcal {L}}_A)$ need not be subanalytic.

Proof

For the first assertion, simply argue as in the proof of [5, Theorem 6.1] replacing the linear section provided by Schwarz and Mather with that of Theorem 10.5.

As for the second part of the statement, notice first that it follows easily from Theorem 10.5 when $\psi$ is compactly supported; since the image of $\Phi$ is closed, it follows by approximation in the general case. $\square$

In the following result, we give a simple but very useful application of Theorem 10.6.

Corollary 10.7

Let V and W be two finite-dimensional vector spaces, C a subanalytic closed convex cone in V, and L a linear mapping of V into W which is proper on C. Take $m_1\in {\mathcal {S}}(V)$, and assume that there is $m_2:W\rightarrow \mathbb {C}$ such that $m_1=m_2\circ L$ on C. Then, there is $m_3\in {\mathcal {S}}(W)$ such that $m_1=m_3\circ L$ on C.

Proof

Observe first that we may assume that C has non-empty interior and vertex 0. In addition, observe that L(C) is subanalytic (cf. [10, Theorem 0.1 and Proposition 3.13]), hence Nash subanalytic. Now, fix $x\in C$. Since the interior of C is not empty, it is clear that C is a total subset of V, so that we may find a free family $(v_j)_{j\in J}$ in C which generates an algebraic complement $V'$ of $\ker L$ in V. In addition, since either $x=0$ or $x\not \in \ker L$, we may assume that $x\in V'$. Let $L':W\rightarrow V$ be a linear mapping such that $L'\circ L$ is the identity on $V'$ and such that $L\circ L'$ is the identity on L(V).

Define $m'{:}{=}m_1\circ L'$, so that $m'\in {\mathcal {E}}(W)$. Next, define $C'{:}{=}V'\cap C$, so that $C'$ is a closed convex cone with non-empty interior in $V'$, since it contains the non-empty open set $\sum _{j\in J} \mathbb {R}_+^* v_j$. Take $z\in C'$ and any $y\in C\cap (x+\ker L)$. Then, $x+z=(L'\circ L) (x+z)=(L' \circ L )(y+z)$, so that $m_1=m'\circ L$ on $y+C'$. Since $m_1$ is constant on the intersections of C with the translates of $\ker L$, the same holds on $C\cap (y+C'+\ker L)$. Now, denote by $C'^\circ$ the interior of $C'$ in $V'$. Then, $y+C'^\circ +\ker L$ is an open convex set and y is adherent to $C\cap (y+C'^\circ +\ker L)$, which has non-empty interior since it is not empty and C is the closure of its interior (cf. [12, Corollary 1 to Proposition 16 of Chapter II, §2, No. 6]). Hence, the Taylor polynomials of every fixed order of $m_1$ and $m'\circ L$ about y coincide on $C\cap (y+C'^\circ +\ker L)$, hence on V. Since this holds for every $y\in C\cap [x+\ker L]$, Theorem 10.6 implies that there is $m_3\in {\mathcal {S}}(W)$ such that $m_1=m_3\circ L$ on C. $\square$