Generic norm growth of powers of homogeneous unimodular Fourier multipliers

Abstract

For an integer $d\ge 2$, $t\in \mathbb {R}$, and a 0-homogeneous function $\Phi \in C^{\infty }(\mathbb {R}^{d}{\setminus }\{0\},\mathbb {R})$, we consider the family of Fourier multiplier operators $T_{\Phi }^t$ associated with symbols $\xi \mapsto \exp (it\Phi (\xi ))$ and prove that for a generic phase function $\Phi$, one has the estimate $\Vert T_{\Phi }^t\Vert _{L^p\rightarrow L^p} \gtrsim _{d,p, \Phi }\langle t\rangle ^{d|\frac{1}{p}-\frac{1}{2}|}$. That is the maximal possible order of growth in $t\rightarrow \pm \infty$, according to the previous work by V. Kovač and the author and the result shows that the two special examples of functions $\Phi$ that induce the maximal growth, given by V. Kovač and the author and independently by D. Stolyarov, to disprove a conjecture of Maz’ya actually exhibit the same general phenomenon.

1 Introduction

For $d\in \mathbb {N}$, $d\ge 2$, and $\Phi \in C^{\infty }(\mathbb {R}^{d}{\setminus }\{0\}, \mathbb {R})$, a homogeneous function of degree 0, we consider a family of Fourier multiplier operators indexed by $t\in \mathbb {R}$, defined on the set of Schwartz functions ${\mathcal {S}}(\mathbb {R}^d)$ with

$$\begin{aligned} T_{\Phi }^t f(x):=\int \limits _{\mathbb {R}^d}e^{it\Phi (\xi )+i\xi \cdot x}{\widehat{f}}(\xi )d\xi . \end{aligned}$$

(1.1)

Since $\Phi$ is a smooth and real-valued homogeneous function of degree 0, by [11, §III.3.5, Theorem 6], we know that $T_{\Phi }^t$ has a representation in the form

$$\begin{aligned} T_{\Phi }^tf=a_\Phi ^t f + S_{\Phi }^tf, \quad f\in {\mathcal {S}}(\mathbb {R}^d), \end{aligned}$$

where, denoting a spherical measure on $S^{d-1}$ with $\sigma _{d-1}$, $a_\Phi ^t$ is a constant given by

$$\begin{aligned} a_\Phi ^t:=\frac{1}{\sigma _{d-1}(S^{d-1})}\int \limits _{S^{d-1}}e^{it\Phi (\xi )} d\sigma _{d-1}(\xi ), \end{aligned}$$

and $S_{\Phi }^t$ is a singular integral operator defined on ${\mathcal {S}}(\mathbb {R}^d)$ as

$$\begin{aligned} S_{\Phi }^tf(x):=\lim _{\varepsilon \rightarrow 0}\int \limits _{|y|\ge \varepsilon } \frac{\Omega _\Phi ^t(\frac{y}{|y|})}{|y|^d}f(x-y)dy, \end{aligned}$$

for some function $\Omega _\Phi ^t\in C^{\infty }(S^{d-1},\mathbb {R})$ that satisfies $\int _{S^{d-1}}\Omega _{\Phi }^t d\sigma _{d-1} =0$.

It is obvious from the definition that $|a_\Phi ^t|\le 1$ and by the Calderon–Zygmund theory (see [2, Theorem 1] or [11, §II.4.2, Theorem 3]), we know that the singular integral operator $S_{\Phi }^t$ extends to an $L^p\rightarrow L^p$ bounded operator for all $p\in (1,\infty )$.

The question of the asymptotic behavior of $t\mapsto |T_\Phi ^t|_{L^p\rightarrow L^p}$ as $t\rightarrow \pm \infty$ was implicitly raised in [10] and explicitly posed by V. Maz’ya in [9, §4.2], where he asked whether there exists a finite constant $C_{d,p,\Phi }$ for which the following estimate holds for all homogeneous functions $\Phi \in C^{\infty }(\mathbb {R}^d{\setminus }\{0\}, \mathbb {R})$ of degree 0 and all $t\ge 1$:

$$\begin{aligned} |T_{\Phi }^t|_{L^p\rightarrow L^p}\le C_{d,p,\Phi } |t|^{(d-1)|\frac{1}{2}-\frac{1}{p}|}. \end{aligned}$$

The question was negatively resolved by V. Kovač and the author in [1] for even d by the following theorem. Since sharpness in $p\in (1,\infty )$ was a substantial part of [1], we will denote $p^{*}:=\max \{p,\frac{p}{p-1}\}$ and write the full statement of the theorem. For the definition of the Japanese bracket $\left\langle \cdot \right\rangle$, see Section 1.1 below — we use use it only to suppress writing $|t|\ge C$ in the statements of the theorems.

Theorem 1

([1, Theorem 2]).

• (a) Fix an integer $d\ge 2$ and a homogeneous function $\Phi \in C^{\infty }(\mathbb {R}^{d}{\setminus }\{0\}, \mathbb {R})$ of degree 0. There exists a finite constant $C_{d,\Phi }$ such that for every exponent $p\in (1,\infty )$ and $t\in \mathbb {R}$, we have

  $$\begin{aligned} |T^{t}_{\Phi }|_{L^p\rightarrow L^p}\le C_{d,\Phi } (p^{*}-1)\left\langle t \right\rangle ^{d|\frac{1}{2}-\frac{1}{p}|}. \end{aligned}$$

• (b) Fix an even integer $d\ge 2$. There exists a homogeneous function $\Phi \in C^{\infty }(\mathbb {R}^{d}{\setminus }\{0\},\mathbb {R})$ of degree 0 and a constant $c_{d,\Phi }>0$ such that for every exponent $p\in (1,\infty )$ and every nonzero integer k, we have

  $$\begin{aligned} |T^{k}_{\Phi }|_{L^p\rightarrow L^p}\ge c_{d,\Phi }(p^{*}-1) |k|^{d|\frac{1}{2}-\frac{1}{p}|}. \end{aligned}$$

One has to note that the answer to the particular case $d=2$ of Maz’ya’s question follows from the work of Dragičević, Petermichl, and Volberg [3]. This fact was overlooked in [9], when the question was formulated.

Furthermore, D. Stolyarov [13] independently, and using different techniques, proved part (a) of Theorem 1 and showed the existence of a function $\Phi$ as in part (b) for all $d\ge 2$, but without the sharp dependence on $p\in (1,\infty )$ on both parts.

Both papers, [1] and [13], have short proofs for the upper bound. In [1], the authors used spherical harmonics to reduce the problem to an $L^1\rightarrow L^{1,\infty }$ estimate for singular integrals with rough kernels of Seeger and Tao, while in [13], the author reduced the problem to the sharp version of the Hörmander–Mikhlin multiplier theorem.

However, in [1], we gave an extremely specific class of functions $\Phi$ for which the bound is asymptotically sharp for $t\rightarrow \pm \infty$ and in [13], the author gave one particular function for which the bound is asymptotically sharp. Both papers have relatively long proofs from which the cause of the exact worst asymptotics was not apparent. In this paper, we give a short proof of the existence of a general phenomenon that drives the growth of powers of norms for a generic symbol, proving that all, when $d=2$, and “almost all”, when $d\ge 3$, unimodular homogeneous symbols of degree 0 are counterexamples to the asymptotic order of growth conjectured by V. Maz’ya in [9, §4.2] and, in fact, exhibit the worst possible asymptotics. More precisely, we prove the following theorem — for the definition of Whitney topology, see Section 2.1 below.

Theorem 2

Let $d\in \mathbb {N}$, $d\ge 2$, $p\in (1,\infty )$, and $t\in \mathbb {R}$.

• (a) For $d=2$ and any nonconstant homogeneous function $\Phi \in C^{\infty }(\mathbb {R}^2{\setminus }\{0\},\mathbb {R})$ of degree 0, there exists a constant $c_{d,p,\Phi }>0$ such that

  $$\begin{aligned} |T^{t}_{\Phi }|_{L^p\rightarrow L^p}\ge c_{d,p,\Phi } \left\langle t \right\rangle ^{2|\frac{1}{2}-\frac{1}{p}|}. \end{aligned}$$

• (b) For $d\ge 3$, there is a dense open set ${\mathcal {G}}$ in the Whitney topology on $C^{\infty }(S^{d-1},\mathbb {R})$ such that for all $\phi \in {\mathcal {G}}$, there exists a constant $c_{d,p,\phi }>0$ for which the 0-homogeneous extension $\Phi \in C^{\infty }(\mathbb {R}^d{\setminus }\{0\},\mathbb {R})$ of $\phi$, defined as $\Phi (\xi ):=\phi (\frac{\xi }{|\xi |})$, satisfies

  $$\begin{aligned} |T^{t}_{\Phi }|_{L^p\rightarrow L^p}\ge c_{d,p,\phi } \left\langle t \right\rangle ^{d|\frac{1}{2}-\frac{1}{p}|}. \end{aligned}$$

In topology, the property that holds on a dense open set (or, more generally, on the complement of a countable union of nowhere dense sets) is called generic. Therefore, since a 0-homogeneous function on $\mathbb {R}^{d}{\setminus }\{0\}$ is uniquely defined by its restriction on the sphere $S^{d-1}$, the previous theorem says that multipliers associated with powers of a generic 0-homogeneous unimodular symbol exhibit the asymptotically maximal possible order of growth of $L^p\rightarrow L^p$ norms.

Part (b) of Theorem 2 will be a consequence of the following theorem, which can be of its own interest, so we state it here. For the definition of nondegeneracy of a critical point, see Section 2.1 below.

Theorem 3

Let $d\in \mathbb {N}$, $d\ge 2$, $p\in (1,\infty )$, and $t\in \mathbb {R}$. For a homogeneous function $\Phi \in C^{\infty }(\mathbb {R}^d{\setminus }\{0\},\mathbb {R})$ of degree 0 such that $\Phi \vert _{S^{d-1}}$ has a nondegenerate local minimum or maximum, there exists a constant $c_{d,p,\Phi }>0$ such that

$$\begin{aligned} |T^t_{\Phi }|_{L^p(\mathbb {R}^d)\rightarrow L^p(\mathbb {R}^d)} \ge c_{d,p,\Phi }\left\langle t \right\rangle ^{d|\frac{1}{2}-\frac{1}{p}|}. \end{aligned}$$

Choosing $\Phi (\xi )= \frac{\xi _1}{|\xi |}$ in Theorem 3, one gets the asymptotics for the so-called Riesz group (a.k.a. the Poincaré–Riesz–Sobolev group) that appears, when $d=3$, in the analysis of the Navier–Stokes equations in a rotating frame; see [5, §2], [6, §4], [7, Eq. (1.3)], or [4, Eq. (23)], while the asymptotics of the same symbol when $d=2$ was studied in [3]. In fact, the lower bound for $d\ge 3$ established by Stolyarov [13] is equivalent to the particular case of Theorem 3 for this particular choice of $\Phi$.

Finally, for the sake of completeness, we mention that the asymptotics of $t\mapsto |T_{\Phi }^t|_{L^p\rightarrow L^p}$ is an uninteresting problem when $d=1$ because for any $\Phi$ as before and $t\in \mathbb {R}$, one can write $T_{\Phi }^t$ as a bounded linear combination of the identity and the Hilbert transform to get the bound $|T_{\Phi }^t|_{L^p\rightarrow L^p}\lesssim 1$ for all $t\in \mathbb {R}$.

1.1 Notation

For $t\in \mathbb {R}$, we denote $\left\langle t \right\rangle :=(1+|t|^2)^{\frac{1}{2}}$. For $x\in \mathbb {R}^d$, $x=(x_1,\dots , x_d)$, we use $|x|$ to denote the standard Euclidean norm and we use the notation $x_{-}:=(x_1,\dots , x_{d-1})$. We will sometimes write “−” in subscript to emphasize that we are working with the first $d-1$ coordinates only. When working with matrices, we identify vectors in $\mathbb {R}^d$ with matrices of size $d\times 1$. For a matrix $A\in M_d(\mathbb {R})$, we write $A>0$ to denote that it is positive definite. For a function $f:\mathbb {R}^d\rightarrow \mathbb {R}$, we use the notation $\nabla f$ to denote the $1\times d$ matrix $\nabla _x f:=\left[ \frac{\partial f}{\partial x_j}\right] _j$ and Hf to denote the $d\times d$ Hessian matrix $H_x f:= \left[ \frac{\partial ^2f}{\partial x_j \partial x_k}\right] _{j,k}$. We will suppress x in subscript if the variables upon which we differentiate are clear from the context. We write $B_{d}(c,r)$ to denote the ball in $\mathbb {R}^d$ with center c and radius r and use the notation $S^{d-1}$ to denote the unit sphere around 0 in $\mathbb {R}^d$. If A, B are some expressions, then for some (possibly empty) set of indices P, the expression $A\lesssim _P B$ means that there is a constant $C>0$ depending on P such that $A\le CB$. The expression $A\sim _P B$ means that $A\lesssim _P B$ and $B\lesssim _P A$. Finally, since we care the most about the phase of the oscillatory integral, to suppress writing $2\pi$ in the exponent, we use the following normalization of the Fourier transform of a function $f\in L^1(\mathbb {R}^d)$:

$$\begin{aligned} {\widehat{f}}(\xi ):= \frac{1}{(2\pi )^d}\int \limits _{\mathbb {R}^d}f(x)e^{-i\xi \cdot x}dx. \end{aligned}$$

2 Idea of the proof and preliminaries

Statements of the theorems are interesting only when $|t|$ is large, so the idea of the proof is to follow the approach from [14, Exercise 2.34], used to study the asymptotics of $L^p$ behavior of the Schrödinger propagator (defined by (1.1) with $\Phi (\xi )=|\xi |^2$), that relies on the following observation. When $p\in (1,2]$, and $t>0$, the correct asymptotics can be written as $t^{-\frac{d}{2}}\times t^{\frac{d}{p}}$, what can be interpreted as base $\times$ height approximation of the $L^p$ norm of a function that resembles a bump of height $t^{-\frac{d}{2}}$ on the ball of radius t with a rapidly decreasing tail, both in x and t, outside of it. In the case of the Schrödinger propagator, Young’s inequality for convolutions with an explicit calculation of the kernel gives $|T_{|\cdot |^2}^tf|_{L^{\infty }}\lesssim t^{-\frac{d}{2}}$ and the method of nonstationary phase applied to (1.1) for x outside of the ball of radius $\gtrsim t$ proves that the function has a rapidly decreasing tail, both in x and t. The two estimates imply $|T_{|\cdot |^2}^tf|_{L^p}\lesssim t^{\frac{d}{p}-\frac{d}{2}}$ as $t\rightarrow \infty$ for any $p\in (1,\infty )$ and then the log-convexity of $L^p$ norms applied to p and $p' = \frac{p}{p-1}$, together with the fact that $T_{|\cdot |^2}$ is a unitary operator on $L^2$, transfers the upper bounds to lower bounds.

On the contrary, in the case of the homogeneous multipliers of degree 0, the kernel is singular, so one cannot use Young’s convolution inequality to control the $L^{\infty }$ size of $T_{\Phi }^tf$ and needs a different approach. The obvious method to try is the method of (non-)stationary phase, but since $\Phi$ is homogeneous of degree 0, it follows that for all $\xi \in , \mathbb {R}^d{\setminus }\{0\}$,

$$\begin{aligned} 0=\frac{d^2}{d h^2}(\Phi (\xi +h \xi ))\Big \vert _{h=0} = \xi ^{\top }H \Phi (\xi )\xi , \end{aligned}$$

implying, together with the fact that the Hessian of the function $\xi \mapsto \langle \xi ,x\rangle$ is equal to 0 for all $x,\xi \in \mathbb {R}^d$, that all stationary points of the phase are degenerate, so they do not fall under the scope of the classical method of stationary phase.

We are able to circumvent this problem and reduce the problem to the classical method of stationary phase by transforming the integral representation in the case of an appropriately localized function ${\widehat{f}}$ using the change of variables in the integral (see (3.2) below). The reason why the method works better when $d=2$ is the fact that regularity of the Hessian of the phase in the transformed expression does not depend on the second derivative, contrary to the $d\ge 3$ case.

Applying the method of stationary phase to the modified representation (3.2) with a localized function ${\widehat{f}}$, it turns out again that the best $L^{\infty }$ bound for $T_{\Phi }^tf$ is bigger than $t^{-\frac{d}{2}}$, so the function $T_{\Phi }^t f$ does not resemble a bump function of height $t^{-\frac{d}{2}}$ as it did in the case of the Schrödinger propagator.

However, using implicit and inverse function theorems, we are able to show the existence of the set of x’s of measure $\sim t^{d}$ on which the modified phase in (3.2)*** is stationary and nondegenerate, so the method of stationary phase gives the required asymptotics $|T^t_{\Phi }f(x)| \sim _{d,p,\Phi } t^{-\frac{d}{2}}$ on the given set. Using the base $\times$ height bound, this implies the required asymptotics, that is, $|T^t_{\Phi }f|_p \gtrsim _{d,p,\Phi } t^{\frac{d}{p} - \frac{d}{2}}$.

In the remainder of the section, we quickly recall the main theorems from the Morse theory and the method of stationary phase that will be used in the proof.

2.1 Preliminaries

An introduction to differential topology and the Morse theory can be found, for example, in [8]. We recall the basic facts needed in this paper. For a manifold M, we say that a function $f:M\rightarrow \mathbb {R}$ has nondegenerate critical point at $p\in M$ if $\nabla (f\circ \psi ^{-1}) (\psi (p))=0$ and $(H(f\circ \psi ^{-1}))(\psi (p))$ is a regular matrix for some local chart $\psi$ at p. Direct calculation shows that the definition is independent of a chosen chart around a critical point. A function $f\in C^2(S^{d-1},\mathbb {R})$ is called a Morse function if all critical points of f are nondegenerate. The Whitney topology on $C^{r}(S^{d-1},\mathbb {R})$ for $r\in \mathbb {N}\cup \{\infty \}$ is defined in the following way. First, for $r\in \mathbb {N}$, $f\in C^{r}(S^{d-1},\mathbb {R})$, chart $(\psi , U)$, $\varepsilon > 0$, and a compact set $K\subset S^{d-1}$, we define a subbasic neighbourhood ${\mathcal {N}}^r(f;(\psi , U), K, \varepsilon )$ as the set of all functions $g\in C^r(S^{d-1},\mathbb {R})$ such that for all $k=1,2,\dots , r$, the following holds:

$$\begin{aligned} \sup _{x\in \psi (K)} |{D^k(f\circ \psi ^{-1})(x) - D^k(g\circ \psi ^{-1})(x)}| <\varepsilon , \end{aligned}$$

where $D^k$ denotes the operator of k-th derivative and the absolute value sign denotes a norm on that space. The Whitney topology on $C^{r}(S^{d-1},\mathbb {R})$ is then the topology generated by the family of all subbasic neighbourhoods, meaning that the neighbourhood of $f\in C^{r}(S^{d-1},\mathbb {R})$ is any set containing a finite intersection of the given collection of sets. The Whitney topology on $C^{\infty }(S^{d-1},\mathbb {R})$ is the union of the topologies induced by the inclusion map $C^{\infty }(S^{d-1},\mathbb {R})\rightarrow C^{r}(S^{d-1},\mathbb {R})$. For more details on the Whitney topology, see [8, §II.1]. We will use a known theorem that the set of Morse functions in $C^{\infty }(S^{d-1},\mathbb {R})$ is a dense open set in the Whitney topology on $C^{\infty }(S^{d-1}, \mathbb {R})$. For the reference, see [8, §VI, Theorem 1.2].

One can find a thorough introduction to oscillatory integrals in [12, §8]. Definition of nondegeneracy of a critical point in the case $M=\mathbb {R}^d$ reduces to the statement that $f:\mathbb {R}^d\rightarrow \mathbb {R}$ has a nondegenerate critical point at $p\in \mathbb {R}^d$ if $\nabla f(p)=0$ and $\det Hf(p)\ne 0$.

We will need the following theorem that follows by combining the method of stationary phase [12, §8, Proposition 6] around a critical point with the method of nonstationary phase [12, §8, Proposition 4] away from it.

Theorem 4

Let $\psi \in C^{\infty }_c(\mathbb {R}^d,\mathbb {R})$ and let $\Phi \in C^2(\mathbb {R}^d,\mathbb {R})$ be a function that has a unique critical point on the support of the function $\psi$, call it $\xi _0$. If $\xi _0$ is a nondegenerate critical point of $\Phi$, the following holds:

$$\begin{aligned} \int \limits _{\mathbb {R}^d}e^{it \Phi (\xi )}\psi (\xi ) d\xi = Ce^{it\Phi (\xi _0)}t^{-\frac{d}{2}} + O(t^{-\frac{d}{2}-1}),\quad t\rightarrow \infty , \end{aligned}$$

where $C =\psi (\xi _0)(2\pi )^{\frac{d}{2}}e^{\frac{i\pi }{4}{\text {sgn}}(H\Phi (\xi _0))}|\det H\Phi (\xi _0)|^{-\frac{1}{2}}$ and ${\text {sgn}}(H\Phi (\xi _0))$ denotes the number of positive eigenvalues minus the number of negative eigenvalues of the matrix $H\Phi (\xi _0)$.

3 Proofs

Theorem 2 will essentially follow from Theorem 3, but before we proceed to the proof of Theorem 3, we prove the following technical lemma that is crucial for the proof of both theorems. It proves the existence of a large set U of x’s for which the modified phase in (3.2) below is stationary and nondegenerate. To help visualize the situation, we note that the set U will be an open set in the vicinity of $(-\nabla \phi (0),0)$ whose projection to the first $d-1$ coordinates does not contain 0.

Lemma 1

Let $d\in \mathbb {N}$, $d\ge 2$, and $\phi \in C^2(\mathbb {R}^{d-1},\mathbb {R})$. Suppose that either $H\phi (0) > 0$ or $d=2$ and $\phi '(0)\ne 0$. For $x,\xi \in \mathbb {R}^d$, define

$$\begin{aligned} \Phi _x(\xi ):=\phi (\xi _{-}) + \xi _d(\langle \xi _{-},x_{-} \rangle +x_d). \end{aligned}$$

(3.1)

There exist an open set $U\in \mathbb {R}^d$ and an open set $V\subset \mathbb {R}^{d-1}\times (\frac{1}{4},4)$ for which there is a unique function $g:U\rightarrow V$ such that for all $x\in U$, it holds $\nabla _{\xi } \Phi _x(g(x)) = 0$ and the matrix $H_{\xi }\Phi _x(g(x))$ is regular.

Proof

Suppose first that $H\phi (0)>0$. Define $F:\mathbb {R}^{2d}\rightarrow \mathbb {R}^d$ with

$$\begin{aligned} F(\xi ,x)= \nabla _\xi \Phi _x(\xi ) = \begin{bmatrix} \nabla \phi (\xi _{-}) +\xi _dx_{-}^{\top }&\xi _1x_1+\dots +\xi _{d-1}x_{d-1}+x_d \end{bmatrix}. \end{aligned}$$

We want to apply the implicit function theorem to prove the existence of a function g such that $F(g(x),x)=0$. First observe that

$$\begin{aligned} \nabla _\xi F(\xi ,x) = \begin{bmatrix} H\phi (\xi _{-}) &{}\quad x_{-}\\ x_{-}^{\top } &{}\quad 0 \end{bmatrix}. \end{aligned}$$

Applying determinant to both sides of the block-matrix identity

$$\begin{aligned} \begin{bmatrix} I_{d-1} &{}\quad 0\\ -x_{-}^{\top }(H\phi (\xi _{-}))^{-1} &{}\quad 1 \end{bmatrix} \begin{bmatrix} H\phi (\xi _{-}) &{}\quad x_{-}\\ x_{-}^{\top } &{}\quad 0 \end{bmatrix}= \begin{bmatrix} H\phi (\xi _{-}) &{}\quad x_{-}\\ 0 &{}\quad -x_{-}^{\top }(H\phi (\xi _{-}))^{-1}x_{-} \end{bmatrix}, \end{aligned}$$

it follows that

$$\begin{aligned} \det \nabla _\xi F(\xi ,x) = -\langle (H\phi (\xi _{-}))^{-1}x_{-},x_{-}\rangle \det H\phi (\xi _{-}). \end{aligned}$$

Since $H\phi (0) > 0$, there exists $\varepsilon >0$ such that $H\phi (\xi _{-})>0$ for all $\xi _{-}\in B_{d-1}(0,\varepsilon )$. Furthermore, since the inverse of a positive definite matrix is positive definite, we conclude that $\nabla _\xi F(\xi ,x)$ is regular whenever $\xi _{-}\in B_{d-1}(0,\varepsilon )$ and $x_{-}\ne 0$.

In order to apply the implicit function theorem, we need to show that there exist $\xi ^0\in B_d(0,\varepsilon )\times (\frac{1}{4},4)$ and $x^0\in \mathbb {R}^d$ with $x^0_{-}\ne 0$ such that $F(\xi ^0,x^0)=0$.

Since $H\phi (0)$ is regular, because of the inverse function theorem, there exist an open set $A\subset B_{d-1}(0,\varepsilon )$ and an open set $B\subset \mathbb {R}^{d-1}$ such that $\nabla \phi : A\rightarrow B$ is bijective. Taking $x^0_{-}\in (-B){\setminus }\{0\}$ and $\xi ^0_{-}$ such that $\nabla \phi (\xi ^0_{-})=-x^0_{-}$ and defining

$$\begin{aligned} \xi ^0 = (\xi ^0_{-},1), \quad x^0 = \left( x^0_{-}, -\langle \xi _{-}^{0},x^{0}_{-}\rangle \right) , \end{aligned}$$

we can see that $F(\xi ^0,x^0)=0$.

Therefore, the implicit function theorem implies the existence of open sets $U'\ni x^0$ and $V\subset B(0,\varepsilon )\times (\frac{1}{4},4)$ (the second inclusion follows from the fact that $\xi ^0_d=1$ by shrinking $U'$ if necessary) and a unique function $g:U'\rightarrow V$ such that $F(g(x),x)=0$ for every $x\in U'$. If we choose U to be a subset of $U'$ such that all $x\in U$ satisfy $x_{-}\ne 0$, the regularity of $H_\xi \Phi _x$ follows from the fact that $H_{\xi }\Phi _y(\xi ) = \nabla _{\xi } F(\xi ,x)$ and the previous conclusion of regularity of $\nabla _{\xi } F(\xi ,x)$.

When $d=2$ and $\phi '(0)\ne 0$, solving

$$\begin{aligned} F(\xi , x)= \begin{bmatrix} \phi '(\xi _1)+\xi _2x_1&\xi _1x_1+x_2 \end{bmatrix} = 0, \end{aligned}$$

one can see that the unique function $g:\mathbb {R}^2{\setminus }(\{0\}\times \mathbb {R})\rightarrow \mathbb {R}^2$ for which $F(g(x),x)=0$ is given by

$$\begin{aligned} g(x_1,x_2) = \left( -\frac{x_2}{x_1}, -\frac{\phi '(-\frac{x_2}{x_1})}{x_1}\right) . \end{aligned}$$

Also, observe that the matrix

$$\begin{aligned}\nabla _{\xi }F(\xi ,x) = \begin{bmatrix} \phi ''(\xi _1) &{}\quad x_1\\ x_1 &{}\quad 0 \end{bmatrix} \end{aligned}$$

is regular whenever $x_1\ne 0$, regardless of $\phi ''(\xi _1)$. However, to satisfy the assumption that $V\subset \mathbb {R}^{d-1}\times (\frac{1}{4},4)$, one has to restrict x to a smaller set. Without loss of generality, we may assume that $c:=\phi '(0) > 0$. By continuity, there exists $\delta >0$ such that $\phi '(\xi _1)\in (\frac{c}{2},2c)$ for all $\xi _1\in (-\delta , \delta )$. Therefore, if $x_1\in (-2c, -\frac{c}{2})$ and $x_2\in (-\frac{c\delta }{2}, \frac{c\delta }{2})$, then one has $-\frac{x_2}{x_1}\in (-\delta , \delta )$ and $\frac{-\phi '(-\frac{x_2}{x_1})}{x_1}\in (\frac{1}{4},4)$, giving the proof of the theorem with $U=(-2c, -\frac{c}{2})\times (-\frac{c\delta }{2}, \frac{c\delta }{2})$ and $V=\mathbb {R}\times (\frac{1}{4},4)$. $\square$

Proof of Theorem 3

Using duality of $L^p$ spaces and the fact that

$$\begin{aligned} \langle T_{\Phi }^t u,v \rangle = \langle u, T^{-t}_{\Phi }v\rangle = \langle T_{\Phi }^t \tilde{v},\tilde{u} \rangle , \end{aligned}$$

where $\tilde{u}(x):=\overline{u(-x)}$, we can, without loss of generality, assume that $t\ge 0$ and $p\in (1,2]$.

By composing the function $\Phi$ with the rotation, if necessary, we can assume that the function $\Phi \vert _{S^{d-1}}$ has a local minimum at $e_d:=(0,\dots , 0,1)\in \mathbb {R}^d$. From the fact that it is nondegenerate, we know that the function $\phi :\mathbb {R}^{d-1}\rightarrow \mathbb {R}$ defined as restriction of $\Phi$ to the hyperplane $\langle \xi ,e_d \rangle = 1$,

$$\begin{aligned} \phi (\xi _1,\dots , \xi _{d-1}):=\Phi (\xi _1,\dots , \xi _{d-1},1), \end{aligned}$$

satisfies $H\phi (0) >0$. Indeed, observing that

$$\begin{aligned} \phi (\xi _{-})= \Phi \left( \xi _{-}, 1 \right) =\Phi \vert _{S^{d-1}}\left( \frac{\xi _{-}}{ \sqrt{1+|\xi _{-}|^2}}, \frac{1}{\sqrt{1+|\xi _{-}|^2}} \right) , \end{aligned}$$

the statement follows by the definition of nondegeneracy of $\Phi \vert _{S^{d-1}}$ and the fact that the function $\xi _{-}\mapsto \frac{1}{\sqrt{1+|\xi _{-}|^2}}(\xi _{-},1)$ is the inverse of the chart of $S^{d-1}$ at $e_d$ given as $\xi \mapsto \frac{1}{\xi _d}\xi _{-}$.

To reduce the integral to the correct form for application of the method of stationary phase, observe that

$$\begin{aligned} T_{\Phi }^tf(tx) =\int \limits _{\mathbb {R}^d}e^{it(\Phi (\xi )+i\xi \cdot x)}{\widehat{f}}(\xi )d\xi . \end{aligned}$$

Furthermore, observe that for any $\xi \in \mathbb {R}^{d-1}\times (0,\infty )$, one has $\Phi (\xi ) = \phi (\frac{1}{\xi _d}\xi _{-})$. Since the function $\Lambda (\xi ):=\xi _d(\xi _{-},1)$ is a $C^{\infty }$ diffeomorphism from $\mathbb {R}^{d-1}\times \mathbb {R}_{+}$ onto itself, for any f such that ${\text {supp}}{\widehat{f}} \subset \mathbb {R}^{d-1}\times \mathbb {R}_{+}$, the change of variables $\xi = \Lambda (\xi ')$ gives

$$\begin{aligned} T_{\Phi }^tf(tx) = \int \limits _{\mathbb {R}^d} e^{it(\phi (\xi _{-}) + \xi _d(\langle \xi _{-},x_{-} \rangle +x_d))}{\widehat{f}}(\xi _d\xi _{-},\xi _d)\xi _d^{d-1}d\xi . \end{aligned}$$

(3.2)

Let U,V and g be as in Lemma 1. Since $\Lambda$ is a $C^{\infty }$ diffeomorphism on V and V is open, there exist $\xi _0\in \Lambda (g(U))$ and a ball $B_d(\xi _0,\varepsilon )\subset \Lambda (V)$. We choose a function $f\in {\mathcal {S}}(\mathbb {R}^d)$ such that ${\text {supp}}{\widehat{f}}\subset B_d(\xi _0,\varepsilon )$ and ${\widehat{f}}(\xi )=1$ for $\xi \in B_d(\xi _0,\frac{\varepsilon }{2})$. Denoting $F(\xi ):={\widehat{f}}(\Lambda (\xi ))\xi _d^{d-1}$ and

$$\begin{aligned} U_1=\{x\in U: \Lambda (g(x))\in B_d(\xi _0,\frac{\varepsilon }{2})\}, \end{aligned}$$

the continuity of $\Lambda \circ g$ and the fact that $\xi _d\sim 1$ on V imply that $U_1$ is an open set such that $|F\circ g\vert _{U_1}|\gtrsim 1$. Denoting $\Phi _x(\xi )$ as in (3.1), the existence and uniqueness of the function g in Lemma 1 imply that for any $x\in U$, the function $\Phi _x$ has a unique stationary point in $V\supset {\text {supp}}F$. Therefore, from (3.2), Theorem 4, and the lower bound for $F\circ g$ on $U_1$, we have

$$\begin{aligned} \begin{aligned} \int \limits _{\mathbb {R}^d} |T_{\Phi }^tf(tx)|^p dx&\ge \int \limits _{U}|T_{\Phi }^tf(tx)|^p dx\\&= \int \limits _{U} \left| \,\,{\int \limits _{\mathbb {R}^d} e^{it\Phi _x(\xi )}F(\xi )d\xi }\right| ^p dx\\&= \int \limits _{U} \left| {t^{-\frac{d}{2}}(2\pi )^{\frac{d}{2}}F(g(x))|\det H\Phi (g(x))|^{-\frac{1}{2}} + O_x(t^{-\frac{d}{2}-1})}\right| ^p dx\\&\gtrsim _{d,p} t^{-\frac{dp}{2}}\int \limits _{U_1}\left| {|\det H\Phi (g(x))|^{-\frac{1}{2}} + O_x(t^{-1})}\right| ^p dx \end{aligned} \end{aligned}$$

(3.3)

Taking the p-th root and applying Fatou’s lemma, we have

$$\begin{aligned} \liminf _{t\rightarrow \infty } \frac{|T_{\Phi }^tf|_{L^p}}{t^{\frac{d}{p}-\frac{d}{2}}}&= \liminf _{t\rightarrow \infty } \frac{|T_{\Phi }^tf(t\cdot )|_{L^p}}{t^{-\frac{d}{2}}} \\&\gtrsim _{d,p} \liminf _{t\rightarrow \infty } \left( \,\, \int \limits _{U_1}\left| {|\det H\Phi (g(x))|^{- \frac{1}{2}} + O_x(t^{-1})}\right| ^p dx\right) \\&\gtrsim _{d,p} \left( \,\, \int \limits _{U_1}\left| {|\det H\Phi (g(x))|^{-\frac{1}{2}}}\right| ^p dx \right) ^{\frac{1}{p}}\\&\gtrsim _{d,p,\Phi } 1. \end{aligned}$$

Since f was fixed, one has $|f|_{L^p}\sim _p 1$, so the calculation implies that

$$\begin{aligned} \liminf _{t\rightarrow \infty }\frac{|T_{\Phi }^t|_{L^p\rightarrow L^p}}{t^{\frac{d}{p}-\frac{d}{2}}}\ge \liminf _{t\rightarrow \infty }\frac{|T_{\Phi }^tf|_{L^p}}{|f|_{L^p} t^{\frac{d}{p}-\frac{d}{2}}}\gtrsim _{d,p,\Phi } 1, \end{aligned}$$

giving the proof of the theorem for $t\ge M$, where $M\in \mathbb {R}_{+}$ is an absolute constant.

The case $t\in [0,M]$ can be proved using soft methods. Fix any nonzero function $f\in {\mathcal {S}}(\mathbb {R}^d)$. The fact that the operator $T_{\Phi }^t$ is a unitary operator on $L^2$ for any $t\in \mathbb {R}$ implies that $T_{\Phi }^tf$ is not the zero function for any $t\in \mathbb {R}$. Therefore, for all $t\in \mathbb {R}$, one has $|T_{\Phi }^tf|_{L^p}>0$. Furthermore, for any fixed $x\in \mathbb {R}^d$ and $t_0\in \mathbb {R}$, from (1.1) and Lebesgue’s dominated convergence theorem, we have

$$\begin{aligned} \lim _{t\rightarrow t_0} T_{\Phi }^tf(x) = T_{\Phi }^{t_0}f(x). \end{aligned}$$

Fatou’s lemma then implies that

$$\begin{aligned} \liminf _{t\rightarrow t_0} |T_{\Phi }^tf|_{L^p} \ge |T_{\Phi }^{t_0}f|_{L^p}, \end{aligned}$$

meaning that the function $t\mapsto |T_{\Phi }^tf|_{L^p}$ is lower semicontinuous. As a lower semicontinuous function, it attains a minimum on the compact interval [0, M] and it must be positive by the previous observation. Finally, using the fact that $\left\langle t \right\rangle ^{\frac{d}{p}-\frac{d}{2}} \sim _{M} 1$ for $t\in [0,M]$, one gets the required lower bound in the range [0, M], giving the proof of the theorem. $\square$

Finally, we prove Theorem 2.

Proof of Theorem 2

Let us prove part (a). Since $\Phi \vert _{S^{d-1}}$ is not constant, there exist a point $\xi _0\in S^{d-1}$ and a chart $\psi$ at $\xi _0$ for which $\nabla (\Phi \vert _{S^{d-1}}\circ \psi ^{-1}) (\psi (\xi _0))\ne 0$. By composing the function $\Phi$ with rotation, if necessary, we can assume that $\xi _0= e_2$ implying that $\partial _1\Phi (e_2)\ne 0$. Defining $\phi$ as in the proof of Theorem 3, Lemma 1 gives the same conclusion needed to repeat the proof of Theorem 3 verbatim, thus proving part (a).

We continue with proof of part (b). Let $\phi :S^{d-1}\rightarrow \mathbb {R}$ be any Morse function on the sphere. Since the sphere is a compact set, it has a minimum and the fact that the function is Morse implies that the minimum is nondegenerate. Applying Theorem 3 to the function $\Phi (\xi ):=\phi (\frac{\xi }{|\xi |})$, we can see that all Morse functions satisfy the required asymptotics. From [8, §VI, Theorem 1.2], we know that the set of Morse functions is an open dense set in the standard topology on $C^{\infty }(S^{d-1},\mathbb {R})$, so the statement follows. $\square$

4 Closing remarks

Remark 1

The following question remains open and it could be interesting.

When $d\ge 3$, does there exist a nonconstant unimodular homogeneous function $\Phi \in C^{\infty }(\mathbb {R}^d{\setminus }\{0\},\mathbb {R})$ of degree 0 such that

$$\begin{aligned} \liminf _{t\rightarrow \infty } \frac{|T_\Phi ^t|_{L^p\rightarrow L^p}}{t^{d|\frac{1}{2}-\frac{1}{p}|}}=0 ? \end{aligned}$$

Remark 2

When $d=2$, one can modify the approach from the proof of [1, Theorem 3] to prove the sharp estimate both in p and t for all nonconstant 0-homogeneous unimodular functions $\Phi \in C^{\infty }(S^1,\mathbb {R})$, but it is not clear how to extend that approach to $d \ge 3$ nor it is apparent how to modify the current approach to imply the sharpness of the estimate both in p and t.