Blow-up and lifespan estimate for generalized Tricomi equations related to Glassey conjecture

Abstract

We study in this paper the small data Cauchy problem for the semilinear generalized Tricomi equations with a nonlinear term of derivative type

$$\begin{aligned} u_{tt}-t^{2m}\Delta u=|u_t|^p \end{aligned}$$

for $m\ge 0$. Blow-up result and lifespan estimate from above are established for $1<p\le 1+\frac{2}{(m+1)(n-1)-m}$. If $m=0$, our results coincide with those of the semilinear wave equation. The novelty consists in the construction of a new test function, by combining cut-off functions, the modified Bessel function of second kind and a (generalized) eigenfunction of the Laplacian. Interestingly, if $n=2$ the blow-up power is independent of m. We also furnish a local existence result, which implies the optimality of lifespan estimate at least in the 1-dimensional case.

Appendices

Appendices

A Some formulas for the modified Bessel functions

For the reader’s convenience, here we collect some formulas from Section 9.6 and Section 9.7 of the handbook by Abramowitz and Stegun [1].

• The solutions to the differential equation

  $$\begin{aligned} z^2 \frac{d^2}{dz^2} w(z) + z \frac{d}{dz} w(z) - \left( z^2 + \nu ^2 \right) w(z) = 0 \end{aligned}$$

  are the modified Bessel functions $I_{\pm \nu }(z)$ and $K_{\nu }(z)$. $I_\nu (z)$ and $K_\nu (z)$ are real and positive when $\nu >-1$ and $z>0$.

• Relations between solutions:

  $$\begin{aligned} K_\nu (z) = K_{-\nu }(z) = \frac{\pi }{2} \frac{I_{-\nu }(z)-I_{\nu }(z)}{\sin (\nu \pi )}. \end{aligned}$$

  (A. 1)

  When $\nu \in \mathbb {Z}$, the right hand-side of this equation is replaced by its limiting value.

• Recurrence relations:

  $$\begin{aligned} \partial _z I_{\nu }(z)&= I_{\nu +1}(z) + \frac{\nu }{z} I_{\nu }(z),&\partial _z K_{\nu }(z)&= -K_{\nu +1}(z) + \frac{\nu }{z} K_{\nu }(z), \end{aligned}$$

  (A. 2)
  $$\begin{aligned} \partial _z I_{\nu }(z)&= I_{\nu -1}(z) - \frac{\nu }{z} I_{\nu }(z),&\partial _z K_{\nu }(z)&= -K_{\nu -1}(z) - \frac{\nu }{z} K_{\nu }(z). \end{aligned}$$

  (A. 3)

• Limiting forms for fixed $\nu$ and $z \rightarrow 0$:

  $$\begin{aligned} I_{\nu }(z)&\sim \frac{1}{\Gamma (\nu +1)} \left( \frac{z}{2}\right) ^{\nu }&\quad&\text {for } \nu \ne -1,-2,\dots \end{aligned}$$

  (A. 4)
  $$\begin{aligned} K_0(z)&\sim -\ln (z),&\quad&\end{aligned}$$

  (A. 5)
  $$\begin{aligned} K_\nu (z)&\sim \frac{\Gamma (\nu )}{2} \left( \frac{z}{2}\right) ^{-\nu },&\quad&\text {for } \Re \nu >0. \end{aligned}$$

  (A. 6)

• Asymptotic expansions for fixed $\nu$ and large arguments:

  $$\begin{aligned}&I_\nu (z) = \frac{1}{\sqrt{2\pi }} z^{-1/2} e^z \times (1+O(z^{-1})),&{\text {for } |z| \text { large and } |\arg z| < \frac{\pi }{2},} \end{aligned}$$

  (A. 7)
  $$\begin{aligned}&K_\nu (z) = \sqrt{\frac{\pi }{2}} z^{-1/2} e^{-z} \times (1+O(z^{-1})),&\text {for } |z| \text { large and } |\arg z| < \frac{3}{2}\pi . \end{aligned}$$

  (A. 8)

B Proof of Theorem 3

In this appendix we prove the $L^p-L^q$ estimates on the conjugate line for $W_1(t,s,D_x)$, $W_2(t,s,D_x)$ and their derivatives respect to time collected in Theorem 3. The argument is adapted from the proof of Theorem 3.3 by Yagdjian [47] (see also [33] and [8,  Chapter 16]), where similar estimates for $V_1(t,D_x)$ and $V_2(t,D_x)$ are presented. Note that in [47] the additional hypothesis $\sigma \ge 0$ is supposed, but this can be dropped, as we will show.

Before to proceed, we recall the following key lemmata.

Definition 2

Denote by $L^q_p \equiv L^q_p(\mathbb {R}^n)$ the space of tempered distributions T such that

$$\begin{aligned} \left\Vert T * f\right\Vert _{L^q} \le C \left\Vert f\right\Vert _{L^p} \end{aligned}$$

for all Schwartz functions $f\in \mathcal {S}(\mathbb {R}^n)$ and a suitable positive constant C independent of f.

Denote instead with $M^q_p \equiv M^q_p(\mathbb {R}^n)$ the set of multiplier of type (p, q), i.e. the set of Fourier transforms $\mathcal {F}(T)$ of distributions $T \in L^q_p$.

Lemma 4

[19,  Theorem 1.11] Let f be a measurable function such that for all positive ${\lambda }$, we have

$$\begin{aligned} \mathop {\mathrm {meas}}\limits \{\xi \in \mathbb {R}^n :|f(\xi )| \le {\lambda }\} \le C {\lambda }^{-b} \end{aligned}$$

for some suitable $b \in (1,\infty )$ and positive C. Then, $f \in M^q_p$ if $1<p\le 2\le q < \infty$ and $1/p-1/q=1/b$.

Lemma 5

[6,  Lemma 2] Fix a nonnegative smooth function $\chi \in C^\infty _0([0,\infty ))$ with compact support $\mathop {\mathrm {supp}}\limits \chi \subset \{x \in \mathbb {R}^n :1/2 \le |x| \le 2 \}$ such that $\sum _{k=-\infty }^{\infty } \chi (2^{-k} x)=1$ for $x\ne 0$. Set $\chi _k(x) := \chi (2^{-k}x)$ for $k\ge 1$ and $\chi _0(x) := 1 - \sum _{k=1}^{\infty } \chi _k(x)$, so that $\mathop {\mathrm {supp}}\limits \chi _0 \subset \{x \in \mathbb {R}^n :|x| \le 2 \}$.

Let $a\in L^\infty (\mathbb {R}^n)$, $1<p\le 2$ and assume that

$$\begin{aligned} \left\Vert \mathcal {F}^{-1}(a \chi _k {\widehat{v}})\right\Vert _{L^{p'}} \le C \left\Vert v\right\Vert _{L^p} \quad \text { for } k\ge 0. \end{aligned}$$

Then for some constant A independent of a we have

$$\begin{aligned} \left\Vert \mathcal {F}^{-1}(a {\widehat{v}})\right\Vert _{L^{p'}} \le AC \left\Vert v\right\Vert _{L^{p}}. \end{aligned}$$

Lemma 6

(Littman type Lemma, see Lemma 4 in [6]) Let P be a real function, smooth in a neighbourhood of the support of $v \in C_0^\infty (\mathbb {R}^n)$. Assume that the rank of the Hessian matrix $(\partial ^2_{\eta _j \eta _k} P(\eta ))_{j,k\in \{1,\dots ,n\}}$ is at least $\rho$ on the support of v. Then for some integer N the following estimate holds:

$$\begin{aligned} \left\Vert \mathcal {F}^{-1}(e^{itP(\eta )}v(\eta ))\right\Vert _{L^\infty } \le C (1+|t|)^{-\rho /2} \sum _{|\alpha | \le N} \left\Vert \partial _\eta ^\alpha v\right\Vert _{L^1}. \end{aligned}$$

We will prove now only estimate (iii) of Theorem 3, since the computation for estimates (i) and (ii) are completely analogous; about estimate (iv), we will sketch the proof since it could be strange to the reader that this is the only case where the range of $\sigma$ collapses to be only a value.

First of all, let us set $\tau := t/s \ge 1$, $z=2i\phi (t)\xi$, $\zeta =2i\phi (s)\xi$ and let us introduce the smooth functions $X_0, X_1, X_2 \in C^\infty (\mathbb {R}^n;[0,1])$ satisfying

$$\begin{aligned} X_0(x)&= {\left\{ \begin{array}{ll} 1 &{} \text { for } |x| \le 1/2, \\ 0 &{} \text { for } |x| \ge 3/4 , \end{array}\right. } \\ X_2(x)&= {\left\{ \begin{array}{ll} 1 &{} \text { for } |x| \ge 1, \\ 0 &{} \text { for } |x| \le 3/4, \end{array}\right. } \\ X_1(x)&= 1- X_0(\tau ^{m+1} x) - X_2(x). \end{aligned}$$

In particular, observe that

$$\begin{aligned} X_0(\phi (t)\xi ) + X_1(\phi (s)\xi ) + X_2(\phi (s)\xi ) \equiv 1 \end{aligned}$$

for $0<s\le t$ and $\xi \in \mathbb {R}^n$.

By relations (3.3) and (3.4), it is straightforward to get

$$\begin{aligned} \begin{aligned} \partial _t V_1(t,|\xi |) =&\, \frac{m+1}{2} t^{-1} z e^{-z/2} [\Phi (\mu +1,2\mu +1;z)-\Phi (\mu ,2\mu ;z)]\\ \partial _t V_2(t,|\xi |) =&\, e^{-z/2} \left[ \Phi (1-\mu ,1-2\mu ;z) - \frac{m+1}{2} z \Phi (1-\mu ,2(1-\mu );z)\right] . \end{aligned} \end{aligned}$$

Thus one can check, using identity (3.6), that

$$\begin{aligned} \partial _t W_1(t,s,|\xi |)&=\, i s t^{m} e^{-(z+\zeta )/2} |\xi | [\Phi (\mu +1,2\mu +1;z)-\Phi (\mu ,2\mu ;z)] \Phi (1-\mu ,2(1-\mu );\zeta ) \\&=\, i s t^{m} e^{-\zeta /2} |\xi | [e^{z/2} {H}^0_+(z) + e^{-z/2} {H}^0_-(z) +] \Phi (1-\mu ,2(1-\mu );\zeta ) \\&=\, i s t^{m} |\xi | \left[ e^{[1+\tau ^{m+1}]\zeta /2} {H}^0_+(z) {H}^1_+(\zeta ) + e^{-[1-\tau ^{m+1}]\zeta /2} {H}^0_+(z) {H}^1_-(\zeta ) \right. \\&\quad \left. + e^{[1-\tau ^{m+1}]\zeta /2} {H}^0_-(z) {H}^1_+(\zeta ) + e^{-[1+\tau ^{m+1}]\zeta /2} {H}^0_-(z) {H}^1_-(\zeta ) \right] \\ \partial _t W_2(t,s,|\xi |)&=\, e^{-(z+\zeta )/2} \Phi (\mu ,2\mu ;\zeta )\left[ \Phi (1-\mu ,1-2\mu ;z)\right. \\&\quad \left. - i t^{m+1} |\xi | \Phi (1-\mu ,2(1-\mu );z)\right] \\&=\, e^{-\zeta /2} \Phi (\mu ,2\mu ;\zeta ) [e^{z/2} H^2_+(z) + e^{-z/2} H^2_-(z)] \\&=\, e^{[1+\tau ^{m+1}]\zeta /2} {H}^2_+(z) {H}^3_+(\zeta ) + e^{-[1-\tau ^{m+1}]\zeta /2} {H}^2_+(z) {H}^3_-(\zeta ) \\&\quad + e^{[1-\tau ^{m+1}]\zeta /2} {H}^2_-(z) {H}^3_+(\zeta ) + e^{-[1+\tau ^{m+1}]\zeta /2} {H}^2_-(z) {H}^3_-(\zeta ) \end{aligned}$$

where for the simplicity we set

$$\begin{aligned} {H}^0_\pm (z)&:=\, \frac{\Gamma (2\mu +1)}{\Gamma \left( \mu +\frac{1}{2} \pm \frac{1}{2} \right) } H_\pm (\mu +1,2\mu +1;z) - \frac{\Gamma (2\mu )}{\Gamma (\mu )} H_\pm (\mu ,2\mu ;z) , \\ {H}^1_{\pm }(\zeta )&:=\, \frac{\Gamma (2(1-\mu ))}{\Gamma (1-\mu )} {H}_{\pm }(1-\mu ,2(1-\mu );\zeta ), \end{aligned}$$

and

$$\begin{aligned} H^2_\pm (z)&:=\, \frac{\Gamma (1-2\mu )}{\Gamma \left( \frac{1}{2}\pm \frac{1}{2}-\mu \right) } H_\pm (1-\mu ,1-2\mu ;z) \\&\quad -i t^{m+1} |\xi | \frac{\Gamma (2(1-\mu ))}{\Gamma (1-\mu )} H_\pm (1-\mu ,2(1-\mu );z) , \\ H^3_\pm (\zeta )&:=\, \frac{\Gamma (2\mu )}{\Gamma (\mu )} H_\pm (\mu ,2\mu ;\zeta ). \end{aligned}$$

1.1 Estimates at low frequencies for $\partial _t W_1(t,s,D_x)$

Let us consider the Fourier multiplier

$$\begin{aligned} \mathcal {F}^{-1}_{\xi \rightarrow x}\left( X_0(\phi (t)\xi ) |\xi |^{-\sigma } \partial _t W_1(t,s,|\xi |) {\widehat{\psi }}\right) . \end{aligned}$$

By the change of variables $\eta := \phi (t)\xi$ and $x:=\phi (t)y$ we get

$$\begin{aligned}&\left\Vert F^{-1}_{\xi \rightarrow x} \left( X_0(\phi (t)\xi ) |\xi |^{-\sigma } \partial _t W_1(t,s,|\xi |) {\widehat{\psi }} \right) \right\Vert _{L^{q'}} \\&\quad \lesssim \tau ^{-1} t^{(n/q'-n+\sigma )(m+1)} \left\Vert T_0 * \mathcal {F}^{-1}_{\eta \rightarrow y}\left( {\widehat{\psi }}(\eta /\phi (t))\right) \right\Vert _{L^{q'}} \end{aligned}$$

where

$$\begin{aligned} T_0&:=\, \mathcal {F}^{-1}_{\eta \rightarrow y} \left( X_0(\eta ) |\eta |^{1-\sigma } e^{-i[1+1/\tau ^{m+1}]|\eta |} {\Phi }_0(\mu ;\tau ;|\eta |) \right) \\ {\Phi }_0(\mu ;\tau ;|\eta |)&:=\, [\Phi (\mu +1,2\mu +1;2i|\eta |)-\Phi (\mu ,2\mu ;2i|\eta |)] \\&\quad \times \Phi (1-\mu ,2(1-\mu );2i|\eta |/\tau ^{m+1}) \\&=\, O(|\eta |)[1+\tau ^{-(m+1)}O(|\eta |)]. \end{aligned}$$

The last equality above is implied by (3.2), from which we deduce $|{\Phi }_0(\mu ;\tau ;|\eta |)|\lesssim 1$ if $|\eta |\le 3/4$. So, for any ${\lambda }>0$, we obtain

$$\begin{aligned} \mathop {\mathrm {meas}}\limits \{ \eta \in \mathbb {R}^n :|\mathcal {F}_{y \rightarrow \eta }(T_0)| \ge {\lambda }\}&\le \mathop {\mathrm {meas}}\limits \{ \eta \in \mathbb {R}^n :|\eta | \le 3/4\text { and } |\eta |^{1-\sigma } > rsim {\lambda }\} \\&\lesssim {\left\{ \begin{array}{ll} 1 &{} \text { if } 0 < {\lambda }\le 1, \\ 0 &{} \text { if } {\lambda }\ge 1 \text { and } \sigma \le 1, \\ {\lambda }^{-\frac{n}{\sigma -1}} &{} \text { if } {\lambda }\ge 1 \text { and } \sigma > 1, \end{array}\right. } \\&\lesssim {\lambda }^{-b}, \end{aligned}$$

where $1<b<\infty$ if $\sigma \le 1$ and $1 <b \le \frac{n}{\sigma -1}$ if $\sigma >1$. Hence by Lemma 4, we get $T_0 \in L^{q'}_q$ for $1<q\le 2 \le q'<\infty$ and $\sigma \le 1 + n (\frac{1}{q}-\frac{1}{q'})$.

Then we obtain the Hardy–Littlewood type inequality

$$\begin{aligned} \left\Vert F^{-1}_{\xi \rightarrow x} \left( X_0(\phi (t)\xi ) |\xi |^{-\sigma } \partial _t W_1(t,s,|\xi |) {\widehat{\psi }} \right) \right\Vert _{L^{q'}} \lesssim \tau ^{-1} t^{\left[ \sigma -n \left( \frac{1}{q} - \frac{1}{q'} \right) \right] (m+1)} \left\Vert \psi \right\Vert _{L^q}. \end{aligned}$$

Observing that, by the assumption on the range of $\sigma$,

$$\begin{aligned} \tau ^{-1} t^{\left[ \sigma -n \left( \frac{1}{q} - \frac{1}{q'} \right) \right] (m+1)}&= \tau ^{-\left[ 1-\mu + n \left( \frac{1}{q} - \frac{1}{q'} \right) -\sigma \right] (m+1)} \tau ^{m/2} s^{\left[ \sigma -n \left( \frac{1}{q} - \frac{1}{q'} \right) \right] (m+1)} \\ {}&\le \tau ^{m/2} s^{\left[ \sigma -n \left( \frac{1}{q} - \frac{1}{q'} \right) \right] (m+1)}, \end{aligned}$$

we finally get

$$\begin{aligned} \left\Vert F^{-1}_{\xi \rightarrow x} \left( X_0(\phi (t)\xi ) |\xi |^{-\sigma } \partial _t W_1(t,s,|\xi |) {\widehat{\psi }} \right) \right\Vert _{L^{q'}} \lesssim \tau ^{m/2} s^{\left[ \sigma -n \left( \frac{1}{q} - \frac{1}{q'} \right) \right] (m+1)} \left\Vert \psi \right\Vert _{L^q}. \end{aligned}$$

(B. 1)

1.2 Estimates at intermediate frequencies for $\partial _t W_1(t,s,D_x)$

We proceed similarly as before. Let us consider now the Fourier multiplier

$$\begin{aligned} \mathcal {F}^{-1}_{\xi \rightarrow x}\left( X_1(\phi (s)\xi ) |\xi |^{-\sigma } \partial _t W_1(t,s,|\xi |) {\widehat{\psi }}\right) . \end{aligned}$$

Exploiting this time the change of variables $\eta := \phi (s)\xi$ and $x:=\phi (s)y$, we get

$$\begin{aligned}&\left\Vert F^{-1}_{\xi \rightarrow x} \left( X_1(\phi (s)\xi ) |\xi |^{-\sigma } \partial _t W_1(t,s,|\xi |) {\widehat{\psi }} \right) \right\Vert _{L^{q'}} \\&\quad \lesssim \tau ^{m/2} s^{(n/q'-n+\sigma )(m+1)} \left\Vert T_1 * \mathcal {F}^{-1}_{\eta \rightarrow y}\left( {\widehat{\psi }}(\eta /\phi (s))\right) \right\Vert _{L^{q'}} \end{aligned}$$

where

$$\begin{aligned} T_1&:= \mathcal {F}^{-1}_{\eta \rightarrow y} \left( X_1(\eta ) |\eta |^{1-\sigma } e^{-i[1+\tau ^{m+1}]|\eta |} {\Phi }_1(\mu ;\tau ;|\eta |) \right) \\ {\Phi }_1(\mu ;\tau ;|\eta |)&:= \tau ^{m/2} [\Phi (\mu +1,2\mu +1;2i\tau ^{m+1}|\eta |)-\Phi (\mu ,2\mu ;2i\tau ^{m+1}|\eta |)] \\&\quad \times \Phi (1-\mu ,2(1-\mu );2i|\eta |). \end{aligned}$$

Taking in account (3.2) and (3.5), we infer that

$$\begin{aligned} |\Phi _1(\mu ;\tau ;|\eta |)| \lesssim \tau ^{m/2} (\tau ^{m+1}|\eta |)^{-\mu } = |\eta |^{-\mu } \quad \text { on } \mathop {\mathrm {supp}}\limits X_1(\eta ) \subseteq \left[ (2\tau ^{m+1})^{-1},1 \right] , \end{aligned}$$

and thus, for any ${\lambda }>0$, we obtain

$$\begin{aligned} \mathop {\mathrm {meas}}\limits \{ \eta \in \mathbb {R}^n :|\mathcal {F}_{y \rightarrow \eta }(T_1)| \ge {\lambda }\}&\le \mathop {\mathrm {meas}}\limits \{ \eta \in \mathbb {R}^n :|\eta | \le 1 \text { and } |\eta |^{1-\mu -\sigma } > rsim {\lambda }\} \\&\lesssim {\left\{ \begin{array}{ll} 1 &{} \text {if } 0 < {\lambda }\le 1, \\ 0 &{} \text {if } {\lambda }\ge 1 \text { and } \sigma \le 1-\mu ,\\ {\lambda }^{-\frac{n}{\sigma -1+\mu }} &{} \text {if } {\lambda }\ge 1 \text { and } \sigma > 1-\mu , \end{array}\right. } \\&\lesssim {\lambda }^{-b}, \end{aligned}$$

where $1<b<\infty$ if $\sigma \le 1-\mu$ and $1<b \le \frac{n}{\sigma -1+\mu }$ if $\sigma >1-\mu$. Hence by Lemma 4, we get $T \in L^{q'}_q$ for $1<q\le 2 \le q'<\infty$ and $\sigma \le 1-\mu + n (\frac{1}{q}-\frac{1}{q'})$.

Then we reach

$$\begin{aligned} \left\Vert F^{-1}_{\xi \rightarrow x} \left( X_1(\phi (s)\xi ) |\xi |^{-\sigma } \partial _t W_1(t,s,|\xi |) {\widehat{\psi }} \right) \right\Vert _{L^{q'}} \lesssim \tau ^{m/2} s^{\left[ \sigma -n \left( \frac{1}{q} - \frac{1}{q'} \right) \right] (m+1)} \left\Vert \psi \right\Vert _{L^q}. \end{aligned}$$

(B. 2)

1.3 Estimates at high frequencies for $\partial _t W_1(t,s,D_x)$

Finally, we want to estimate the Fourier multiplier

$$\begin{aligned} \mathcal {F}^{-1}_{\xi \rightarrow x}\left( X_2(\phi (s)\xi ) |\xi |^{-\sigma } \partial _t W_1(t,s,|\xi |) {\widehat{\psi }}\right) . \end{aligned}$$

We choose a set of functions $\{ \chi _k \}_{k\ge 0}$ as in the statement of Lemma 5.

1.3.1 $L^1-L^\infty$ estimates

We claim that, for $k\ge 0$,

$$\begin{aligned} \left\Vert \mathcal {F}^{-1}_{\xi \rightarrow x}\left( X_2(\phi (s)\xi ) \, \chi _k(\phi (s) \xi ) \, |\xi |^{-\sigma } \partial _t W_1(t,s,|\xi |) \right) \right\Vert _{L^\infty } \lesssim 2^{k\left( n-\sigma \right) } \tau ^{m/2} s^{(\sigma -n)(m+1)}. \end{aligned}$$

(B. 3)

Exploiting the change of variables $\phi (s)\xi =2^k \eta$ and $2^k x=\phi (s)y$, by the expression of the symbol $\partial _t W_1(t,s,|\xi |)$, we obtain

$$\begin{aligned} \left\Vert \mathcal {F}^{-1}_{\xi \rightarrow x}\left( X_2(\phi (s)\xi ) \, \chi _k(\phi (s) \xi ) \, |\xi |^{-\sigma } \partial _t W_1(t,s,|\xi |) \right) \right\Vert _{L^\infty } \nonumber \\ \lesssim 2^{k(n-\sigma +1)} \tau ^m s^{(\sigma -n)(m+1)} [A^+_+ + A^+_- + A^-_+ + A^-_-] \end{aligned}$$

(B. 4)

where

$$\begin{aligned} A^+_\pm :=&\, \left\Vert \mathcal {F}^{-1}_{\eta \rightarrow y} \left( e^{i[ \pm 1 + \tau ^{m+1} ]2^k|\eta |} v^{+,\pm }_k(\eta ) \right) \right\Vert _{L^\infty }, \\ A^-_\pm :=&\, \left\Vert \mathcal {F}^{-1}_{\eta \rightarrow y} \left( e^{i[ \pm 1 - \tau ^{m+1}]2^k|\eta |} v^{-,\pm }_k(\eta ) \right) \right\Vert _{L^\infty }, \end{aligned}$$

and

$$\begin{aligned} v^{+,\pm }_k(\eta ) :=&\, X_2(2^k \eta )\chi (\eta ) |\eta |^{1-\sigma } {H}^0_+(2i\tau ^{m+1}2^{k}|\eta |) {H}^1_\pm (2i2^{k}|\eta |), \\ v^{-,\pm }_k(\eta ) :=&\, X_2(2^k \eta )\chi (\eta ) |\eta |^{1-\sigma } {H}^0_-(2i\tau ^{m+1}2^{k}|\eta |) {H}^1_\pm (2i2^{k}|\eta |). \end{aligned}$$

The functions $v^{\pm ,\pm }_k(\eta )$ are smooth and compactly supported on $\{\eta \in \mathbb {R}^n :1/2 \le |\eta | \le 2\}$.

When $k=0$, it is easy to see by estimates (3.7)–(3.8) that

$$\begin{aligned} \left\Vert \mathcal {F}^{-1}_{\eta \rightarrow y} \left( e^{i[\pm 1+\tau ^{m+1}]|\eta |} v^{+,\pm }_0(\eta ) \right) \right\Vert _{L^\infty } \le \left\Vert v^{+,\pm }_0\right\Vert _{L^1} \lesssim \tau ^{-(m+1)\mu } \left\Vert X_2(\eta )\chi (\eta )|\eta |^{-\sigma }\right\Vert _{L^1} \lesssim \tau ^{- m/2}. \end{aligned}$$

For $k\ge 1$, by Lemma 6 we have, for some integer $N>0$, that

$$\begin{aligned} \left\Vert \mathcal {F}^{-1}_{\eta \rightarrow y} \left( e^{i[\pm 1+\tau ^{m+1}]2^k|\eta |} v^{+,\pm }_k(\eta ) \right) \right\Vert _{L^\infty } \lesssim (1+[\pm 1 + \tau ^{m+1}]2^k)^{-\frac{n-1}{2}} \sum _{|\alpha |\le N} \left\Vert \partial _\eta ^\alpha v^{+,\pm }_k\right\Vert _{L^1}. \end{aligned}$$

(B. 5)

Since $X_2(2^k\eta )\chi (\eta )=\chi (\eta )$ for $k\ge 1$, by estimates (3.7)–(3.8) and Leibniz rule we infer

$$\begin{aligned} |\partial ^\alpha _\eta v^{+,\pm }_k(\eta )|&= \left|\sum _{\gamma \le \beta \le \alpha } \left( {\begin{array}{c}\alpha \\ \beta \end{array}}\right) \left( {\begin{array}{c}\beta \\ \gamma \end{array}}\right) \partial ^{\alpha -\beta }_\eta \left( \chi (\eta )|\eta |^{1-\sigma } \right) \partial ^{\beta -\gamma }_\eta {H}^0_+(2i\tau ^{m+1}2^{k}|\eta |) \partial _\eta ^\gamma {H}^1_\pm (2i2^{k}|\eta |) \right|\\&\lesssim \tau ^{-m/2} 2^{-k} \sum _{\beta \le \alpha } C_{\mu ,\alpha ,\beta } {\mathbf {1}}_{\left[ {1}/{2},2\right] }(\eta ) |\eta |^{-1-|\beta |} \end{aligned}$$

where ${\mathbf {1}}_{\left[ {1}/{2},2\right] }(\eta )=1$ for $1/2\le |\eta | \le 2$ and ${\mathbf {1}}_{\left[ {1}/{2},2\right] }(\eta )=0$ otherwise. From the latter estimate and (B. 5), we get

$$\begin{aligned} A^+_\pm \lesssim \tau ^{-m/2} 2^{-k} (1+[\pm 1+\tau ^{m+1}])^{-\frac{n-1}{2}} \le \tau ^{-m/2} 2^{-k}. \end{aligned}$$

Similarly we obtain also that $A^-_\pm \lesssim \tau ^{-m/2} 2^{-k}$. Thus, inserting in (B. 4) we obtain (B. 3), which combined with the Young inequality give us the $L^1-L^\infty$ estimate

$$\begin{aligned}&\left\Vert \mathcal {F}^{-1}_{\xi \rightarrow x}\left( X_2(\phi (s)\xi ) \, \chi _k(\phi (s) \xi ) \, |\xi |^{-\sigma } \partial _t W_1(t,s,|\xi |) {\widehat{\psi }}\right) \right\Vert _{L^\infty } \nonumber \\&\quad \lesssim 2^{k\left( n-\sigma \right) } \tau ^{m/2} s^{(\sigma -n)(m+1)} \left\Vert \psi \right\Vert _{L^1}. \end{aligned}$$

(B. 6)

1.3.2 $L^2 - L^2$ estimates

By the Plancherel formula, Hölder inequality, estimate (3.5) and the substitution $\phi (s)\xi =2^k\eta$, we obtain

$$\begin{aligned}&\left\Vert \mathcal {F}^{-1}_{\xi \rightarrow x}\left( X_2(\phi (s)\xi ) \, \chi _k(\phi (s) \xi ) \, |\xi |^{-\sigma } \partial _t W_1(t,s,|\xi |) {\widehat{\psi }}\right) \right\Vert _{L^2} \nonumber \\&\quad \le \left\Vert X_2(\phi (s)\xi ) \, \chi _k(\phi (s) \xi ) \, |\xi |^{-\sigma } \partial _t W_1(t,s,|\xi |) \right\Vert _{L^\infty } \left\Vert \psi \right\Vert _{L^2} \\&\quad \lesssim 2^{-k\sigma } \tau ^{m/2} s^{\sigma (m+1)} \left\Vert \psi \right\Vert _{L^2}.\nonumber \end{aligned}$$

(B. 7)

1.3.3 $L^q-L^{q'}$ estimates

The interpolation between (B. 6) and (B. 7) give us the estimates on the conjugate line

$$\begin{aligned}&\left\Vert \mathcal {F}^{-1}_{\xi \rightarrow x}\left( X_2(\phi (s)\xi ) \, \chi _k(\phi (s) \xi ) \, |\xi |^{-\sigma } \partial _t W_1(t,s,|\xi |) {\widehat{\psi }}\right) \right\Vert _{L^{q'}} \nonumber \\&\quad \lesssim 2^{k\left[ n \left( \frac{1}{q}-\frac{1}{q'} \right) -\sigma \right] } \tau ^{m/2} s^{\left[ \sigma - n \left( \frac{1}{q}-\frac{1}{q'}\right) \right] (m+1)} \left\Vert \psi \right\Vert _{L^q}, \end{aligned}$$

(B. 8)

where $1<q\le 2$. Now, choosing $n \left( \frac{1}{q}-\frac{1}{q'} \right) \le \sigma$, putting together (B. 1), (B. 2) and (B. 8) with an application of Lemma 5, we finally obtain the $L^q-L^{q'}$ estimate for $\partial _t W_1(t,s,D_x)$.

1.4 Estimates for $\partial _t W_2(t,s,D_x)$

For the intermediate and high frequencies cases, proceeding as above we straightforwardly obtain, under the constrains $\sigma \le \mu + n (\frac{1}{q}-\frac{1}{q'})$ and $n \left( \frac{1}{q}-\frac{1}{q'} \right) \le \sigma$ respectively, that

$$\begin{aligned} \left\Vert F^{-1}_{\xi \rightarrow x} \left( X_j(\phi (s)\xi ) |\xi |^{-\sigma } \partial _t W_2(t,s,|\xi |) {\widehat{\psi }} \right) \right\Vert _{L^{q'}} \lesssim \tau ^{m/2} s^{\left[ \sigma -n \left( \frac{1}{q} - \frac{1}{q'} \right) \right] (m+1)} \left\Vert \psi \right\Vert _{L^q}. \end{aligned}$$

(B. 9)

for $j\in \{1,2\}$ and $1<q\le 2 \le q'<\infty$.

At low frequencies, by computations similar to that for $\partial _tW_1(t,s,D_x)$, we obtain

$$\begin{aligned}&\left\Vert F^{-1}_{\xi \rightarrow x} \left( X_0(\phi (t)\xi ) |\xi |^{-\sigma } \partial _t W_2(t,s,|\xi |) {\widehat{\psi }} \right) \right\Vert _{L^{q'}} \\&\quad \lesssim t^{(n/q'-n+\sigma )(m+1)} \left\Vert T_0 * \mathcal {F}^{-1}_{\eta \rightarrow y}\left( {\widehat{\psi }}(\eta /\phi (t))\right) \right\Vert _{L^{q'}} \end{aligned}$$

where this time

$$\begin{aligned} T_0 :=&\, \mathcal {F}^{-1}_{\eta \rightarrow y} \left( X_0(\eta ) |\eta |^{-\sigma } e^{-i[1+1/\tau ^{m+1}]|\eta |} {\Phi }_0(\mu ;\tau ;|\eta |) \right) \\ {\Phi }_0(\mu ;\tau ;|\eta |) :=&\, \Phi (\mu ,2\mu ;2i|\eta |/\tau ^{m+1}) \\&\times [\Phi (1-\mu ,1-2\mu ;2i|\eta |)-i(m+1)|\eta |\Phi (1-\mu ,2(1-\mu );2i|\eta |)] \\ =&\, [1+\tau ^{-(m+1)}O(|\eta |)][1+O(|\eta |)], \end{aligned}$$

and hence again $|{\Phi }_0(\mu ;\tau ;|\eta |)|\lesssim 1$ if $|\eta |\le 3/4$. For any ${\lambda }>0$, we get

$$\begin{aligned} \mathop {\mathrm {meas}}\limits \{ \eta \in \mathbb {R}^n :|\mathcal {F}_{y \rightarrow \eta }(T_0)| \ge {\lambda }\} \le \mathop {\mathrm {meas}}\limits \{ \eta \in \mathbb {R}^n :|\eta | \le 3/4 \text { and } |\eta |^{-\sigma } > rsim {\lambda }\} \lesssim {\lambda }^{-b}, \end{aligned}$$

where $1<b<\infty$ if $\sigma \le 0$ and $1 <b \le \frac{n}{\sigma }$ if $\sigma >0$. Another application of Lemma 4 tell us that $T_0 \in L^{q'}_q$ for $1<q\le 2 \le q'<\infty$ with the condition on $\sigma$ given by $\sigma \le n (\frac{1}{q}-\frac{1}{q'})$. Finally, similarly as in the case of $\partial _t W_1(t,s,D_x)$ we conclude that (B. 9) holds true also for $j=0$.

The proof of estimates (iv) in Theorem 3 is thus reached combining (B. 9) for $j\in \{0,1,2\}$, and putting together all the constrains on the range of $\sigma$, we are forced to choose $\sigma = n (\frac{1}{q}-\frac{1}{q'})$.