Note that for any
\(\varphi\in \mathfrak{F}_{1}\) or
\(\mathfrak{F}_{2}\), there exist
\(C_{1}, C_{2}>0\) such that
\(C_{1}<\frac{\varphi(2r)}{\varphi(r)}<C_{2}\). Thus, we only prove the case
\(\varphi, \psi\in\mathfrak{F}_{1}\), and other cases are analogous. Estimate (
2.9) is obvious. By the change of variables,
$$\begin{aligned} & \bigl|\widehat{\sigma_{\mu;r,s}^{\iota,\kappa}}(\xi,\eta) \bigr|^{2} \\ &\quad= \biggl| \iint_{S^{m-1}\times S^{n-1}}\exp\bigl(-2\pi i\bigl(\xi\cdot \Gamma_{\iota}\bigl(\varphi(r)u'\bigr)\bigr)\\ &\qquad{}+\eta\cdot \Upsilon_{\kappa}\bigl(\psi(s)v'\bigr)\bigr)\Omega _{\mu}\bigl(u',v'\bigr)\,d \sigma_{m}\bigl(u'\bigr)\,d\sigma_{n} \bigl(v'\bigr) \biggr|^{2} \\ &\quad= \iint_{(S^{m-1}\times S^{n-1})^{2}} \exp\bigl(-2\pi i\bigl(\xi\cdot\bigl( \Gamma_{\iota}\bigl(\varphi(r)u'\bigr)-\Gamma_{\iota}\bigl(\varphi (r)\theta\bigr)\bigr)\bigr)\bigr) \\ &\qquad{}\times\exp\bigl(-2\pi i\eta\cdot\bigl( \Upsilon_{\kappa}\bigl(\psi(s)v'\bigr)-\Upsilon_{\kappa}\bigl(\psi (s)\omega\bigr)\bigr)\bigr) \\ &\qquad{} \times \Omega_{\mu}\bigl(u',v' \bigr)\overline{\Omega_{\mu}(\theta,\omega )}\,d\sigma_{m} \bigl(u'\bigr)\,d\sigma_{n}\bigl(v'\bigr)\,d \sigma_{m}(\theta)\,d\sigma_{n}(\omega). \end{aligned}$$
It follows that
$$\begin{aligned} & \int_{a_{\mu}^{j}}^{a_{\mu}^{j+1}} \int_{a_{\mu}^{k}}^{a_{\mu}^{k+1}} \bigl|\widehat{\sigma_{\mu;r,s}^{\iota,\kappa}}( \xi,\eta) \bigr|^{2}\frac {dr\,ds}{rs} \\ &\quad\leq \iint_{(S^{m-1}\times S^{n-1})^{2}}\bigl|H_{k,\mu}\bigl(u',\theta ,\xi \bigr)\bigr|\bigl|J_{j,\mu}\bigl(v',\omega,\eta\bigr)\bigr| \\ &\qquad{}\times\bigl| \Omega_{\mu}\bigl(u',v'\bigr)\overline{ \Omega_{\mu}(\theta,\omega)}\bigr| \,d\sigma_{m} \bigl(u'\bigr)\,d\sigma_{n}\bigl(v'\bigr)\,d \sigma_{m}(\theta)\,d\sigma_{n}(\omega), \end{aligned}$$
(2.14)
where
$$\begin{aligned}& H_{k,\mu}\bigl(u',\theta,\xi\bigr):= \int_{a_{\mu}^{k}}^{a_{\mu}^{k+1}}\exp\bigl(-2\pi i\xi \cdot\bigl( \Gamma_{\iota}\bigl(\varphi(r)u'\bigr)- \Gamma_{\iota}\bigl(\varphi(r)\theta\bigr)\bigr)\bigr)\frac{dr}{r}, \\& J_{j,\mu}\bigl(v',\omega,\eta\bigr):= \int_{a_{\mu}^{j}}^{a_{\mu}^{j+1}}\exp\bigl(-2\pi i\xi \cdot\bigl( \Upsilon_{\kappa}\bigl(\psi(s)v'\bigr)- \Upsilon_{\kappa}\bigl(\psi(s)\omega\bigr)\bigr)\bigr)\frac{ds}{s}. \end{aligned}$$
By the change of variables and (
2.7) we have
$$\begin{aligned} &\bigl|H_{k,\mu}\bigl(u',\theta,\xi\bigr)\bigr| \\ &\quad= \Biggl|\sum _{l=0}^{\mu } \int_{2^{l}a_{\mu}^{k}}^{2^{l+1}a_{\mu}^{k}} \exp\bigl(-2\pi i\xi\cdot\bigl( \Gamma_{\iota}\bigl(\varphi(r)u'\bigr)- \Gamma_{\iota}\bigl(\varphi(r)\theta\bigr)\bigr)\bigr)\frac{dr}{r} \Biggr| \\ &\quad=\sum_{l=0}^{\mu} \biggl| \int_{\varphi(2^{l}a_{\mu}^{k})}^{\varphi(2^{l+1}a_{\mu}^{k})} \exp\bigl(-2\pi i\xi\cdot\bigl( \Gamma_{\iota}\bigl(ru'\bigr)- \Gamma_{\iota}(r \theta)\bigr)\bigr)\frac{dr}{r} \biggr| \\ &\quad=\sum_{l=0}^{\mu} \biggl| \int_{1}^{\frac{\varphi (2^{l+1}a_{\mu}^{k})}{\varphi(2^{l}a_{\mu}^{k})}} \exp\bigl(-2\pi i\xi\cdot\bigl( \Gamma_{\iota}\bigl(\varphi\bigl(2^{l}a_{\mu}^{k} \bigr)ru'\bigr)- \Gamma_{\iota}\bigl(\varphi \bigl(2^{l}a_{\mu}^{k}\bigr)r\theta\bigr)\bigr) \bigr)\frac{dr}{r} \biggr| \\ &\quad=\sum_{l=0}^{\mu} \biggl| \int_{1}^{\frac{\varphi (2^{l+1}a_{\mu}^{k})}{\varphi(2^{l}a_{\mu}^{k})}} \exp \Biggl(-2\pi i\sum _{j=1}^{\iota}\varphi\bigl(2^{l}a_{\mu}^{k} \bigr)^{d\alpha _{j}}r^{d\alpha_{j}} R_{j}\bigl(\xi^{j}\bigr) \cdot\bigl(\tilde{\Phi}^{j}\bigl(u'\bigr)-\tilde{ \Phi}^{j}(\theta)\bigr) \Biggr)\frac {dr}{r} \biggr|. \end{aligned}$$
(2.15)
Note that
\(\frac{\varphi(2^{l+1}a_{\mu}^{k})}{\varphi(2^{l}a_{\mu}^{k})} \leq c_{\varphi}\). Combining (
2.15) with Lemma
2.2 yields that
$$\begin{aligned} \bigl|H_{k,\mu}\bigl(u',\theta,\xi\bigr)\bigr| \leq{}& \sum_{l=0}^{\mu}\min \bigl\{ 1,\bigl|\varphi \bigl(2^{l}a_{\mu}^{k}\bigr)^{d\alpha_{j}} R_{\iota}\bigl(\xi^{\iota}\bigr)\cdot\bigl(\tilde{ \Phi}^{\iota}\bigl(u'\bigr)-\tilde{\Phi}^{\iota}( \theta )\bigr)\bigr|^{-1/\iota}\bigr\} \\ \leq{}& C(\mu+1)\min\bigl\{ 1,\max\bigl\{ \varphi\bigl(a_{\mu}^{k} \bigr)^{d\alpha _{j}},\varphi\bigl(2^{\mu}a_{\mu}^{k} \bigr)^{d\alpha_{j}}\bigr\} ^{-\epsilon_{1}} \\ &{}\times\bigl|R_{\iota}\bigl( \xi^{\iota}\bigr)\cdot\bigl(\tilde{\Phi}^{\iota}\bigl(u'\bigr)-\tilde{\Phi}^{\iota}(\theta) \bigr)\bigr|^{-\epsilon _{1}}\bigr\} \end{aligned}$$
(2.16)
for any
\(0<\epsilon_{1}\leq1/\lambda_{1}\). Similarly, we have
$$\begin{aligned} \bigl|J_{j,\mu}\bigl(v',\omega,\eta\bigr)\bigr|\leq{}& C(\mu+1)\min \bigl\{ 1,\max\bigl\{ \psi\bigl(a_{\mu}^{j} \bigr)^{v_{\beta_{\kappa}}},\psi\bigl(2^{\mu}a_{\mu}^{j} \bigr)^{v_{\beta_{\kappa}}}\bigr\} ^{-\epsilon_{2}} \\ &{}\times\bigl| H_{\kappa}\bigl( \eta^{\kappa}\bigr)\cdot\bigl(\tilde{\Psi}^{\kappa}\bigl(v'\bigr)-\tilde{\Psi}^{\kappa}(\omega) \bigr)\bigr|^{-\epsilon_{2}}\bigr\} \end{aligned}$$
(2.17)
for any
\(0<\epsilon_{2}\leq1/\lambda_{2}\). For any
\(1\leq\iota \leq\lambda_{1}\) and
\(x\in S^{o_{\iota}-1}\), since
\(\{\Phi_{\iota,1},\ldots,\Phi_{\iota,o_{\iota}}\}\) is linear independent,
\(x\cdot\tilde{\Phi}^{\iota}(\cdot)\) is a nonzero real-analytic function. Invoking Lemma
2.3, there exists
\(\delta_{1}>0\) such that
$$ \sup_{x\in S^{o_{\iota}-1}} \iint_{(S^{m-1})^{2}}\bigl|x\cdot\bigl(\tilde{\Phi }^{\iota}\bigl(u'\bigr) -\tilde{\Phi}^{\iota}(\theta) \bigr)\bigr|^{-\delta_{1}}\,d\sigma_{m}\bigl(u'\bigr)\,d \sigma_{m}(\theta )< \infty. $$
(2.18)
Similarly, for any
\(1\leq\kappa\leq\lambda_{2}\), there exists
\(\delta_{2}>0\) such that
$$ \sup_{y\in S^{\varpi_{\kappa}-1}} \iint_{(S^{n-1})^{2}}\bigl|y\cdot\bigl(\tilde {\Psi}^{\kappa}\bigl(u'\bigr) -\tilde{\Psi}^{\kappa}(\theta) \bigr)\bigr|^{-\delta_{2}}\,d\sigma_{n}\bigl(u'\bigr)\,d \sigma_{n}(\theta )< \infty. $$
(2.19)
From (
2.14) and (
2.16)-(
2.17) we have
$$\begin{aligned} & \int_{a_{\mu}^{j}}^{a_{\mu}^{j+1}} \int_{a_{\mu}^{k}}^{a_{\mu}^{k+1}} \bigl|\widehat{\sigma_{\mu;r,s}^{\iota,\kappa}}( \xi,\eta) \bigr|^{2}\frac {dr\,ds}{rs} \\ &\quad\leq C(\mu+1)^{2}\min\bigl\{ 1,\max\bigl\{ \varphi \bigl(a_{\mu}^{k}\bigr)^{d\alpha_{j}},\varphi \bigl(2^{\mu}a_{\mu}^{k}\bigr)^{d\alpha_{j}}\bigr\} ^{-\epsilon_{1}}\bigl|R_{\iota}\bigl(\xi^{\iota}\bigr)\bigr|^{-\epsilon_{1}} \bigr\} \\ &\qquad{}\times \min\bigl\{ 1,\max\bigl\{ \psi\bigl(a_{\mu}^{j} \bigr)^{v_{\beta_{\kappa}}},\psi\bigl(2^{\mu}a_{\mu}^{j} \bigr)^{v_{\beta_{\kappa}}}\bigr\} ^{-\epsilon_{2}}\bigl|H_{\kappa}\bigl(\eta ^{\kappa}\bigr)\bigr|^{-\epsilon_{2}}\bigr\} \\ &\qquad{}\times\iint_{(S^{m-1}\times S^{n-1})^{2}} \biggl|\frac {R_{\iota}(\xi^{\iota})}{|R_{\iota}(\xi^{\iota})|}\cdot\bigl(\tilde{ \Phi}^{\iota}\bigl(u'\bigr) -\tilde{\Phi}^{\iota}( \theta)\bigr) \biggr|^{-\epsilon_{1}} \biggl|\frac{H_{\kappa}(\eta^{\kappa})}{|H_{\kappa}(\eta ^{\kappa})|}\cdot\bigl(\tilde{\Psi}^{\kappa}\bigl(v'\bigr) -\tilde{\Psi}^{\kappa}(\omega) \bigr)\biggr|^{-\epsilon_{2}} \\ &\qquad{}\times \bigl|\Omega_{\mu}\bigl(u',v' \bigr)\overline{\Omega_{\mu}(\theta,\omega)}\bigr|\,d\sigma_{m} \bigl(u'\bigr) \,d\sigma_{n}\bigl(v'\bigr) \,d\sigma_{m}(\theta)\,d\sigma_{n}(\omega) \end{aligned}$$
(2.20)
for any
\(0<\epsilon_{1}\leq1/\lambda_{1}\) and
\(0<\epsilon_{2}\leq1/\lambda_{2}\). Take
\(\epsilon_{1}=\min\{1/\lambda_{1},\delta_{1}/2\}\) and
\(\epsilon_{2}=\min\{1/\lambda_{2},\delta_{2}/2\}\). Using (
2.3), (
2.18)-(
2.19) and Hölder’s inequality along with (
2.20), we obtain
$$\begin{aligned} & \int_{a_{\mu}^{j}}^{a_{\mu}^{j+1}} \int_{a_{\mu}^{k}}^{a_{\mu}^{k+1}} \bigl|\widehat{\sigma_{\mu;r,s}^{\iota,\kappa}}( \xi,\eta) \bigr|^{2}\frac {dr\,ds}{rs} \\ &\quad\leq Ca_{\mu}^{4}(\mu+1)^{2}\min\bigl\{ 1, \max\bigl\{ \varphi\bigl(a_{\mu}^{k}\bigr)^{d\alpha _{j}}, \varphi\bigl(2^{\mu}a_{\mu}^{k} \bigr)^{d\alpha_{j}}\bigr\} ^{-\epsilon_{1}}\bigl|R_{\iota}\bigl( \xi^{\iota}\bigr)\bigr|^{-\epsilon_{1}}\bigr\} \\ &\qquad{}\times \min\bigl\{ 1,\max\bigl\{ \psi\bigl(a_{\mu}^{j} \bigr)^{v_{\beta_{\kappa}}},\psi\bigl(2^{\mu}a_{\mu}^{j} \bigr)^{v_{\beta_{\kappa}}}\bigr\} ^{-\epsilon_{2}}\bigl|H_{\kappa}\bigl( \eta^{\kappa}\bigr)\bigr|^{-\epsilon _{2}}\bigr\} . \end{aligned}$$
(2.21)
We can easily check that
$$\begin{aligned}& \int_{a_{\mu}^{j}}^{a_{\mu}^{j+1}} \int_{a_{\mu}^{k}}^{a_{\mu}^{k+1}} \bigl|\widehat{\sigma_{\mu;r,s}^{\iota,\kappa}}( \xi,\eta) \bigr|^{2}\frac{dr\,ds}{rs} \leq C(\mu+1)^{2}, \end{aligned}$$
(2.22)
$$\begin{aligned}& \int_{a_{\mu}^{j}}^{a_{\mu}^{j+1}} \int_{a_{\mu}^{k}}^{a_{\mu}^{k+1}} \bigl|\widehat {\sigma_{\mu;r,s}^{\iota,\kappa}} (\xi,\eta)-\widehat{\sigma_{\mu;r,s}^{\iota-1,\kappa}}(\xi,\eta) \bigr|^{2}\frac{dr\,ds}{rs}\leq C(\mu+1)^{2}. \end{aligned}$$
(2.23)
Interpolating between (
2.21) and (
2.22) yields (
2.12). We now prove (
2.10) and (
2.11). By the change of variables,
$$\begin{aligned} & \bigl|\widehat{\sigma_{\mu;r,s}^{\iota,\kappa}}(\xi,\eta)-\widehat { \sigma_{\mu;r,s}^{\iota-1,\kappa}}(\xi,\eta) \bigr|^{2} \\ &\quad= \biggl| \iint_{S^{m-1}\times S^{n-1}}\bigl(\exp\bigl(-2\pi i\xi\cdot \Gamma_{\iota}\bigl(\varphi(r)u'\bigr)\bigr) -\exp\bigl(-2 \pi i\xi\cdot\Gamma_{\iota-1}\bigl(\varphi(r)u'\bigr)\bigr) \bigr) \\ &\qquad{} \times\exp\bigl(-2\pi i\eta\cdot\Upsilon_{\kappa}\bigl( \psi(s)v'\bigr)\bigr) \Omega_{\mu}\bigl(u',v' \bigr)\,d\sigma_{m}\bigl(u'\bigr)\,d\sigma_{n} \bigl(v'\bigr) \biggr|^{2} \\ &\quad= \iint_{(S^{m-1}\times S^{n-1})^{2}}\bigl(\exp\bigl(-2\pi i\xi\cdot \Gamma_{\iota}\bigl(\varphi(r)u'\bigr)\bigr)-\exp\bigl(-2\pi i\xi\cdot\Gamma_{\iota-1}\bigl(\varphi (r)u'\bigr)\bigr) \bigr) \\ &\qquad{}\times \bigl(\exp\bigl(2\pi i\xi\cdot\Gamma_{\iota}\bigl( \varphi(r)\theta\bigr)\bigr)-\exp \bigl(-2\pi i\xi\cdot\Gamma_{\iota-1} \bigl(\varphi(r)\theta\bigr)\bigr)\bigr) \\ &\qquad{}\times\exp\bigl(-2\pi i\eta\cdot\bigl( \Upsilon_{\kappa}\bigl(\psi(s)v'\bigr)-\Upsilon_{\kappa}\bigl(\psi (s)\omega\bigr)\bigr)\bigr)\\ &\qquad{}\times \Omega_{\mu}\bigl(u',v' \bigr)\overline{\Omega_{\mu}(\theta,\omega)}\,d\sigma _{m} \bigl(u'\bigr)\,d\sigma_{n}\bigl(v'\bigr)\,d \sigma_{m}(\theta)\,d\sigma_{n}(\omega), \end{aligned}$$
which, together with (
2.7), implies that
$$\begin{aligned} & \int_{a_{\mu}^{j}}^{a_{\mu}^{j+1}} \int_{a_{\mu}^{k}}^{a_{\mu}^{k+1}} \bigl|\widehat{\sigma_{\mu;r,s}^{\iota,\kappa}} (\xi,\eta)-\widehat{\sigma_{\mu;r,s}^{\iota-1,\kappa}}(\xi,\eta) \bigr|^{2}\frac{dr\,ds}{rs} \\ &\quad\leq C(\mu+1)\min\bigl\{ 1,\bigl|\varphi\bigl(a_{\mu}^{k} \bigr)^{d_{\alpha_{\iota}}} R_{\iota}\bigl(\xi^{\iota}\bigr)\bigr|\bigr\} \iint_{(S^{m-1}\times S^{n-1})^{2}}\bigl|J_{j,\mu }\bigl(v',\omega,\eta \bigr)\bigr| \\ &\qquad{} \times \bigl|\Omega_{\mu}\bigl(u',v' \bigr) \overline{\Omega_{\mu}(\theta,\omega)}\bigr|\,d\sigma_{m} \bigl(u'\bigr)\,d\sigma_{n}\bigl(v'\bigr)\,d \sigma _{m}(\theta)\,d\sigma_{n}(\omega). \end{aligned}$$
(2.24)
Using (
2.3), (
2.17), (
2.19), (
2.24), and Hölder’s inequality, we have
$$ \begin{aligned}[b] & \int_{a_{\mu}^{j}}^{a_{\mu}^{j+1}} \int_{a_{\mu}^{k}}^{a_{\mu}^{k+1}} \bigl|\widehat{\sigma_{\mu;r,s}^{\iota,\kappa}} (\xi,\eta)-\widehat{\sigma_{\mu;r,s}^{\iota-1,\kappa}}(\xi,\eta) \bigr|^{2}\frac{dr\,ds}{rs}\\ &\quad\leq Ca_{\mu}^{4}(\mu+1)^{2}\min\bigl\{ 1,\bigl| \varphi\bigl(a_{\mu}^{k}\bigr)^{d_{\alpha_{\iota}}} R_{\iota}\bigl(\xi^{\iota}\bigr)\bigr|\bigr\} \\ &\qquad{}\times\min\bigl\{ 1,\max\bigl\{ \psi\bigl(a_{\mu}^{j}\bigr)^{v_{\beta_{\kappa}}},\psi \bigl(2^{\mu}a_{\mu}^{j}\bigr)^{v_{\beta_{\kappa}}}\bigr\} ^{-\epsilon_{2}}\bigl|H_{\kappa}\bigl(\eta^{\kappa}\bigr)\bigr|^{-\epsilon_{2}}\bigr\} . \end{aligned} $$
(2.25)
Estimate (
2.10) follows form (
2.23) and (
2.25). Similarly, we can get (
2.11). Estimate (
2.13) follows from the inequality
$$\begin{aligned} & \bigl|\widehat{\sigma_{\mu;r,s}^{\iota,\kappa}}(\xi,\eta)-\widehat { \sigma_{\mu;r,s}^{\iota-1,\kappa}}(\xi,\eta) -\widehat{\sigma_{\mu;r,s}^{\iota,\kappa-1}}( \xi,\eta) +\widehat{\sigma_{\mu;r,s}^{\iota-1,\kappa-1}}(\xi,\eta) \bigr| \\ &\quad= \biggl| \iint_{S^{m-1}\times S^{n-1}}\exp\bigl(-2\pi i\bigl(\xi\cdot \Gamma_{\iota-1} \bigl(\varphi(r)u'\bigr) +\eta\cdot\Upsilon_{\kappa-1}\bigl( \psi(s)v'\bigr)\bigr)\bigr) \\ &\qquad{} \times\bigl(\exp\bigl(-2\pi i\bigl(\xi\cdot\bigl(\Gamma_{\iota}\bigl(\varphi(r)u'\bigr)-\Gamma _{\iota-1}\bigl( \varphi(r)u'\bigr)\bigr)\bigr)\bigr)-1\bigr) \\ &\qquad{} \times \bigl(\exp\bigl(-2\pi i\bigl(\eta\cdot\bigl( \Upsilon_{\kappa}\bigl(\psi (s)v'\bigr)-\Upsilon_{\kappa-1} \bigl(\psi(s)v'\bigr)\bigr)\bigr)\bigr)-1\bigr)\\ &\qquad{}\times \Omega \bigl(u',v'\bigr)J_{m}\bigl(u' \bigr)J_{n}\bigl(v'\bigr)\,d\sigma_{m} \bigl(u'\bigr)\,d\sigma_{n}\bigl(v'\bigr) \biggr| \\ &\quad\leq C\min\bigl\{ 1,\bigl|\varphi(r)^{d_{\alpha_{\iota}}} R_{\iota}\bigl( \xi^{\iota}\bigr)\bigr|\bigr\} \min\bigl\{ 1,\bigl|\psi(s)^{v_{\beta_{\kappa}}}H_{\kappa}\bigl(\eta ^{\kappa}\bigr)\bigr|\bigr\} . \end{aligned}$$
This proves Lemma
2.4. □