The proof of Theorem
1.1 mainly depends on the approaches employed in the proof of [
11, Theorem 1.1] and [
4, Theorem 1.6]. By duality, for
\(1<\gamma\leq 2\), we get
$$\begin{aligned} \mathcal{M}_{P,\varOmega, \phi}^{(\gamma)}(f) (x,x+1)= \biggl( \int_{0}^{\infty} \biggl\vert \int_{\mathbf{S}^{n-1}}e^{iP(ru)}f\bigl(x-r u,x_{n+1}-\phi(r) \bigr)\varOmega(u)\,d\sigma(u) \biggr\vert ^{\gamma'}\,\frac{dr}{r} \biggr) ^{1/\gamma'}, \end{aligned}$$
which gives
$$\begin{aligned} \bigl\Vert \mathcal{M}_{P,\varOmega, \phi}^{(\gamma)}(f) \bigr\Vert _{L^{p}(\textbf{R}^{n+1})}= \bigl\Vert N(f) \bigr\Vert _{L^{p}(L^{\gamma'}(\textbf{R}^{+},\frac{dr}{r}),\textbf {R}^{n+1})}, \end{aligned}$$
(3.1)
where
\(N:L^{p}(\textbf{R}^{n+1})\rightarrow L^{p}(L^{\gamma'}(\textbf{R}^{+},\frac{dr}{r}),\textbf{R}^{n+1})\) is a linear operator defined by
$$\begin{aligned} N(f) (x,x_{n+1},r)= \int_{\mathbf{S}^{n-1}}e^{iP(u)}f\bigl(x-r u,x_{n+1}-\phi(r) \bigr)\varOmega(u)\,d\sigma(u). \end{aligned}$$
Now if we assume that
$$\begin{aligned} \bigl\Vert \mathcal{M}_{P,\varOmega, \phi}^{(2)}(f) \bigr\Vert _{L^{p}(\textbf{R}^{n+1})}= \bigl\Vert N(f) \bigr\Vert _{L^{p}(L^{2}(\textbf{R}^{+},\frac{dr}{r}),\textbf {R}^{n+1})}\leq C_{p,q}(1+\beta_{\varOmega})^{1/2} \Vert f \Vert _{L^{p}( \mathbf{R} ^{n+1})} \end{aligned}$$
for
\(2\leq p<\infty\); and
$$\begin{aligned} \bigl\Vert \mathcal{M}_{P,\varOmega, \phi}^{(1)}(f) \bigr\Vert _{L^{\infty}(\textbf{R}^{n+1})}= \bigl\Vert N(f) \bigr\Vert _{L^{\infty}(L^{\infty}(\textbf{R}^{+},\frac {dr}{r}),\textbf{R}^{n+1})}\leq C \Vert f \Vert _{L^{\infty}( \mathbf{R} ^{n+1})}, \end{aligned}$$
then by applying the interpolation theorem for the Lebesgue mixed normed spaces to the last two inequalities, we directly obtain
$$\begin{aligned} \bigl\Vert \mathcal{M}_{P,\varOmega, \phi}^{(\gamma)}(f) \bigr\Vert _{L^{p}(\textbf{R}^{n+1})}\leq C_{p,q}(1+\beta_{\varOmega})^{1/\gamma'} \Vert f \Vert _{L^{p}( \mathbf{R} ^{n+1})} \end{aligned}$$
(3.2)
for
\(\gamma'\leq p<\infty\) with
\(1<\gamma\leq2\); and
\(\Vert \mathcal{M}_{P,\varOmega, \phi}^{(1)}(f) \Vert _{L^{\infty}(\textbf{R}^{n+1})}\leq C \Vert f \Vert _{L^{\infty}( \mathbf{R} ^{n+1})}\). Thus, to prove our theorem, it is enough to prove it only for the cases
\(\gamma=1\) and
\(\gamma=2\).
Case 2 (if
\(\gamma=2\)). We use the induction on the degree of the polynomial
P. If the degree of
P is 0, then by Lemma
2.3 we get that, for all
\(p\geq2\),
$$\begin{aligned} \bigl\Vert \mathcal{M}^{(2)}_{P,\varOmega,\phi}(f) \bigr\Vert _{L^{p}(\mathbf{ \mathbf{R} }^{n+1})} \leq&C_{p,q} ( 1+\beta_{\varOmega})^{1/2} \Vert f \Vert _{L^{p}( \mathbf{R} ^{n+1})}. \end{aligned}$$
(3.3)
Now, assume that (
1.3) is satisfied for any polynomial of degree less than or equal to
m with
\(m\geq1\). We need to show that (
1.3) is still true if
\(\operatorname{deg}(P)=m+1\). Let
$$ P(x)=\sum_{|\alpha|\leq m+1}a_{\gamma}x^{\gamma} $$
be a polynomial of degree
\(m+1\). Without loss of generality, we may assume that
\(\sum_{ \vert \gamma \vert =m+1} \vert a_{\gamma} \vert =1\), and also we may assume that
P does not contain
\(\vert x \vert ^{m+1}\) as one of its terms. Let
\(\{ \varphi_{k} \}_{k\in\mathbf{Z}} \) be a collection of
\(\mathcal{C}^{\infty}(0,\infty)\) functions satisfying the following conditions:
$$\begin{aligned} &\operatorname{supp}\varphi_{k} \subseteq\mathcal{I}_{k,\beta_{\varOmega }}= \bigl[ 2^{-(k+1)\beta_{\varOmega}},2^{-(k-1)\beta_{\varOmega}} \bigr];\quad 0\leq\varphi_{k} \leq1; \\ &\sum_{k\in\mathbf{Z}}\varphi_{k} ( r ) =1;\quad \mbox{and} \quad \biggl\vert \,\frac{d^{k}\varphi_{k} ( r ) }{du^{r}} \biggr\vert \leq \frac{C_{k}}{r^{k}}. \end{aligned}$$
Define the multiplier operators
\(S_{k}\) in
\(\textbf{R}^{n+1}\) by
$$\begin{aligned} \widehat{{(S_{k}f)}}(\xi,\eta)=\varphi_{k}\bigl( \vert \xi \vert \bigr)\widehat{f}(\xi,\eta)\quad \mbox{for } (\xi,\eta)\in \mathbf{R}^{n}\times\mathbf{R}, \end{aligned}$$
and set
$$ \varGamma_{\infty}(r)=\sum_{k=-\infty}^{0} \varphi_{k}(r), \qquad \varGamma_{0}(r)=\sum _{k=1}^{\infty}\varphi _{k}(r). $$
Thanks to Minkowski’s inequality, we have
$$ \mathcal{M}^{(2)}_{P,\varOmega,\phi}(f) (x,x_{n+1})\leq \mathcal{M}^{(2)}_{P,\varOmega,\phi,\infty }(f) (x,x_{n+1})+ \mathcal{M}^{(2)}_{P,\varOmega ,\phi,0}(f) (x,x_{n+1}), $$
(3.4)
where
$$\begin{aligned} &\mathcal{M}^{(2)}_{P,\varOmega,\phi,\infty}(f) (x,x_{n+1})\\ &\quad = \biggl( \int_{2^{-\beta_{\varOmega}}}^{\infty} \biggl\vert \varGamma_{\infty}(r) \int_{\mathbf{S}^{n-1}}e^{iP(r u)}f\bigl(x-r u,x_{n+1}-\phi(r) \bigr)\varOmega(u)\,d\sigma(u) \biggr\vert ^{2}\,\frac{dr}{r} \biggr) ^{1/2}, \end{aligned}$$
and
$$\begin{aligned} &\mathcal{M}^{(2)}_{P,\varOmega,\phi,0}(f) (x,x_{n+1})\\ &\quad = \biggl( \int_{0}^{1} \biggl\vert \varGamma _{0}(r) \int_{\mathbf{S} ^{n-1}}e^{iP(r u)}f\bigl(x-ru,x_{n+1}-\phi(r) \bigr)\varOmega(u)\,d\sigma(u) \biggr\vert ^{2}\,\frac{dr}{r} \biggr) ^{1/2}. \end{aligned}$$
Let us first estimate
\(L^{p}\)-norm of
\(\mathcal{M}^{(2)}_{P,\varOmega ,\phi,\infty}(f)\). Define
$$\begin{aligned} &\mathcal{M}^{(2)}_{P,\varOmega,\phi,\infty,k}(f) (x,x_{n+1})\\ &\quad = \biggl( \int_{2^{-(k+1)\beta_{\varOmega}}}^{2^{-(k-1)\beta_{\varOmega }}} \biggl\vert \int_{\mathbf{S}^{n-1}}e^{iP(r u)}f\bigl(x-r u,x_{n+1}-\phi(r) \bigr)\varOmega(u)\,d\sigma(u) \biggr\vert ^{2}\,\frac{dr}{r} \biggr) ^{1/2}. \end{aligned}$$
Hence, by generalized Minkowski’s inequality, it is easy to show that
$$\begin{aligned} \mathcal{M}^{(2)}_{P,\varOmega,\phi,\infty}(f) (x,x_{n+1}) \leq &\sum _{k=-\infty}^{0}\mathcal{M}^{(2)}_{P,\varOmega ,\phi,\infty,k}(f) (x,x_{n+1}). \end{aligned}$$
(3.5)
If
\(p=2\), then by a simple change of variables, Plancherel’s theorem, Fubini’s theorem, and Lemma
2.1, we get that
$$\begin{aligned} \bigl\Vert \mathcal{M}^{(2)}_{P,\varOmega,\phi,\infty ,k}(f) \bigr\Vert _{L^{2}( \mathbf{R} ^{n+1})} =& \biggl( \int_{ \mathbf{R} ^{n+1}} \bigl\vert \widehat{f}(\zeta,\eta ) \bigr\vert ^{2} \mathcal{J}_{k,\varOmega,\phi}(\zeta,\eta )\,d\zeta \,d\eta \biggr) ^{1/2} \\ \leq &C2^{\frac{(k+1)}{8q'}} (1+\beta_{\varOmega})^{1/2} \Vert f \Vert _{L^{2}( \mathbf{R} ^{n+1})}. \end{aligned}$$
(3.6)
However, if
\(p>2\), then by the duality, there exists
\(\varPsi\in L^{(p/2)^{\prime}}( \mathbf{R} ^{n+1})\) with
\(\Vert \varPsi \Vert _{L^{(p/2)^{\prime}}( \mathbf{R} ^{n+1})}=1\) such that
$$\begin{aligned} & \bigl\Vert \mathcal{M}^{(2)}_{P,\varOmega,\phi,\infty ,k}(f) \bigr\Vert _{L^{p}( \mathbf{R} ^{n+1})}^{2} \\ &\quad = \int_{ \mathbf{R} ^{n+1}} \int_{1}^{2^{2\beta_{\varOmega}}} \biggl\vert \int_{\mathbf{S}^{n-1}}\mathcal{G}_{k,\varOmega,P }(r,u,0,0)f \bigl(x-2^{-(k+1)\beta_{\varOmega}}r u,x_{n+1}\\ &\qquad {}-\phi\bigl(2^{-(k+1)\beta_{\varOmega}}r\bigr) \bigr)\,d\sigma(u) \biggr\vert ^{2}\,\frac{dr}{r} \\ &\qquad {}\times \bigl\vert \varPsi(x,x_{n+1}) \bigr\vert \,dx\,dx_{n+1}. \end{aligned}$$
So, by Hölder’s inequality and Lemma
2.2, we conclude that
$$\begin{aligned} &\bigl\Vert \mathcal{M}^{(2)}_{P,\varOmega,\phi,\infty ,k}(f) \bigr\Vert _{L^{p}( \mathbf{R} ^{n+1})}^{2}\\ &\quad \leq C \int_{\mathbf{R} ^{n+1}} \bigl\vert f(z,z_{n+1}) \bigr\vert ^{2} \int_{1}^{2^{2 \beta_{\varOmega}}} \int_{\mathbf{S}^{n-1}} \bigl\vert \varOmega(u) \bigr\vert \\ &\qquad {}\times \bigl\vert \varPsi\bigl(z+2^{-(k+1)\beta_{\varOmega}}r u,z_{n+1}+\phi \bigl(2^{-(k+1)\beta_{\varOmega}}r\bigr) \bigr) \bigr\vert \,d\sigma (u)\,\frac{dr}{r}\,dz\,dz_{n+1} \\ &\quad \leq C_{p} (1+\beta_{\varOmega}) \bigl\Vert \vert f \vert ^{2} \bigr\Vert _{L^{(p/2)}( \mathbf{R} ^{n+1})} \bigl\Vert \mathcal{M}_{\varOmega,\phi}( \widetilde{\varPsi}) \bigr\Vert _{L^{(p/2)^{\prime}}( \mathbf{R} ^{n+1})} \\ &\quad \leq C_{p} (1+\beta_{\varOmega}) \Vert f \Vert _{L^{p}( \mathbf{R} ^{n+1})}^{2} \Vert \widetilde{\varPsi} \Vert _{L^{(p/2)^{\prime}}( \mathbf{R} ^{n+1})} \Vert \varOmega \Vert _{L^{1}(\mathbf{S}^{n-1})}, \end{aligned}$$
where
\(\widetilde{\varPsi}(z,z_{n+1})=\varPsi(-z,-z_{n+1})\). Thus,
$$ \bigl\Vert \mathcal{M}^{(2)}_{P,\varOmega,\phi,\infty ,k}(f) \bigr\Vert _{L^{p}( \mathbf{R} ^{n+1})}\leq C_{p} (1+\beta_{\varOmega})^{1/2} \Vert f \Vert _{L^{p}( \mathbf{R} ^{n+1})}, $$
which when combined with (
3.6) gives that there is
\(0<\nu<1\) so that
$$ \bigl\Vert \mathcal{M}^{(2)}_{P,\varOmega,\phi,\infty ,k}(f) \bigr\Vert _{L^{p}( \mathbf{R} ^{n+1})}\leq C_{p} 2^{\nu (k+1)/8} (1+\beta_{\varOmega})^{1/2} \Vert f \Vert _{L^{p}( \mathbf{R} ^{n+1})} $$
(3.7)
for all
\(p\geq2\). Therefore, by (
3.5) and (
3.7), we obtain
$$ \bigl\Vert \mathcal{M}^{(2)}_{P,\varOmega,\phi,\infty}(f) \bigr\Vert _{L^{p}( \mathbf{R} ^{n+1})}\leq C_{p,q} (1+\beta_{\varOmega})^{1/2} \Vert f \Vert _{L^{p}( \mathbf{R} ^{n+1})}. $$
(3.8)
Now, let us estimate the
\(L^{p}\)-norm of
\(\mathcal{M}^{(2)}_{P,\varOmega ,\phi,0 }(f)\). Let
\(Q(x)=\sum_{ \vert \gamma \vert \leq m}a_{\gamma}x^{\gamma}\). Define
\(\mathcal{M}^{(2)}_{Q,\varOmega ,\phi,0 }(f)\) and
\(\mathcal{M}^{(2)}_{P,Q,\varOmega,\phi,0 }(f)\) by
$$\begin{aligned} &\mathcal{M}^{(2)}_{Q,\varOmega,\phi,0 }(f) (x,x_{n+1})= \biggl( \int_{0}^{1} \biggl\vert \int_{\mathbf{S} ^{n-1}}e^{iQ(r u)}f\bigl(x-r u,x_{n+1}-\phi(r) \bigr)\varOmega(u)\,d\sigma (u) \biggr\vert ^{2}\,\frac{dr}{r} \biggr) ^{1/2} , \\ &\mathcal{M}^{(2)}_{P,Q,\varOmega,\phi,0 }(f) (x,x_{n+1})\\ &\quad = \biggl( \int_{0}^{1} \biggl\vert \int_{\mathbf{S} ^{n-1}} \bigl(e^{iP(r u)}-e^{iQ(r u)} \bigr)f \bigl(x-r u,x_{n+1}-\phi(r) \bigr)\varOmega(u)\,d\sigma(u) \biggr\vert ^{2}\,\frac{dr}{r} \biggr) ^{1/2}. \end{aligned}$$
Thus, by Minkowski’s inequality, we deduce
$$ \mathcal{M}^{(2)}_{P,\varOmega,\phi,0 }(f) (x,x_{n+1})\leq \mathcal{M}^{(2)}_{Q,\varOmega,\phi,0 }(f) (x,x_{n+1})+ \mathcal{M}^{(2)}_{P,Q,\varOmega,\phi,0 }(f) (x,x_{n+1}). $$
(3.9)
On the one hand, since
\(\deg(Q)\leq m\), then by our assumption,
$$ \bigl\Vert \mathcal{M}^{(2)}_{Q,\varOmega,\phi,0 }(f) \bigr\Vert _{L^{p}( \mathbf{R} ^{n+1})}\leq C_{p,q} (1+\beta_{\varOmega})^{1/2} \Vert f \Vert _{L^{p}( \mathbf{R} ^{n+1})} $$
(3.10)
for all
\(p\geq2\). On the other hand, since we have
$$\begin{aligned} \bigl\vert e^{iP(r u)}-e^{iQ(r u)} \bigr\vert \leq&r ^{(m+1)} \biggl\vert \sum_{ \vert \gamma \vert =m+1}a_{\gamma}(u)^{\gamma} \biggr\vert \leq r ^{(m+1)}, \end{aligned}$$
then by the Cauchy–Schwarz inequality, we reach that
$$\begin{aligned} &\mathcal{M}^{(2)}_{P,Q,\varOmega,\phi,0 }(f) (x,x_{n+1}) \\ &\quad \leq C \biggl( \int_{0}^{1} \int_{\mathbf{S}^{n-1}} r ^{2(m+1)} \bigl\vert \varOmega(u) \bigr\vert \bigl\vert f\bigl(x-r u,x_{n+1}-\phi(r) \bigr) \bigr\vert ^{2}\,d\sigma(u) \,\frac{dr}{r} \biggr) ^{1/2} \\ &\quad \leq \Biggl( \sum_{j=1 }^{\infty}2^{-j({2(m+1)})} \int_{2^{{-j}}}^{2{-j+1}} \int _{\mathbf{S} ^{n-1}} \bigl\vert \varOmega(u) \bigr\vert \bigl\vert f \bigl(x-ru,x_{n+1}-\phi(r)\bigr) \bigr\vert ^{2}\,d\sigma (u) \,\frac{dr}{r} \Biggr) ^{1/2} \\ &\quad \leq C \bigl( \mathcal{M}_{\varOmega,\phi } \bigl( \vert f \vert ^{2} \bigr) \bigr) ^{1/2}. \end{aligned}$$
Hence, by Lemma
2.2, we get that
$$\begin{aligned} \bigl\Vert \mathcal{M}^{(2)}_{P,Q,\varOmega,\phi,0 }(f) \bigr\Vert _{L^{p}( \mathbf{R} ^{n+1})} \leq& C_{p} \Vert \varOmega \Vert _{L^{1}(\mathbf{S} ^{n-1})} \bigl\Vert \vert f \vert ^{2} \bigr\Vert _{L^{p/2}( \mathbf{R} ^{n+1})}^{1/2} \\ \leq& C_{p} \Vert f \Vert _{L^{p}( \mathbf{R} ^{n+1})}\leq C_{p} (1+\beta_{\varOmega})^{1/2} \Vert f \Vert _{L^{p}( \mathbf{R} ^{n+1})} \end{aligned}$$
(3.11)
for all
\(p\geq2\). Therefore, by (
3.9)–(
3.11), we obtain
$$ \bigl\Vert \mathcal{M}^{(2)}_{P,\varOmega,\phi,0 }(f) \bigr\Vert _{L^{p}( \mathbf{R} ^{n})}\leq C_{p} (1+\beta_{\varOmega})^{1/2} \Vert f \Vert _{L^{p}( \mathbf{R} ^{n})}. $$
(3.12)
Consequently, by (
3.4), (
3.8), and (
3.12), we finish the proof of Theorem
1.1. □