1 Introduction

Let \(\mathbf{R}^{n}\) (\(n\geq 2\)) be the n-dimensional Euclidean space and \(\mathbf{S}^{n-1}\) denote the unit sphere in \(\mathbf{R}^{n}\) equipped with the induced Lebesgue measure dσ. Let \(K(\cdot )\) be a kernel of Calderón–Zygmund type on \(\mathbf{R}^{n}\) given by

$$ K(y)=\frac{\varOmega (y)h( \vert y \vert )}{ \vert y \vert ^{n}}, $$

where h is a suitable function defined on \(\mathbf{R}_{+}:= (0, \infty )\) and Ω is homogeneous of degree zero, with \(\varOmega \in L^{1}(\mathbf{S}^{n-1})\) and

$$ \int _{\mathbf{S}^{n-1}}\varOmega (u)\,d\sigma (u)=0. $$
(1.1)

Suppose that P is a real polynomial on R of degree N and satisfies \(P(0)=0\). For a suitable function φ defined on \(\mathbf{R}_{+}\), we consider that the singular integral operator \(T_{h,\varOmega ,P,\varphi }\) along the “polynomial compound curve” \(P(\varphi (|y|))y'\) on \(\mathbf{R}^{n}\) is defined by

$$ T_{h,\varOmega ,P,\varphi }f(x):=\mathrm{p.v.} \int _{\mathbf{R}^{n}} f\bigl(x-P\bigl( \varphi \bigl( \vert y \vert \bigr)\bigr)y'\bigr)K(y)\,dy, $$
(1.2)

where \(y'=\frac{y}{|y|}\) for any nonzero point \(y\in \mathbf{R}^{n}\).

For the sake of simplicity, we denote \(T_{h,\varOmega ,P,\varphi }= T _{h,\varOmega }\) if \(P(t)\equiv t\) and \(\varphi (t)\equiv t\) and \(T_{h,\varOmega }=T_{\varOmega }\) if \(h(t)\equiv 1\). In their fundamental work on singular integrals, Calderón and Zygmund [1] proved that \(T_{\varOmega }\) is bounded on \(L^{p}(\mathbf{R}^{n})\) for \(1< p<\infty \), provided that \(\varOmega \in L\log L(\mathbf{S}^{n-1})\). The same conclusion under the less restrictive condition that \(\varOmega \in H ^{1}(\mathbf{S}^{n-1})\) was later obtained independently by Coifman and Weiss [2] and Connett [3]. It should be pointed out that the condition that \(\varOmega \in H^{1} (\mathbf{S}^{n-1})\) turns out to be the most desirable size condition for the \(L^{p}\) boundedness of \(T_{\varOmega }\). For the singular integral operator with radial kernel, Fefferman [4] in 1978 first introduced the singular integral operator \(T_{h,\varOmega }\) and established the \(L^{p}\) (\(1< p<\infty \)) boundedness of \(T_{h,\varOmega }\), provided that \(\varOmega \in \operatorname{Lip}_{\alpha }(\mathbf{S}^{n-1})\) for some \(\alpha >0\) and \(h\in L^{\infty }(\mathbf{R}_{+})\). Later on, Namazi [5] improved Fefferman’s result by assuming \(\varOmega \in L^{q}(\mathbf{S} ^{n-1})\) for some \(q>1\) instead of \(\varOmega \in \operatorname{Lip}_{\alpha }( \mathbf{S}^{n-1})\). In 1986, Duoandikoetxea and Rubio de Francia [6] used the Littlewood–Paley theory to improve the above results to the case \(\varOmega \in L^{q}(\mathbf{S}^{n-1})\) for any \(q>1\) and \(h\in \Delta _{2}(\mathbf{R}_{+})\). Here \(\Delta _{\gamma } (\mathbf{R} _{+})\) (\(\gamma >0\)) is the set of all measurable functions \(h: \mathbf{R}_{+}\rightarrow \mathbf{R}\) satisfying

$$ \Vert h \Vert _{\Delta _{\gamma }(\mathbf{R}_{+})}:=\sup_{R>0} \biggl(R^{-1} \int _{0}^{R} \bigl\vert h(t) \bigr\vert ^{\gamma }\,dt \biggr)^{1/\gamma }< \infty . $$

It is clear that

$$ L^{\infty }(\mathbf{R}_{+})=\Delta _{\infty }( \mathbf{R}_{+})\subsetneq \Delta _{\gamma _{2}}( \mathbf{R}_{+})\subsetneq \Delta _{\gamma _{1}}( \mathbf{R}_{+}), \quad \forall 1\leq \gamma _{1}< \gamma _{2}< \infty $$
(1.3)

and

$$ \operatorname{Lip}_{\alpha }\bigl(\mathbf{S}^{n-1}\bigr)\subsetneq L^{q}\bigl(\mathbf{S} ^{n-1}\bigr)\subsetneq L\log L\bigl( \mathbf{S}^{n-1}\bigr)\subsetneq H^{1}\bigl( \mathbf{S}^{n-1}\bigr)\subsetneq L^{1}\bigl( \mathbf{S}^{n-1}\bigr), \quad \forall \alpha >0 \text{ and } q>1. $$

In both \(T_{\varOmega }\) and \(T_{h,\varOmega }\), the singularity is along the diagonal \(\{x=y\}\). However, many problems in analysis have led one to consider singular integral operators with singularity along more general sets. One of the principal motivations for the study of such operators is the requirements of several complex variables and large classes of “subelliptic” equations. We refer the readers to Stein’s survey articles [7, 8] for more background information. During the last several years the \(L^{p}\) mapping properties for singular integral operators with singularity along various sets and with rough kernel in \(H^{1}(\mathbf{S}^{n-1})\) have been actively studied by many authors. For example, see [9] for polynomial mappings, [10] for real-analytic submanifolds, [11, 12] for homogeneous mappings, [13] for polynomial curves. For further results on the singular integral operators with singularity along the above curves or surfaces, we refer the readers to consult [14,15,16,17], among others. Particularly, Fan and Pan [9] established the \(L^{p}\) boundedness of the singular integral operator along polynomial mappings for p with \(|1/p-1/2|<\max \{1/2,1/\gamma '\}\) under the conditions that \(\varOmega \in H^{1}(\mathbf{S}^{n-1})\) and \(h\in \Delta _{\gamma }(\mathbf{R}_{+})\) for some \(\gamma >1\). In this paper we focus on the \(L^{p}\)-boundedness of the singular integral operator along the polynomial compound curves with rough kernels \(\varOmega \in H^{1}(\mathbf{S}^{n-1})\) and \(h\in \Delta _{\gamma }(\mathbf{R} _{+})\) for some \(\gamma >1\). Recently, Fan and Pan [13] proved the following result.

Theorem A

([13])

Let \(\varphi (t) \equiv t\). If \(\varOmega \in H^{1}(\mathbf{S}^{n-1})\) satisfies (1.1) and \(h\in L^{\infty }(\mathbf{R}_{+})\), then the operator \(T_{h,\varOmega ,P,\varphi }\) is bounded on \(L^{2}(\mathbf{R}^{n})\). More precisely, we have

$$ \Vert T_{h,\varOmega ,P,\varphi }f \Vert _{L^{2}(\mathbf{R}^{n})}\leq C \Vert h \Vert _{L ^{\infty }(\mathbf{R}_{+})} \Vert \varOmega \Vert _{H^{1}(\mathbf{S}^{n-1})} \Vert f \Vert _{L^{2}(\mathbf{R}^{n})}. $$

Here \(C>0\) is independent of h, Ω, f and the coefficients of P.

Before stating our main results, let us introduce two classes of functions.

Definition 1.1

(\(\mathcal{G}\) class)

Let \(\mathcal{G}\) be the set of all nonnegative (or non-positive) and monotonic \(\mathcal{C}^{1}(\mathbf{R}_{+})\) functions φ such that \(\varUpsilon _{\varphi }(t):=\frac{\varphi (t)}{t\varphi '(t)}\) with \(|\varUpsilon _{\varphi }(t)|\leq C_{\varphi }\), where \(C_{\varphi }\) is a positive constant that depends only on φ.

Definition 1.2

(\(\mathfrak{F}\) class)

We denote by \(\mathfrak{F}\) the set of all functions ϕ satisfying one of the following conditions:

  1. (i)

    φ is a positive increasing \(\mathcal{C}^{1} ( \mathbf{R}_{+})\) function such that \(t\varphi '(t)\geq C_{\varphi } \varphi (t)\) and \(\varphi (2t)\leq c_{\varphi }\varphi (t)\) for all \(t>0\), where \(C_{\varphi }\) and \(c_{\varphi }\) are independent of t.

  2. (ii)

    φ is a positive decreasing \(\mathcal{C}^{1} ( \mathbf{R}_{+})\) function such that \(t\varphi '(t)\leq -C_{\varphi } \varphi (t)\) and \(\varphi (t)\leq c_{\varphi }\varphi (2t)\) for all \(t>0\), where \(C_{\varphi }\) and \(c_{\varphi }\) are independent of t.

Remark 1.1

It is clear that \(\mathfrak{F} \subsetneq \mathcal{G}\). There are some model examples for the class \(\mathfrak{F}\), such as \(t^{\alpha }\) (\(\alpha \neq 0\)), \(t^{\alpha }( \ln (1+t))^{\beta }\) (\(\alpha ,\beta >0\)), \(t\ln \ln (\mathbf{e}+t)\), \(t^{-1}\ln (1+t^{-1})\), real-valued polynomials P on R with positive coefficients and \(P(0)=0\), and so on. Note that the following facts are valid (see [18]): if \(\varphi \in \mathcal{G}\), then

  1. (a)

    \(\lim_{t\rightarrow 0}\varphi (t)=0\) and \(\lim_{t\rightarrow \infty }|\varphi (t)|=\infty \) if φ is nonnegative and increasing, or non-positive and decreasing;

  2. (b)

    \(\lim_{t\rightarrow 0}|\varphi (t)|=\infty \) and \(\lim_{t \rightarrow \infty }\varphi (t)=0\) if φ is nonnegative and decreasing, or non-positive and increasing.

In this paper we shall establish the following.

Theorem 1.1

Let \(T_{h,\varOmega ,P, \varphi }\) be the singular integral operator defined by (1.2) and \(\varphi \in \mathcal{G}\). If \(\varOmega \in H^{1}(\mathbf{S}^{n-1})\) satisfies (1.1) and \(h\in \Delta _{\gamma }(\mathbf{R}_{+})\) for some \(\gamma \in (1,\infty ]\), then the operator \(T_{h,\varOmega ,P, \varphi }\) is bounded on \(L^{2}(\mathbf{R}^{n})\). More precisely, we have

$$ \Vert T_{h,\varOmega ,P,\varphi }f \Vert _{L^{2}(\mathbf{R}^{n})}\leq C \Vert h \Vert _{ \Delta _{\gamma }(\mathbf{R}_{+})} \Vert \varOmega \Vert _{H^{1}(\mathbf{S}^{n-1})} \Vert f \Vert _{L^{2}(\mathbf{R}^{n})}, $$

where \(C>0\) is independent of h, γ, Ω, f and the coefficients of P, but depends on φ and \(\deg (P)\).

Remark 1.2

By (1.3), Theorem 1.1 generalizes and improves Theorem A.

The second type of our operators we consider is the parametric Marcinkiewicz integral operator along polynomial compound curves. More precisely, let h, Ω, P, φ be given as in (1.2). For a complex number \(\rho =\tau +i\vartheta \) (\(\tau , \vartheta \in \mathbf{R}\) with \(\tau >0\)), we consider the parametric Marcinkiewicz integral operator \(\mathfrak{M}_{h,\varOmega ,P,\varphi , \rho }\) along “polynomial compound curve” \(P(\varphi (|y|))y'\) on \(\mathbf{R}^{n}\) by

$$ \mathfrak{M}_{h,\varOmega ,P,\varphi ,\rho }f(x)= \biggl( \int _{0}^{\infty } \biggl\vert \frac{1}{t^{\rho }} \int _{ \vert y \vert \leq t}f\bigl(x-P\bigl(\varphi \bigl( \vert y \vert \bigr)\bigr)y'\bigr)\frac{h( \vert y \vert ) \varOmega (y')}{ \vert y \vert ^{n-\rho }}\,dy \biggr\vert ^{2}\frac{dt}{t} \biggr)^{1/2}. $$
(1.4)

If \(\rho =1\), \(P(t)\equiv t\), and \(\varphi (t)\equiv t\), \(\mathfrak{M}_{h,\varOmega ,P,\varphi ,\rho }\) is just the classical Marcinkiewicz integral operator, which is denoted by \(\mathfrak{M} _{h,\varOmega }\). The \(L^{p}\) mapping properties of \(\mathfrak{M}_{h, \varOmega }\) and other extensions have been studied by many authors extensively (see [19,20,21,22,23,24,25,26,27,28] for example). Particularly, Ding et al. [21] proved that the parametric Marcinkiewicz integral operator along polynomial mappings is of type \((p, p)\) for \(1< p<\infty \) if \(\varOmega \in H^{1}(\mathbf{S}^{n-1})\) and \(h\in L^{\infty }(\mathbf{R}_{+})\). In this paper we focus on the parametric Marcinkiewicz integral operator along polynomial compound curves. More precisely, we shall establish the following result.

Theorem 1.2

Let \(\mathfrak{M}_{h, \varOmega , P,\varphi ,\rho }\) be the Marcinkiewicz integral operator defined by (1.4) and \(\varphi \in \mathfrak{F}\). If \(\varOmega \in H ^{1} (\mathbf{S}^{n-1})\) satisfies (1.1) and \(h\in \Delta _{\gamma }( \mathbf{R}_{+})\) for some \(\gamma \in (1,\infty ]\), then the operator \(\mathfrak{M}_{h,\varOmega ,P,\varphi ,\rho }\) is bounded on \(L^{2}( \mathbf{R}^{n})\). More precisely, we have

$$ \Vert \mathfrak{M}_{h,\varOmega ,P,\varphi ,\rho }f \Vert _{L^{2}(\mathbf{R}^{n})} \leq C \frac{\gamma }{\gamma -1} \Vert h \Vert _{\Delta _{\gamma }(\mathbf{R}_{+})} \Vert \varOmega \Vert _{H^{1}(\mathbf{S}^{n-1})} \Vert f \Vert _{L^{2}(\mathbf{R}^{n})}, $$

where \(C>0\) is independent of h, γ, Ω, f and the coefficients of P, but depends on φ, ρ and \(\deg (P)\).

Remark 1.3

  1. (i)

    It was shown in [22] that the operator \(\mathfrak{M}_{h,\varOmega ,P,\varphi , \rho }\) with \(\rho >0\) and \(P(t)\equiv t\) is bounded on \(L^{p}( \mathbf{R}^{n})\) for p with \(|1/p-1/2|<\min \{1/2,1/\gamma '\}\) under the same conditions of Theorem 1.2 (see also [27] for more generalization results).

  2. (ii)

    Theorem 1.2 is new even in the special case \(\rho =1\), \(h(t)\equiv 1\), and \(\varphi (t)\equiv t\).

In 2009, Sato [29] introduced a class of functions \(\mathcal{N}_{\alpha }(\mathbf{R}_{+})\), which may be the most desirable size condition on the radial kernel of rough singular integrals so far. Here \(\mathcal{N}_{\alpha }(\mathbf{R}_{+})\) for \(\alpha >0\) is the set of all measurable functions \(h:\mathbf{R}_{+}\rightarrow \mathbf{R}\) satisfying

$$ N_{\alpha }(h)=\sum_{m=1}m^{\alpha }2^{m} \sup_{k\in \mathbb{Z}}2^{-k} \bigl\vert E(k,m) \bigr\vert < \infty , $$

where \(E(k,1)=\{t\in (2^{k},2^{k+1}]; |h(t)|\leq 2\}\) and

$$ E(k,m)=\bigl\{ t\in (2^{k},2^{k+1}]; 2^{m-1}< \bigl\vert h(t) \bigr\vert \leq 2^{m}\bigr\} \quad \text{for } m\geq 2. $$

It is easy to check that

$$ \Delta _{\gamma }(\mathbf{R}_{+})\subsetneq \mathcal{N}_{\alpha }( \mathbf{R}_{+}), \quad \forall \gamma >1 \text{ and } \alpha >0. $$

Applying Theorems 1.1–1.2 and the extrapolation arguments following from [29], we have the following result.

Corollary 1.1

Let \(T_{h,\varOmega ,P, \varphi }\) and \(\mathfrak{M}_{h,\varOmega ,P,\varphi ,\rho }\) be defined by (1.2) and (1.4), respectively. Let Ω satisfy (1.1) and \(\varOmega \in H^{1}(\mathbf{S}^{n-1})\). Then

  1. (i)

    If \(h\in \mathcal{N}_{\alpha }( \mathbf{R}_{+})\) for some \(\alpha >0\) and \(\varphi \in \mathcal{G}\), then the operator \(T_{h, \varOmega ,P,\varphi }\) is bounded on \(L^{2}(\mathbf{R}^{n})\). More precisely, we have

    $$ \Vert T_{h,\varOmega ,P,\varphi }f \Vert _{L^{2}(\mathbf{R}^{n})}\leq C\bigl(1+N_{ \alpha }(h) \bigr) \Vert \varOmega \Vert _{H^{1}({ {\mathbf{S}}}^{n-1})} \Vert f \Vert _{L^{2}(\mathbf{R}^{n})}, $$

    where \(C>0\) is independent of h, α, Ω, f and the coefficients of P, but depends on φ and \(\deg (P)\).

  2. (ii)

    If \(h\in \mathcal{N}_{1}( \mathbf{R}_{+})\) and \(\varphi \in \mathfrak{F}\), then the operator \(\mathfrak{M}_{h, \varOmega ,P,\varphi ,\rho }\) is bounded on \(L^{2}( \mathbf{R}^{n})\). More precisely, we have

    $$ \Vert \mathfrak{M}_{h,\varOmega ,P,\varphi ,\rho }f \Vert _{L^{2}(\mathbf{R}^{n})} \leq C \bigl(1+N_{1}(h)\bigr) \Vert \varOmega \Vert _{H^{1}({ {\mathbf{S}}}^{n-1})} \Vert f \Vert _{L^{2}(\mathbf{R}^{n})}, $$

    where \(C>0\) is independent of h, Ω, f and the coefficients of P, but depends on φ, ρ and \(\deg (P)\).

The third type of our operators is the maximal functions related to the singular integrals and Marcinkiewicz integrals along polynomial compound curves. More precisely, let \(T_{h,\varOmega ,P, \varphi }\) and \(\mathfrak{M}_{h,\varOmega ,P,\varphi ,\varrho }\) be defined as in (1.2) and (1.4), respectively. We define the maximal operators \(\mathcal{S} _{\varOmega ,P,\varphi }\) and \(\mathcal{M}_{\varOmega , P,\varphi ,\rho }\) along the “polynomial compound curve” \(P(\varphi (|y|))y'\) on \(\mathbf{R}^{n}\) by

$$\begin{aligned}& \mathcal{S}_{\varOmega ,P,\varphi }f(x)=\sup_{h\in \mathcal{K} _{2}} \bigl\vert T_{h,\varOmega ,P,\varphi }f(x) \bigr\vert , \end{aligned}$$
(1.5)
$$\begin{aligned}& \mathcal{M}_{\varOmega ,P,\varphi ,\rho }f(x)=\sup_{h\in \mathcal{K}_{2}} \bigl\vert \mathfrak{M}_{h,\varOmega ,P,\varphi , \rho }f(x) \bigr\vert , \end{aligned}$$
(1.6)

where \(\mathcal{K}_{2}\) is the set of all measurable functions \(h:\mathbf{R}_{+}\rightarrow \mathbf{R}\) with \(\|h\|_{L^{2} ( \mathbf{R}_{+},r^{-1}\,dr)}\leq 1\). Here \(L^{\gamma }(\mathbf{R}_{+}, r ^{-1}\,dr)\) (\(\gamma >0\)) is the set of all measurable functions \(h:\mathbf{R}_{+}\rightarrow \mathbf{R}\) that satisfies

$$ \Vert h \Vert _{L^{\gamma }(\mathbf{R}_{+},r^{-1}\,dr)}:= \biggl( \int _{0}^{\infty } \bigl\vert h(r) \bigr\vert ^{ \gamma }r^{-1}\,dr \biggr)^{1/\gamma }< \infty . $$

Clearly, \(L^{\gamma }(\mathbf{R}_{+},r^{-1}\,dr)\subsetneq \Delta _{ \gamma }(\mathbf{R}_{+})\) for \(\gamma >0\).

The rest of the main results can be formulated as follows.

Theorem 1.3

Let \(\mathcal{S}_{\varOmega , P,\varphi }\) and \(\mathcal{M}_{\varOmega ,P,\varphi ,\rho }\) be the maximal operators defined by (1.5) and (1.6), respectively. If \(\varOmega \in H^{1}(\mathbf{S}^{n-1})\) satisfies (1.1) and \(\varphi \in \mathcal{G}\), then both the operators \(\mathcal{S}_{\varOmega ,P, \varphi }\) and \(\mathcal{M}_{\varOmega ,P, \varphi ,\rho }\) are bounded on \(L^{2}(\mathbf{R}^{n})\). More precisely, we have

$$\begin{aligned}& \Vert \mathcal{S}_{\varOmega ,P,\varphi }f \Vert _{L^{2}(\mathbf{R}^{n})}\leq C \Vert \varOmega \Vert _{H^{1}(\mathbf{S}^{n-1})} \Vert f \Vert _{L^{2}(\mathbf{R}^{n})}, \\& \Vert \mathcal{M}_{\varOmega ,P,\varphi ,\rho }f \Vert _{L^{2}(\mathbf{R}^{n})} \leq C \Vert \varOmega \Vert _{H^{1}(\mathbf{S}^{n-1})} \Vert f \Vert _{L^{2}(\mathbf{R} ^{n})}, \end{aligned}$$

where \(C>0\) is independent of Ω, f and the coefficients of P, but depends on φ and \(\deg (P)\).

Remark 1.4

The maximal operator related to singular integrals, which is denoted by \(\mathcal{S}_{\varOmega }\) and corresponds to the special case of \(\mathcal{S}_{\varOmega ,P, \varphi }\) with \(P(t)=\varphi (t)=t\), was first introduced by Chen and Lin [30]. Chen and Lin proved that if \(\varOmega \in \mathcal{C}( \mathbf{S}^{n-1})\), then \(\mathcal{S}_{\varOmega }\) is of type \((p, p)\) for any \(p>2n/(2n-1)\) and the range of p is best possible. Subsequently, the \(L^{p}\) mapping properties of \(\mathcal{S}_{\varOmega }\) have been discussed extensively by many authors. Particularly, Xu et al. [31] established the \(L^{p}(\mathbf{R}^{n})\) bounds for \(\mathcal{S}_{\varOmega }\) with \(2\leq p<\infty \), provided that \(\varOmega \in H^{1}(\mathbf{S}^{n-1})\) satisfying (1.1). It should be pointed out that Theorem 1.3 is new even in the special case \(\varphi (t)\equiv t\).

The rest of this paper is organized as follows. After recalling some preliminary notations and lemmas in Sect. 2, we prove the main results in Sect. 3. Finally, we present certain Hardy type inequalities related to the parametric Marcinkiewicz integral operators and maximal operators related to singular integrals along polynomial compound curves in Sect. 4. We would like to remark that our main results and proofs are inspired by the work in [13], but our main results and proofs are more delicate and complex than those of [13]. Some ideas in our proofs are taken from [18, 29, 32, 33]. Throughout the paper, we denote by \(p'\) the conjugate index of p, which satisfies \({1}/{p}+ {1}/{p'}=1\). The letter C or c, sometimes with certain parameters, will stand for positive constants that are not necessarily the same ones at each occurrence, but are independent of the essential variables. In what follows, we denote \(\mathbf{e}=\lim_{x\rightarrow \infty }(1+ \frac{1}{x})^{x}\).

2 Preliminary definitions and lemmas

Recall that the Hardy space \(H^{1}(\mathbf{S}^{n-1})\) is the set of all functions \(\varOmega \in L^{1}(\mathbf{S}^{n-1})\) satisfying the condition

$$ \Vert \varOmega \Vert _{H^{1}(\mathbf{S}^{n-1})} := \biggl\Vert \sup _{0\leq r< 1} \biggl\vert \int _{\mathbf{S}^{n-1}}\varOmega (\theta )P_{r(\cdot )} (\theta )\,d \sigma (\theta ) \biggr\vert \biggr\Vert _{L^{1}(\mathbf{S}^{n-1})}< \infty , $$

where \(P_{rw}(\theta )=\frac{1-r^{2}}{|rw-\theta |^{n}}\) for \(0\leq r<1\) and \(\theta , w\in {\mathbf{S}}^{n-1}\) denotes the Poisson kernel on \(\mathbf{S}^{n-1}\).

Definition 2.1

(\(H^{1}(\mathbf{S}^{n-1})\) atoms)

A function \(a: \mathbf{S}^{n-1}\rightarrow \mathbf{C}\) is a \((1,\infty )\) atom if there exist \(\vartheta \in {\mathbf{S}}^{n-1}\) and \(\varrho \in (0,2]\) such that

$$\begin{aligned}& \operatorname{supp}(a)\subset {\mathbf{S}}^{n-1}\cap B(\vartheta ,\varrho ), \quad \text{where } B(\vartheta ,\varrho )=\bigl\{ y\in \mathbf{R}^{n}: \vert y- \vartheta \vert < \varrho \bigr\} ; \end{aligned}$$
(2.1)
$$\begin{aligned}& \Vert a \Vert _{L^{\infty }(\mathbf{S}^{n-1})}\leq \varrho ^{-n+1}; \end{aligned}$$
(2.2)
$$\begin{aligned}& \int _{\mathbf{S}^{n-1}}a(y)\,d\sigma (y)=0. \end{aligned}$$
(2.3)

An important property of \(H^{1}(\mathbf{S}^{n-1})\) is the atomic decomposition, which is listed as follows:

Lemma 2.1

([34, 35])

If \(\varOmega \in H^{1}(\mathbf{S}^{n-1})\) satisfies the cancelation condition (1.1), then there exist a sequence of complex numbers \(\{c_{j}\}\) and a sequence of \((1,\infty )\) atoms \(\{\varOmega _{j}\}\) such that \(\varOmega =\sum_{j}c_{j}\varOmega _{j}\) and \(\|\varOmega \|_{H^{1}(\mathbf{S} ^{n-1})}\approx \sum_{j}|c_{j}|\).

The following results are known (see [36]).

Lemma 2.2

([36])

Suppose that \(n\geq 3\) and \(b(\cdot )\) is a \((1,\infty )\) atom on \(\mathbf{S}^{n-1}\) supported in \(\mathbf{S}^{n-1}\cap B(\zeta , \varrho )\), where \(\zeta \in {\mathbf{S}}^{n-1}\) and \(\varrho \in (0,2]\). Let

$$ F_{b}(s)=\bigl(1-s^{2}\bigr)^{{(n-3)}/{2}}\chi _{(-1,1)}(s) \int _{\mathbf{S}^{n-2}}b\bigl(s,\bigl(1-s ^{2} \bigr)^{{1}/{2}}\tilde{y}\bigr)\,d\sigma (\tilde{y}). $$

Then there exists a constant C, independent of b, such that

$$\begin{aligned}& \operatorname{supp}(F_{b})\subset \bigl(\zeta _{1}-2r(\zeta ),\zeta _{1}+2r(\zeta )\bigr); \end{aligned}$$
(2.4)
$$\begin{aligned}& \Vert F_{b} \Vert _{L^{\infty }(\mathbf{R})}\leq \frac{C}{r(\zeta )}; \end{aligned}$$
(2.5)
$$\begin{aligned}& \int _{\mathbf{R}}F_{b}(s)\,ds=0, \end{aligned}$$
(2.6)

where \(\zeta =(\zeta _{1},\ldots ,\zeta _{n})\), \(r(\zeta )=|L_{\varrho }(\zeta )|\), and \(L_{\varrho }(\zeta )=(\varrho ^{2}\zeta _{1},\varrho \zeta _{2},\ldots , \varrho \zeta _{n})\).

Lemma 2.3

([36])

Suppose that \(n=2\) and \(b(\cdot )\) is a \((1,\infty )\) atom on \(\mathbf{S}^{1}\) supported in \(\mathbf{S}^{1}\cap B(\zeta ,\varrho )\), where \(\zeta \in {\mathbf{S}}^{1}\) and \(\varrho \in (0,2]\). Let

$$ F_{b}(s)=\bigl(1-s^{2}\bigr)^{-{1}/{2}}\chi _{(-1,1)}(s) \bigl(b\bigl(s,\bigl(1-s^{2}\bigr)^{{1}/ {2}} \bigr)+b\bigl(s,-\bigl(1-s^{2}\bigr)^{{1}/{2}}\bigr)\bigr). $$

Then \(F_{b}(\cdot )\) satisfies (2.4), (2.6) and

$$ \Vert F_{b} \Vert _{L^{q}(\mathbf{R})}\leq C \bigl\vert L_{\varrho }(\zeta ) \bigr\vert ^{-1+{1}/ {q}} $$

for some \(q\in (1,2)\), where \(\zeta =(\zeta _{1},\zeta _{2})\), \(r(\zeta )=|L_{\varrho }(\zeta )|\), and \(L_{\varrho }(\zeta )=(\varrho ^{2}\zeta _{1},\varrho \zeta _{2})\).

The following lemmas will play key roles in the proof of Theorem 1.1.

Lemma 2.4

([18])

Let \(\varphi \in \mathcal{G}\) and \(h\in \Delta _{\gamma }(\mathbf{R}_{+})\) for some \(\gamma >1\), then

$$ \bigl\Vert h\bigl(\varphi ^{-1}\bigr)\varUpsilon _{\varphi } \bigl(\varphi ^{-1}\bigr) \bigr\Vert _{\Delta _{\gamma }(\mathbf{R}_{+})}\leq C \Vert h \Vert _{\Delta _{\gamma }(\mathbf{R}_{+})}, $$

where the constant \(C>0\) depends only on φ.

Lemma 2.5

Let \(\varphi \in \mathcal{G}\) and \(\varUpsilon _{\varphi }(t)=\frac{\varphi (t)}{t\varphi '(t)}\). Then

  1. (i)

    if φ is nonnegative and increasing, \(T_{h,\varOmega ,P,\varphi }f=T_{h(\varphi ^{-1})\varUpsilon _{\varphi }( \varphi ^{-1}),\varOmega ,P}f\);

  2. (ii)

    if φ is nonnegative and decreasing, \(T_{h,\varOmega ,P,\varphi }f=-T_{h(\varphi ^{-1})\varUpsilon _{\varphi }( \varphi ^{-1}),\varOmega ,P}f\);

  3. (iii)

    if φ is non-positive and decreasing, \(T_{h,\varOmega ,P,\varphi }f=T_{h(\varphi ^{-1})\varUpsilon _{\varphi }( \varphi ^{-1}),\tilde{\varOmega },P}f\);

  4. (iv)

    if φ is non-positive and increasing, \(T_{h,\varOmega ,P,\varphi }f=-T_{h(\varphi ^{-1})\varUpsilon _{\varphi }( \varphi ^{-1}),\tilde{\varOmega },P}f\),

where \(\tilde{\varOmega }(y)=\varOmega (-y)\).

Lemma 2.5 can be proved by similar arguments as in the proof of [18, Lemma 2.3], we omit the details.

3 Proofs of the main results

In this section we shall prove Theorems 1.1–1.3. In what follows, we let \(\deg (P)=N\) and \(P(t)=\sum_{k=1}^{N}a_{k}t^{k}\) and assume that \(\varOmega \in H^{1}(\mathbf{S}^{n-1})\) satisfies (1.1).

Proof of Theorem 1.1

By Lemmas 2.4 and 2.5, to prove Theorem 1.1, it suffices to show that

$$ \Vert T_{h,\varOmega ,P}f \Vert _{L^{2}(\mathbf{R}^{n})}\leq C \Vert h \Vert _{\Delta _{ \gamma }(\mathbf{R}_{+})} \Vert \varOmega \Vert _{H^{1}(\mathbf{S}^{n-1})} \Vert f \Vert _{L^{2}(\mathbf{R}^{n})}, $$
(3.1)

where \(C>0\) is independent of h, Ω, f and the coefficients of P, but depends on \(\deg (P)\). It is clear that \(T_{h,\varOmega ,P}f(x)=K _{h,\varOmega ,P}*f(x)\), where the function \(K_{h,\varOmega ,P}\) is defined by

$$ \widehat{K_{h,\varOmega ,P}}(\xi )= \int _{\mathbf{R}^{n}}{\mathbf{e}}^{-2 \pi i P( \vert y \vert )y'\cdot \xi }\frac{h( \vert y \vert )\varOmega (y')}{ \vert y \vert ^{n}}\,dy. $$

By Plancherel’s theorem, (3.1) reduces to the following:

$$ \Vert \widehat{K_{h,\varOmega ,P}} \Vert _{L^{\infty }(\mathbf{R}^{n})}\leq C \Vert h \Vert _{\Delta _{\gamma }(\mathbf{R}_{+})} \Vert \varOmega \Vert _{H^{1}(\mathbf{S} ^{n-1})}, $$
(3.2)

where \(C>0\) is independent of h, Ω and the coefficients of P. Invoking Lemma 2.1, we can write \(\varOmega =\sum_{j}\lambda _{j} a _{j}\) satisfying \(\sum_{j}|\lambda _{j}|\leq C\|\varOmega \|_{H^{1} ( \mathbf{S}^{n-1})}\), where each \(a_{j}\) is a \((1,\infty )\) atom and each \(\lambda _{j}\) is a complex number. Therefore, to prove (3.2), it is enough to prove that

$$ \Vert \widehat{K_{h,\varOmega ,P}} \Vert _{L^{\infty }(\mathbf{R}^{n})}\leq C \Vert h \Vert _{\Delta _{\gamma }(\mathbf{R}_{+})}, $$
(3.3)

where Ω is a \((1,\infty )\) atom and \(C>0\) is independent of h, Ω and the coefficients of P.

We now prove (3.3). Let Ω be a \((1,\infty )\) atom satisfying (2.1)–(2.3). Without loss of generality, we may assume that \(\vartheta =(1,0,\ldots ,0)\). We only prove the case \(n\geq 3\), since the proof for \(n=2\) is essentially the same (using Lemma 2.3 instead of Lemma 2.2). Fix \(\xi \neq (0,0,\ldots ,0)\) and write \(\xi '={\xi }/ {|\xi |}=(\xi _{1}',\ldots ,\xi _{n}')\). We can choose a rotation \(\mathcal{O}\) such that \(\mathcal{O}(\xi ')=\vartheta \). By the change of variables, we write

$$ \widehat{K_{h,\varOmega ,P}}(\xi )= \int _{0}^{\infty }h(t) \int _{\mathbf{S}^{n-1}}{\mathbf{e}}^{-2\pi i P(t)|\xi |y'\cdot \iota }\varOmega \bigl( \mathcal{O}^{-1}y'\bigr)\,d\sigma \bigl(y' \bigr)\frac{dt}{t}. $$

Let \(b(y')=\varOmega (\mathcal{O}^{-1}y')\). It is easy to see that b is a regular \((1,\infty )\) atom satisfying (2.2)–(2.3) and \(\operatorname{supp}(b) \subset B(\xi ',\varrho )\cap {\mathbf{S}}^{n-1}\). By the change of variables, we have

$$ \widehat{K_{h,\varOmega ,P}}(\xi )= \int _{0}^{\infty }h(t) \int _{ \mathbf{R}}F_{b}(s)\mathbf{e}^{-2\pi i P(t)|\xi |s}\,ds \frac{dt}{t}, $$

where \(F_{b}\) is the function defined in Lemma 2.2. We know by Lemma 2.2 that \(\operatorname{supp}(F_{b})\subset (\xi _{1}'-2r(\xi '),\xi _{1}'+ 2r(\xi '))\) and \(F_{b}\) satisfies (2.5)–(2.6), where \(r(\xi ')= |(\varrho ^{2}\xi _{1}', \varrho \xi _{2}',\ldots ,\varrho \xi _{n}')|\). We set \(N_{b}(s)=r( \xi ')F_{b}(r(\xi ')s+\xi _{1}')\). One can easily check that \(\operatorname{supp}(N_{b})\subset (-2,2)\) and \(\|N_{b}\|_{L^{\infty }(\mathbf{R})} \leq C\) and \(\int _{\mathbf{R}}N_{b}(s)\,ds=0\). By the change of variables again,

$$ \widehat{K_{h,\varOmega ,P}}(\xi )= \int _{0}^{\infty }h(t) \int _{ \mathbf{R}}N_{b}(s)\mathbf{e}^{-2\pi i P(t)|\xi |r(\xi ')s}\,ds \frac{dt}{t}. $$

For convenience, we set

$$ b_{k}=2\pi a_{k} \vert \xi \vert r\bigl(\xi '\bigr), \qquad \vert \beta _{\kappa } \vert ^{{1}/{\kappa }}=\max_{1\leq k\leq N} \vert b_{k} \vert ^{ {1}/{k}}, \quad \text{and} \quad \beta = \vert \beta _{\kappa } \vert ^{-{1}/{\kappa }}. $$

We can write

$$\begin{aligned} \widehat{K_{h,\varOmega ,P}}(\xi ) =& \int _{0}^{\beta }h(t) \int _{\mathbf{R}}N_{b}(s)\mathbf{e}^{-2\pi i P(t)| \xi |r(\xi ')s}\,ds \frac{dt}{t} \\ &{} + \int _{\beta }^{\infty }h(t) \int _{\mathbf{R}}N_{b}(s)\mathbf{e}^{-2 \pi i P(t)|\xi |r(\xi ')s}\,ds \frac{dt}{t} \\ =:&I_{1}+I_{2}. \end{aligned}$$
(3.4)

For \(I_{1}\), let us choose an integer \(K_{0}\) such that \(2^{K_{0}} \leq \beta <2^{K_{0}+1}\). By the cancelation condition of \(N_{b}\) and Hölder’s inequality, we have

$$\begin{aligned} \vert I_{1} \vert =& \biggl\vert \int _{0}^{\beta }h(t) \int _{\mathbf{R}}N_{b}(s) \bigl(\mathbf{e}^{-2 \pi i P(t) \vert \xi \vert r(\xi ')s}-1 \bigr)\,ds\frac{dt}{t} \biggr\vert \\ \leq& \sum_{j=-\infty }^{K_{0}} \int _{2^{j}}^{2^{j+1}} \bigl\vert h(t) \bigr\vert \int _{\mathbf{R}} \bigl\vert N_{b}(s) \bigl( \mathbf{e}^{-2\pi i P(t) \vert \xi \vert r(\xi ')s}-1\bigr) \bigr\vert \,ds \frac{dt}{t} \\ \leq& C \Vert h \Vert _{\Delta _{\gamma }(\mathbf{R}_{+})}\sum_{j=-\infty }^{K _{0}} \Biggl( \int _{2^{j}}^{2^{j+1}} \Biggl(\sum _{k=1}^{N} \vert b_{k} \vert t ^{k} \Biggr)^{\gamma '}\frac{dt}{t} \Biggr)^{{1}/{\gamma '}} \\ \leq& C \Vert h \Vert _{\Delta _{\gamma }(\mathbf{R}_{+})}\sum_{j=-\infty }^{K _{0}} \sum_{k=1}^{N} \vert b_{k} \vert \biggl( \int _{2^{j}}^{2^{j+1}}t^{k \gamma '-1}\,dt \biggr)^{{1}/{\gamma '}} \\ =& C \Vert h \Vert _{\Delta _{\gamma }(\mathbf{R}_{+})}\sum_{j=-\infty }^{K _{0}} \sum_{k=1}^{N} \vert b_{k} \vert \bigl(k\gamma '\bigr)^{-{1}/{\gamma '}}\bigl(2^{k \gamma '}-1 \bigr)^{{1}/{\gamma '}}2^{kj} \\ \leq& C \Vert h \Vert _{\Delta _{\gamma }(\mathbf{R}_{+})}, \end{aligned}$$
(3.5)

where in the last inequality of (3.5) we have used the fact that \(\alpha ^{\alpha }\leq 1\) for all \(\alpha \in (0,1]\) and \(\max_{1\leq k\leq N}|b_{k}|2^{K_{0}k}\leq 1\). Here \(C>0\) is independent of f, ζ and the coefficients of P.

For \(I_{2}\), let \(\varPhi \in \mathcal{C}_{0}^{\infty }(\mathbf{R})\) such that \(\varPhi (t)\equiv 1\) if \(|t|\leq 1\) and \(\varPhi (t)\equiv 0\) if \(|t|\geq 2\). For any \(j\in \mathbf{Z}\), we set \(R_{j}=[2^{j},2^{j+1})\) and define the operator \(T_{j}\) by

$$ T_{j}f(t)=\chi _{R_{j}}(t) \int _{\mathbf{R}}\varPhi (s)f(s)\mathbf{e}^{-2 \pi i P(t)|\xi |r(\xi ')s}\,ds. $$

From the estimate on page 60 in [32], there exists a large integer \(\varLambda >0\) independent of j such that

$$ \Vert T_{j}f \Vert _{L^{2}(\mathbf{R})}\leq C2^{{j}/{2}} \vert \beta _{k} \vert ^{-{1}/ {(2\varLambda )}}2^{-{j\kappa }/{(2\varLambda )}} \Vert f \Vert _{L^{2}(\mathbf{R})}, $$
(3.6)

where \(C>0\) is independent of f, ξ and the coefficients of P. By (3.6) and Hölder’s inequality,

$$\begin{aligned} \vert I_{2} \vert \leq& \sum _{j=K_{0}}^{\infty } \int _{2^{j}}^{2^{j+1}} \bigl\vert h(t) \bigr\vert \biggl\vert \int _{\mathbf{R}}N_{b}(s)\mathbf{e}^{-2\pi i P(t) \vert \xi \vert r(\xi ')s}\,ds \biggr\vert \frac{dt}{t} \\ \leq& 2 \Vert h \Vert _{\Delta _{\gamma }(\mathbf{R}_{+})}\sum_{j=K_{0}}^{ \infty } \biggl( \int _{2^{j}}^{2^{j+1}} \biggl\vert \int _{\mathbf{R}}N_{b}(s) \mathbf{e}^{-2\pi i P(t) \vert \xi \vert r(\xi ')s}\,ds \biggr\vert ^{\gamma '} \frac{dt}{t} \biggr)^{{1}/{\gamma '}} \\ \leq& C \Vert h \Vert _{\Delta _{\gamma }(\mathbf{R}_{+})}\sum_{j=K_{0}}^{ \infty } \biggl( \int _{2^{j}}^{2^{j+1}} \biggl\vert \int _{\mathbf{R}}N_{b}(s) \mathbf{e}^{-2\pi i P(t) \vert \xi \vert r(\xi ')s}\,ds \biggr\vert ^{2}\frac{dt}{t} \biggr)^{ {1}/{\tilde{\gamma }}} \\ \leq& C \Vert h \Vert _{\Delta _{\gamma }(\mathbf{R}_{+})}\sum_{j=K_{0}}^{ \infty }2^{-{j}/{\tilde{\gamma }}} \biggl( \int _{\mathbf{R}} \biggl\vert \chi _{R _{j}}(t) \int _{\mathbf{R}}N_{b}(s)\mathbf{e}^{-2\pi i P(t) \vert \xi \vert r( \xi ')s}\,ds \biggr\vert ^{2}\,dt \biggr)^{{1}/{\tilde{\gamma }}} \\ \leq& C \Vert h \Vert _{\Delta _{\gamma }(\mathbf{R}_{+})}\sum_{j=K_{0}}^{ \infty }2^{-{j}/{\tilde{\gamma }}} \bigl(2^{j} \vert \beta _{\kappa } \vert ^{-{1}/ {\varLambda }}2^{-{j\kappa }/{\varLambda }} \Vert N_{b} \Vert _{L^{2}(\mathbf{R})} ^{2} \bigr)^{{1}/{\tilde{\gamma }}} \\ \leq& C \Vert h \Vert _{\Delta _{\gamma }(\mathbf{R}_{+})}, \end{aligned}$$
(3.7)

where in the last inequality of (3.7) we have used the fact that \(2^{K_{0}\kappa }\geq 2^{-\kappa }\beta _{\kappa }^{-1}\). Here \(\tilde{\gamma }=\max \{2,\gamma '\}\) and the constant \(C>0\) is independent of h, b, ξ and the coefficients of P. We get from (3.4)–(3.5) and (3.7) that

$$ \bigl\vert \widehat{K_{h,\varOmega ,P}}(\xi ) \bigr\vert \leq C \Vert h \Vert _{\Delta _{\gamma }( \mathbf{R}_{+})}, $$

where \(C>0\) is independent of h, Ω, ξ and the coefficients of P. This yields (3.3) and completes the proof of Theorem 1.1. □

Proof of Theorem 1.2

Let h, Ω, φ be given as in Theorem 1.2. We only prove Theorem 1.2 for the case \(\varphi \in \mathfrak{F}\) satisfying condition (a), and another case is discussed similarly. For \(t>0\), we define the measure \(\sigma _{h,\varOmega ,P,\varphi ,t}\) by

$$ \widehat{\sigma _{h,\varOmega ,P,\varphi ,t}}(x)=\frac{1}{t^{\rho }} \int _{{t}/{2}< \vert y \vert \leq t}{\mathbf{e}}^{-2\pi iP(\varphi ( \vert y \vert ))x\cdot y'}\frac{h( \vert y \vert ) \varOmega (y')}{ \vert y \vert ^{n-\rho }} \,dy. $$
(3.8)

By Minkowski’s inequality and the change of variables, one can easily verify that

$$ \mathfrak{M}_{h,\varOmega ,P,\varphi ,\rho }f(x) \leq \frac{1}{1-2^{- \tau }} \biggl( \int _{0}^{\infty } \bigl\vert \sigma _{h,\varOmega ,P,\varphi ,t}*f(x) \bigr\vert ^{2} \frac{dt}{t} \biggr)^{{1}/{2}}. $$
(3.9)

By Plancherel’s theorem and Fubini’s theorem, to prove Theorem 1.2, we only need to show that

$$ \biggl\Vert \biggl( \int _{0}^{\infty } \bigl\vert \widehat{\sigma _{h,\varOmega ,P,\varphi ,t}}(\cdot ) \bigr\vert ^{2}\frac{dt}{t} \biggr)^{1/2} \biggr\Vert _{L^{\infty }(\mathbf{R}^{n})}\leq C \Vert h \Vert _{ \Delta _{\gamma }(\mathbf{R}_{+})} \Vert \varOmega \Vert _{H^{1}(\mathbf{S}^{n-1})}, $$
(3.10)

where \(C>0\) is independent of h, Ω and the coefficients of P, but depends on φ and N.

Invoking Lemma 2.1, (3.10) reduces to the following:

$$ \biggl\Vert \biggl( \int _{0}^{\infty } \bigl\vert \widehat{\sigma _{h,\varOmega ,P,\varphi ,t}}(\cdot ) \bigr\vert ^{2}\frac{dt}{t} \biggr)^{{1}/{2}} \biggr\Vert _{L^{\infty }(\mathbf{R}^{n})}\leq C \Vert h \Vert _{ \Delta _{\gamma }(\mathbf{R}_{+})}, $$
(3.11)

where Ω is a \((1,\infty )\) atom and \(C>0\) is independent of h, Ω and the coefficients of P, but depends on φ and N.

Given \(\xi \neq (0,0,\ldots ,0)\). To prove (3.11), we want to show that

$$ \int _{0}^{\infty } \bigl\vert \widehat{\sigma _{h,\varOmega ,P,\varphi ,t}}(\xi ) \bigr\vert ^{2} \frac{dt}{t}\leq C \Vert h \Vert _{\Delta _{\gamma }(\mathbf{R}_{+})}^{2}, $$
(3.12)

where Ω is a \((1,\infty )\) atom and \(C>0\) is independent of h, Ω, ξ and the coefficients of P.

We now prove (3.12). Without loss of generality we may assume that Ω is a \((1,\infty )\) atom satisfying (2.1)–(2.3) with \(\vartheta =(1,0,\ldots ,0)\) and only prove the case \(n\geq 3\). Let b and \(N_{b}\) be given as in the proof of Theorem 1.1. By some change of variables, we write

$$ \widehat{\sigma _{h,\varOmega ,P,\varphi ,t}}(\xi )=\frac{1}{t^{\rho }} \int _{{t}/{2}}^{t}h(r) \int _{\mathbf{R}}N_{b}(s)\mathbf{e}^{-2\pi i P( \varphi (r))|\xi |r(\xi ')s}\,ds \frac{dr}{r^{1-\rho }}. $$
(3.13)

By (3.13) and Hölder’s inequality, we obtain that

$$ \bigl\vert \widehat{\sigma _{h,\varOmega ,P,\varphi ,t}}(\xi ) \bigr\vert \leq C \Vert h \Vert _{ \Delta _{\gamma }(\mathbf{R}_{+})} \biggl( \int _{t/2}^{t} \biggl\vert \int _{ \mathbf{R}}N_{b}(s)\mathbf{e}^{-2\pi i P(\varphi (r)) \vert \xi \vert r(\xi ')s}\,ds \biggr\vert ^{\gamma '}\frac{dr}{r} \biggr)^{{1}/{\gamma '}}. $$

Hence, to prove (3.12), it suffices to show that

$$ \int _{0}^{\infty } \biggl( \int _{t/2}^{t} \biggl\vert \int _{\mathbf{R}}N_{b}(s) \mathbf{e}^{-2\pi i P(\varphi (r)) \vert \xi \vert r(\xi ')s}\,ds \biggr\vert ^{\gamma '} \frac{dr}{r} \biggr)^{{2}/{\gamma '}} \frac{dt}{t}\leq C, $$
(3.14)

where \(C>0\) is independent of h, Ω, ξ and the coefficients of P.

Next we shall prove (3.14). By a change of variable and the properties of φ,

$$\begin{aligned}& \int _{0}^{\infty } \biggl( \int _{{t}/{2}}^{t} \biggl\vert \int _{\mathbf{R}}N_{b}(s) \mathbf{e}^{-2\pi i P(\varphi (r)) \vert \xi \vert r(\xi ')s}\,ds \biggr\vert ^{\gamma '} \frac{dr}{r} \biggr)^{{2}/{\gamma '}} \frac{dt}{t} \\& \quad \leq C_{\varphi } \int _{0}^{\infty } \biggl( \int _{\varphi ({t}/{2})}^{\varphi (t)} \biggl\vert \int _{\mathbf{R}}N_{b}(s)\mathbf{e}^{-2\pi i P(r) \vert \xi \vert r( \xi ')s}\,ds \biggr\vert ^{\gamma '}\frac{dr}{r} \biggr)^{{2}/{\gamma '}} \frac{dt}{t}. \end{aligned}$$
(3.15)

Let β be given as in the proof of Theorem 1.1 and \(\delta = \varphi ^{-1}(\beta )\). We write

$$\begin{aligned}& \int _{0}^{\infty } \biggl( \int _{\varphi ({t}/{2})}^{\varphi (t)} \biggl\vert \int _{\mathbf{R}}N_{b}(s)\mathbf{e}^{-2\pi i P(r) \vert \xi \vert r(\xi ')s}\,ds \biggr\vert ^{\gamma '}\frac{dr}{r} \biggr)^{{2}/{\gamma '}} \frac{dt}{t} \\& \quad = \int _{0}^{\delta } \biggl( \int _{\varphi ({t}/{2})}^{\varphi (t)} \biggl\vert \int _{\mathbf{R}}N_{b}(s)\mathbf{e}^{-2\pi i P(r) \vert \xi \vert r(\xi ')s}\,ds \biggr\vert ^{\gamma '}\frac{dr}{r} \biggr)^{{2}/{\gamma '}} \frac{dt}{t} \\& \qquad {} + \int _{\delta }^{\infty } \biggl( \int _{\varphi ({t}/{2})}^{\varphi (t)} \biggl\vert \int _{\mathbf{R}}N_{b}(s)\mathbf{e}^{-2\pi i P(r) \vert \xi \vert r(\xi ')s}\,ds \biggr\vert ^{\gamma '}\frac{dr}{r} \biggr)^{{2}/{\gamma '}} \frac{dt}{t} \\& \quad =:J_{1}+J_{2}. \end{aligned}$$
(3.16)

For \(J_{1}\), by the fact that \(\int _{\mathbf{R}}N_{b}(s)\,ds=0\) and the change of variables, we have

$$\begin{aligned} J_{1} =& \int _{0}^{\delta } \biggl( \int _{\varphi ({t}/{2})}^{\varphi (t)} \biggl\vert \int _{\mathbf{R}}N_{b}(s) \bigl(\mathbf{e}^{-2\pi i P(r) \vert \xi \vert r(\xi ')s}-1 \bigr)\,ds \biggr\vert ^{\gamma '}\frac{dr}{r} \biggr)^{{2}/{\gamma '}} \frac{dt}{t} \\ \leq& C\sum_{k=1}^{N} \vert b_{k} \vert ^{2} \int _{0}^{\delta } \biggl( \int _{\varphi ({t}/{2})}^{\varphi (t)}r^{k\gamma '-1}\,dr \biggr)^{{2}/ {\gamma '}}\frac{dt}{t} \\ \leq& C\sum_{k=1}^{N} \vert b_{k} \vert ^{2} \int _{0}^{\delta }\bigl(\varphi (t) \bigr)^{2k} \frac{dt}{t} \\ \leq& C\sum_{k=1}^{N} \vert b_{k} \vert ^{2} \int _{0}^{\varphi (\delta )}t^{2k-1}\,dt \\ \leq& C\sum_{k=1}^{N} \vert b_{k} \vert ^{2}\beta ^{2k}\leq C, \end{aligned}$$
(3.17)

where \(C>0\) is independent of h, Ω, ξ and the coefficients of P, but depends on φ and N.

For \(J_{2}\). Fix \(t\geq \delta \), there exists an integer \(j_{0}\) such that \(2^{j_{0}}\leq \varphi ({t}/{2})<2^{j_{0}+1}\). By the fact that \(\varphi (2t)\leq c_{\varphi }\varphi (t)\), there exists an integer \(k_{0}\) such that \(2^{k_{0}}\leq c_{\varphi }< 2^{k_{0}+1}\) and then \(\varphi (t)\leq 2^{k_{0}+j_{0}+2}\). From (3.6) we have

$$\begin{aligned}& \int _{\varphi ({t}/{2})}^{\varphi (t)} \biggl\vert \int _{\mathbf{R}}N_{b}(s) \mathbf{e}^{-2\pi i P(r) \vert \xi \vert r(\xi ')s}\,ds \biggr\vert ^{2}\frac{dr}{r} \\& \quad \leq \sum_{j=j_{0}}^{k_{0}+j_{0}+1} \int _{2^{j}}^{2^{j+1}} \biggl\vert \int _{\mathbf{R}}N_{b}(s)\mathbf{e}^{-2\pi i P(r) \vert \xi \vert r(\xi ')s}\,ds \biggr\vert ^{2}\frac{dr}{r} \\& \quad \leq \sum_{j=j_{0}}^{k_{0}+j_{0}+1}2^{-j}2^{j} \vert \beta _{\kappa } \vert ^{- {1}/{\varLambda }}2^{-{j\kappa }/{\varLambda }} \Vert N_{b} \Vert _{L^{2}(\mathbf{R})} ^{2} \\& \quad \leq C\sum_{j=j_{0}}^{k_{0}+j_{0}+1} \vert \beta _{\kappa } \vert ^{-{1}/{\varLambda }}2^{-{j\kappa }/{\varLambda }} \\& \quad \leq C \vert \beta _{\kappa } \vert ^{-{1}/{\varLambda }}\varphi (t)^{-{\kappa }/{\varLambda }}, \end{aligned}$$
(3.18)

where \(C>0\) is independent of Ω, ξ and the coefficients of P, but depends on φ. (3.18) together with Hölder’s inequality and the change of variable shows that

$$\begin{aligned} J_{2} \leq& C(\varphi ) \int _{\delta }^{\infty } \biggl( \int _{\varphi ({t}/{2})}^{\varphi (t)} \biggl\vert \int _{\mathbf{R}}N_{b}(s)\mathbf{e}^{-2\pi i P(r) \vert \xi \vert r(\xi ')s}\,ds \biggr\vert ^{2}\frac{dr}{r} \biggr)^{{2}/{\tilde{\gamma }}} \frac{dt}{t} \\ \leq& C \int _{\delta }^{\infty } \bigl( \vert \beta _{\kappa } \vert ^{-{1}/{\varLambda }} \varphi (t)^{-{\kappa }/{\varLambda }} \bigr)^{{2}/{\tilde{\gamma }}} \frac{dt}{t} \\ \leq& C \vert \beta _{\kappa } \vert ^{-{2}/{(\varLambda \tilde{\gamma })}} \int _{\delta } ^{\infty }\varphi (t)^{-{2\kappa }/{(\varLambda \tilde{\gamma })}} \frac{dt}{t} \\ \leq& C \vert \beta _{\kappa } \vert ^{-{2}/{(\varLambda \tilde{\gamma })}} \int _{\beta } ^{\infty }t^{-{2\kappa }/{(\varLambda \tilde{\gamma })}-1}\,dt \\ \leq& C\frac{\varLambda \tilde{\gamma }}{2\kappa }. \end{aligned}$$
(3.19)

Here \(C>0\) is independent of h, Ω, ξ and the coefficients of P, but depends on φ. Then (3.14) follows from (3.15)–(3.17) and (3.19). This proves Theorem 1.2. □

Proof of Theorem 1.3

Let Ω, φ be given as in Theorem 1.3. By arguments similar to those used in deriving (3.14) and (3.16) in [37], one can easily get that

$$ \mathcal{M}_{\varOmega ,P,\varphi ,\rho }f(x)\leq C(\rho )\mathcal{S} _{\varOmega ,P,\varphi }f(x). $$

Thus, we only prove Theorem 1.3 for the operator \(\mathcal{S}_{\varOmega ,P,\varphi }\). Define the measure \(\sigma _{\varOmega ,P,\varphi ,t}\) by

$$ \widehat{\sigma _{\varOmega ,P,\varphi ,t}}(\xi )= \int _{\mathbf{S}^{n-1}} {\mathbf{e}}^{-2\pi i P(\varphi (t))y'\cdot \xi }\varOmega \bigl(y'\bigr)\,d\sigma \bigl(y'\bigr). $$
(3.20)

By duality we can write

$$ \mathcal{S}_{\varOmega ,P,\varphi }f(x)= \biggl( \int _{0}^{\infty } \bigl\vert \sigma _{\varOmega ,P,\varphi ,t}*f(x) \bigr\vert ^{2}\frac{dt}{t} \biggr)^{1/2}. $$
(3.21)

By (3.21) and the same arguments as in the proof of Theorem 1.2, to prove Theorem 1.3 for the operator \(\mathcal{S}_{\varOmega , P,\varphi }\), it suffices to show that

$$ \biggl\Vert \biggl( \int _{0}^{\infty } \bigl\vert \widehat{\sigma _{\varOmega ,P,\varphi ,t}}(\cdot ) \bigr\vert ^{2}\frac{dt}{t} \biggr)^{1/2} \biggr\Vert _{L^{\infty }(\mathbf{R}^{n})}\leq C \Vert \varOmega \Vert _{H ^{1}(\mathbf{S}^{n-1})}, $$
(3.22)

where \(C>0\) is independent of Ω and the coefficients of P. By Lemma 2.1 and Minkowski’s inequality, (3.22) reduces to the following:

$$ \biggl\Vert \biggl( \int _{0}^{\infty } \bigl\vert \widehat{\sigma _{\varOmega ,P,\varphi ,t}}(\cdot ) \bigr\vert ^{2}\frac{dt}{t} \biggr)^{1/2} \biggr\Vert _{L^{\infty }(\mathbf{R}^{n})}\leq C, $$
(3.23)

where Ω is a \((1,\infty )\) atom and \(C>0\) is independent of Ω and the coefficients of P.

We now prove (3.23). We only assume that Ω is a \((1, \infty )\) atom satisfying (2.1)–(2.3) with \(\vartheta =(1,0, \ldots ,0)\) and consider the case \(n\geq 3\). Let ξ, b, and \(N_{b}\) be given as in the proof of Theorem 1.1. By the change of variables, we have

$$ \int _{0}^{\infty } \bigl\vert \widehat{\sigma _{\varOmega ,P,\varphi ,t}}(\xi ) \bigr\vert ^{2} \frac{dt}{t}= \int _{0}^{\infty } \biggl\vert \int _{\mathbf{R}}N_{b}(s) \mathbf{e}^{-2\pi i P(\varphi (t)) \vert \xi \vert r(\xi ')s}\,ds \biggr\vert ^{2} \frac{dt}{t}. $$

By a change of variable and the properties of φ, we obtain that

$$ \int _{0}^{\infty } \bigl\vert \widehat{\sigma _{\varOmega ,P,\varphi ,t}}(\xi ) \bigr\vert ^{2} \frac{dt}{t}\leq C( \varphi ) \int _{0}^{\infty } \biggl\vert \int _{\mathbf{R}}N _{b}(s)e^{-2\pi i P(t) \vert \xi \vert r(\xi ')s}\,ds \biggr\vert ^{2}\frac{dt}{t}. $$
(3.24)

Let \(b_{k}\), β be given as in the proof Theorem 1.1. We write

$$\begin{aligned}& \int _{0}^{\infty } \biggl\vert \int _{\mathbf{R}}N_{b}(s)\mathbf{e}^{-2\pi i P(t) \vert \xi \vert r(\xi ')s}\,ds \biggr\vert ^{2}\frac{dt}{t} \\& \quad = \int _{0}^{\beta } \biggl\vert \int _{\mathbf{R}}N_{b}(s)\mathbf{e}^{-2\pi i P(t) \vert \xi \vert r(\xi ')s}\,ds \biggr\vert ^{2}\frac{dt}{t}+ \int _{\beta }^{\infty } \biggl\vert \int _{\mathbf{R}}N_{b}(s)\mathbf{e}^{-2\pi i P(t) \vert \xi \vert r(\xi ')s}\,ds \biggr\vert ^{2}\frac{dt}{t} \\& \quad =:L_{1}+L_{2}. \end{aligned}$$
(3.25)

For \(L_{1}\), by the cancelation condition of \(N_{b}\), we have

$$ L_{1}= \int _{0}^{\beta } \biggl\vert \int _{\mathbf{R}}N_{b}(s) \bigl(\mathbf{e}^{-2 \pi i P(t) \vert \xi \vert r(\xi ')s}-1 \bigr)\,ds \biggr\vert ^{2}\frac{dt}{t}\leq C\sum _{k=1}^{N} \vert b_{k} \vert ^{2} \int _{0}^{\beta }t^{-1+2k}\,dt\leq C, $$
(3.26)

where \(C>0\) is independent of b, ξ and the coefficients of P.

For \(L_{2}\), let \(K_{0}\) be given as in the proof of Theorem 1.1. Applying (3.6), we get

$$\begin{aligned} L_{2} \leq& \sum_{j=K_{0}}^{\infty } \int _{2^{j}}^{2^{j+1}} \biggl\vert \int _{\mathbf{R}}N_{b}(s)\mathbf{e}^{-2\pi i P(t) \vert \xi \vert r(\xi ')s}\,ds \biggr\vert ^{2}\frac{dt}{t} \\ \leq& \sum_{j=K_{0}}^{\infty }2^{-j} \int _{R} \biggl\vert \chi _{R_{j}}(t) \int _{\mathbf{R}}N_{b}(s)\mathbf{e}^{-2\pi i P(t) \vert \xi \vert r(\xi ')s}\,ds \biggr\vert ^{2}\,dt \\ \leq& C\sum_{j=K_{0}}^{\infty }2^{-j}2^{j} \vert \beta _{\kappa } \vert ^{-{1}/ {\varLambda }}2^{-{j\kappa }/{\varLambda }} \Vert N_{b} \Vert _{L^{2}(\mathbf{R})} ^{2}\leq C, \end{aligned}$$
(3.27)

where in the last inequality of (3.27) we have used the fact that \(2^{K_{0}\kappa }\geq 2^{-\kappa }\beta _{\kappa }^{-1}\) and \(C>0\) is independent of b, ξ and the coefficients of P. We get from (3.24)–(3.27) that

$$ \int _{0}^{\infty } \bigl\vert \widehat{\sigma _{\varOmega ,P,\varphi ,t}}(\xi ) \bigr\vert ^{2} \frac{dt}{t}\leq C, $$

where \(C>0\) is independent of Ω, ξ and the coefficients of P. This yields (3.23) and completes the proof of Theorem 1.3. □

4 Hardy-type inequalities

In this section we shall establish the following Hardy-type inequalities.

Theorem 4.1

Let P be a real polynomial on R satisfying \(P(0)=0\) and \(\varOmega \in H^{1}( {\mathbf{S}}^{n-1})\) satisfying (1.1). Then we have

  1. (i)

    If \(\varphi \in \mathfrak{F}\), then

    $$ \biggl( \int _{\mathbf{R}^{n}} \biggl\vert \frac{1}{ \vert x \vert ^{\rho }} \int _{{ \vert x \vert }/{2}\leq \vert y \vert < \vert x \vert }{\mathbf{e}}^{-2\pi i P(\varphi ( \vert y \vert ))x' \cdot y'}\frac{\varOmega (y)}{ \vert y \vert ^{n-\rho }}\,dy \biggr\vert ^{2} \frac{dx}{ \vert x \vert ^{n}} \biggr)^{1/2}\leq C \Vert \varOmega \Vert _{H^{1}({\mathbf{S}} ^{n-1})}, $$
    (4.1)

    where C is a positive constant independent of Ω and the coefficients of P, but depends on ρ, φ, and \(\deg (P)\).

  2. (ii)

    If \(\varphi \in \mathcal{G}\), then

    $$ \biggl( \int _{\mathbf{R}^{n}} \biggl\vert \int _{{\mathbf{S}}^{n-1}}{\mathbf{e}} ^{-2\pi i P(\varphi ( \vert x \vert ))x'\cdot y'}\varOmega \bigl(y'\bigr)\,d\sigma \bigl(y'\bigr) \biggr\vert ^{2} \frac{dx}{ \vert x \vert ^{n}} \biggr)^{1/2}\leq C \Vert \varOmega \Vert _{H^{1}({\mathbf{S}} ^{n-1})}, $$
    (4.2)

    where C is a positive constant independent of Ω and the coefficients of P, but depends on φ and \(\deg (P)\).

Proof of Theorem 4.1

We first prove (i). Using Lemma 2.1, (4.1) reduces to the following:

$$ \biggl( \int _{\mathbf{R}^{n}} \biggl\vert \frac{1}{ \vert x \vert ^{\rho }} \int _{{ \vert x \vert }/{2}\leq \vert y \vert < \vert x \vert }{\mathbf{e}}^{-2\pi i P(\varphi ( \vert y \vert ))x' \cdot y'}\frac{\varOmega (y)}{ \vert y \vert ^{n-\rho }}\,dy \biggr\vert ^{2} \frac{dx}{ \vert x \vert ^{n}} \biggr)^{1/2}\leq C, $$
(4.3)

where Ω is a \((1,\infty )\) atom and C is a positive constant independent of Ω and the coefficients of P. By the polar coordinates,

$$\begin{aligned}& \int _{\mathbf{R}^{n}} \biggl\vert \frac{1}{ \vert x \vert ^{\rho }} \int _{{ \vert x \vert }/{2}\leq \vert y \vert < \vert x \vert }{\mathbf{e}}^{-2\pi i P(\varphi ( \vert y \vert ))x' \cdot y'}\frac{\varOmega (y)}{ \vert y \vert ^{n-\rho }}\,dy \biggr\vert ^{2} \frac{dx}{ \vert x \vert ^{n}} \\& \quad = \int _{\mathbf{S}^{n-1}} \int _{0}^{\infty } \biggl\vert \frac{1}{t^{\rho }} \int _{{t}/{2}}^{t} \int _{\mathbf{S}^{n-1}}{\mathbf{e}}^{-2\pi i P( \varphi (r))x'\cdot y'}\varOmega \bigl(y'\bigr)\,d\sigma \bigl(y'\bigr)\frac{dr}{r^{1-\rho }} \biggr\vert ^{2}\frac{dt}{t}\,d\sigma \bigl(x' \bigr) \\& \quad = \int _{\mathbf{S}^{n-1}} \int _{0}^{\infty } \bigl\vert \widehat{\sigma _{b,\varOmega ,P,\varphi ,t}}\bigl(x'\bigr) \bigr\vert ^{2} \frac{dt}{t}\,d\sigma \bigl(x'\bigr) \\& \quad \leq \omega _{n-1} \biggl\Vert \biggl( \int _{0}^{\infty } \bigl\vert \widehat{\sigma _{b,\varOmega ,P,\varphi ,t}}(\cdot ) \bigr\vert ^{2}\frac{dt}{t} \biggr)^{1/2} \biggr\Vert _{L^{\infty }(\mathbf{S}^{n-1})}^{2}. \end{aligned}$$
(4.4)

Here \(\omega _{n-1}\) is the surface area of the unit \(\mathbf{S}^{n-1}\) and \(\sigma _{b,\varOmega ,P,\varphi ,t}\) is defined as in (3.8) with \(b(\cdot )\equiv 1\). Then (4.3) follows from (4.4) and (3.11).

It remains to prove (ii). To prove (4.2), by Lemma 2.1 it suffices to show that

$$ \biggl( \int _{\mathbf{R}^{n}} \biggl\vert \int _{\mathbf{S}^{n-1}}{\mathbf{e}} ^{-2\pi i P(\varphi ( \vert x \vert ))x'\cdot y'}\varOmega \bigl(y'\bigr)\,d\sigma \bigl(y'\bigr) \biggr\vert ^{2} \frac{dx}{ \vert x \vert ^{n}} \biggr)^{1/2}\leq C, $$
(4.5)

where Ω is a \((1,\infty )\) atom and C is a positive constant independent of Ω and the coefficients of P. Using the polar coordinates, we can obtain

$$\begin{aligned}& \int _{\mathbf{R}^{n}} \biggl\vert \int _{\mathbf{S}^{n-1}}{\mathbf{e}}^{-2 \pi i P(\varphi ( \vert x \vert ))x'\cdot y'}\varOmega \bigl(y'\bigr)\,d\sigma \bigl(y'\bigr) \biggr\vert ^{2} \frac{dx}{ \vert x \vert ^{n}} \\& \quad = \int _{\mathbf{S}^{n-1}} \int _{0}^{\infty } \biggl\vert \int _{\mathbf{S}^{n-1}} {\mathbf{e}}^{-2\pi i P(\varphi (t))x'\cdot y'}\varOmega \bigl(y'\bigr)\,d\sigma \bigl(y'\bigr) \biggr\vert ^{2}\frac{dt}{t}\,d\sigma \bigl(x'\bigr) \\& \quad = \int _{\mathbf{S}^{n-1}} \int _{0}^{\infty } \bigl\vert \widehat{\sigma _{\varOmega ,P,\varphi ,t}}\bigl(x'\bigr) \bigr\vert ^{2} \frac{dt}{t}\,d\sigma \bigl(x'\bigr) \\& \quad \leq \omega _{n-1} \biggl\Vert \biggl( \int _{0}^{\infty } \bigl\vert \widehat{\sigma _{\varOmega ,P,\varphi ,t}}\bigl(x'\bigr) \bigr\vert ^{2} \frac{dt}{t} \biggr)^{1/2} \biggr\Vert _{L^{\infty }(\mathbf{S}^{n-1})}^{2}, \end{aligned}$$
(4.6)

where \(\widehat{\sigma _{\varOmega ,P,\varphi ,t}}\) is defined as in (3.20). (4.6) together with (3.22) yields (4.5). Theorem 4.1 is proved. □