1 Introduction and the Main Result

Let A be an \(n\times n\) matrix, and \(|\det A| = b\). We say that A is a dilation matrix if all the eigenvalues \(\lambda \) of A satisfy \(|\lambda |>1\). Let \(\lambda _1,\ldots ,\lambda _n\) be the eigenvalues of A, ordered by their norm from smallest to largest. Define \(\lambda _-\) and \(\lambda _+\), such that \(1<\lambda _-<|\lambda _1|\) and \(|\lambda _n|<\lambda _+\). Then \(\ln \lambda _{\pm }/\ln b\) are called the eccentricities of dilation A. We point out that, if A is diagonalizable, we may let \(\lambda _-:=|\lambda _1|\) and \(\lambda _+:=|\lambda _n|\). Otherwise, we may choose them sufficiently close to these equalities in accordance with what we need in our arguments. In addition, there is a sequence of nested ellipsoids \(\{B_j\}_{j\in {\mathbb Z}}\) associated with A such that

$$\begin{aligned} B_{j+1} = A(B_j)\ \ \ \text {and}\ \ \ |B_0 |= 1. \end{aligned}$$

If \(A^*\) is the adjoint of A, then \(A^*\) is also a dilation matrix with the same determinant b and eigenvalues as well as eccentricities, with its own nested ellipsoids \(\{B_j^*\}_{j\in {\mathbb Z}}\). We refer the reader to [6] for more properties about the dilation.

Let \({\mathcal S}({{\mathbb {R}}^n})\) be the Schwartz space, and let \({\mathcal S}'({{\mathbb {R}}^n})\) be the space of tempered distributions. Given a multi-index \(\vec {p}:=(p_1,\ldots ,p_n)\) with \(p_i\in (0,\infty )\) for any \(1\le i\le n\), the mixed-norm Lebesgue space \({L^{\vec {p}}({{\mathbb {R}}^n})}\) consists of all measurable functions f, for which

$$\begin{aligned} \Vert f\Vert _{{L^{\vec {p}}({{\mathbb {R}}^n})}}:=&\left\| \ldots \left\| f\right\| _{L_{x_1}^{p_1}}\ldots \right\| _{L_{x_n}^{p_n}}\\ :=&\left\{ \int _{{\mathbb R}}\ldots \left[ \int _{{\mathbb R}}|f(x_1,\ldots ,x_n)|^{p_1} \,dx_1\right] ^{\frac{p_2}{p_1}}\ldots \, dx_n\right\} ^{\frac{1}{p_n}}<\infty . \end{aligned}$$

If \(p_1=\ldots =p_n=p\), then the space \({L^{\vec {p}}({{\mathbb {R}}^n})}\) reduces to the classical Lebesgue space \(L^p({{\mathbb {R}}^n})\). The anisotropic mixed-norm Hardy space \({H_A^{\vec {p}}({{\mathbb {R}}^n})}\) associated with dilation matrix A is defined as

$$\begin{aligned} {H_A^{\vec {p}}({{\mathbb {R}}^n})}:=\left\{ f\in {\mathcal S}'({{\mathbb {R}}^n}):\ \Vert f\Vert _{{H_A^{\vec {p}}({{\mathbb {R}}^n})}}:=\left\| \sup _{k\in {\mathbb {Z}}}|f*\varphi _k|\right\| _{{L^{\vec {p}}({{\mathbb {R}}^n})}}<\infty \right\} , \end{aligned}$$

where \(\varphi \in {\mathcal S}({{\mathbb {R}}^n})\) satisfies \(\int _{{{\mathbb {R}}^n}}\varphi (x)\,dx\ne 0\) and \(\varphi _k(x):=b^{k}\varphi (A^k x)\) for any \(k\in {\mathbb {Z}}\). If \(p_1=\ldots =p_n=p\) and the dilation matrix

$$\begin{aligned} A:=\left( \begin{array}{cccc} 2~ &{} ~0~ &{} ~\ldots ~ &{} ~0\\ 0~ &{} ~2~ &{} ~\ldots ~ &{} ~0\\ \vdots &{} \vdots &{} &{}\vdots \\ 0~&{} ~0~ &{} ~\ldots ~ &{} ~2 \\ \end{array} \right) , \end{aligned}$$
(1.1)

then the anisotropic mixed-norm Hardy space \({H_A^{\vec {p}}({{\mathbb {R}}^n})}\) coincides with the classical Hardy space \(H^p({{\mathbb {R}}^n})\) of Fefferman-Stein [20].

The mixed-norm Lebesgue space \({L^{\vec {p}}({{\mathbb {R}}^n})}\) was systematically studied by Benedek-Panzone in [5], which goes back to Hörmander [28]. After that, many works on these spaces have been done due to the importance of \({L^{\vec {p}}({{\mathbb {R}}^n})}\), not only in harmonic analysis but also in partial differential equations and geometric inequalities. For instance, in a series of recent papers of Chen-Sun [9,10,11], they studied the Hardy-Littlewood-Sobolev inequalities on \({L^{\vec {p}}({{\mathbb {R}}^n})}\) and characterized the boundedness of multilinear fractional integral operators on \({L^{\vec {p}}({{\mathbb {R}}^n})}\). When A is an anisotropic diagonal matrix, precisely,

$$\begin{aligned} A:=\left( \begin{array}{cccc} 2^{a_1} &{} 0 &{} \ldots &{} 0\\ 0 &{} 2^{a_2} &{} \ldots &{} 0\\ \vdots &{} \vdots &{} &{}\vdots \\ 0 &{} 0 &{} \ldots &{} 2^{a_n} \\ \end{array} \right) \end{aligned}$$
(1.2)

with \(1\le a_i<\infty \) for \(1\le i\le n\), the anisotropic mixed-norm Hardy space \({H_A^{\vec {p}}({{\mathbb {R}}^n})}\) was first introduced and studied by Cleanthous-Georgiadis-Nielsen [15], and further developed by the author and his collaborators in [29,30,31, 33, 35]. This anisotropic mixed-norm Hardy space \({H_A^{\vec {p}}({{\mathbb {R}}^n})}\) associated with the diagonal matrix (1.2) was later extended to the general dilation matrix A (no need to be diagonal matrix or even no need to be diagonalizable) by the author and his collaborators in [32]. Here we refer to [13, 14, 17, 22,23,24, 26, 27, 34, 36, 37, 43, 44] for more detials on (anisotropic) mixed-norm function spaces and their applications.

This paper is devoted to studying the Mihlin multiplier theorem on the Hardy space \({H_A^{\vec {p}}({{\mathbb {R}}^n})}\). To state the multiplier theorem, let \({\hat{f}}\) and \({\check{f}}\) denote the Fourier transform and inverse Fourier transform of f, respectively. To be exact, when \(f\in {\mathcal S}({{\mathbb {R}}^n})\), then

$$\begin{aligned} {{\hat{f}}}(\xi ) := \int _{{{\mathbb {R}}^n}} f(x) e^{-2\pi \imath x \cdot \xi } \, dx\ \ \text {and}\ \ {\check{f}}(\xi ):={{\hat{f}}}(-\xi )= \int _{{{\mathbb {R}}^n}} f(x) e^{2\pi \imath x \cdot \xi } \, dx,\ \ \forall \xi \in {{\mathbb {R}}^n}, \end{aligned}$$

where \(\imath :=\sqrt{-1}\); when \(f\in {\mathcal S}'({{\mathbb {R}}^n})\), \(\langle {{\hat{f}}}, \phi \rangle :=\langle f, {\hat{\phi }}\rangle \) for any \(\phi \in {\mathcal S}({{\mathbb {R}}^n})\). Let \(m\in L^{\infty }({{\mathbb {R}}^n})\). We say the measurable function m is a Fourier multiplier on \({H_A^{\vec {p}}({{\mathbb {R}}^n})}\) if its associated Fourier multiplier operator \(T_m\), initially defined by

$$\begin{aligned} T_m f(x):= (m{{\hat{f}}})^\vee (x)=\int _{{{\mathbb {R}}^n}}m(\xi ){{\hat{f}}}(\xi )e^{2\pi \imath x \cdot \xi }\,d\xi ,\ \ \forall x\in {{\mathbb {R}}^n}, \end{aligned}$$

for \(f\in L^2({{\mathbb {R}}^n})\cap {H_A^{\vec {p}}({{\mathbb {R}}^n})}\), is bounded on \({H_A^{\vec {p}}({{\mathbb {R}}^n})}\). For a dilation matrix A, define the dilation operator \(D_A\) by

$$\begin{aligned} D_A f(x)=f(Ax), \ \ \forall x\in {{\mathbb {R}}^n}. \end{aligned}$$

For any \(\Omega \in {{\mathbb {R}}^n}\) and \(N\in {\mathbb N}\cup \{0\}\), denote by \(C^N(\Omega )\) the set of all functions on \(\Omega \) whose derivatives with order no greater than N exist and are continuous. Then the following anisotropic Mihlin condition was introduced in [4, 47]. Let A be a dilation matrix and \(m\in C^N({{\mathbb {R}}^n}{\setminus }\{0\})\) with \(N\in {\mathbb N}\cup \{0\}=:{\mathbb Z}_+\). We say m satisfies the anisotropic Mihlin condition of order N if there exists a constant \(C:=C_N\) such that for any multi-indices \(\alpha \) with \(|\alpha |\le N\),

$$\begin{aligned} \left| D^{-j}_{A^*}\partial ^{\alpha }_{\xi } D^{j}_{A^*} m(\xi )\right| \le C,\ \ \ \forall \xi \in B^*_{j+1}\setminus B^*_{j}, \ j\in {\mathbb Z}, \end{aligned}$$
(1.3)

where, for any \(j\in {\mathbb Z}\), \(D_{A^*}^j m(\xi ):= m((A^*)^j \xi )\). Henceforth, we always use C to denote a positive constant which may depend on the dilation matrix A and scalar parameters such as n and \(\vec {p}\), and may vary from line to line, but independent of the main parameters such as \(f\in {H_A^{\vec {p}}({{\mathbb {R}}^n})}\). Given a vector \(\vec {p}:=(p_1,\ldots ,p_n)\), let \(p_-:=\min \{p_1,\ldots ,p_n\}\) and \(p_+:=\max \{p_1,\ldots ,p_n\}\). For any \(s\in {{\mathbb {R}}}\), we always use \(\lfloor s\rfloor \) to denote the largest integer no greater than s.

Now we can state the Mihlin multiplier theorem on the Hardy space \({H_A^{\vec {p}}({{\mathbb {R}}^n})}\) as follows.

Theorem 1.1

Let A be a dilation matrix, \(\vec {p}\in (0,1]^n\), \(N\in {\mathbb N}\) and

$$\begin{aligned} M:=\left( N\frac{\ln \lambda _-}{\ln b}-1\right) \frac{\ln b}{\ln \lambda _+}. \end{aligned}$$

If m satisfies the anisotropic Mihlin condition of order N and \(T_m\) is the Fourier multiplier operator, then \(T_m: H_A^{\vec {p}}({{\mathbb {R}}^n})\rightarrow H_A^{\vec {p}}({{\mathbb {R}}^n})\) is bounded, provided \(\vec {p}\) satisfies

$$\begin{aligned} 0\le \frac{1}{p_-}-1<\lfloor M\rfloor \frac{(\ln \lambda _-)^2}{\ln b \ln \lambda _+}. \end{aligned}$$

Recall that the study of the Fourier multiplier theory was initiated by Mihlin [42] and Hörmander [28] in the late 1950 s. Then the multiplier theory for Triebel-Lizorkin spaces and Besov-Lipschitz spaces was considered by Peetre [45] in 1975; for classical Hardy spaces was studied by Taibleson-Weiss [46] and Baernstein-Sawyer [3]; for Hardy spaces in the parabolic setting was inversitaged by Calderón-Torchinsky [7, 8]; for anisotropic Hardy spaces was obtained by Wang [47]. Additionally, Fourier multipliers (or more general operators) on the anisotropic mixed-norm setting were well studied by Cleanthous et al. in [16] as well as by Georgiadis et al. in [21, 23, 25], and the extensions on manifolds, Lie groups or discrete settings were considered in [1, 2, 12, 18, 19, 21, 38, 39].

Next we give some remarks on Theorem 1.1.

Remark 1.2

Let \(p_1=\ldots =p_n=p\) and the dilation matrix A be as in (1.1). Then \(p_-=p\), \(\lambda _-=\lambda _+=2\), \(b=|\det A|=2^n\) and the Hardy space \({H_A^{\vec {p}}({{\mathbb {R}}^n})}\) goes back to the classical Hardy space \(H^p({{\mathbb {R}}^n})\), and hence \(M=N-n\) and \(\frac{n}{N}<p\le 1\). This theorem, in this case, recovers the classical case.

Remark 1.3

Let \(p_1=\ldots =p_n=p\). Then the anisotropic mixed-norm Hardy space \({H_A^{\vec {p}}({{\mathbb {R}}^n})}\) reduces to the anisotropic Hardy space \(H^{p}_A({{\mathbb {R}}^n})\), and hence Theorem 1.1 coincides with the result in anisotropic Hardy space setting.

Remark 1.4

Let the dilation matrix A be as in (1.1). Then \(\lambda _-=\lambda _+=2\), \(b=|\det A|=2^n\) and the Hardy space \({H_A^{\vec {p}}({{\mathbb {R}}^n})}\) goes back to the isotropic mixed-norm Hardy space \(H^{\vec {p}}({{\mathbb {R}}^n})\), and hence \(M=N-n\) and \(\frac{n}{N}<p_-\le 1\). We point out that, even in this case, Theorem 1.1 is also new.

Finally, we make some conventions on notation. The notation \(f\lesssim g\) means \(f\le Cg\) and, if \(f\lesssim g\lesssim f\), then we write \(f\sim g\). We also use the following convention: If \(f\le Cg\) and \(g=h\) or \(g\le h\), we then write \(f\lesssim g\sim h\) or \(f\lesssim g\lesssim h\), rather than \(f\lesssim g=h\) or \(f\lesssim g\le h\).

2 Proof of the Main Theorem

To prove Theorem 1.1, the main ingredients are the atoms of \({H_A^{\vec {p}}({{\mathbb {R}}^n})}\) introduced in [32] and the criterion on the boundedness of sublinear operators on \({H_A^{\vec {p}}({{\mathbb {R}}^n})}\) established in [32]. Moreover, the Calderón-Zygmund operator theory on anisotropic mixed-norm Hardy spaces \({H_A^{\vec {p}}({{\mathbb {R}}^n})}\) also plays an important role in our proof.

We begin with giving the following notion of the homogeneous quasi-norm.

Definition 2.1

For any given dilation A, a homogeneous quasi-norm, with respect to A, is a measurable mapping \(\rho :\ {{\mathbb {R}}^n}\rightarrow [0,\infty )\) satisfying

  1. (i)

    if \(x\ne 0\), then \(\rho (x)\in (0,\infty )\);

  2. (ii)

    for any \(x\in {{\mathbb {R}}^n}\), \(\rho (Ax)=b\rho (x)\);

  3. (iii)

    there exists some \(R\in [1,\infty )\) such that

    $$\begin{aligned} \rho (x+y)\le R[\rho (x)+\rho (y)],\ \ \forall x,y\in {{\mathbb {R}}^n}. \end{aligned}$$

For a fixed dilation A, the associated homogeneous quasi-norms are non-unique. But they are equivalent to each other (see [6, p. 6 Lemma 2.4]). Thus, in what follows, we may use the following step homogeneous quasi-norm \(\rho \) defined by setting

$$\begin{aligned} \rho (x):=\sum _{j\in {\mathbb {Z}}} b^j{\textbf{1}}_{B_{j+1}\setminus B_j}(x) \textrm{when}\ x\in {{\mathbb {R}}^n}\setminus \{0\}, \mathrm{or\ else} \rho (0):=0 \end{aligned}$$

for both simplicity and convenience. In addition, if \(A^*\) is the adjoint of a given dilation matrix A, then \(A^*\) is also a dilation matrix with the same determinant and eigenvalues, with its own nested ellipsoids \(\{B_j^*\}_{j\in {\mathbb Z}}\) and step homogeneous quasi-norms \(\rho _{*}\). Given a dilation A, we say that \((\vec {p}, r, s)\) is an admissible triplet if \(\vec {p}\in (0,1]^n\), \(r\in (1,\infty ]\) and

$$\begin{aligned} s\in \left[ \left\lfloor \left( \dfrac{1}{p_-}-1\right) \dfrac{\ln b}{\ln \lambda _-}\right\rfloor ,\infty \right) \cap {\mathbb Z}_+. \end{aligned}$$

We now present the definition of \((\vec {p},r,s)\)-atom from [32, Definition 4.1] as follows.

Definition 2.2

Let \((\vec {p}, r, s)\) be admissible. A measurable function a on \({{\mathbb {R}}^n}\) is called a \((\vec {p},r,s)\)-atom if

  1. (i)

    \(\mathop \mathrm {\,supp\,}a \subset x+B_k\) for some \(x\in {{\mathbb {R}}^n}\) and \(k\in {\mathbb Z}\);

  2. (ii)

    \(\Vert a\Vert _{L^r({{\mathbb {R}}^n})}\le \frac{|B_k|^{1/r}}{\Vert {\textbf{1}}_{x+B_k}\Vert _{{L^{\vec {p}}({{\mathbb {R}}^n})}}}\);

  3. (iii)

    for any \(\gamma \in {\mathbb Z}_+^n\) with \(|\gamma |\le s\), \(\int _{\mathbb R^n}a(x)x^\gamma \,dx=0\).

We now recall the following notion from [47, Definition 3.1].

Definition 2.3

Let \(R\in {\mathbb Z}_+\) and \(K\in C^{R}({{\mathbb {R}}^n}\setminus \{0\})\). We say that K is a Calderón-Zygmund convolution kernel of order R if there exists a constant C such that for all multi-indices \(\alpha \) with \(|\alpha |\le R\), and for any \(j\in {\mathbb Z}\) and \(x\in B_{j+1}{\setminus } B_j\),

$$\begin{aligned} \left| D^{-j}_{A}\partial ^{\alpha }_{\xi } D^{j}_{A} K(x)\right| \le \frac{C}{\rho (x)}. \end{aligned}$$

If K is such a kernel, we say K satisfies CZC-R and its associated singular integral operator T is defined by \(T f:= K*f \), which is called a Calderón-Zygmund operator of order R.

For the CZC-R kernel and the Mihlin condition (1.3), we have the following key lemma (see [47, Lemma 3.2]).

Lemma 2.4

Let \(N\in {\mathbb Z}_+\) and \(m\in L^1_{\mathrm{{loc}}} ({{\mathbb {R}}^n}\setminus \{0\})\). Suppose m satisfies the Mihlin condition of order N as in (1.3), and define K by \(K:={\check{m}}\). Then K is a Calderón-Zygmund convolution kernel of order R provided \(R\in {\mathbb N}\) and

$$\begin{aligned} 0\le R<\left( N\frac{\ln \lambda _-}{\ln b}-1\right) \frac{\ln b}{\ln \lambda _+}. \end{aligned}$$

To prove our main theorem, the following lemma plays an important role.

Lemma 2.5

Let \((\vec {p},\infty ,s)\) be an admissible triplet, \(m\in L^{\infty }({{\mathbb {R}}^n})\) and \(T_m\) the associated Fourier multiplier operator initially defined on \(L^2({{\mathbb {R}}^n})\cap {H_A^{\vec {p}}({{\mathbb {R}}^n})}\). If there exists a positive constant C such that, for any \((\vec {p},\infty ,s)\)-atom a, \(\Vert T_ma\Vert _{{H_A^{\vec {p}}({{\mathbb {R}}^n})}}\le C\), then \(T_m\) has a unique bounded extension \({\widetilde{T}}_m:H_A^{\vec {p}}({{\mathbb {R}}^n})\rightarrow H_A^{\vec {p}}({{\mathbb {R}}^n})\).

To prove Lemma 2.5, we need the following Lemma 2.6 from [32, Corollary 4], which gives the boundedness criterion about sublinear operators on \({H_A^{\vec {p}}({{\mathbb {R}}^n})}\). To state it, we first recall that a complete vector space \({\mathcal {B}}\), equipped with a quasi-norm \(\Vert \cdot \Vert _{{\mathcal {B}}}\), is called a quasi-Banach space if

  1. (i)

    \(\Vert \varphi \Vert _{{\mathcal {B}}}=0\) if and only if \(\varphi \) is the zero element of \({\mathcal {B}}\);

  2. (ii)

    there exists a positive constant \(C\in [1,\infty )\) such that, for any \(\varphi ,\,\phi \in {\mathcal {B}}\), \(\Vert \varphi +\phi \Vert _{{\mathcal {B}}} \le C(\Vert \varphi \Vert _{{\mathcal {B}}}+\Vert \phi \Vert _{{\mathcal {B}}}).\)

In addition, for any given \(\gamma \in (0,1]\), a \(\gamma \)-quasi-Banach space \({{\mathcal {B}}_{\gamma }}\) is a quasi-Banach space equipped with a quasi-norm \(\Vert \cdot \Vert _{{\mathcal {B}}_{\gamma }}\) satisfying that there exists a constant \(C\in [1,\infty )\) such that, for any \(K\in {\mathbb N}\) and \(\{\varphi _i\}_{i=1}^{K}\subset {\mathcal {B}}_{\gamma }\),

$$\begin{aligned} \left\| \sum _{i=1}^K \varphi _i\right\| _{{\mathcal {B}}_{\gamma }}^{\gamma } \le C \sum _{i=1}^K \left\| \varphi _i\right\| _{{\mathcal {B}}_{\gamma }}^{\gamma }. \end{aligned}$$

Let \({\mathcal {B}}_{\gamma }\) be a \(\gamma \)-quasi-Banach space with \(\gamma \in (0,1]\) and \({\mathcal {Y}}\) a linear space. An operator T from \({\mathcal {Y}}\) to \({\mathcal {B}}_{\gamma }\) is said to be \({\mathcal {B}}_{\gamma }\)-sublinear if there exists a positive constant \(\widetilde{C}\) such that, for any \(K\in {\mathbb N}\), \(\{\mu _{i}\}_{i=1}^K\subset {\mathbb {C}}\) and \(\{\varphi _{i}\}_{i=1}^K\subset {\mathcal {Y}}\),

$$\begin{aligned} \left\| T\left( \sum _{i=1}^K \mu _i \varphi _{i}\right) \right\| _{{\mathcal {B}}_{\gamma }}^{\gamma }\le \widetilde{C}\sum _{i=1}^K |\mu _i|^{\gamma }\left\| T(\varphi _{i})\right\| _{{\mathcal {B}}_{\gamma }}^{\gamma } \end{aligned}$$

and, for any \(\varphi ,\,\phi \in {\mathcal {Y}}\), \(\Vert T(\varphi )-T(\phi )\Vert _{{\mathcal {B}}_{\gamma }} \le \widetilde{C}\Vert T(\varphi -\phi )\Vert _{{\mathcal {B}}_{\gamma }}\) (see [30, 48]).

Lemma 2.6

Let \((\vec {p},\infty ,s)\) be an admissible triplet, \(\gamma \in (0,1]\) and \({\mathcal {B}}_{\gamma }\) a \(\gamma \)-quasi-Banach space. If T is a \({\mathcal {B}}_{\gamma }\)-sublinear operator defined on all continuous \((\vec {p},\infty ,s)\)-atoms satisfying

$$\begin{aligned} \sup \left\{ \Vert T(a)\Vert _{{\mathcal {B}}_{\gamma }}:\ a\ is\ any\ continuous\ (\vec {p},\infty ,s){\text{- }}atom\right\} <\infty , \end{aligned}$$

then T uniquely extends to a bounded \({\mathcal {B}}_{\gamma }\)-sublinear operator from \({H_A^{\vec {p}}({{\mathbb {R}}^n})}\) into \({\mathcal {B}}_{\gamma }\).

Additionally, the following property of Fourier transform of elements in Hardy spaces \({H_A^{\vec {p}}({{\mathbb {R}}^n})}\) is required in the proof of Lemma 2.5 (see [41, Theorem 3.1]).

Lemma 2.7

Let \(\vec {p}\in (0,1]^n\). Then, for any \(f\in {H_A^{\vec {p}}({{\mathbb {R}}^n})}\), there exists a continuous function g on \({{\mathbb {R}}^n}\) such that \({\widehat{f}}=g\) in \({\mathcal S}'({{\mathbb {R}}^n}),\) and there exists a positive constant C such that

$$\begin{aligned} \left| g(x)\right| \le C\Vert f\Vert _{{H_A^{\vec {p}}({{\mathbb {R}}^n})}}\max \left\{ \left[ \rho _{*}(x)\right] ^{\frac{1}{p_+}-1},\, \left[ \rho _{*}(x)\right] ^{\frac{1}{p_-}-1}\right\} ,\ \ \ \forall x\in {{\mathbb {R}}^n}. \end{aligned}$$

With the help of Lemmas 2.6 and 2.7, we next show Lemma 2.5.

Proof of Lemma 2.5

From [32, Lemma 3.4], we infer that, for any \(\{f_i\}_{i\in {\mathbb N}}\subset {H_A^{\vec {p}}({{\mathbb {R}}^n})}\),

$$\begin{aligned} \left\| \sum _{i=1}^K f_{i}\right\| _{{H_A^{\vec {p}}({{\mathbb {R}}^n})}}^{p_-} \le C\sum _{i=1}^K \left\| f_{i}\right\| _{{H_A^{\vec {p}}({{\mathbb {R}}^n})}}^{p_-}, \end{aligned}$$

which implies that \(({H_A^{\vec {p}}({{\mathbb {R}}^n})},\Vert \cdot \Vert _{{H_A^{\vec {p}}({{\mathbb {R}}^n})}})\) is a \(p_-\)-quasi-Banach space. By Lemma 2.7, we find that \({{\hat{f}}}\) agrees with a continuous function in the sense of distribution for any \(f\in {H_A^{\vec {p}}({{\mathbb {R}}^n})}\). Moreover, applying [32, Lemma 3.4] again, we know that there exists a positive constant C such that, for any \(K\in {\mathbb N}\), \(\{\mu _{i}\}_{i=1}^K\subset {\mathbb {C}}\) and \(\{\varphi _{i}\}_{i=1}^K\subset L^2({{\mathbb {R}}^n})\cap {H_A^{\vec {p}}({{\mathbb {R}}^n})}\),

$$\begin{aligned} \left\| T_m\left( \sum _{i=1}^K \mu _i \varphi _{i}\right) \right\| _{{H_A^{\vec {p}}({{\mathbb {R}}^n})}}^{p_-} =\left\| \sum _{i=1}^K \mu _i T_m\varphi _{i}\right\| _{{H_A^{\vec {p}}({{\mathbb {R}}^n})}}^{p_-}\le C\sum _{i=1}^K |\mu _i|^{p_-}\left\| T_m\varphi _{i}\right\| _{{H_A^{\vec {p}}({{\mathbb {R}}^n})}}^{p_-}, \end{aligned}$$

which implies that the Fourier multiplier \(T_m\) is a \({H_A^{\vec {p}}({{\mathbb {R}}^n})}\)-sublinear operator. Then combining the assumption \(\Vert T_ma\Vert _{{H_A^{\vec {p}}({{\mathbb {R}}^n})}}\le C\) for all \((\vec {p},\infty ,s)\)-atom a and the criterion on the boundedness of sublinear operators on \({H_A^{\vec {p}}({{\mathbb {R}}^n})}\) (see Lemma 2.6), we conclude that the operator \(T_m\) has a unique bounded extension \({\widetilde{T}}_m:H_A^{\vec {p}}({{\mathbb {R}}^n})\rightarrow H_A^{\vec {p}}({{\mathbb {R}}^n})\). This hence completes the proof. \(\square \)

When applying Lemma 2.5, we must first show that the condition in Lemma 2.5 is satisfied. Thus, we need the following result, which is just a consequence of [40, Theorem 3]. In what follows, for any \(s\in {\mathbb N}\), an operator T is said to have the vanishing moments up to order s if, for any \(f\in L^2({{\mathbb {R}}^n})\) with compact support and satisfying that, for any \(\alpha \in {\mathbb Z}_+^n\) with \(|\alpha |\le s\), \(\int _{{{\mathbb {R}}^n}}x^{\alpha }f(x)\,dx=0\), it holds true that \(\int _{{{\mathbb {R}}^n}}x^{\alpha }Tf(x)\,dx=0\).

Lemma 2.8

Let \((\vec {p},\infty ,s)\) be an admissible triplet and \(\ell \in {\mathbb N}\) with

$$\begin{aligned} 0\le \frac{1}{p_-}-1<\ell \frac{(\ln \lambda _-)^2}{\ln b\ln \lambda _+}. \end{aligned}$$

Assume that T is a Calderón-Zygmund operator of order \(\ell \) and has the vanishing moment conditions up to order \(s_0:=\lfloor (1/{p_-}-1)\ln b/\ln \lambda _-\rfloor \). Then there exists a positive constant C such that for any \((\vec {p},\infty ,s)\)-atom a,

$$\begin{aligned} \Vert Ta\Vert _{{H_A^{\vec {p}}({{\mathbb {R}}^n})}}\le C. \end{aligned}$$

Using Lemmas 2.5 and 2.8, we now show Theorem 1.1.

Proof of Theorem 1.1

Let m satisfy the anisotropic Mihlin condition of order N as in (1.3). Without loss of generality, we may assume \(M\notin {\mathbb N}\). Otherwise, if \(M\in {\mathbb N}\), then let \({\widetilde{\lambda }}_-\) and \({\widetilde{\lambda }}_+\) be defined as

$$\begin{aligned} 1<\lambda _-<{\widetilde{\lambda }}_-<|\lambda _1|\le \ldots \le |\lambda _n|<{\widetilde{\lambda }}_+<\lambda _+ \end{aligned}$$

such that the new

$$\begin{aligned} {\widetilde{M}}:=\left( N\frac{\ln {\widetilde{\lambda }}_-}{\ln b}-1\right) \frac{\ln b}{\ln {\widetilde{\lambda }}_+}, \end{aligned}$$

defined in terms of the new eccentricities \(\ln {\widetilde{\lambda }}_{\pm }/\ln b\), is slightly larger and no longer an integer satisfying \(\lfloor {\widetilde{M}}\rfloor =\lfloor M\rfloor \). Notice that, applying Lemma 2.4, we conclude that \(K:={\check{m}}\) is a Calderón-Zygmund convolution kernel of order R provided \(R\in {\mathbb Z}_+\) and

$$\begin{aligned} 0\le R<\left( N\frac{\ln \lambda _-}{\ln b}-1\right) \frac{\ln b}{\ln \lambda _+}. \end{aligned}$$

Thus, we may take

$$\begin{aligned} R:=\left\lfloor \left( N\frac{\ln \lambda _-}{\ln b}-1\right) \frac{\ln b}{\ln \lambda _+}\right\rfloor \end{aligned}$$

and then, from the assumption of Theorem 1.1, it follows that \(R=\lfloor M\rfloor \).

We now show that there exists a positive constant C such that, for all \((\vec {p},\infty ,s)\)-atom a, the singular integral operator T associated with kernel K defined by \(T f:= K*f \) satisfying

$$\begin{aligned} \Vert Ta\Vert _{{H_A^{\vec {p}}({{\mathbb {R}}^n})}}\le C \end{aligned}$$

when

$$\begin{aligned} 0\le \frac{1}{p_-}-1<\lfloor M\rfloor \frac{(\ln \lambda _-)^2}{\ln b\ln \lambda _+}. \end{aligned}$$

Indeed, we first note that T is a Calderón-Zygmund operator of order \(\lfloor M\rfloor \). Moreover, by the definition of operator T and the vanishing moments condition of \((\vec {p},\infty ,s)\)-atom a, we know that, for any \(\gamma \in {\mathbb Z}_+^n\) with \(|\gamma |\le s\),

$$\begin{aligned} \int _{{{\mathbb {R}}^n}}Ta(x)x^{\gamma }\,dx =&\int _{{{\mathbb {R}}^n}}\int _{{{\mathbb {R}}^n}}a(x-y)K(y)\,dy\,x^{\gamma }\,dx\\ =&\int _{{{\mathbb {R}}^n}}\int _{{{\mathbb {R}}^n}}a(x-y)x^{\gamma }\,dx\,K(y)\,dy =0, \end{aligned}$$

which implies that T has the vanishing moment conditions up to order s. Therefore, the operator T satisfies all assumptions of Lemma 2.8 and hence, from Lemma 2.8, we infer that \(\Vert Ta\Vert _{{H_A^{\vec {p}}({{\mathbb {R}}^n})}}\le C.\) By this and the fact that \(T=T_m \), we find that \(\Vert T_ma\Vert _{{H_A^{\vec {p}}({{\mathbb {R}}^n})}}\le C\). Combining this and Lemma 2.5, we finally conclude that \(T_m\) has a unique bounded extension \({\widetilde{T}}_m:H_A^{\vec {p}}({{\mathbb {R}}^n})\rightarrow H_A^{\vec {p}}({{\mathbb {R}}^n})\) and hence Theorem 1.1 is proved. \(\square \)