1 Introduction

The goal of this work is to establish a correspondence between two scales of spaces: the product real Hardy spaces and certain variation type spaces. To make this sentence meaningful, we give certain basics on each of the scales in an easy manner, while more details will be given in the sequel. We begin with the former scale. Defining the Fourier transform in a standard way

$$\begin{aligned} \widehat{f}(x)={\mathcal {F}}f(x)=\int _{{\mathbb {R}}^n} f(u)e^{-i\langle x,u\rangle }du, \end{aligned}$$

where \(\langle x,u\rangle =x_1u_1+\cdots +x_nu_n\), we may use it for assigning the Riesz transforms \(R^j\), \(j=1,2,\ldots ,n\),

$$\begin{aligned} \widehat{R^j f}(x):=\frac{ix_j}{|x|^n}\widehat{f}(x). \end{aligned}$$

The class of integrable functions f for which all the Riesz transforms are integrable is the classical real Hardy space \(H^1({\mathbb {R}}^n)\). One can split all the n coordinates into groups and define a real Hardy space for each group, with the other variables fixed. Taking superpositions of the Riesz transforms with respect to these groups (variables) and assuming integrability of every superposition over \({\mathbb {R}}^n\), we arrive at the product Hardy space with respect to the chosen splitting. For a different way of splitting, the above procedure gives a different product Hardy space. Recall that if a group contains only one variable, then the only Riesz transform reduces to the classical Hilbert transform. Therefore, if we split \({\mathbb {R}}^n\) into n groups, each with respect to a single variable, the corresponding product Hardy space, traditionally denoted by \(H^1({\mathbb {R}}\times \cdots \times {\mathbb {R}})\), is defined by various combinations of the Hilbert transform.

In [1], \(H^1({\mathbb {R}}^n)\) is related to a space resembling the classical Tonelli variation. Recall that a function is of bounded variation in the sense of Tonelli if the classical one-dimensional variations of the sections in each variable are bounded, with the rest variables fixed, and all these variations are Lebesgue integrable over \({\mathbb {R}}^{n-1}\). Even earlier, in [12], it was shown what kind of variation exactly fits \(H^1({\mathbb {R}}\times \cdots \times {\mathbb {R}})\). It turned out to be the classical Hardy variation. Recall that, as in dimension one, the Jordan variation is defined by means of sums of differences of first order, the Hardy variation is defined similarly via the sums of differences of higher order with respect to every group of variables, while the rest of the variables are fixed.

For all the other product Hardy spaces intermediate between \(H^1({\mathbb {R}}^n)\) and \(H^1({\mathbb {R}}\times \cdots \times {\mathbb {R}})\), we introduce certain spaces by means of the properties resembling the idea of variation. These scales are equal in number, with a unique correspondence between their elements. The way how the elements of these two scales are related to one another via the Fourier transform is our main result. It can roughly be formulated as follows; for the precise formulation, see Theorem 2.

If f belongs to a space defined by means of certain conditions, where the absolute continuity of lower derivatives is involved, and the corresponding n-th derivative belongs to the related product Hardy space, then the Fourier transform of f is Lebesgue integrable.

The main peculiarity here is that the derivatives are considered with respect to the same variables in which the product Hardy space is designated.

The outline of the paper is as follows. In the next section, we give some preliminaries. More precisely, we first present the above scheme in the one-dimensional setting hoping that this will simplify the understanding of the multivariate case. We then introduce a so-called indicator notation which should help to write down complicated relations in a relatively compact form. In Sect. 3, the main ingredients of our construction are given in detail. First, we give precise definitions of product Hardy spaces. They are related to the Fourier transform via the so-called Fourier-Hardy inequalities. Further, we present appropriate information about absolute continuity. In several dimensions, it is as diverse and sometimes complicated as the notion of variation. Finally, we conclude the section with the definitions for product variation type spaces. More precisely, we define the spaces of functions with suitable conditions concerning the absolute continuity of the derivatives. In view of the well-known relation between the concepts of absolute continuity and variation, it is natural to label them as ”variation type spaces”. In the last section, we formulate in a precise form our main result roughly given above and prove it. The proof could be longer but much work is done in the preceding sections.

By C we denote absolute constants, that may be different in different occurrences. Notations \(\lesssim \) and \(\gtrsim \) mean \(\le C\) and \(\ge C,\) respectively, when we do not wish to indicate the constants explicitly.

2 Preliminaries

For better understanding of various features of the chosen topic, we present the one-dimensional setting. After that, we prepare for the multidimensional work by introducing appropriate notations.

2.1 One-dimensional scheme

We consider functions f of bounded variation on \({\mathbb {R}}\), and vanishing at infinity, \(\lim \nolimits _{|t|\rightarrow \infty }f(t)=0\), written \(f\in BV_0({\mathbb {R}})\), and locally absolutely continuous on \({\mathbb {R}}\), written \(f\in AC_{loc}({\mathbb {R}})\). All in one, we will denote these by

$$\begin{aligned} f\in BV_0({\mathbb {R}})\cap AC_{loc}({\mathbb {R}}). \end{aligned}$$

Integrating by parts and applying Hardy’s inequality (see, e.g., [8, (7.24)]; in [13], it is suggested to call it the Fourier-Hardy inequality as distinct from the weighted inequality for Hardy’s operator)

$$\begin{aligned} \int _{{\mathbb {R}}}\frac{|\widehat{g}(x)|}{|x|}\,dx \lesssim \Vert g\Vert _{H^1({\mathbb {R}})}, \end{aligned}$$
(1)

we arrive at a pleasant fact that \(f'\in H^1({\mathbb {R}})\) ensures the integrability of the Fourier transform of f. We are going to generalize this situation to the multidimensional case in a variety of ways. One more circumstance to be taken into account is that any function of bounded variation possesses a derivative (almost everywhere), which is Lebesgue integrable. On the other hand, the direct assumption that f is of bounded variation is not necessary, since the integrability of the derivative is provided by the belonging of this derivative to \(H^1({\mathbb {R}})\). In fact, replacing too wide \(L^1({\mathbb {R}})\) by not much restrictive but very convenient \(H^1({\mathbb {R}})\) is a typical scenario in a variety of problems. Here \(H^1({\mathbb {R}})\) is understood as the space of integrable functions with integrable Hilbert transform. The point why the notion of variation is just around the corner here consists in the fact that an absolutely continuous function is in particular a function of bounded variation whose variation can be computed as the \(L^1-\)norm of its derivative.

2.2 Basic notation

As is generally accepted, the success of many of multidimensional results (more precisely, clarity of their formulations and likeness of the proof to that in dimension one) strongly depends on appropriate notation. We suggest universal indicator type notation that easily allows one to distinct different phenomena on certain groups of variables and in many cases minimize the number of indices. The main tool for this are zero–one vectors \(\eta \) and \(\eta ^1\),..., \(\eta ^m\). We define the first one, while the rest will always be of the same meaning.

Let \(\eta =(\eta _1,\ldots ,\eta _n)\) be an n-dimensional vector with the entries either 0 or 1 only. Its main task is to indicate the variables in which that or another action to be fulfilled. Correspondingly, we will use the notation \(|\eta |=\eta _1+\cdots +\eta _n\). The inequality of vectors is meant coordinate wise. If the only 1 entry is on the j-th place, while the rest are zeros, such a (basis) vector will be denoted by \(e_j\). On the other hand, if all the entries are 1, then such a vector will be denoted by \(\textbf{1}\); of course, \(|\textbf{1}|=n\). If all the entries are 0, then such a vector will be denoted by \(\textbf{0}\). By \({x}_\eta \) and \(d{x}_\eta \) we denote the \(|\eta |\)-tuple consisting only of \(x_j\) such that \(\eta _j=1\) and

$$\begin{aligned} d{x}_\eta :=\prod \limits _{j:\eta _j=1}dx_j, \end{aligned}$$

respectively. Correspondingly, the Euclidean space in \({x}_\eta \) will be denoted by \({\mathbb {R}}_\eta \).

When we apply the superpositions of the Hilbert transform with respect to \(\eta \), naturally,

$$\begin{aligned} {\mathcal {H}}_\eta :=\prod \limits _{j:\eta _j=1} {\mathcal {H}}_j. \end{aligned}$$

Any other operator with subindex \(\eta \) will be understood in the same manner. For instance, if we wish to apply the Fourier transform with respect to certain variables, the notation will be similar to the above: \({\mathcal {F}}_\eta f\) or \({\mathcal {F}}_{\eta ^{j_1},\ldots ,\eta ^{j_k}}f\), with integration over \({\mathbb {R}}_\eta \). Naturally, \({\mathcal {F}}f={\mathcal {F}}_\textbf{1}f\); if we wish to take the Fourier transform with respect to the j-th variable only, it will be denoted by

$$\begin{aligned} {\mathcal {F}}_{e_j}f(x_j, u_{\textbf{1}-e_j})=\int _{{\mathbb {R}}} f(u_j,u_{\textbf{1}-e_j})e^{-ix_ju_j}\,du_j. \end{aligned}$$

Let

$$\begin{aligned} {[}a,b]=[a_1,b_1]\times [a_2,b_2]\times \cdots \times [a_n,b_n] \end{aligned}$$

denote an n-dimensional parallelepiped.

For brevity, in addition to the full notation, we shall denote partial derivatives with respect to the j-th variable by \(\partial _j\), \(\partial _j^2\), etc.

3 Multidimensional spaces

In this section, we recall or introduce the needed notions of function spaces. As mentioned, one of them is the notion of product Hardy space. A variety of the corresponding Fourier-Hardy inequalities will be proved for they. The rest of the section will be devoted to absolute continuity and variation type functionals.

3.1 Product Hardy spaces

We will deal with the product Hardy spaces

$$\begin{aligned} H^1_m({\mathbb {R}}^n):=H^1_{\eta ^1,\ldots ,\eta ^m}:=H^1({\mathbb {R}}^{n_1}\times \cdots \times {\mathbb {R}}^{n_m}), \end{aligned}$$

with \(n_1+n_2+\cdots +n_m=n\), \(\eta ^1+\cdots +\eta ^m=\textbf{1}\) and \(|\eta ^j|=n_j\), \(j=1,2,\ldots ,m\). Trivially, \(H^1_\textbf{1}({\mathbb {R}}^n)=H^1({\mathbb {R}}^n).\) These spaces are of interest and importance in certain questions of Fourier Analysis (see, e.g., [6, 7, 10, 15]).

Let us give one of the definitions. For an appropriate function or tempered distribution f,  the Riesz operators (transforms) can be defined as follows. Let

$$\begin{aligned} \mu _{q_i}(x_{\eta ^i})=\frac{ix_{q_i}}{|x_{\eta ^i}|}, \qquad q_i: \eta ^i_{q_i}=1, \end{aligned}$$

be \(n_i\) multipliers corresponding to the variables \(x_{\eta ^i}\). We define the corresponding Riesz transforms by means of their Fourier transforms via

$$\begin{aligned} \widehat{R_{\eta ^i}^{q_i} f} (x)=\mu _{q_i}(x_{\eta ^i})\widehat{f}(x). \end{aligned}$$

The Riesz transforms defined in the introduction are those for the case \(m=1\). We have

$$\begin{aligned} \qquad \Vert f\Vert _{H^1_m}&\sim \sum \limits _{k=1}^m \sum \limits _{\begin{array}{c} j_1,\ldots ,j_k:\\ 1\le j_1<\cdots <j_k\le m \end{array}} \sum \limits _{\begin{array}{c} q_{j_1},\ldots , q_{j_k}: \\ \eta _{q_{j_1}}^{j_1}=1,\ldots , \eta _{q_{j_k}}^{j_k}=1 \end{array}} \Vert R_{\eta ^{j_1}}^{q_{j_1}}\cdots R_{\eta ^{j_k}}^{q_{j_k}} f\Vert _{L^1({\mathbb {R}}^n)}\nonumber \\&\quad + \Vert f\Vert _{L^1({\mathbb {R}}^n)}. \end{aligned}$$
(2)

In the case of the product Hardy space with \(m=n\) and, correspondingly, \(n_1=\cdots =n_n=1\), we can use the notation

$$\begin{aligned} H^1({\mathbb {R}}\times \cdots \times {\mathbb {R}}):=H_n^1 ({\mathbb {R}}\times \cdots \times {\mathbb {R}}). \end{aligned}$$

Since each of the components is one-dimensional, the Riesz transforms become the Hilbert ones; thus the combinations of the Hilbert transforms replace the Riesz transforms in (2). By this, the norm of \(f \in H^1({\mathbb {R}}\times \cdots \times {\mathbb {R}})\) on the basis of the Hilbert transforms, applied to corresponding variables as in [10], will be

$$\begin{aligned} \Vert f\Vert _{H^1({\mathbb {R}}\times \cdots \times {\mathbb {R}})} =\sum \limits _{\textbf{0}\le \eta \le \textbf{1}} \Vert {\mathcal {H}}_{\eta }f\Vert _{L^1({\mathbb {R}}^n)}. \end{aligned}$$
(3)

We note that

$$\begin{aligned} H^1({\mathbb {R}}\times \cdots \times {\mathbb {R}})\subset H^1({\mathbb {R}}^n), \end{aligned}$$
(4)

see, e.g., [10, Th.1]. In the same paper [10], one can find A. Uchiyama’s example that this inclusion is proper.

Roughly speaking, the product Hardy spaces appear when all the n variables are split in groups and the function is in the real Hardy space with respect to each of the group and every combination of groups still claims for certain Hardy space conditions with respect to the rest of the variables. A natural question arises, how many such spaces exist for the fixed dimension. It is clear that if we do not take into account the case \(m=1\), that is the case of the classical real Hardy space \(H^1({\mathbb {R}}^n)\), the only case with one option is that where \(m=n\) and, correspondingly, \(n_1=\cdots =n_n=1.\) In all other cases there are plenty of product Hardy spaces. Denoting its number by S(n, m), one can easily find out that

$$\begin{aligned} S(n,2)=2^{n-1}-1 \end{aligned}$$

and

$$\begin{aligned} S(n,n-1)=\frac{n(n-1)}{2}. \end{aligned}$$

For the rest of S(n, m) calculations are not that simple and turn out to be the classical Stirling set numbers (or the Stirling numbers of the second kind), see, e.g., [11] or more recent [14]. For every \(m=2,3,\ldots ,n-1\), we have

$$\begin{aligned} S(n,m)=\frac{1}{m!}\sum \limits _{k=0}^m (-1)^{m-k} {m\atopwithdelims ()k} k^n. \end{aligned}$$

Of course, the cases \(m=1\) and \(m=n\) are subject to this formula as well. It is worth mentioning that knowing these numbers for dimension n, one can find them for the next dimension by means of the relation

$$\begin{aligned} S(n+1,m)=mS(n,m)+S(n,m-1). \end{aligned}$$

Finally, to know the number of all possible product Hardy spaces on \({\mathbb {R}}^n\), one has to sum up S(n, m) in m:

$$\begin{aligned} \sum \limits _{m=2}^n S(n,m)&=\sum \limits _{m=2}^n \frac{1}{m!} \sum \limits _{k=0}^m (-1)^{m-k} {m\atopwithdelims ()k} k^n\\&=\sum \limits _{k=1}^n (-1)^k k^n \sum \limits _{m=k}^n \frac{(-1)^m}{m!} {m\atopwithdelims ()k}-1. \end{aligned}$$

By the way, funny questions may be asked about the construction of certain product Hardy spaces, for example, those related to the Goldbach problem.

3.2 Fourier–Hardy type inequalities

In Sect. 2.1, the one-dimensional Fourier-Hardy inequality (1) established the needed connections between all the elements of the construction. We present three multivariate scenarios where such inequalities come into play.

First, recall that (1) has been referred to the general multidimensional formula for \(H^1({\mathbb {R}}^n)\) (in [8, (7.24)])

$$\begin{aligned} \int _{{\mathbb {R}}^n}\frac{|\widehat{g}(x)|}{|x|^n} \,dx\lesssim \Vert g\Vert _{H^1({\mathbb {R}}^n)}. \end{aligned}$$
(5)

On the other hand, for the product Hardy space with \(m=n\) and, correspondingly, \(n_1=\cdots =n_n=1\), the Fourier-Hardy inequality can naturally be rewritten as (see [12] or [13])

$$\begin{aligned} \int _{{\mathbb {R}}^n}\frac{|\widehat{g}(x)|}{|x_1|\cdots |x_n|}\,dx \lesssim \Vert g\Vert _{H^1({\mathbb {R}}\times \cdots \times {\mathbb {R}})}. \end{aligned}$$
(6)

In fact, these two versions are particular cases of the following one, more general and unifying. We naturally assume \(\eta ^1\),...,\(\eta ^m\) to be such that \(|\eta ^1|=n_1\),..., \(|\eta ^m|=n_m\), and \(n_1+\cdots +n_m=n\) and \(\eta ^1+\cdots +\eta ^m=\textbf{1}\).

Proposition 1

There holds

$$\begin{aligned} \int _{{\mathbb {R}}^n}\frac{|\widehat{g}(x)|}{|x_{\eta ^1}|^{n_1}\cdots |x_{\eta ^m}|^{n_m}}\,dx \lesssim \Vert g\Vert _{H^1({\mathbb {R}}^{n_1}\times \cdots \times {\mathbb {R}}^{n_m})}. \end{aligned}$$
(7)

Proof

We will just apply (5) repeatedly for each \({\mathbb {R}}^{n_j}\). We have, by (5) applied to the Fourier transform of g just with respect to the \(\eta ^m\) coordinates,

$$\begin{aligned}&\int _{{\mathbb {R}}^{n-n_m}} \frac{1}{|x_{\eta ^1}|^{n_1}\cdots | x_{\eta ^{m-1}}|^{n_{m-1}}}\, \biggl [\int _{{\mathbb {R}}^{n_m}} \frac{|\widehat{g}(x)|}{|x_{\eta ^m}|^{n_m}}\,dx_{\eta ^m}\biggr ] \,dx_{\eta ^1}\cdots dx_{\eta ^{m-1}}\\&\quad \lesssim \int _{{\mathbb {R}}^{n-n_m}} \frac{\sum \limits _{q_m: \eta _{q_m}^m=1} \Vert R_{\eta ^m}^{q_m}{\mathcal {F}}_{\textbf{1}-\eta ^m} g(x_{\textbf{1}-\eta ^m},\cdot )\Vert _{L^1({\mathbb {R}}^{n_m})}}{|x_{\eta ^1}|^{n_1}\cdots |x_{\eta ^{m-1}}|^{n_{m-1}}}\, dx_{\textbf{1}-\eta ^m}. \end{aligned}$$

Since \({\mathcal {F}}_{\textbf{1}-\eta ^m}g\) and the involved Riesz transforms are taken with respect to different variables, they commute. We can now apply the above procedure to

$$\begin{aligned} \int _{{\mathbb {R}}^{n_1}}\cdots \int _{{\mathbb {R}}^{n_{m-1}}} \frac{\,|{\mathcal {F}}_{\textbf{1}-\eta ^m}R_{\eta ^m}^{q_m} g(x_{\textbf{1}-\eta ^m},\cdot )|}{|x_{\eta ^1}|^{n_1}\cdots | x_{\eta ^{m-1}}|^{n_{m-1}}}\,dx_{\eta ^1}\cdots dx_{\eta ^{m-1}}, \end{aligned}$$

repeatedly for each integral. This completes the proof. \(\square \)

It is clear that we get (5) as a partial case where \(m=1\) and \(n_1=n\), while (6) follows, as mentioned, for \(m=n\) and, correspondingly, \(n_1=\cdots =n_n=1.\)

3.3 Absolute continuity

We discuss a multidimensional notion of absolute continuity; serious study of this notion in the multivariate case may be traced back to the classical monograph [17]. Later on, certain results can be found in, e.g., [5, 9, 18]). In [1], the authors had to exploit the absolute continuity of a different form. It suggests a universal approach. Since in our considerations we are going to deal with derivatives of higher order, and each of them should be absolutely continuous, we will make use of the absolute continuity in a single variable for almost every value of the rest of the variables (see, e.g., [2,3,4, 16, 19]) On the other hand, our definition and subsequent operations will be of two kinds: one for usual multivariate functions and the other for vector-valued functions. The latter will be the gradient of a function in our study. We hope that certain “abuse” of notation will result in no confusion, while the similarity in notation will make the presentation more “homogeneous”.

Definition 1

A function \(f:{\mathbb {R}}^n\longrightarrow \mathbb {C}\) is said to be locally absolutely continuous, written \(f\in AC_{loc}({\mathbb {R}}^n)\), if it is locally absolutely continuous in each variable for almost every value of the rest of the variables, that is, for every interval \(\prod \nolimits _{i=1}^n [a_i,b_i]\) and for every \(j=1,2,\dots ,n\), the \(j-\)th section \(f(x_j,x_{\textbf{1}-e_j}):[a_j,b_j]\longrightarrow \mathbb {C}\) is absolutely continuous for almost every \(x_{\textbf{1}-e_j}\in \prod \nolimits _{i\ne j}[a_i,b_i]\). For a function (mapping) \(f=(f_1,\ldots ,f_n):{\mathbb {R}}^n\longrightarrow \mathbb {C}^n\), we write \(f\in AC_{loc}({\mathbb {R}}^n)\) if each \(f_1\),...,\(f_n\) is locally absolutely continuous in the j-th variable for almost every value of the rest of the variables.

The main property of absolutely continuous functions is the possibility to restore the function from its derivative. In symbols, this can be written as

$$\begin{aligned} f(x)&-f(x_1,\ldots ,x_{j-1},a_j,x_{j+1},\ldots ,x_n)\nonumber \\&=\int _{a_j}^{x_j} \partial _j f(x_1,\ldots ,x_{j-1},u_j, x_{j+1},\ldots ,x_n)\,du_j, \end{aligned}$$
(8)

\(a_j\le x_j\le b_j\), for almost every \(x_{\textbf{1}-e_j}\in \prod \nolimits _{i\ne j}[a_i,b_i]\).

We will denote by \(AC_0({\mathbb {R}}^n)\) the subspace of \(AC_{loc}({\mathbb {R}}^n)\) of the functions that vanish at infinity, namely,

$$\begin{aligned} AC_0({\mathbb {R}}^n):=\{f\in AC_{loc}({\mathbb {R}}^n): \ \lim _{|x|\rightarrow \infty } f(x)=0\}. \end{aligned}$$

We point out that the absolute continuity could be relaxed at the origin, assuming local absolute continuity on each interval not containing 0. However, this is a slight improvement on which we do not wish to concentrate and thus omit the details.

3.4 Product variation type spaces

As mentioned, Hardy’s variation, at least in our study, is closely related with the product Hardy space with \(m=n\) and, correspondingly, \(n_1=\cdots =n_n=1.\) On the other hand, the case of \(n_1=n\), that is, \(H^1({\mathbb {R}}^n)\), is related to Tonelli’s variation in [1]. In fact, as outlined above, we exploit absolute continuity at every step, while the integrability of the “last” derivative will be provided by belonging to a certain Hardy space. Let us treat all the cases in their full generality. However, before going further, we recall the class we dealt with in [1]. For \(f:{\mathbb {R}}^n\rightarrow {\mathbb {C}}\) and \(k=0,1,2,\ldots \), we define the following operator:

$$\begin{aligned} D^mf(x)= {\left\{ \begin{array}{ll} \Delta ^kf(x), &{} \hbox {for}\ m=2k,\\ \nabla \Delta ^{k}f(x), &{} \hbox {for}\ m=2k+1. \end{array}\right. } \end{aligned}$$

Here \(\Delta \) denotes the Laplace operator, \(\Delta f(x)=\partial ^2_1 f(x)+\cdots +\partial ^2_n f(x)\), and \(\nabla \) denotes the gradient, \(\nabla f(x)=\left( \partial _1f(x),\ldots ,\partial _n f(x)\right) \). It follows from the definition that \(D^0f(x)=f(x)\), \(D^1 f(x)=\nabla f(x)\), \(D^2 f(x)=\Delta f(x)\), \(D^3 f(x) =\nabla \Delta f(x)\), and so on. Of course, the essential difference between the cases of even and odd dimensions is evident. The operator \(D^m\) turns out then to be of the scalar or the vector form, respectively. On the other hand, such a definition allows one to have a consistent approach to the results as well as consistent definitions of the classes of functions to be dealt with.

Definition 2

We say that \(f\in AC_m\), \(m=1,2,\ldots \), if \(D^lf\in AC_0({\mathbb {R}}^n)\) for every \(0\le l\le m\).

Recall that in the case of \(l=2k+1\), that is, for l odd, \(D^l f =\nabla \Delta ^{k} f\in AC_0({\mathbb {R}}^n)\) means that \(\partial _j \Delta ^{k}f\) is locally absolutely continuous in the j-th variable and vanishes at infinity for every \(j=1,\ldots , n\).

Definition 2 was applied to the case where \(m=1\) (or, equivalently, \(n_m=n\)) in order to realize the mentioned strategy of getting absolute continuity for every derivative (whatever this means) of order up to \(n-1\) and posing special assumptions, including integrability, on the last derivative. However, we cannot use the same definition for any case, since our variables are split into several groups. We thus adjust our definition to such cases. In fact, it is very probable that the above definition does work in more general cases, with an appropriate assumption on the “last derivative”. However, it a priori seems redundant. Indeed, it takes into account all the variables on each step, while in the product case only specific variables are crucial for every group.

For \(f:{\mathbb {R}}^n\rightarrow {\mathbb {C}}\) and \(j=0,1,2,\ldots \), we define the following operator:

$$\begin{aligned} D^l_\eta f(x)= {\left\{ \begin{array}{ll} \Delta ^j_\eta f(x), &{} \hbox {for}\ l=2j,\\ \nabla _\eta \Delta _\eta ^{j} f(x), &{} \hbox {for}\ l=2j+1. \end{array}\right. } \end{aligned}$$

Here \(\Delta _\eta \) denotes the Laplace operator with respect to these \(x_\eta \) variables,

$$\begin{aligned} \Delta _\eta f(x)=\sum \limits _{i: \eta _i=1} \partial ^2_i f(x), \end{aligned}$$

and \(\nabla _\eta \) denotes the gradient,

$$\begin{aligned} \nabla _\eta f(x)=\left( \partial _i f(x_\eta , x_{\textbf{1}-\eta }), \quad i: \eta _i=1\right) . \end{aligned}$$

Definition 3

We say that \(f\in AC_\eta \), if \(D^l_\eta f\in AC_0({\mathbb {R}}^n)\) for every \(0\le l\le |\eta |\).

Some important remarks are in order. If \(\eta =e_j\), then the above definition means that the function and its derivative in the j-th variable are (locally) absolutely continuous (in the above defined sense). Further, in the definition of \(D^l_\eta \) we can restrict ourselves only to the \(\nabla _\eta \) operator, by writing \(D^l_\eta =\nabla ^l_\eta \), where \(\nabla _\eta \,\nabla _\eta \) is understood as the scalar product \(\langle \nabla _\eta ,\nabla _\eta \rangle \), that is, the Laplacian.

We now proceed to superpositions of the introduced definitions. Let \(\eta ^1\),...,\(\eta ^m\) be the same vectors as \(\eta \) such that \(\eta ^1+\cdots +\eta ^m \le \textbf{1}\), \(m=2,\ldots ,n\). With the definition of \(f\in AC_{\eta ^i}\), \(i=1,\ldots ,m\), in hand, the following one will be inductive.

Definition 4

We say that \(f\in AC_{\eta ^1,\ldots ,\eta ^m}\), \(m=2,\ldots ,n\), if \(f\in AC_{\eta ^1,\ldots ,\eta ^{m-1}}\) and \(D^{l_m}_{\eta ^m} \left( D^{l_1}_{\eta ^1}\cdots D^{l_{m-1}}_{\eta ^{m-1}}f\right) \in AC_0({\mathbb {R}}^n)\) for every \(0\le l_m\le |\eta ^m|\).

In words, the definition of the first group given above suggests all the derivatives except the last ones (the corresponding Laplace operator or each of the last partial derivatives) to be locally absolutely continuous. However, the next group adds absolute continuity to them. Only the action of the very last group leaves the last derivatives not necessarily absolutely continuous.

Here we have managed the inductive process in the convenient order \(1,2,\ldots ,m\). However, it is obvious that the result does not depend on the order, since all our derivatives are continuous. It is worth reminding that if certain \(l_i\) is odd, then the next step is applied to all the entries of the corresponding \(\nabla \), and the whole process ramifies. This may happen several times, whereas each path should be verified.

4 Integrability of the Fourier transform

In the previous section, all the preliminaries are given for establishing sufficient conditions for the integrability of the Fourier transform. It is straight-forward for the even dimensions: we just integrate by parts so that \(\frac{1}{|x|^n}\) appears and then apply the Fourier-Hardy inequality. For the odd dimensions, one has to work harder: one more step is needed, where the integrability of the corresponding derivatives is explicitly involved. However, in general the result is, roughly speaking, the same in each dimension: belonging of the “n-th derivative” to the Hardy space ensures the integrability of the Fourier transform. In these special conditions, we will deal with the multidimensional Fourier transform defined in the improper sense, i.e.,

$$\begin{aligned} \int _{{\mathbb {R}}^n}f(u)e^{-i\langle x,u\rangle }du =\lim _{M\rightarrow +\infty }\int _{[-M,M]^n} f(u)e^{-i\langle x,u\rangle }du. \end{aligned}$$

In dimension one, this is typical for functions of bounded variation; it is no wonder that the same approach is applicable in several dimensions (cf., e.g., [20]).

Theorem 2

Let \(|\eta ^i|=n_i\), \(i=1,\ldots ,m\) and \(n_1+n_2+\cdots +n_m=n.\) Let \(j^*\) be such that \(\eta ^m_{j^*}=1\). If \(f\in AC_{\eta ^1,\ldots ,\eta ^{m-j^*}}\) and \(D^{n_m}_{\eta ^m}\cdots D^{n_1}_{\eta ^1}f\in H^1_{\eta ^1,\ldots ,\eta ^m}\), then \(\widehat{f}\in L^1({\mathbb {R}}^n)\).

Proof

We will integrate by parts so that the function or its preceding derivatives will be differentiated. Let us start with this procedure with respect to \(D^{n_1}_{\eta ^1}\). For the case where \(n_1=2k\), we act as follows. Since, by assumption, \(f\in AC_0({\mathbb {R}}^n)\), integrating by parts with respect to \(u_j\) such that \(\eta ^1_j=1\), we obtain for \(x_j\ne 0\),

$$\begin{aligned} \widehat{f}(x)&= \lim _{M\rightarrow +\infty }\int _{[-M,M]^{n-1}} \left( \lim _{M\rightarrow +\infty }\int _{[-M,M]} f(u_j,u_{\textbf{1}-\eta ^1_j}) e^{-i\langle u , x\rangle }\,du_j\right) \,du_{\textbf{1}-\eta ^1_j}\\&=\lim _{M\rightarrow +\infty }\int _{[-M,M]^{n-1}} \biggl [ \lim _{M\rightarrow +\infty } f(u_j,u_{\textbf{1}-\eta ^1_j}) \frac{e^{-i\langle u , x\rangle }}{-i x_j} \Big |_{-M}^M\\&\quad -\frac{i}{x_j} \lim _{M\rightarrow +\infty }\int _{[-M,M]} \partial _j f (u_j,u_{\textbf{1}-\eta ^1_j}) e^{-i\langle u,x\rangle }\,du_j\biggr ] \,du_{\textbf{1}-\eta ^1_j}\\&= - {i\over x_j} \lim _{M\rightarrow +\infty }\int _{[-M,M]^{n}} \partial _j f (u) e^{-i\langle u,x\rangle }\,du. \end{aligned}$$

Integrating by parts again in a similar way, since by assumption \(\nabla _{\eta ^1} f\in AC_0({\mathbb {R}}^n)\), we get

$$\begin{aligned} \widehat{f}(x)=-\frac{1}{x_j^2}\lim _{M\rightarrow +\infty }\int _{[-M,M]^{n}} \partial ^2_j f(u)e^{-i\langle u,x\rangle } \,du. \end{aligned}$$

Summing up in j such that \(\eta ^1_j=1\), we have, for a.e. \(x\in {\mathbb {R}}^n\),

$$\begin{aligned} -|x_{\eta ^1}|^2 \widehat{f}(x)= \lim _{M\rightarrow +\infty } \int _{[-M,M]^{n}} \Delta _{\eta ^1} f(u)e^{-i\langle x,u\rangle }\,du. \end{aligned}$$

We can repeat this procedure replacing f by \(\Delta _{\eta ^1} f\), then \(\Delta _{\eta ^1} f\) by \(\Delta ^2_{\eta ^1} f\), and so on, as long as we obtain, for a.e. \(x\in {\mathbb {R}}^n\),

$$\begin{aligned} (-1)^{k} |x_{\eta ^1}|^{n_1} \widehat{f}(x) =\lim _{M\rightarrow +\infty }\int _{[-M,M]^{n}} \Delta ^{k}_{\eta ^1} f(u)e^{-i\langle x,u\rangle }\,du =\widehat{\Delta ^{k}_{\eta ^1} f}(x). \end{aligned}$$
(9)

Therefore

$$\begin{aligned} |x_{\eta ^1}|^{n_1}|\widehat{f}(x)|= |\widehat{\Delta ^{k}_{\eta ^1} f}(x)| = |\widehat{(\nabla ^{k}_{\eta ^1}\nabla ^{k}_{\eta ^1} ) f}(x)| = |\widehat{\nabla ^{n_1}_{\eta ^1} f}(x)|. \end{aligned}$$
(10)

Let now \(n_1=2k+1\). The absolute continuity assumptions allow one to repeat the integration by parts and summation procedure k times. This yields (10), for a.e. \(x\in {\mathbb {R}}^n\), but we must continue rather than proceed to a different group.

Again integrating by parts, we obtain, for the same j,

$$\begin{aligned} (-1)^{k}|x_{\eta ^1}|^{2k} \widehat{f}(x)=-{i\over x_j} \lim _{M\rightarrow +\infty }\int _{[-M,M]^{n}} \partial _j \Delta ^{k}_{\eta ^1} f(u) e^{-i\langle x,u\rangle }du, \end{aligned}$$

and so

$$\begin{aligned} |x_j|\,|x_{\eta ^1}|^{2k}|\widehat{f}(x)|=\biggl |\lim _{M\rightarrow +\infty } \int _{[-M,M]^{n}} {\partial _j}\Delta ^{k}_{\eta ^1} f(u) e^{-i\langle x,u\rangle }du\biggr |. \end{aligned}$$
(11)

Hence,

$$\begin{aligned} \sum \limits _{j: \eta _j^1=1}|x_j|\,|x_{\eta ^1}|^{2k}|\widehat{f}(x)| = \sum \limits _{j: \eta _j^1=1} \biggl |\widehat{\partial _j\Delta ^{k}_{\eta ^1} f}(x)\biggr |. \end{aligned}$$
(12)

Since \(|x_{\eta ^1}|\le \sum \nolimits _{j: \eta _j^1=1}|x_j|\), we get

$$\begin{aligned} |\widehat{f}(x)|\le \frac{1}{|x_{\eta ^1}|^{n_1}} \sum \limits _{j: \eta _j^1=1} \biggl |\widehat{\partial _j \Delta ^{k}_{\eta ^1} f}(x)\biggr |. \end{aligned}$$
(13)

Moreover we can write

$$\begin{aligned} \sum \limits _{j: \eta _j^1=1} \biggl |\widehat{\partial _j \Delta ^{k}_{\eta ^1} f}(x)\biggr |= & {} \sum \limits _{j: \eta _j^1=1} \biggl |{\mathcal {F}}\left( \partial _j (\nabla ^{k}_{\eta ^1} \nabla ^{k}_{\eta ^1} )f\right) (x)\biggr |\\= & {} \sum \limits _{j: \eta _j^1=1} \biggl |\widehat{\partial _j \nabla ^{2k}_{\eta ^1} f}(x)\biggr | =\biggl |\widehat{\nabla ^{2k+1}_{\eta ^1} f}(x)\biggr |. \end{aligned}$$

Finally, both (10) and (13) can be combined in one universal estimate

$$\begin{aligned} |\widehat{f}(x)|\le \frac{1}{|x_{\eta ^1}|^{n_1}} \biggl |\widehat{\nabla ^{n_1}_{\eta ^1} f}(x)\biggr |. \end{aligned}$$
(14)

We now can continue integrating by parts in the same manner with respect to the variables corresponding \(\eta ^2\), but applied to \(\widehat{\nabla ^{n_1}_{\eta ^1} f}\) rather than to \(\widehat{f}\). Naturally, the step after that will be the same but corresponding to \(\eta ^3\), and so on till those corresponding to \(\eta ^{m-1}\). Analogously to the above, we obtain the estimate

$$\begin{aligned} |\widehat{f}(x)|\le \frac{1}{|x_{\eta ^1}|^{n_1}}\cdots \frac{1}{|x_{\eta ^{m-1}}|^{n_{m-1}}} \biggl |{\mathcal {F}}\left( \nabla ^{n_{m-1}}_{\eta ^{m-1}}\cdots \nabla ^{n_1}_{\eta ^1} f\right) (x)\biggr |. \end{aligned}$$
(15)

We are still able to repeat the above process applied to \({\mathcal {F}}\left( \nabla ^{n_{m-1}}_{\eta ^{m-1}}\cdots \nabla ^{n_1}_{\eta ^1} f\right) \) in the completely same mode up to

$$\begin{aligned} {\mathcal {F}}\left( \nabla ^{n_{m-e_{j^*}}}_{\eta ^{m}} \nabla ^{n_{m-1}}_{\eta ^{m-1}}\cdots \nabla ^{n_1}_{\eta ^1} f\right) . \end{aligned}$$
(16)

We need to make one more step in the integration by parts, this time taking into account the last absolute continuity assumption on the transformed function in (16) and that the final function \(\nabla ^{n_{m-e_{j^*}}}_{\eta ^{m}}\nabla ^{n_{m-1}}_{\eta ^{m-1}}\cdots \nabla ^{n_1}_{\eta ^1} f\) is Lebesgue integrable. We then arrive at the estimate

$$\begin{aligned} |\widehat{f}(x)|\le \frac{1}{|x_{\eta ^1}|^{n_1}}\cdots \frac{1}{|x_{\eta ^{m-1}}|^{n_{m-1}}}\frac{1}{|x_{\eta ^{m}}|^{n_{m}}} \biggl |{\mathcal {F}}\left( \nabla ^{n_{m}}_{\eta ^{m}}\cdots \nabla ^{n_1}_{\eta ^1} f\right) (x)\biggr |. \end{aligned}$$
(17)

Integrating both parts over \({\mathbb {R}}^n\), using the assumption of the theorem and applying (7) to a single transformed function on the right-hand side of (17) or, as discussed above, to each summand of the transformed vector-function, we complete the proof. \(\square \)

5 Conclusion

The impact for this work is twofold. On the one hand, in the one-dimensional setting the real Hardy space is directly related to the space of (locally) absolutely continuous functions, which is the subspace of the space of functions with bounded variation. On the other hand, in the multivariate setting two classical real Hardy spaces \(H^1({\mathbb {R}}^n)\) and \(H^1({\mathbb {R}}\times \cdots \times {\mathbb {R}})\) are related to different spaces of functions with (different) bounded variations along with absolute continuity. In each of these case the bridge is the integrability of the Fourier transform. These suggested the authors an idea that there should exist a correspondence between the rich scale of product Hardy spaces, for which the two aforementioned classes are, in a sense, the endpoints, and a certain scale of spaces with properties resembling the boundedness of variation and absolute continuity. This idea proved to hold true. Such a parallel scale is established, and as in all the preceding cases it is tied to the product Hardy spaces by means of the integrability of the Fourier transform.