1. Introduction
Within the study of the Fourier transform, itself a vast and important area, the integrability of the Fourier transform is a topic of particular interest. In the literature, one can find numerous results in which such a property of the Fourier transform is pivotal; see, for example, the books [Reference Duoandikoetxea3, Reference Stein and Weiss22, Reference Trigub and Belinsky24] and, more recently, [Reference Grafakos5].
In dimension one, there is a nice and quite strong condition (see, e.g., [Reference Liflyand8] or [Reference Iosevich and Liflyand6, Ch.3]) that guarantees the integrability of the Fourier transform
\begin{equation} \widehat {f}(x)=\int _{\mathbb R} f(t)e^{-ixt}dt, \quad x\in \mathbb{R}, \end{equation}
of a function
$f\;:\; \mathbb R\to \mathbb C$
of bounded variation. The function
$f$
is also assumed to be (locally) absolutely continuous and vanishing at infinity, but these assumptions can be considered as minimal and almost necessary. The condition itself is
$f'\in H^1(\mathbb R)$
, where the space is the real Hardy space (recall that the derivative of a function of bounded variation exists almost everywhere and is Lebesgue integrable). Indeed, we just integrate by parts in (1) and apply the one-dimensional version of Fourier-Hardy’s inequality (see, e.g., [Reference Garcia-Cuerva and Rubio de Francia4, (7.24)])
\begin{equation} \int _{\mathbb R^n}\frac {|\widehat {g}(x)|}{|x|^n}\,dx\lesssim \|g\|_{H^1(\mathbb R^n)}. \end{equation}
Here, the Fourier transform is multivariate, defined as
\begin{equation} \widehat {g}(x)=\int _{\mathbb{R}^n} g(u)e^{-i\langle x,u\rangle }\,du, \quad x\in \mathbb{R}^n, \end{equation}
where
$\langle x,u\rangle =x_1u_1+\ldots +x_nu_n$
. At times, we use the notation
$Fg$
instead of
$\widehat g$
, particularly when the notation for the transformed function is too long. Of course, it is necessary to specify in what sense the Fourier transform is understood; for example, in the above one-dimensional consideration, it is understood in the improper sense. In the sequel, we shall mostly understand
$g$
(or a substitute for it) to be Lebesgue integrable, which is the most convenient setting for the Fourier transform.
We use the notation “
$ \lesssim $
” as an abbreviation for “
$\le C $
”, where
$C$
is a positive absolute constant that may vary from one occurrence to another.
It is worth mentioning that stopping the process even earlier, before applying the Fourier-Hardy’s inequality, leads to the necessary and sufficient condition for the integrability of the Fourier transform of functions in the considered class. The finiteness of the integral on the left-hand side of (2) in the case
$n=1$
leads to the space
\begin{equation*} Q\,:\!=\Big \{g\in L^1(\mathbb R)\; :\; \int _{\mathbb R}\frac {|\widehat {g}(x)|}{|x|}\,dx\lt \infty \Big \} ,\end{equation*}
introduced in [Reference Johnson and Warner7], where the necessary and sufficient condition is
$f'\in Q$
. However, checking whether a function belongs to
$Q$
, that is, whether the integral on the left-hand side of (2) is finite, is considerably more difficult than verifying the
$H^1$
condition. Discussions on this and related issues can be found in [Reference Liflyand10].
Equivalent norms in
$H^1(\mathbb R)$
may be introduced in various ways. For example, it can be understood as the sum of the
$L^1(\mathbb R)$
norms of the function and its Hilbert transform. A multidimensional analog of this characterization, the one via the Riesz transforms, will be explained below. To be precise, under the assumption that
$f'\in H^1(\mathbb R)$
, there is no need to assume bounded variation. On the one hand, this assumption itself provides
$f'\in L^1(\mathbb R)$
and, on the other hand, the integrability of the derivative together with the absolute continuity of the function makes it to be of bounded variation (at least, that it is equal almost everywhere to such a function; see, e.g., [Reference Bochner2, Ch. 1] or [Reference Iosevich and Liflyand6, 3.5.1]).
Realizing this scheme in higher dimensions, one unavoidably encounters certain difficulties. Attempts have been made in [Reference Angeloni, Liflyand and Vinti1] and in [Reference Liflyand9], where some notions of multidimensional variation were involved, which led to the use of repeated partial derivatives and their absolute continuity. However, (2) strongly suggests applying the Riesz fractional derivative, denoted by
$D^\alpha f$
,
$\alpha \gt 0$
, which is defined at least for “nice functions”, via the Fourier transforms as
\begin{equation} \widehat {D^\alpha f}(x)=|x|^\alpha \widehat {f}(x). \end{equation}
For “more general” functions, this relation holds under an appropriate definition of the Riesz derivative. On the other hand,
$D^\alpha f$
can be interpreted as the fractional power
$({-}\Delta )^{\frac {\alpha }2} f$
, where
$\Delta$
denotes the Laplace operator.
One might think that the problem just reduces to establishing (4) under appropriate assumptions. This is true only to a certain extent. Of course, we wish to define the function
$f$
on the right-hand side of (4) on the largest possible space. However, numerous questions arise concerning the left-hand side. Moreover, even if it can be guaranteed that the left-hand side is the Fourier transform of an appropriate function, to be called the Riesz derivative, it is natural to seek a direct way to compute this derivative. These considerations inevitably lead not only to certain issues in Fourier analysis, but also to the study of Riesz (and related) potentials and derivatives, as well as hypersingular integrals. It is known that the idea of multidimensional potentials is due to M. Riesz, see, for example [Reference Riesz15]. The topic of hypersingular integrals can be traced back, in dimension one, to the seminal work of Marchaud [Reference Marchaud14]. There are numerous books in which one-dimensional theories and results are considered. However, there are very few books that provide more or less complete information in the multivariate case; we mention [Reference Rubin16–Reference Samko, Kilbas and Marichev18]. Of course, the names of the authors whose results were pioneering in this area should be mentioned: Stein (see, e.g., [Reference Stein19]) and Lizorkin (see, e.g., [Reference Lizorkin12, Reference Lizorkin13]). In Theorem 1, our main result, a suitable choice of function space and fractional derivative achieves a balance between assumptions and conclusions, as reflected in the corollaries and potentially to be clarified by future research.
Returning to the space
$Q$
, there is also a converse aspect to this approach. If one assumes that the integral
\begin{eqnarray} \int _{\mathbb R}\frac {|\widehat {g}(x)|}{|x|}\,dx \end{eqnarray}
is finite, then
$g$
is the derivative of a function
$f$
of bounded variation which is absolutely continuous,
$\lim \limits _{|t|\to \infty }f(t)=0$
and its Fourier transform is integrable. See [Reference Liflyand10, Ch.1, Prop.1.18].
We are going to generalize this within the framework of the present setting, where the property of being the derivative of a function in a certain class will be replaced by a relation between the Riesz fractional derivative and the corresponding Riesz and Bessel potentials.
It is worth comparing, from a general point of view, the derivative
$f'$
in the condition
$f'\in H^1(\mathbb R)$
with the Riesz derivative of order one in the one-dimensional setting. First of all, the function itself is assumed to be locally absolutely continuous in the former setting and Lebesgue integrable in the latter. Recall that a function of bounded variation, though possessing the (integrable) derivative almost everywhere, need not, in general, be Lebesgue integrable. The first two derivatives are by no means equivalent. On the one hand, the Riesz derivative requires a more complicated procedure. On the other hand, this allows us to avoid the assumption of absolute continuity. In both cases, we obtain the integrability of the Fourier transform. In addition, in the multivariate case, applying ordinary derivatives up to order
$n$
is more tedious than using hypersingular integral representations, which are uniform across dimensions and orders.
The outline of the paper is as follows. In Section 2, we consider the classes of functions involved and certain corresponding notions. In Section 3, we establish our main result and some corollaries. Finally, in Section 4, we discuss characterizations of the function space used in establishing the results obtained. This is, in a sense, a substitute for (5) and related results.
2. Classes of functions
In this section, we introduce the main classes of functions that we will consider. We allow for the use of both centered and non-centered differences.
\begin{align} \big(\Delta _h^l\, f\big)(x)&=\big(\tau _{-\frac {h}2}-\tau _{\frac {h}{2}}\big)^l\, f(x)\nonumber \\ &=\sum \limits _{k=0}^l ({-}1)^k {\left({l}\atop{k}\right)}\, f\Big (x+\Big (\frac {l}{2-k}\Big )\Big ), \end{align}
where
$(\tau _h\, f)(x)=f(x-h)$
, with
$x$
being the center of this difference. The non-centered difference is
\begin{equation} \big(\Delta _h^l\, f\big)(x)=(I-\tau _h)f(x)=\sum \limits _{k=0}^l ({-}1)^k {\left({l}\atop{k}\right)}\, f(x-kh), \end{equation}
where
$I$
is the identity operator. To avoid complications, we adopt the same notation
$\big(\Delta _h^l\, f\big)(x)$
for both types of differences, where the choice of difference can be specified when essential. The reason for considering both types of differences lies in the fact that it is not always possible to restrict oneself to just one type of difference in applications. For the relations between the differences, see [Reference Samko17, Ch. 3, Sect. 1].
2.1. Riesz derivative
We shall rely on the constructions in [Reference Samko17, Ch. 3]. We formally define the Riesz fractional derivative of order
$\alpha \gt 0$
by
\begin{equation} D^\alpha f(x)\,:\!=\frac 1{d_{n,l}(\alpha )}\int _{\mathbb R^n} \frac {\big(\Delta _y^l\, f\big)(x)} {|y|^{n+\alpha }}\,dy. \end{equation}
The value of
$d_{n,l}(\alpha )$
is chosen so that the right-hand side in (8) does not depend on
$l$
, for
$l\gt \alpha$
.
Of course, all these statements hold provided the class of functions and the mode of convergence of the integral are properly chosen. We will now proceed to this task. The integral
\begin{equation} D^\alpha _\delta\, f(x)\,:\!=\frac 1{d_{n,l}(\alpha )}\int _{|y|\gt \delta } \frac {\big(\Delta _y^l\, f\big)(x)} {|y|^{n+\alpha }}\,dy \end{equation}
will be called the truncated hypersingular integral.
Of course, the situation is much simpler for “very good (smooth)” functions
$f$
. For example, in the one-dimensional case, we have
\begin{equation*} D^1f(x)=\frac{1}{\pi}\frac{d}{dx}\int_{\mathbb R}\frac{f(t)}{x-t}\,dt=\frac{1}{\pi}\int_{\mathbb R}\frac{f'(t)}{x-t}\,dt,\end{equation*}
where each of the integrals is understood in the principal value sense.
However, we shall make use of the space of Lebesgue integrable functions for which the
$\alpha$
-th Riesz derivative exists in the sense that the truncated hypersingular integrals converge in the
$L^1$
norm as
$\delta \to 0$
. We shall continue denoting this limit by
$D^\alpha f$
, while the space of such functions will be denoted by
$L^{1,\alpha }$
. We will discuss this space later on, in particular the case
$\alpha \ge n$
, which is more problematic in various respects. Meanwhile we note that
\begin{equation} \|\, f\|_{L^{1,\alpha }}\,:\!=\|\,f\|_{L^1}+\|D^\alpha f\|_{L^1}. \end{equation}
Here, the subscript
$1$
indicates that we are working in
$L^1$
. If
$p$
were used instead, this would correspond to
$L^p$
actions.
On the other hand, we introduce a different subspace of the class of integrable functions. For
$g\in L^1(\mathbb R^n)$
, let
\begin{equation} Q_\alpha =Q_\alpha (\mathbb R^n) \,:\!=\Big \{g\in L^1(\mathbb R^n)\; :\; \int _{\mathbb R^n} \frac {|\widehat {g}(x)|}{|x|^\alpha }\,dx\lt \infty \Big \}. \end{equation}
For
$\alpha =n$
, this space was introduced in [Reference Johnson and Warner7] for different purposes and was later widely used in problems of integrability of the Fourier transforms; see, e.g., [Reference Liflyand10] or [Reference Angeloni, Liflyand and Vinti1].
The spaces introduced in (10) and (11) will play a crucial role in our considerations. In fact, they arise in a natural way along the chosen lines of reasoning. More precisely, since the goal was to derive certain conditions for integrability of the Fourier transform in terms of the Riesz derivatives,
$L^1$
turned out to be the optimal space. This inevitably leads to reduction of
$L^1$
to
$L^{1,\alpha }$
. Further, Fourier-Hardy’s inequality (2) serves as one of the main tools. Inspired by the space
$Q$
and its properties, we arrive at the extended notion of
$Q_\alpha$
spaces. It is, in a sense, pleasing to observe how these spaces correspond to one another in later results.
2.2. Real Hardy space
Using tools from real Hardy space theory, we find it necessary to present at least one of the equivalent definitions. Among the numerous characterizations of the real Hardy space, a particularly natural one is given in terms of the Riesz transforms –
$n$
singular integral operators – serving as an analog of the one-dimensional definition based on the unique singular operator, the Hilbert transform. The latter was the basis for all the previous constructions mentioned in the introduction. The
$j$
-th Riesz transform, for
$j=1,2,\ldots ,n,$
can be defined either explicitly
\begin{equation*} R_j\,f(x)=\frac {\Gamma \big(\frac {n+1}2\big)}{\pi ^{\frac {n+1}2}}\int _{\mathbb R^n}\frac {u_j}{|u|^{n+1}}f(x-u)\,du,\end{equation*}
or via the Fourier transform
\begin{equation*}\widehat {R_j\,f}(x)=i\frac {x_j}{|x|}\,\hat f(x).\end{equation*}
One can see that taking
$n=1$
gives just the Hilbert transform. It is well known (see, e.g., [Reference Stein21, Ch.VII, §3]) that
\begin{equation*} \|\,f\|_{H^1({\mathbb R^n})}\sim \|\,f\|_{L^1({\mathbb R^n})} +\sum \limits _{j=1}^n \|R_j\,f\|_{L^1({\mathbb R^n})}.\end{equation*}
For different characterizations, such as via maximal function or atomic, see, for example [Reference Stein21].
3. Integrability of the Fourier transform
We study conditions for the integrability of the Fourier transform of a function in
$L^{1,\alpha }$
.
Theorem 1.
Let
$\alpha \gt 0$
and
$f\in L^{1,\alpha }$
. Then we have
$\widehat {f}\in L^1({\mathbb R^n})$
if and only if
$D^\alpha f\in Q_\alpha$
.
Proof. In fact, we will prove (4). This relation is known for smooth enough functions, see (3.11) in [Reference Samko17, Ch. 3]. There are certain alterations and peculiarities in our setting.
First, we observe that
$D^\alpha f\in L^1({\mathbb R^n})$
, since it is the limit of
$L^1$
functions. Note that the integrability of the truncated hypersingular integrals (9) is obvious. For definiteness, we suppose that the difference
$\big(\Delta _y^l\, f\big)(x)$
is non-centered. Then
\begin{align*} \widehat {D^\alpha _\delta\, f}(x)&=\sum \limits _{k=0}^l ({-}1)^k {\left({l}\atop{k}\right)}\frac 1{d_{n,l}(\alpha )}\int _{|y|\gt \delta } \frac {dy}{|y|^{n+\alpha }}\,\int _{\mathbb R^n} e^{-i\langle x,u\rangle }f(u-ky)\,du\\ &=\widehat {f}(x)\sum \limits _{k=0}^l ({-}1)^k {\left({l}\atop{k}\right)}\frac 1{d_{n,l}(\alpha )}\int _{|y|\gt \delta } \frac {e^{-ik\langle x,y\rangle }}{|y|^{n+\alpha }}\,dy. \end{align*}
Hence,
\begin{equation} \widehat {D^\alpha _\delta\, f}(x)=\widehat {f}(x) \frac 1{d_{n,l}(\alpha )}\int _{|y|\gt \delta } \frac {(1-e^{-i\langle x,y\rangle })^l}{|y|^{n+\alpha }}\,dy. \end{equation}
The passage to the limit is in order. Taking
$D^\alpha f$
to be the mentioned
$L^1$
limit of
$D^\alpha _\delta f$
as
$\delta \to 0$
, existing by assumption, we have
\begin{equation*} \widehat {D^\alpha _\delta\, f}(x)=\widehat {D^\alpha f}(x)+F\big (\big[D^\alpha _\delta -D^\alpha \big]\, f\big )(x).\end{equation*}
Obviously,
\begin{equation*} \big |\,F\big(\big[D^\alpha _\delta -D^\alpha \big]\, f\big )(x)\,\big |\le \int _{\mathbb R^n} \big |\,\big[D^\alpha _\delta -D^\alpha \big]\, f(u)\big |\,du.\end{equation*}
The right-hand side tends to zero as
$\delta \to 0$
by assumption. Concluding, we can pass to the limit as
$\delta \to 0$
pointwise in both sides of (13). This results in
\begin{align} \widehat {D^\alpha f}(x)&=\widehat {f}(x) \frac 1{d_{n,l}(\alpha )}\int _{\mathbb R^n} \frac {(1-e^{-i\langle x,y\rangle })^l}{|y|^{n+\alpha }}\,dy\nonumber \\ &=|x|^\alpha \widehat {f}(x) \frac 1{d_{n,l}(\alpha )}\int _{\mathbb R^n} \frac {(1-e^{-i\langle \frac {x}{|x|},y\rangle })^l}{|y|^{n+\alpha }}\,dy. \end{align}
With a proper choice of
$l$
, the last integral is exactly
$d_{n,l}(\alpha )$
; for discussion on such a choice, see [Reference Samko17, Ch.3].
By these, we have arrived at (4) for the chosen class of functions. It remains to divide both sides by
$|x|^\alpha$
and take the
$L^1$
norm.
The following consequence is less general. However, this direct generalization of the scheme introduced at the beginning of this work is certainly reasonable, since using the real Hardy space – well established and admitting various characterizations – is much more convenient than dealing with
$Q_\alpha$
spaces.
Corollary 2.
If
$\alpha \gt 0$
and
$f\in L^{1,\alpha }$
, then
\begin{equation*} \int _{\mathbb R^n} \,|x|^{\alpha -n}|\,\widehat {f}(x)|\,dx\le C\|D^{\alpha }f\|_{H^1}.\end{equation*}
Proof. To prove this, we continue from where the proof of Theorem 1 left off. This can be rewritten as
\begin{equation*} \int _{\mathbb R^n} \,|x|^{\alpha -n}|\,\widehat {f}(x)|\,dx=\int _{\mathbb R^n} \, \frac {|\widehat {D^\alpha f}(x)|}{|x|^n}\,dx.\end{equation*}
By (2), the right-hand side is dominated by the
$H^1$
norm of
$D^\alpha f$
, as desired.
It turns out that we can derive one more direct consequence of Theorem 1. It concerns the so-called Wiener algebra
$W_0$
defined as follows. We say that
$f$
belongs to
$W_0=W_0(\mathbb R^n),$
written
$f\in W_0(\mathbb R^n),$
if
\begin{equation} f(x)=\int _{\mathbb R^n}g(u)e^{-i\langle x,u \rangle }\,du,\qquad g\in L^1(\mathbb R^n). \end{equation}
A comprehensive survey on this space can be found in [Reference Liflyand, Samko and Trigub11].
Denote
$f_\alpha (x)=|x|^\alpha\, f(x)$
.
Corollary 3.
If
$g\in L^{1,\alpha }$
, then
$f_\alpha \in W_0({\mathbb R^n})$
.
Proof.
Indeed, it follows from
$g\in L^{1,\alpha }$
and (14) that
\begin{equation*} |x|^\alpha\, f(x)=\int _{\mathbb R^n}(D^\alpha g)(u)e^{-i\langle x,u \rangle }\,du.\end{equation*}
Recalling that, by definition,
$D^\alpha g\in L^1(\mathbb R^n)$
, completes the proof.
4. Characterization of
$\boldsymbol{L}^{\boldsymbol{1,\alpha} }$
Naturally, the above study assumes a deep knowledge of the space
$L^{1,\alpha }$
,
$\alpha \gt 0$
. Much is known about this space, and this section is primarily devoted to discussing and comparing previously established results.
It is desirable to characterize
$L^{1,\alpha }$
in terms of the Riesz potential operator, which is known to be defined as
\begin{equation} I^\alpha \varphi (x)=\int _{\mathbb R^n} k_\alpha (x-y)\varphi (y)\,dy, \end{equation}
where
\begin{equation} k_\alpha (x)=\frac 1{\gamma _n(\alpha )}\begin{cases} |x|^{\alpha -n}, & \alpha \ne n+2j,\\ |x|^{\alpha -n}\ln \frac 1{|x|}, & \alpha =n+2j, \end{cases} \end{equation}
assuming
$j=0,1,2,\ldots$
, with
\begin{equation} \gamma _n(\alpha )=\begin{cases} 2^\alpha \pi ^{\frac {n}2}\frac {\Gamma \big (\frac \alpha 2\big )} {\Gamma \big (\frac {n-\alpha }2\big )}, & \alpha \ne n+2j,\\ ({-}1)^{\frac {n-\alpha }2}\pi ^{\frac {n}2}2^{\alpha -1}\big (\frac {\alpha -n}2\big )! \,\Gamma \big (\frac {\alpha }2\big ), & \alpha =n+2j. \end{cases} \end{equation}
For this and further facts, see [Reference Samko17, Ch.2 and Ch.7]. We shall cite results and formulas in this source in the particular case of
$L^1$
, although most of them are stated for more general Lebesgue spaces. Of course, there are many other relevant sources, see e.g., [Reference Stein20, Ch.5].
However, it turns out that
$L^{1,\alpha }$
can be completely characterized in terms of Bessel potentials rather than the Riesz potentials. For
$\alpha \gt 0$
, the Bessel potential is defined as
\begin{equation} B^\alpha \varphi (x)=\int _{\mathbb R^n} G_\alpha (x-y)\varphi (y)\,dy, \end{equation}
where
$G_\alpha$
is the Bessel kernel defined via its Fourier transform:
\begin{equation} \widehat {G_\alpha }(x)=(1+|x|^2)^{-\frac {\alpha }2}. \end{equation}
Of course, there is a representation of the kernel itself via a special function, see, e.g., [Reference Samko17, (7.30)]. Thus, the space
$B^\alpha (L^1)$
is defined as the space of functions
$f$
representable as
$f=B^\alpha \varphi$
in the sense of (18) such that
$\varphi \in L^1(\mathbb R^n)$
. In this case, [Reference Samko17, (7.34)] (cf. also [Reference Trebels23]) can be formulated as
$f\in B^\alpha (L^1)$
if and only if
$f\in L^1(\mathbb R^n)$
and the hypersingular integral
$D^\alpha f$
converges in the
$L^1$
-norm.
This description coincides with the space
$L^{1,\alpha }$
and is given in [Reference Samko17] as Theorem 7.16 in Subsection 2.2 of Chapter 7. In its complete form, it is due to Lizorkin, while an earlier variant for
$0\lt \alpha \lt 2$
is due to Stein; relevant references are given in [Reference Samko17]. We note that Lizorkin’s results were established for centered differences, whereas Stein used non-centered differences. The former requires
$l\gt \alpha$
, while the latter permits a smaller value of
$l$
, in particular,
$l=1$
for
$0\lt \alpha \lt 2$
. Both cases are treated in [Reference Samko17].
Nevertheless, much can be expressed in terms of the Riesz potential operator. Theorem 7.18 with its Corollary 1 in Chapter 7 of [Reference Samko17], when correctly interpreted, asserts that for
$0\lt \alpha \lt n$
, we have
\begin{equation} L^{1,\alpha }=I^\alpha (L^1) \bigcap L^1(\mathbb R^n). \end{equation}
Recall that the case
$L^1$
is problematic in many results concerning hypersingular integrals and potentials. Moreover, the case
$\alpha \ge n$
is no less problematic. Therefore, it is not surprising that in the doubly problematic case
$L^1$
and
$\alpha \ge n$
relation (20) fails to hold. Analytically, one can say that in this case, the kernel (or its Fourier transform) in (16) does not exhibit appropriate behavior. In contrast, the corresponding Bessel kernel does.
In conclusion, one is essentially led to work with functions in
$B^\alpha (L^1)$
in the problem under consideration. This is particularly relevant in the main case
$\alpha =n$
, which provides a direct generalization of the problem introduced at the beginning of the introduction.
Acknowledgements
The authors are grateful to the referees for their thorough reading and valuable suggestions.
Competing interests
There is no competing interests.
