1 Introduction

In harmonic analysis the uncertainty principles play an important role, which states that a function f and its Fourier transform \(\mathcal Ff\) cannot be at the same time simultaneously and sharply localized [19, 25]. One of the most important of these uncertainty principles is the well-known Beckner’s logarithmic inequality [5], which is closely related to the logarithmic Sobolev inequality and which implies, in particular, to the well-known Heisenberg-Pauli-Weyl uncertainty principle [26].

W. Beckner in [5] showed that for every \(f\in {\mathcal {S}}(\mathbb {R}^d)\)

$$\begin{aligned} \int _{\mathbb {R}^d}\ln (|x|)|f(x)|^2dx+\int _{\mathbb {R}^d}\ln (|y|)|\mathcal Ff(y)|^2dy\ge \Big (\psi \Big (\frac{d}{4}\Big )+\ln 2\Big )\int _{\mathbb {R}^d}|f(x)|^2dx,\nonumber \\ \end{aligned}$$
(1.1)

where \(\mathcal Ff\) is the classical Fourier transform of f defined through

$$\begin{aligned} \mathcal Ff(y)=(2\pi )^{-d/2}\int _{\mathbb {R}^d}f(x)e^{-i\langle x,y\rangle }dx, \end{aligned}$$

with \(\langle x,y\rangle :=\sum _{j=1}^dx_jy_j\), \(\psi \) is the logarithmic derivative of the Gamma function \(\Gamma \), and \({\mathcal {S}}(\mathbb {R}^d)\) denotes the Schwartz space.

The key ingredient to prove Beckner’s logarithmic inequality (1.1) is the following Pitt’s inequality for the Fourier transform [5]

$$\begin{aligned} \Big \Vert |\cdot |^{-\beta }\mathcal Ff\Big \Vert _2\le c(\beta )\Big \Vert |\cdot |^{\beta }f\Big \Vert _2 \end{aligned}$$
(1.2)

for \(f\in {\mathcal {S}}(\mathbb {R}^d)\), \(0\le \beta <d/2\), with sharp constant

$$\begin{aligned} c(\beta )=2^{-\beta }\frac{\Gamma \Big (\frac{1}{2}\Big (\frac{d}{2}-\beta \Big )\Big )}{\Gamma \Big (\frac{1}{2}\Big (\frac{d}{2}+\beta \Big )\Big )}. \end{aligned}$$

It it noted that by Parseval’s identity, Pitt’s inequality can be viewed as a Hardy-Rellich inequality

$$\begin{aligned} \Big \Vert |\cdot |^{-\beta }f\Big \Vert _2\le c(\beta )\Big \Vert |\cdot |^{\beta }\mathcal Ff\Big \Vert _2=c(\beta )\Big \Vert (-\Delta )^{\beta /2}f\Big \Vert _2, \end{aligned}$$

whose proofs and extensions can be found in [18, 33].

The original proof of (1.2) by Beckner in [5] is based on an equivalent integral realization as a Stein-Weiss fractional integral on \(\mathbb {R}^d\). In [33], Yafaev used the following decomposition of \(L^2(\mathbb {R}^d)\) [32] to study inequality (1.2) on the subsets of \(L^2(\mathbb {R}^d)\) which are invariant under the Fourier transform:

$$\begin{aligned} L^2(\mathbb {R}^d)=\sum _{n=0}^\infty \oplus {\mathcal {R}}_n, \end{aligned}$$
(1.3)

where \({\mathcal {R}}_0\) denotes the space of radial functions, and \({\mathcal {R}}_n:={\mathcal {R}}_0\otimes {\mathcal {H}}_n\) denotes the space of functions on \(\mathbb {R}^d\) which are products of radial functions and spherical harmonics of degree n.

Following Yafaev’s idea, Gorbachev et al. in [22, 23] recently proved the sharp Pitt’s inequalities for the Hankel transform [11, 13, 17], Dunkl transform [14, 16] and (k, a)-generalized Fourier transform [6].

In this paper, following the idea in [22, 23], and using the decomposition (1.3) of the space \(L^2(\mathbb {R}^d)\), we prove the sharp Pitt’s inequality for a generalized Clifford-Fourier transform which is given by a similar operator exponential as the classical Fourier transform but containing generators of Lie superalgebra.

In signal processing, multiplexing which is originated in telegraphy and now widely applied in the areas of electronic, telecommunications, digital video and computer net works motivates us to develop the function theory for multivector-valued functions \(f=(f_1,\dots ,f_n)\). From a mathematical point of view, the idea of multiplexing is to encode n independent functions \(f_j\in L^2(\mathbb {R}^d;\mathbb {C})\), \(j=1,\dots ,n\), as a single function f that captures the information of each component \(f_j\)( [1, 3, 4, 24]). It is well-known that Clifford algebra \(\mathcal Cl_{0,d}\), a noncommutative complex \(2^d\)-dimensional universal algebra generated by the orhthonormal basis \(\{e_1,\dots ,e_d\}\), provides an explicit way to present the multivector-valued functions. And Clifford analysis [7, 15] is a refinement of harmonic analysis in \(\mathbb {R}^d\), in the sense that Lie algebra \({\mathfrak {s}}{\mathfrak {l}}_2\) generated by the Laplace operator \(\Delta \) and the norm squared of a vector \(|x|^2\), \(x\in \mathbb {R}^d\), in harmonic analysis (see, e.g. [28]) is refined to the Lie superalgebra \(\mathfrak {osp}(1|2)\) (containing \({\mathfrak {s}}{\mathfrak {l}}_2\) as its even subalgebra).

Several attempts have been considered to introduce the generalizations of the classical Fourier transform to the setting of Clifford analysis (see [2, 9, 10, 20, 21, 27, 29] and the references therein). In this paper we consider the so-called Clifford-Fourier transform in literature first introduced in [8] and further developed in [12], because it is given by a similar operator exponential as the classical Fourier transform but now containing generators of \({\mathfrak {osp}}(1|2)\). More precisely, denote \(\partial _y:=\sum _{j=1}^de_j\partial _{y_j}\) to be the Dirac operator in Clifford analysis. The Clifford-Fourier transform can be written as the following integral from

$$\begin{aligned} \mathcal F_{\pm }(f)(y)=(2\pi )^{-\frac{d}{2}}\int _{\mathbb {R}^d}K_{\pm }(x,y)f(x)dx, \end{aligned}$$
(1.4)

here the kernel function \(K_{\pm }(x,y)\) is given by

$$\begin{aligned} K_{\pm }(x,y)=e^{\mp i\frac{\pi }{2}\Gamma _{y}}e^{-i\langle x,y\rangle }, \end{aligned}$$

with \(\Gamma _y:=(\partial _yy-y\partial _y)/2+d/2\). In [12], the authors give the algebra background and full discussion about the kernel and the corresponding Clifford-Fourier transform. They obtain a completely explicit description of the kernel in terms of a finite sum of Bessel functions when d is even, and for the odd case they show that it is enough to identify the kernel in dimension 3, form which kernels in higher odd dimensions can be deduced by taking suitable derivatives. Moreover, they also express the kernel in dimension 3 as a single integral of a combination of Bessel functions.

Our main goal is to study the Pitt’s inequality for the Clifford-Fourier transform \(\mathcal F_-f\) of function f on Clifford-valued Schwartz space \({\mathcal {S}}(\mathbb {R}^d;\mathcal Cl_{0,d})\)

$$\begin{aligned} \Big \Vert |\cdot |^{-\beta }\mathcal F_-f\Big \Vert _2\le c(\beta )\Big \Vert |\cdot |^{\beta }f\Big \Vert _2 \end{aligned}$$
(1.5)

with sharp constant

$$\begin{aligned} c(\beta )=2^{-\beta }\frac{\Gamma \Big (\frac{1}{2}\Big (\frac{d}{2}-\beta \Big )\Big )}{\Gamma \Big (\frac{1}{2}\Big (\frac{d}{2}+\beta \Big )\Big )}, \end{aligned}$$

and the Beckner’s logarithmic uncertainty principle

$$\begin{aligned} \int _{\mathbb {R}^d}\ln (|x|)|f(x)|^2dx+\int _{\mathbb {R}^d}\ln (|y|)|\mathcal F_-f(y)|^2dy\ge \Big (\psi \Big (\frac{d}{4}\Big )+\ln 2\Big )\int _{\mathbb {R}^d}|f(x)|^2dx,\nonumber \\ \end{aligned}$$
(1.6)

provided that

$$\begin{aligned}0\le \beta <d/2.\end{aligned}$$

This paper is organized as follows. The next section is devoted to recalling some definitions and basic properties of the Clifford analysis and the Clifford-Fourier transform. In Sect. 3, based on the direct sum decomposition (1.3) whose subspaces are also invariant under the Clifford-Fourier transform, we prove the sharp Pitt’s inequality (1.5) and the Beckner’s logarithmic uncertainty principle (1.6) for the Clifford-Fourier transform.

2 Preliminaries

Let \(\{e_1, e_2, \dots , e_d\}\) be an orthonormal basis of \(\mathbb {R}^d\) satisfying the anti-commutation relationship

$$\begin{aligned} e_je_k+e_ke_j=-2\delta _{jk}, \end{aligned}$$
(2.1)

where \(\delta _{jk}\) is the Kronecker symbol. The complex universal Clifford algebra \(\mathcal Cl_{0,d}\) is defined as the \(2^d\)-dimensional associative algebra with basis given by \(e_0=1\) and \(e_A=e_{h_1}e_{h_2}\dots e_{h_n}\), where \(A=\{h_1, h_2, \dots , h_n\}\subset \{1,2,\dots ,d\}\), for \(1\le h_1<h_2<\cdots <h_n\le d\). Hence, each element \(x\in \mathcal Cl_{0,d}\) will be represented by \(x=\sum _Ax_Ae_A\), \(x_A\in \mathbb {C}\), here \(\mathbb {C}\) denotes the complex plane. The complex Clifford algebra \(\mathcal Cl_{0,d}\) is a complex linear, associate, but non-commutative algebra.

The typical element of \(\mathbb {R}^d\) is denoted by vector \(x=x_1e_1+x_2e_2+\cdots +x_de_d\), \(x_j\in \mathbb {R}\), \(j=1, 2, \dots , d\). The inner product and the wedge product of two vectors x and y in \(\mathbb {R}^d\) are defined as follows:

$$\begin{aligned}&\langle x,y\rangle :=\sum _{j=1}^dx_jy_j=-\frac{1}{2}(xy+yx),\\&x\wedge y:=\sum _{j<k}e_je_k(x_jy_k-x_ky_j)=\frac{1}{2}(xy-yx). \end{aligned}$$

Here the multiplication of vectors x and y is defined using the relation (2.1) of the Clifford algebra. If we define the Clifford conjugate of any number \(a\in \mathcal Cl_{0,d}\) as

$$\begin{aligned} {\bar{a}}:=\sum _{|A|=n}{\bar{a}}_A{\bar{e}}_A, \ {\bar{e}}_A=(-1)^{\frac{n(n+1)}{2}}e_A,\ a_A\in \mathbb {C}, \end{aligned}$$

where \({\bar{a}}_A\) denotes the conjugate of a complex number, then \(\overline{ab}={\bar{b}}{\bar{a}}\) for any \(a,b\in \mathcal Cl_{0,d}\) and \(\overline{x\wedge y}=y\wedge x\) for any vectors x and y in \(\mathbb {R}^d\).

The above conjugate leads to the scalar part of the product \(f{\bar{g}}\) for \(f,g\in \mathcal Cl_{0,d}\) given by

$$\begin{aligned}{}[f{\bar{g}}]_0:=\sum _{A}f_A{\bar{g}}_A, \ f_A, g_A\in \mathbb {C}. \end{aligned}$$
(2.2)

For \(f=g\) in (2.2), we have the modulus |f| of any \(f\in \mathcal Cl_{0,d}\) defined as

$$\begin{aligned} |f|:=\sqrt{[f{\bar{f}}]_0}=\sqrt{\sum _{A}|f_A|^2}=|{\bar{f}}|. \end{aligned}$$
(2.3)

We note that the square of a vector \(x\in \mathbb {R}^d\) is scalar-valued and equals to the norm squared up to a minus sign, i.e., \(x^2=-|x|^2\). For any \(a,b\in \mathcal Cl_{0,d}\), there has \(|ab|\le 2^d|a||b|\) and \(|a+b|\le |a|+|b|\). But if \(a\in \mathbb {R}^d\) and \(b\in \mathcal Cl_{0,d}\), it holds( [34]):

$$\begin{aligned} |ab|=|a||b|. \end{aligned}$$
(2.4)

Denote the space \(L^p(\mathbb {R}^d;\mathcal Cl_{0,d})\) as the module of all Clifford-valued functions \(f:\ \mathbb {R}^d\rightarrow \mathcal Cl_{0,d}\) with finite norm

$$\begin{aligned} \Vert f\Vert _{p}=\left\{ \begin{array}{ll} \big (\int _{\mathbb {R}^d}|f(x)|^pdx\big )^{\frac{1}{p}}, \ 1\le p< \infty ,\\ \\ \mathrm{ess\ sup}_{x\in \mathbb {R}^d}|f(x)|,\ p=\infty , \end{array} \right. \end{aligned}$$
(2.5)

where \(dx=dx_1\dots dx_d\) represents the usual Lebesgue measure in \(\mathbb {R}^d\). In particular, the \(L^2\)-norm for \(L^2(\mathbb {R}^d;\mathcal Cl_{0,d})\) is introduced by the scalar inner product

$$\begin{aligned} \langle f,g\rangle :=\int _{\mathbb {R}^d}[f(x)\overline{g(x)}]_0dx. \end{aligned}$$
(2.6)

We remark here that the definition of the scalar inner product, defined by (2.6), is reduced to the standard one in \(L^2(\mathbb {R}^d;\mathbb {C})\) which is a subset of \(L^2(\mathbb {R}^d;\mathcal Cl_{0,d})\), where \(f,g\in L^2(\mathbb {R}^d;\mathbb {C})\) leads to \([f(x)\overline{g(x)}]_0=f(x)\overline{g(x)}\).

Finally, we recall some definitions and results of the Clifford-Fourier transform from [12]:

Definition 2.1

On the Schwartz class of Clifford-valued functions \({\mathcal S}(\mathbb {R}^d;\mathcal Cl_{0,d})\), we define the Clifford-Fourier transform as

$$\begin{aligned} \mathcal F_{\pm }f(y):=(2\pi )^{-\frac{d}{2}}\int _{\mathbb {R}^d}K_{\pm }(x,y)f(x)dx, \end{aligned}$$
(2.7)

and their inverses as

$$\begin{aligned} \mathcal F^{-1}_{\pm }f(y):=(2\pi )^{-\frac{d}{2}}\int _{\mathbb {R}^d}{\widetilde{K}}_{\pm }(x,y)f(x)dx, \end{aligned}$$
(2.8)

where

$$\begin{aligned} K_{\pm }(x,y):=e^{\mp i\frac{\pi }{2}{\Gamma _{y}}}e^{-i\langle x,y\rangle }, \end{aligned}$$
(2.9)

and

$$\begin{aligned} {\widetilde{K}}_{\pm }(x,y):=e^{\pm i\frac{\pi }{2}{\Gamma _{y}}}e^{i\langle x,y\rangle }, \end{aligned}$$
(2.10)

are the corresponding kernel functions, \(\Gamma _y:=(\partial _yy-y\partial _y)/2+d/2=-\sum _{j<k}e_je_k(y_j\partial _{y_k}-y_k\partial _{y_j})\) is the so-called Gamma operator([15]).

We note that the above kernel functions are not symmetric, in the sense that for example \(K_-(x,y)\ne K_-(y,x)\) (see [12]). Hence, we adopt the convention that we always integrate over the first variable in the kernels. Throughout this paper, we only focus on the derivation of the kernel \(K_{-}(x,y)=e^{i\frac{\pi }{2}{\Gamma _{y}}}e^{-i\langle x,y\rangle }\). The other ones can be derived similarly.

From [12], we have the following explicit form of the kernel by using Gegenbauer polynomials \(C_k^{\lambda }(\omega )\) and Bessel functions \(J_{\alpha }(t)\):

$$\begin{aligned} K_-(x,y)=A_{\lambda }+B_{\lambda }+(x\wedge y)C_{\lambda }, \end{aligned}$$
(2.11)

with

$$\begin{aligned}&A_{\lambda }=2^{\lambda -1}\Gamma (\lambda +1)\sum _{k=0}^{\infty }(i^d+(-1)^k)(|x||y|)^{-\lambda }J_{k+\lambda }(|x||y|)C^{\lambda }_k(\langle x',y'\rangle ),\\&B_{\lambda }=-2^{\lambda -1}\Gamma (\lambda )\sum _{k=0}^{\infty }(i^d-(-1)^k)(|x||y|)^{-\lambda }J_{k+\lambda }(|x||y|)C^{\lambda }_k(\langle x',y'\rangle ),\\&C_{\lambda }=-(2\lambda )2^{\lambda -1}\Gamma (\lambda )\sum _{k=0}^{\infty }(i^d+(-1)^k)(|x||y|)^{-\lambda -1}J_{k+\lambda }(|x||y|)C^{\lambda +1}_{k-1}(\langle x',y'\rangle ), \end{aligned}$$

where \(x'=x/|x|\), \(y'=y/|y|\), and \(\lambda =(d-2)/2\).

Note that when the dimension d is even, \(A_{\lambda }\), \(B_{\lambda }\) and \(C_{\lambda }\) are real-valued. In particular, when \(d=2\), the Clifford-Fourier kernel is given by

$$\begin{aligned} K_-(x,y)=e^{i\frac{\pi }{2}\Gamma _{y}}e^{-i\langle x,y\rangle }=\cos (x_1y_2-x_2y_1)+e_1e_2\sin (x_1y_2-x_2y_1). \end{aligned}$$

We note that, as \((e_1e_2)^2=-1\), the above formula implies that, upon substituting \(e_1e_2\) by the imaginary unit i, the kernel is equal to the kernel of classical Fourier transform. But this is clearly not the case for higher dimensions. For instance, the authors in [12] show that \(K_-(x,z)K_-(y,z)\ne K_-(x+y,z)\) if the dimension d is even and \(d>2\).

Furthermore, the explicit representation (2.11) of the kernel \(K_-(x,y)\) allows the authors in [12] to study the following Bochner-type identities for the Clifford-Fourier transform:

Proposition 2.2

Let \({\mathcal {M}}_n:=\mathrm{ker}\ \partial _x \cap {{\mathcal {P}}}_n\) denote the space of spherical monogenics of degree n, here \( {{\mathcal {P}}}_n\) is the space of homogeneous polynomials of degree n. Then

(1). for functions of type \(f(x)=M_n(x')f_0(r)\) with \(x=rx'\), \(M_n\in {\mathcal {M}}_n\) and \(f_0(r)\in {\mathcal {S}}(\mathbb {R}^d)\) being real-valued radial function, there has

$$\begin{aligned} \mathcal F_-(f)(y)=(-1)^nM_n(y')\rho ^{n}H_{n+\lambda }(f_0(r)r^{-n})(\rho ),\ y=\rho y'; \end{aligned}$$
(2.12)

(2). for functions of type \(f(x)=x'M_{n-1}(x')f_0(r)\) with \(M_{n-1}\in {\mathcal {M}}_{n-1}\) and \(f_0(r)\in {\mathcal {S}}(\mathbb {R}^d)\), there has

$$\begin{aligned} \mathcal F_-(f)(y)=-i^dy'M_{n-1}(y')\rho ^{n}H_{n+\lambda }(f_0(r)r^{-n})(\rho ), \end{aligned}$$
(2.13)

where \(H_\lambda \) denotes the Hankel transform (see (3.1) in the next section).

3 Pitt’s inequality and logarithmic uncertainty principle for the CFT

Before we prove the Pitt’s inequality for the Clifford-Fourier transform, let us recall some known results for Hankel transform. The Hankel transform is defined through

$$\begin{aligned} H_\lambda (f)(\rho )=\int _{0}^\infty f(r)j_\lambda (\rho r)d\nu _\lambda (r) \end{aligned}$$
(3.1)

where \(j_\lambda (t):=2^\lambda \Gamma (\lambda +1)t^{-\lambda }J_\lambda (t)\) denotes the normalized Bessel function with \(\lambda \ge -1/2\), the normalized Lebesgue measure \(d\nu _\lambda (r):=b_\lambda r^{2\lambda +1}dr\) with constant \(b_\lambda =(2^{\lambda }\Gamma (\lambda +1))^{-1}\). From [22, 31, 33], the Pitt’s inequality for the Hankel transform is given as

$$\begin{aligned} \Vert (\cdot )^{-\beta }H_\lambda f\Vert _{2,d\nu _\lambda }\le c(\beta ,\lambda )\Vert (\cdot )^{\beta }f\Vert _{2,d\nu _\lambda } \end{aligned}$$
(3.2)

for \(f\in {\mathcal {S}}(\mathbb {R}^d)\), \(0<\beta <\lambda +1\) and \(\lambda >-1\), with sharp constant

$$\begin{aligned} c(\beta ,\lambda )=2^{-\beta }\frac{\Gamma \Big (\frac{1}{2}\Big (\lambda +1-\beta \Big )\Big )}{\Gamma \Big (\frac{1}{2}\Big (\lambda +1+\beta \Big )\Big )}, \end{aligned}$$
(3.3)

where the above weight \(L^2\)-norm is defined through

$$\begin{aligned} \Vert f\Vert _{2,d\nu _\lambda }:=\Big (\int _0^\infty |f(r)|^2d\nu _\lambda (r)\Big )^{1/2}. \end{aligned}$$

We are now in a position to prove the Pitt’s inequality (1.5) for the CFT.

Theorem 3.1

Let \(0\le \beta <d/2\). For any \(f\in {\mathcal {S}}(\mathbb {R}^d;\mathcal Cl_{0,d})\), the following Pitt’s inequality

$$\begin{aligned} \Big \Vert |\cdot |^{-\beta }\mathcal F_-f\Big \Vert _2\le c(\beta )\Big \Vert |\cdot |^{\beta }f\Big \Vert _2 \end{aligned}$$
(3.4)

holds with the sharp constant

$$\begin{aligned} c(\beta )=2^{-\beta }\frac{\Gamma \Big (\frac{1}{2}\Big (\frac{d}{2}-\beta \Big )\Big )}{\Gamma \Big (\frac{1}{2}\Big (\frac{d}{2}+\beta \Big )\Big )}. \end{aligned}$$
(3.5)

Proof

For \(\beta =0\) we have \(c(\beta )=1\) and the Pitt’s inequality (3.4) becomes the Parseval’s identity (see [30]). In the rest of the proof, we assume that \(0<\beta <d/2\). From the direct decomposition (1.3), we let \(l_n\) be the dimension of \({\mathcal {H}}_n\), and denote by \(\{Y^j_n: j=1,\ldots ,l_n\}\) the real-valued orthonormal basis \({\mathcal {H}}_n\). Then for \(f\in {\mathcal {S}}(\mathbb {R}^d)\), we have

$$\begin{aligned} f(x)=\sum _{n=0}^\infty \sum _{j=1}^{l_n}f_{nj}(r)Y^j_n(x'), \ x=rx', \end{aligned}$$
(3.6)

where

$$\begin{aligned} f_{nj}(r)=\int _{S^{d-1}}f(rx')Y^j_n(x')d\sigma (x'). \end{aligned}$$

Furthermore, there has

$$\begin{aligned} \int _{S^{d-1}}|f(rx')|^2d\sigma (x')=\sum _{n=0}^\infty \sum _{j=1}^{l_n}|f_{nj}(r)|^2, \end{aligned}$$

and

$$\begin{aligned} \int _{\mathbb {R}^{d}}|x|^{2\beta }|f(x)|^2dx= & {} \int _{0}^\infty r^{2\beta +d-1}\int _{S^{d-1}}|f(rx')|^2d\sigma (x')dr\nonumber \\= & {} \int _{0}^\infty r^{2\beta +d-1}\sum _{n=0}^\infty \sum _{j=1}^{l_n}|f_{nj}(r)|^2dr\nonumber \\= & {} \sum _{n=0}^\infty \sum _{j=1}^{l_n}\int _0^\infty |f_{nj}(r)|^2r^{2\beta }b_{\lambda }^{-1}d\nu _\lambda (r). \end{aligned}$$
(3.7)

Now, due to the following Fisher decomposition:

$$\begin{aligned} {\mathcal {H}}_n={\mathcal {M}}_n\oplus x'{\mathcal {M}}_{n-1}, \end{aligned}$$
(3.8)

the expression (3.6) of f and Bochner-type identities (2.12) and (2.13), there holds for \(y=\rho y'\)

$$\begin{aligned} \mathcal F_-(f)(y)=\sum _{n=0}^\infty \sum _{j=1}^{l_n}\big ((-1)^nZ^j_n(y')+(-i^d)y'W^j_{n-1}(y')\big )\rho ^{n}H_{n+\lambda }(f_{nj}(r)r^{-n})(\rho ), \end{aligned}$$

where \(Z^j_n\in {\mathcal {M}}_n\) and \(W^j_{n-1}\in {\mathcal {M}}_{n-1}\).

Thus, using spherical coordinates, the direct sum decomposition of \(L^2(\mathbb {R}^d)\) (1.3) and the Fisher decomposition (3.8), we have

$$\begin{aligned}&\int _{\mathbb {R}^{d}}|y|^{-2\beta }|\mathcal F_-f(y)|^2dy\nonumber \\&=\int _0^\infty \rho ^{-2\beta +(d-1)}d\rho \int _{S^{d-1}}\bigg |\sum _{n=1}^{n}\sum _{j=1}^{l_n}\rho ^{n}H_{n+\lambda }(f_{nj}(r)r^{-n})(\rho ){\tilde{Y}}^j_n(y')\bigg |^2dy'\nonumber \\&\le \sum _{n=0}^\infty \sum _{j=1}^{l_n}\int _0^\infty |H_{n+\lambda }(f_{nj}(r)r^{-n})(\rho )|^2\rho ^{-2(\beta -n)}b_{\lambda }^{-1}d\nu _\lambda (\rho ), \end{aligned}$$
(3.9)

where \({\tilde{Y}}^j_n(y')=(-1)^nZ^j_n(y')+(-i^d)y'W^j_{n-1}(y')\).

Furthermore, by using the Pitt’s inequality (3.2) for the Hankel transform, we have

$$\begin{aligned}&\int _0^\infty |H_{n+\lambda }(f_{nj}(r)r^{-n})(\rho )|^2\rho ^{-2(\beta -n)}b_{\lambda }^{-1}d\nu _\lambda (\rho )\nonumber \\&\quad =\int _0^\infty |H_{n+\lambda }(f_{nj}(r)r^{-n})(\rho )|^2\rho ^{-2(\beta -n)}\frac{b_{\lambda }}{b_{n+\lambda }}d\nu _{n+\lambda }(\rho )\nonumber \\&\quad \le c^2(\beta ,\lambda )\int _0^\infty |f_{nj}(r)r^{-n}|^2r^{2\beta }\frac{b_{\lambda }}{b_{n+\lambda }}d\nu _{n+\lambda }(r)\nonumber \\&\quad =c^2(\beta ,\lambda )\int _0^\infty |f_{nj}(r)r^{-n}|^2r^{2\beta }d\nu _{\lambda }(r). \end{aligned}$$
(3.10)

Since \(\lambda =(d-2)/2\), then using (3.7), (3.9) and (3.10), we arrive at

$$\begin{aligned} \int _{\mathbb {R}^{d}}|y|^{-2\beta }|\mathcal F_-f(y)|^2dy\le & {} \sum _{n=0}^\infty \sum _{j=1}^{l_n}c^2(\beta ,\lambda )\int _0^\infty |f_{nj}(r)r^{-n}|^2r^{2\beta }d\nu _{\lambda }(r)\nonumber \\= & {} c^2(\beta )\int _{\mathbb {R}^{d}}|x|^{2\beta }|f(x)|^2dx. \end{aligned}$$
(3.11)

\(\square \)

Using the Pitt’s inequality (3.4) we obtain the Beckner’s logarithmic uncertainty principle for the CFT.

Theorem 3.2

Suppose that \(0\le \beta <d/2\). Then the inequality

$$\begin{aligned} \int _{\mathbb {R}^d}\ln (|x|)|f(x)|^2dx+\int _{\mathbb {R}^d}\ln (|y|)|\mathcal F_-f(y)|^2dy\ge \Big (\psi \Big (\frac{d}{4}\Big )+\ln 2\Big )\int _{\mathbb {R}^d}|f(x)|^2dx,\nonumber \\ \end{aligned}$$
(3.12)

holds for any \(f\in {\mathcal {S}}(\mathbb {R}^d)\), here \(\psi (t)=\Gamma '(t)/\Gamma (t)\) being the psi function.

Proof

To simplify our proof, we rewrite the Pitt’s inequality (1.5) in the following form

$$\begin{aligned} \int _{\mathbb {R}^{d}}|y|^{-\beta }|\mathcal F_-f(y)|^2dy\le c^2(\beta /2)\int _{\mathbb {R}^{d}}|x|^{\beta }|f(x)|^2dx, \end{aligned}$$

here \(0\le \beta <d\). Now for \(\beta \in (-d,d)\), we define the function

$$\begin{aligned} \varphi (\beta ):=\int _{\mathbb {R}^{d}}|y|^{-\beta }|\mathcal F_-f(y)|^2dy-c^2(\beta /2)\int _{\mathbb {R}^{d}}|x|^{\beta }|f(x)|^2dx. \end{aligned}$$

The Pitt’s inequality (1.5) and Parseval’s identity for the CFT imply that \(\varphi (\beta )\le 0\) for \(\beta >0\) and \(\varphi (0)=0\), respectively. Hence,

$$\begin{aligned} \varphi '_+(0)=\lim _{\beta \rightarrow 0^+}\frac{\varphi (\beta )-\varphi (0)}{\beta }\le 0. \end{aligned}$$
(3.13)

Since \(f, \mathcal F_-f\in {\mathcal {S}}(\mathbb {R}^d)\), then for any \(|\beta |<d\),

$$\begin{aligned} \int _{|x|>1}|x|^{\beta }\ln (|x|)|f(x)|^2dx \end{aligned}$$

and

$$\begin{aligned} \int _{|y|>1}|y|^{\beta }\ln (|y|)|\mathcal F_-f(y)|^2dy \end{aligned}$$

are well-defined. Furthermore, by spherical coordinates,

$$\begin{aligned} \int _{|x|\le 1}|x|^{\beta }|\ln (|x|)|dx=\int _0^1r^{\beta +d-1}|\ln r|dr\int _{S^{d-1}}dx'<\infty , \end{aligned}$$

which gives

$$\begin{aligned} |x|^{\beta }\ln (|x|)|f(x)|^2\in L^1(\mathbb {R}^d) \qquad \text {and} \qquad |y|^{\beta }\ln (|y|)|\mathcal F_-f(y)|^2\in L^1(\mathbb {R}^d). \end{aligned}$$

Thus,

$$\begin{aligned} \varphi '(\beta )= & {} -\int _{\mathbb {R}^{d}}|y|^{-\beta }\ln (|y|)|\mathcal F_-f(y)|^2dy\nonumber \\&-c^2(\beta /2)\int _{\mathbb {R}^{d}}|x|^{\beta }\ln (|x|)|f(x)|^2dx\nonumber \\&-\frac{dc^2(\beta /2)}{d\beta }\int _{\mathbb {R}^{d}}|x|^{\beta }|f(x)|^2dx. \end{aligned}$$
(3.14)

In addition, from (3.5) we have

$$\begin{aligned} -\frac{dc^2(\beta /2)}{d\beta }=\psi \Big (\frac{d}{4}\Big )+\ln 2. \end{aligned}$$
(3.15)

Combining (3.13), (3.14) and (3.15), we conclude the proof of (3.12). \(\square \)