Introduction

The uncertainty principle (UP) holds significance in quantum mechanics [15] and signal processing [4]. In 1927, a German physicist W. Heisenberg first proposed UP, which asserts that the more precisely the position of a particle is determined, the less precisely its momentum can be known, and vice versa [4]. From a signal processing standpoint, the uncertainty principle dictates that “One cannot sharply localize a signal in both the time domain and frequency domain simultaneously ” [4, 14].

The linear canonical transform (LCT) represents a class of integral transforms with matrix parameter \(m \in SL(2,\mathbb {R})\). The LCT has wide applications in signal processing, radar system analysis, filter design, optics, and many others [3, 17]. It includes many well-known transforms such as the Fourier transform, Hankel transform, Laplace transform and fractional Fourier transform. As is well known, a wide number of papers have been successfully devoted to the extension of the theory of LCT to some other integral transforms such as the Hankel transform, the Dunkl transform and the Fourier-Bessel transform [20]. In the last few years, authors built a class of linear integral transform with a kernel involving a Bessel function and matrix parameter \(m \in SL(2,\mathbb {R})\). In [12], the authors introduced the Dunkl linear canonical transform (DLCT) which is a generalization of the LCT. The DLCT includes many well-known transforms such as the Dunkl transform [8], the fractional Dunkl transform [11] and the canonical Fourier-Bessel transform [7].

In [7], the authors developed a harmonic analysis related to the canonical Fourier-Bessel transform \(\mathcal {F}^m_{\alpha }\). Several properties, such as a Riemann-Lebesgue lemma, inversion formula, operational formulas, Plancherel theorem, and Babenko inequality are established and there are many different kinds of UPs associated with the canonical Fourier-Bessel transform, like Heisenberg’s UP, Hardy’s UP, Nash-type inequality, Carlson type inequality, and global UP.

In this paper, we show similar results to those proved in [7]. Following the framework of Al Subaie and Mourou [1], we introduce a new harmonic analysis associated with the following generalized Bessel type operator

$$\begin{aligned} \Delta ^m_{\alpha ,n}=\dfrac{d^2}{dx^2} +\left( \dfrac{2\alpha +1}{x}-2i\dfrac{a}{b}x\right) \dfrac{d}{dx}-\left( \dfrac{a^2}{b^2}x^2+2i(\alpha +1)\dfrac{a}{b}\right) -\dfrac{4n(\alpha +n)}{x^2}. \end{aligned}$$

\(m=\begin{pmatrix} a& b\\ c& d \end{pmatrix} \in SL(2,\mathbb {R})\) with \(b \ne 0\), and \(n\in \mathbb {N}\),

which is closely related to the generalized canonical Fourier-Bessel transform (GCFBT) \(\mathcal {F}^m_{\alpha ,n}\) given by

$$\begin{aligned} \mathcal {F}^m_{\alpha ,n}f(y)=\dfrac{c_{\alpha +2n}}{(ib)^{\alpha +2n+1}}\int _0^{\infty }K^m_{\alpha ,n}(y,x)f(x)x^{2\alpha +1}dx, \end{aligned}$$
(1)

where

$$\begin{aligned} c_{\alpha +2n}^{-1}=2^{\alpha +2n}\Gamma (\alpha +2n+1) \end{aligned}$$

and

$$\begin{aligned} K^m_{\alpha ,n}(y,x)=x^{2n}K^m_{\alpha +2n}(y,x)=x^{2n}e^{\frac{i}{2}(\frac{a}{b}x^2+\frac{d}{b}y^2)}j_{\alpha +2n}(\frac{xy}{b}). \end{aligned}$$

Here \(j_{\alpha }\) denotes the normalized Bessel function of order \(\alpha >-\frac{1}{2}\), defined by:

$$\begin{aligned} j_{\alpha }(x)=2^{\alpha }\Gamma (\alpha +1)\dfrac{J_{\alpha }(x)}{x^{\alpha }}=\sum \limits _{n=0}^{+\infty }(-1)^n\dfrac{\Gamma (\alpha +1)}{n!\Gamma (n+\alpha +1)}\left( \dfrac{x}{2}\right) ^{2n} \; ;\; x \in \mathbb {C}. \end{aligned}$$

Also, we prove some uncertainty principles for these transform.

This paper is organized as follows. The “Preliminaries” section is devoted to an overview of the canonical Fourier-Bessel transform. In the “Generalized canonical Fourier-Bessel transform” section, we introduce and study the generalized canonical Fourier-Bessel transform \(\mathcal {F}^m_{\alpha ,n}\) on \(\mathbb {R}\) with parameter \(m \in SL(2,\mathbb {R})\), \(n \in \mathbb {N}\), we derive Riemann-Lebesgue lemma, inversion formula, operational formula, Plancherel formula, Paley-Wiener theorem, and Babenko type inequality. The “Heisenberg inequality for \(\mathcal {F}^m_{\alpha ,n}\)” and “Hardy uncertainty principle for \(\mathcal {F}^m_{\alpha ,n}\)” section are devoted to Heisenberg inequality and Hardy theorem. In the “Nash-type inequality” section, we prove a Nash-type inequality. In the “Carlson-type inequality” section, we prove a Carlson-type inequality. In the “Global uncertainty principle” section, we deduce a variation on Heisenberg inequality and we prove another variation of the Heisenberg uncertainty principle, it will be the global uncertainty principle.

Preliminaries

In the present section, we recapitulate some facts about harmonic analysis related to the Bessel type operator \(\Delta ^m_{\alpha }\). We cite here as briefly as possible, only those properties which are actually required for the discussion. For more details, we refer to [7].

Throughout this paper \(\alpha\) denotes a real number such that \(\alpha > -\dfrac{1}{2}\).

Functional spaces

We denote by:

  • \(\mathscr {C}(\mathbb {R})\) the space of even \(C^{\infty }\)-functions on \(\mathbb {R}\). We provide it with the topology of uniform convergence on all compact of \(\mathbb {R}\), for functions and their derivatives.

  • \(\mathscr {C}_{0,e}(\mathbb {R})\) the space of even continuous functions f on \(\mathbb {R}\) such that \(\lim \limits _{\vert x\vert \rightarrow +\infty }f(x)=0\).

  • \(\mathscr {C}_{c,e}(\mathbb {R})\) the subspace of \(\mathscr {C}_{0,e}(\mathbb {R})\) consisting of functions with compact support.

  • \(\mathscr {C}_{*,b}(\mathbb {R})\) the space of even and bounded continuous functions on \(\mathbb {R}\). We provide \(\mathscr {C}_{*,b}(\mathbb {R})\) with the topology of uniform convergence.

  • \(\mathcal {L}_{p,\alpha }\) the Lebesgue space of measurable functions on \([0,\infty [\) such that

    $$\Vert f\Vert _{p,\alpha }= \left\{ \begin{array}{cl} \left( \int _0^{+\infty }\vert f(y)\vert ^p y^{2\alpha +1}dy\right) ^{\frac{1}{p}}< \infty \;\; if\; 1\leqslant p< \infty ,\\ \Vert f \Vert _{\infty }=ess \sup \limits _{y\in [0,+\infty [} \vert f(y)\vert < \infty \;\; if\; p=\infty . \end{array} \right.$$

    We provide \(\mathcal {L}{p,\alpha }\) with the topology defined by the norm \(\Vert .\Vert _{p,\alpha }\).

  • \(\mathcal {L}_{2,\,\alpha }\) the Hilbert space equipped with the inner product \(\langle .\vert .\rangle _{\alpha }\) given by:

    $$\begin{aligned} \langle f\vert g\rangle _{\mu _{\alpha }}=\int _0^{+\infty }f(y)\overline{g(y)}y^{2\alpha +1}dy . \end{aligned}$$

  • \(\mathcal {S}_{*}(\mathbb {R})\) the Schwartz space of even \(C^{\infty }\)-functions on \(\mathbb {R}\) and rapidly decreasing together with their derivatives. We provide \(\mathcal {S}_{*}(\mathbb {R})\) with the topology defined by the semi-norms:

    $$\begin{aligned} q_{n,m}(f)=\sup \limits _{x\ge 0}\,(1+x^2)^n \vert \dfrac{d^m}{dx^m}f(x)\vert ; \;f \in \mathcal {S}_{*}(\mathbb {R}), \;n,\;m\in \mathbb {N}. \end{aligned}$$

  • \(\mathcal {D}_{*}(\mathbb {R})\) the space of even \(C^{\infty }\)-functions on \(\mathbb {R}\) with compact support. We have

    $$\begin{aligned} \mathcal {D}_{*}(\mathbb {R})=\underset{a\ge 0}{\cup }\mathcal {D}_{*,a}(\mathbb {R}), \end{aligned}$$

    where \(\mathcal {D}_{*,a}(\mathbb {R})\) is the space of even \(C^{\infty }\)-functions on \(\mathbb {R}\) with support in the interval \([-a,a]\). We provide \(\mathcal {D}_{*,a}(\mathbb {R})\) with the topology of uniform convergence of functions and their derivatives.

  • \(\mathcal {H}_{*}(\mathbb {C})\) the space of even entire functions on \(\mathbb {C}\) decreasing and of exponential type. We have

    $$\begin{aligned} \mathcal {H}_{*}(\mathbb {C})=\underset{a\ge 0}{\cup }\mathcal {H}_{a}(\mathbb {C}), \end{aligned}$$

    where \(\mathcal {H}_{a}(\mathbb {C})\) is the space of even entire functions \(f:\;\mathbb {C}\rightarrow \mathbb {C}\) and satisfying for all \(m\in \mathbb {N}\)

    $$\begin{aligned} P_m(f)=\sup \limits _{\lambda \in \mathbb {C}}(1+\vert \lambda \vert )^m\vert f(\lambda )\vert e^{-a\vert Im(\lambda )\vert }<\infty . \end{aligned}$$

    We provide \(\mathcal {H}_{a}(\mathbb {C})\) with the topology defined by the semi-norms \(p_m\), \(m\in \mathbb {N}\).

Canonical Fourier-Bessel transform on \(\mathcal {L}_{1,\alpha }\) and \(\mathcal {L}_{2,\alpha }\)

Throughout this paper, we denote by \(m=\begin{pmatrix} a& b\\ c& d \end{pmatrix}\) an arbitrary matrix in \(SL(2,\mathbb {R})\). For \(m \in SL(2,\mathbb {R})\) such that \(b \ne 0\), the canonical Fourier-Bessel transform of a function \(f \in \mathcal {L}_{1,\alpha }\) is defined by

$$\begin{aligned} \mathcal {F}^m_{\alpha }f(y)= \dfrac{C_{\alpha }}{(ib)^{\alpha +1}} \int _0^{\infty } K^m_{\alpha }(y,x) f(x)x^{2\alpha +1}dx, \end{aligned}$$
(2)

where \(C_{\alpha }^{-1}=2^{\alpha }\Gamma (\alpha +1)\) and

$$\begin{aligned} K^m_{\alpha }(y,x)=e^{\frac{i}{2}(\frac{a}{b}x^2+\frac{d}{b}y^2)}j_{\alpha }(\frac{xy}{b}), \end{aligned}$$

here \(j_{\alpha }\) denotes the normalized Bessel function of order \(\alpha >-\dfrac{1}{2}\), defined by:

$$\begin{aligned} j_{\alpha }(x)=2^{\alpha }\Gamma (\alpha +1)\dfrac{J_{\alpha }(x)}{x^{\alpha }}=\sum \limits _{n=0}^{+\infty }(-1)^n\dfrac{\Gamma (\alpha +1)}{n!\Gamma (n+\alpha +1)}\left( \dfrac{x}{2}\right) ^{2n} \; ;\; x \in \mathbb {C}. \end{aligned}$$

For each \(y \in \mathbb {R}\), the kernel \(K^m_{\alpha }(y,.)\) of the canonical Fourier-Bessel transform \(\mathcal {F}^m_{\alpha }\) is the unique solution of:

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta ^m_{\alpha }K^m_{\alpha }(y,.)=\dfrac{-y^2}{b^2}K^m_{\alpha }(y,.), \\ K^m_{\alpha }(y,0)=e^{\frac{i}{2}\frac{d}{b}y^2}\; ;\; \dfrac{d}{dx}K^m_{\alpha }(y,0)=0. \end{array}\right. } \end{aligned}$$
(3)

Where for \(m\in SL(2,\mathbb {R})\) such that \(b \ne 0\), \(\Delta ^m_{\alpha }\) denotes the Bessel type operator:

$$\begin{aligned} \Delta ^m_{\alpha }=\dfrac{d^2}{dx^2}+\left( \dfrac{2\alpha +1}{x}-2i\dfrac{a}{b}x\right) \dfrac{d}{dx}-\left( \dfrac{a^2}{b^2}x^2+2i(\alpha +1)\dfrac{a}{b}\right) . \end{aligned}$$

Proposition 1

Let \(m\in SL(2,\mathbb {R})\). For every f, \(g \in \mathcal {S}_{*}(\mathbb {R})\)

$$\begin{aligned} \int _0^{\infty }\Delta ^m_{\alpha }f(x)\overline{g(x)}x^{2\alpha +1} dx =\int _0^{\infty }f(x)\overline{\Delta ^m_{\alpha }g(x)} x^{2\alpha +1} dx. \end{aligned}$$
(4)

Theorem 2

(Riemann-Lebesgue lemma) Let \(m\in SL(2,\mathbb {R})\). For all \(f \in \mathcal {L}_{1,\alpha }\), the canonical Fourier-Bessel transform \(\mathcal {F}^m_{\alpha }f\) belongs to \(\mathscr {C}_{*,0}(\mathbb {R})\) and verifies

$$\begin{aligned} \Vert \mathcal {F}^m_{\alpha }f \Vert _{\infty }\leqslant C_{\alpha } |b|^{-(\alpha +1)} \Vert f\Vert _{1,\alpha }. \end{aligned}$$

Theorem 3

(Inversion formula) For all \(f \in \mathcal {L}_{1,\alpha }\) with \(\mathcal {F}^m_{\alpha }f \in \mathcal {L}_{1,\alpha }\),

$$\begin{aligned} \left( \mathcal {F}^m_{\alpha } \circ \mathcal {F}^{m^{-1}}_{\alpha }\right) f =\left( \mathcal {F}^{m^{-1}}_{\alpha } \circ \mathcal {F}^m_{\alpha }\right) f=f. \end{aligned}$$
(5)

Theorem 4

(Plancherel Theorem) Let \(m\in SL(2,\mathbb {R})\). For all f and \(g \in \mathcal {L}_{1,\alpha }\), we have

$$\begin{aligned} \int _0^{\infty }\mathcal {F}^m_{\alpha }f(x)\overline{g(x)}x^{2\alpha +1} dx =\int _0^{\infty }f(x)\overline{\mathcal {F}^{m^{-1}}_{\alpha }g(x)} x^{2\alpha +1} dx. \end{aligned}$$
(6)

Theorem 5

(Paley-Wiener Theorem) Let \(m\in SL(2,\mathbb {R})\). The canonical Fourier-Bessel transform \(\mathcal {F}^m_{\alpha }\) is a bijection from \(\mathcal {D}_{*}(\mathbb {R})\) into \(\mathbb {L}_{\frac{d}{b}}(\mathcal {H}_{*}(\mathbb {C}))\),

where \(\mathbb {L}_{\frac{d}{b}}\) is the chirp multiplication operator of index \(\frac{d}{b}\) defined by

$$\begin{aligned} \mathbb {L}_{\frac{d}{b}}f(x)=e^{\frac{id}{2b}x^2}f(x). \end{aligned}$$

Theorem 6

  1. 1.

    For all \(f \in \mathcal {S}_{*}(\mathbb {R})\), we have

    $$\begin{aligned} \Vert \mathcal {F}^m_{\alpha }f \Vert _{2,\alpha } = \Vert f \Vert _{2,\alpha }. \end{aligned}$$
    (7)
  2. 2.

    If \(f \in \mathcal {L}_{1,\alpha }\cap \mathcal {L}_{2,\alpha }\), then \(\mathcal {F}^m_{\alpha }f \in \mathcal {L}_{2,\alpha }\) and \(\Vert \mathcal {F}^m_{\alpha }f \Vert _{2,\alpha } = \Vert f \Vert _{2,\alpha }.\)

  3. 3.

    The canonical Fourier-Bessel transform has a unique extension to an isometric isomorphism of \(\mathcal {L}_{2,\alpha }\). The extension is also denoted by \(\mathcal {F}^m_{\alpha }:\;\mathcal {L}_{2,\alpha }\rightarrow \mathcal {L}_{2,\alpha }\).

Theorem 7

(Babenko-Beckner inequality) Let \(m\in SL(2,\mathbb {R})\). Let p and q be real numbers such that \(1< p\leqslant 2\) and \(\dfrac{1}{p}+\dfrac{1}{q}=1\). Then, \(\mathcal {F}^m_{\alpha }\) extends to a bounded linear operator on \(\mathcal {L}_{p,\alpha }\), and we have

$$\begin{aligned} \Vert \mathcal {F}^m_{\alpha }f \Vert _{q, \alpha }\leqslant |b |^{(\alpha +1)(\frac{2}{q}-1)} \left( \dfrac{(C_{\alpha }p)^{\frac{1}{p}}}{(C_{\alpha }q)^{\frac{1}{q}}}\right) ^{\alpha +1}\Vert f \Vert _{p, \alpha }. \end{aligned}$$
(8)

Uncertainty principles for \(\mathcal {F}^m_{\alpha }f\)

Theorem 8

Let \(m\in SL(2,\mathbb {R})\).

  1. 1.

    (Heisenberg uncertainty principle) For every \(f \in \mathcal {L}_{2,\alpha }\), we have

    $$\begin{aligned} \Vert xf \Vert _{2, \alpha }^2 \Vert x\mathcal {F}^m_{\alpha }f \Vert _{2, \alpha }^2 \geqslant b^2(\alpha +1)^2 \Vert f \Vert _{2, \alpha }^4. \end{aligned}$$
    (9)
  2. 2.

    (Hardy uncertainty principle) Let \(\nu\) and \(\beta\) be positive constants. Suppose f is a measurable function on \([0,\infty [\) satisfying:

    $$\begin{aligned} |f(x) |\leqslant Ce^{-\nu x^2}\qquad and \qquad |\mathcal {F}^m_{\alpha }f(x) |\leqslant Ce^{-\beta x^2}, \end{aligned}$$

    where C is a positive constant. Then, we have the following results:

    1. a.

      If \(\nu \beta =\dfrac{1}{4b^2}\), then \(f(x)= Ae^{-(\nu +\frac{i}{2}\frac{a}{b})x^2}\), where A is an arbitrary constant.

    2. b.

      If \(\nu \beta > \dfrac{1}{4b^2}\), then f = 0.

    3. c.

      If \(\nu \beta < \dfrac{1}{4b^2}\), then there are infinitely many linearly independent functions satisfying the above estimates.

  3. 3.

    (Nash-type inequality) Let \(k>0\), then there exists a constant \(C_{\alpha ,k}>0\) such that for all \(f \in \mathcal {L}_{1,\alpha } \cap \mathcal {L}_{2,\alpha }\)

    $$\begin{aligned} \Vert f \Vert ^2_{2,\alpha }\leqslant C_{\alpha ,k} \Vert f \Vert ^{\frac{2k}{\alpha +k+1}}_{1,\alpha }\Vert x^k \mathcal {F}^m_{\alpha }f \Vert ^{\frac{2\alpha +2}{\alpha +k+1}}_{2,\alpha }. \end{aligned}$$
    (10)
  4. 4.

    (Carlson-type inequality) Let \(k>0\), then there exists a constant \(\nu _{\alpha ,k}>0\) such that for all \(f \in \mathcal {L}_{1,\alpha } \cap \mathcal {L}_{2,\alpha }\)

    $$\begin{aligned} \Vert f \Vert ^{2k+\alpha +1}_{1,\alpha }\leqslant \nu _{\alpha ,k} \Vert f \Vert ^{2k}_{2,\alpha } \Vert x^{2k}f \Vert ^{\alpha +1}_{1,\alpha }, \end{aligned}$$
    (11)

    where \(\nu _{\alpha ,k}\) is given by

    $$\begin{aligned} \nu _{\alpha ,k}=\left( \dfrac{1}{\sqrt{2\alpha +2}}\right) ^{2k}\left[ \left( \dfrac{2k}{\alpha +1}\right) ^{\alpha +1}+\left( \dfrac{2k}{\alpha +1}\right) ^{-2k}\right] . \end{aligned}$$
  5. 5.

    (Global uncertainty principle) Let \(k>0\), then for all \(f \in \mathcal {L}_{1,\alpha } \cap \mathcal {L}_{2,\alpha }\)

    $$\begin{aligned} \Vert f \Vert ^1_{1,\alpha }\Vert f \Vert ^2_{2,\alpha }\leqslant (C_{\alpha ,k})^{\frac{k+\alpha +1}{\alpha +1}}(\nu _{\alpha ,k})^{\frac{1}{\alpha +1}}\Vert x^{2k}f \Vert _{1,\alpha } \Vert x^k \mathcal {F}^m_{\alpha }f \Vert ^{2}_{2,\alpha }. \end{aligned}$$
    (12)

Generalized canonical Fourier-Bessel transform

Note 1

  • Further, we assume that \(\alpha > -\dfrac{1}{2}\) and \(n =0,\, 1,\,2,\,...\). Let \(\mathcal {M}\) be the map defined by

    $$\begin{aligned} \mathcal {M}f(x)=x^{2n}f(x). \end{aligned}$$

  • Let \(\mathcal {L}^p_{\alpha ,n}\), \(1\leqslant p\leqslant \infty\), be the class of mesurable functions f on \([0,+\infty [\) for which

    $$\begin{aligned} \Vert f \Vert _{p,\alpha ,n} = \Vert \mathcal {M}^{-1} f \Vert _{p,\alpha +2n}\,<\,\infty . \end{aligned}$$
    (13)

Remark 1

It is easily seen that \(\mathcal {M}\) is an isometry from \(\mathcal {L}_{p,\alpha +2n}\) onto \(\mathcal {L}^p_{\alpha ,n}\).

For x, \(y \in \mathbb {R}\), we put:

$$\begin{aligned} K^m_{\alpha ,n}(y,x) = x^{2n}K^m_{\alpha +2n}(y,x), \end{aligned}$$
(14)

where \(K^m_{\alpha +2n}\) is the kernel of the canonical Fourier-Bessel transform of index \(\alpha +2n\).

Definition 1

The Generalized canonical Fourier-Bessel transform is defined for a function \(f \in \mathcal {L}^1_{\alpha ,n}\) by:

$$\begin{aligned} \mathcal {F}^m_{\alpha ,n}f(y)=\dfrac{C_{\alpha +2n}}{(ib)^{\alpha +2n+1}}\int _0^{+\infty }K^m_{\alpha ,n}(y,x)f(x)x^{2\alpha +1}dx, \end{aligned}$$
(15)

where \(C_{\alpha +2n}^{-1}=2^{\alpha +2n}\Gamma (\alpha +2n+1)\).

Proposition 9

Let \(m\in SL(2,\mathbb {R})\). We denote by \(\Delta ^m_{\alpha ,n}\) the generalized differential operator

$$\begin{aligned} \Delta ^m_{\alpha ,n}f=\dfrac{d^2}{dx^2}f +\left( \dfrac{2\alpha +1}{x}-2i\dfrac{a}{b}x\right) \dfrac{d}{dx}f-\left( \dfrac{a^2}{b^2}x^2+2i(\alpha +1)\dfrac{a}{b}\right) f-\dfrac{4n(\alpha +n)}{x^2}f. \end{aligned}$$
  1. 1.

    The operator \(\Delta ^m_{\alpha ,n}\) is related to the differential operator \(\Delta ^m_{\alpha }\) by

    $$\begin{aligned} \mathcal {M}^{-1}\circ \Delta ^m_{\alpha ,n} \circ \mathcal {M}=\Delta ^m_{\alpha +2n}. \end{aligned}$$
    (16)
  2. 2.

    For each \(y \in \mathbb {R}\), \(K^m_{\alpha ,n}\) is the unique solution of the problem

    $$\begin{aligned} {\left\{ \begin{array}{ll} \Delta ^m_{\alpha ,n}K^m_{\alpha ,n}(y,.)=\dfrac{-y^2}{b^2}K^m_{\alpha ,n}(y,.), \\ K^m_{\alpha ,n}(y,0)=e^{\frac{i}{2}\frac{d}{b}y^2}\; ;\; \dfrac{d}{dx}K^m_{\alpha ,n}(y,0)=0. \end{array}\right. } \end{aligned}$$

Proof

  1. 1.

    For every \(f \in \mathcal {L}^1_{\alpha +2n}\) and \(x \in \mathbb {R}\). It is easy to see that:

    $$\begin{aligned} \Delta ^m_{\alpha ,n} \circ \mathcal {M}f(x)=\Delta ^m_{\alpha ,n} \left( x^{2n}f(x)\right) . \end{aligned}$$

    We have:

    $$\begin{aligned} \dfrac{d}{dx} \left( x^{2n}f(x)\right) =2n x^{2n-1}f(x)+ x^{2n}\dfrac{d}{dx}f(x) \end{aligned}$$

    and

    $$\begin{aligned} \dfrac{d^2}{dx^2} \left( x^{2n}f(x)\right) =2n(2n-1)x^{2n-2}f(x)+4nx^{2n-1}\dfrac{d}{dx}f(x)+x^{2n}\dfrac{d^2}{dx^2}f(x). \end{aligned}$$

    Now, a simple calculation shows that

    $$\begin{aligned} \Delta ^m_{\alpha ,n} \circ \mathcal {M}f(x)= x^{2n}\left[ \dfrac{d^2}{dx^2}f(x) +\left( \dfrac{2\alpha +4n+1}{x}-2i\dfrac{a}{b}x\right) \dfrac{d}{dx}f-\left( \dfrac{a^2}{b^2}x^2+2i(\alpha +2n+1)\dfrac{a}{b}\right) f(x)\right] , \end{aligned}$$

    then

    $$\begin{aligned} \mathcal {M}^{-1}\circ \Delta ^m_{\alpha ,n} \circ \mathcal {M}=\Delta ^m_{\alpha +2n}. \end{aligned}$$
  2. 2.

    By the transmutation property Eqs. 16 and 3, we have

    $$\begin{aligned} \Delta ^m_{\alpha ,n}K^m_{\alpha ,n}(y,.)&=\left( \mathcal {M}\circ \Delta ^m_{\alpha +2n}\circ \mathcal {M}^{-1}\right) \left( K^m_{\alpha ,n}(y,.)\right) \\&=\mathcal {M}\circ \Delta ^m_{\alpha +2n} K^m_{\alpha +2n}(y,.) \\&=\dfrac{-y^2}{b^2}K^m_{\alpha ,n}(y,.) \end{aligned}$$

Proposition 10

Let \(m\in SL(2,\mathbb {R})\).

  1. 1.

    \(\Delta ^m_{\alpha ,n}\) leaves \(\mathcal {S}_{*}(\mathbb {R})\) invariant.

  2. 2.

    For every f, \(g \in \mathcal {S}_{*}(\mathbb {R})\)

    $$\begin{aligned} \int _0^{\infty }\Delta ^m_{\alpha ,n}f(x)\overline{g(x)}x^{2\alpha +1} dx =\int _0^{\infty }f(x)\overline{\Delta ^m_{\alpha ,n}g(x)} x^{2\alpha +1} dx. \end{aligned}$$
    (17)

Proof

  1. 1.

    By the transmutation property Eq. 16 and the fact that \(\Delta ^m_{\alpha }\) leaves \(\mathcal {S}_{*}(\mathbb {R})\) invariant.

  2. 2.

    Let f, \(g \in \mathcal {S}_{*}(\mathbb {R})\), we have

    $$\begin{aligned} \int _0^{\infty }f(x)\overline{\Delta ^m_{\alpha ,n}g(x)} x^{2\alpha +1} dx&= \int _0^{\infty }f(x)\overline{\mathcal {M}\circ \Delta ^m_{\alpha +2n} \circ \mathcal {M}^{-1}g(x)}x^{2\alpha +1} dx \\&= \int _0^{\infty }\mathcal {M}^{-1}f(x)\overline{\Delta ^m_{\alpha +2n}\left( \mathcal {M}^{-1}g\right) (x)}x^{2\alpha +4n+1} dx \\&= \int _0^{\infty }\Delta ^m_{\alpha +2n}\left( \mathcal {M}^{-1}f\right) (x)\overline{ \mathcal {M}^{-1}g(x)}x^{2\alpha +4n+1} dx \\&= \int _0^{\infty }\mathcal {M}\circ \Delta ^m_{\alpha +2n} \circ \mathcal {M}^{-1}f(x)\overline{g(x)}x^{2\alpha +1} dx \\&=\int _0^{\infty }\Delta ^m_{\alpha ,n}f(x)\overline{g(x)}x^{2\alpha +1} dx. \end{aligned}$$

Particular case In the case n = 0, \(\Delta ^m_{\alpha ,n}\) is reduced to the differential operator \(\Delta ^m_{\alpha }\) and \(\mathcal {F}^m_{\alpha ,n}\) coincides with the canonical Fourier-Bessel transform \(\mathcal {F}^m_{\alpha }\).

Remark 2

Observe that:

$$\begin{aligned} \mathcal {F}^m_{\alpha ,n} =\mathcal {F}^m_{\alpha +2n}\circ \mathcal {M}^{-1}, \end{aligned}$$
(18)

where \(\mathcal {F}^m_{\alpha +2n}\) is the canonical Fourier-Bessel transform of order \(\alpha +2n\).

Generalized canonical Fourier-Bessel transform on \(\mathcal {L}^1_{\alpha ,n}\)

In this subsection, we discuss basic properties of \(\mathcal {F}^m_{\alpha ,n}\), for \(m\in SL(2,\mathbb {R})\).

Theorem 11

(Riemann-Lebesgue lemma) If \(f \in \mathcal {L}^1_{\alpha ,n}\), then \(\mathcal {F}^m_{\alpha }f \in \mathscr {C}_{*,0}(\mathbb {R})\) and:

$$\begin{aligned} \Vert \mathcal {F}^m_{\alpha ,n}f \Vert _{\infty }\leqslant C_{\alpha +2n} |b|^{-(\alpha +2n+1)} \Vert f\Vert _{1,\alpha ,n}. \end{aligned}$$

Proof

Let \(f \in \mathcal {L}^1_{\alpha ,n}\), \(\Vert \mathcal {F}^m_{\alpha ,n}f \Vert _{\infty }=\Vert \mathcal {F}^m_{\alpha +2n}\left( \mathcal {M}^{-1} f\right) \Vert _{\infty }\), we have \(\Vert \mathcal {F}^m_{\alpha }f \Vert _{\infty }\leqslant C_{\alpha } |b|^{-(\alpha +1)} \Vert f\Vert _{1,\alpha },\) then \(\Vert \mathcal {F}^m_{\alpha ,n}f \Vert _{\infty }\leqslant C_{\alpha +2n} |b|^{-(\alpha +2n+1)} \Vert \mathcal {M}^{-1}f\Vert _{1,\alpha +2n}\), which gives:

$$\begin{aligned} \Vert \mathcal {F}^m_{\alpha ,n}f \Vert _{\infty }\leqslant C_{\alpha +2n} |b|^{-(\alpha +2n+1)} \Vert f\Vert _{1,\alpha ,n}. \end{aligned}$$

Theorem 12

(Inversion formula) Let \(f \in \mathcal {L}_{\alpha ,n}^1\) such that \(\mathcal {F}^m_{\alpha }f \in \mathcal {L}_{\alpha ,n}^1\), then

$$\begin{aligned} \left( \mathcal {F}^m_{\alpha ,n} \circ \mathcal {F}^{m^{-1}}_{\alpha ,n}\right) f =\left( \mathcal {F}^{m^{-1}}_{\alpha ,n} \circ \mathcal {F}^m_{\alpha ,n}\right) f=f. \end{aligned}$$
(19)

Proof

Use Eq. 18 and Theorem 3.

Theorem 13

(Plancherel Theorem) For every f and \(g \in \mathcal {L}^1_{\alpha ,n}\), we have the Plancherel formula

$$\begin{aligned} \int _0^{\infty }\mathcal {F}^m_{\alpha ,n}f(x)\overline{g(x)}x^{2\alpha +2n+1} dx =\int _0^{\infty }f(x)\overline{\mathcal {F}^{m^{-1}}_{\alpha ,n}g(x)} x^{2\alpha +2n+1} dx. \end{aligned}$$
(20)

Proof

Let \(f,\;g \in \mathcal {L}^1_{\alpha ,n}\), we have:

$$\begin{aligned} \int _0^{\infty }f(x)\overline{\mathcal {F}^{m^{-1}}_{\alpha ,n}g(x)} x^{2\alpha +2n+1} dx&= \int _0^{\infty } f(x)\overline{\mathcal {F}^{m^{-1}}_{\alpha +2n}\left( \mathcal {M}^{-1} g\right) (x)} x^{2\alpha +2n+1} dx \\&=\int _0^{\infty } \mathcal {M}^{-1}f(x)\overline{\mathcal {F}^{m^{-1}}_{\alpha +2n}\left( \mathcal {M}^{-1} g\right) (x)} x^{2\alpha +4n+1} dx \\&=\int _0^{\infty }\mathcal {F}^{m}_{\alpha +2n}\left( \mathcal {M}^{-1} f\right) (x)\overline{\mathcal {M}^{-1}g(x)} x^{2\alpha +4+1} dx \\&=\int _0^{\infty }\mathcal {F}^{m}_{\alpha ,n} f(x)\overline{g(x)} x^{2\alpha +2n+1} dx. \end{aligned}$$

Theorem 14

(Operational formulas) Let \(f \in \mathcal {S}_{*}(\mathbb {R})\). Then,

  1. 1.
    $$\begin{aligned} \mathcal {F}^{m}_{\alpha ,n}[x^2f(x)](y)=-b^2\Delta ^m_{\alpha ,n}[\mathcal {F}^{m}_{\alpha ,n}f](y). \end{aligned}$$
    (21)
  2. 2.
    $$\begin{aligned} x^2 \mathcal {F}^{m}_{\alpha ,n}f(y)=-b^2\mathcal {F}^{m}_{\alpha ,n}[\Delta ^{m^{-1}}_{\alpha ,n}f](y). \end{aligned}$$
    (22)

Proof

  1. 1.

    Since \(K^m_{\alpha ,n}\) is an eigenfunction of \(\Delta ^m_{\alpha ,n}\) with eigenvalue \(-\dfrac{y^2}{b^2}\), we have the relations

    $$\begin{aligned} \Delta ^m_{\alpha ,n}[\mathcal {F}^{m}_{\alpha ,n}f](y)&=\dfrac{C_{\alpha +2n}}{(ib)^{\alpha +2n+1}}\int _0^{+\infty }\Delta ^m_{\alpha ,n}[x\rightarrow K^m_{\alpha ,n}(y,x)]f(x)x^{2\alpha +1}dx \\&=\dfrac{-1}{b^2}\mathcal {F}^{m}_{\alpha ,n}[x^2f(x)](y). \end{aligned}$$
  2. 2.

    In Eq. 21, replacing m by \(m^{-1}\) and f by \(\mathcal {F}^{m}_{\alpha ,n}f\), we obtain

    $$\begin{aligned} \mathcal {F}^{m^{-1}}_{\alpha ,n}[x^2 f(x)](y)=-b^2\Delta ^{m^{-1}}_{\alpha ,n}f(y), \end{aligned}$$

    taking \(\mathcal {F}^{m}_{\alpha ,n}\) on both sides of the above relation, and therefore, we obtain the desired formula.

Theorem 15

(Paley-Wiener Theorem) Let \(m\in SL(2,\mathbb {R})\). The generalized canonical Fourier-Bessel transform \(\mathcal {F}^m_{\alpha ,n}\) is a bijection from \(\mathcal {D}_{*}(\mathbb {R})\) into \(\mathbb {L}_{\frac{d}{b}}(\mathcal {H}_{*}(\mathbb {C}))\).

Proof

Let f be a function in \(\mathcal {D}_{*,r}(\mathbb {R})\), \(r \geqslant 0\). It is easy to show that \(\mathcal {F}^m_{\alpha ,n}f \in \mathbb {L}_{\frac{d}{b}}(\mathcal {H}_{*}(\mathbb {C}))\).

Now, let \(f \in \mathcal {H}_{*}(\mathbb {C})\). The Paley-Wiener theorem for the canonical Fourier-Bessel transform shows that there exist \(r \geqslant 0\) and \(f_2 \in \mathcal {D}_{*,r}(\mathbb {R})\) such that \(f(\lambda )=\mathcal {F}^m_{\alpha +2n} f_2(\lambda )\). From this, we obtain \(f(\lambda )=\mathcal {F}^m_{\alpha ,n} f_1(\lambda )\), where \(f_1(s)\) is the function of \(\mathcal {D}_{*,r}(\mathbb {R})\) defined by: \(f_1(s)=\mathcal {M}f_2(s)\).

Generalized canonical Fourier-Bessel transform on \(\mathcal {L}^2_{\alpha ,n}\)

This subsection is devoted to extending the generalized canonical Fourier-Bessel transform \(\mathcal {F}^m_{\alpha ,n}\) from \(\mathcal {L}^1_{\alpha ,n}\cap \mathcal {L}^2_{\alpha ,n}\) to \(\mathcal {L}^2_{\alpha ,n}\).

Theorem 16

  1. 1.

    For all \(f \in \mathcal {S}_{*}(\mathbb {R})\), we have

    $$\begin{aligned} \Vert \mathcal {F}^m_{\alpha ,n}f \Vert _{2,\alpha ,n} = \Vert f \Vert _{2,\alpha ,n}. \end{aligned}$$
    (23)
  2. 2.

    If \(f \in \mathcal {L}^1_{\alpha ,n}\cap \mathcal {L}^2_{\alpha ,n}\), then \(\mathcal {F}^m_{\alpha ,n}f \in \mathcal {L}^2_{\alpha ,n}\) and \(\Vert \mathcal {F}^m_{\alpha ,n}f \Vert _{2,\alpha ,n} = \Vert f \Vert _{2,\alpha ,n}.\)

  3. 3.

    The generalized canonical Fourier-Bessel transform has a unique extension to an isometric isomorphism of \(\mathcal {L}^2_{\alpha ,n}\). The extension is also denoted by \(\mathcal {F}^m_{\alpha ,n}:\;\mathcal {L}^2_{\alpha ,n}\rightarrow \mathcal {L}^2_{\alpha ,n}\).

Proof

  1. 1.

    Use Theorem 6 and Eq. 18.

  2. 2.

    Use Theorem 6 and Eq. 18.

  3. 3.

    These follow from the fact that the canonical Fourier-Bessel transform \(\mathcal {F}^{m}_{\alpha ,n}\) has a unique extension to an isometric isomorphism of \(\mathcal {L}_{2,\alpha }\) and Eq. 18.

Generalized canonical Fourier-Bessel transform on \(\mathcal {L}^p_{\alpha ,n}\), \(1< p<2\)

Theorem 17

(Babenko-Beckner inequality) Let p and q be real numbers such that \(1< p\leqslant 2\) and \(\dfrac{1}{p}+\dfrac{1}{q}=1\). Then, \(\mathcal {F}^m_{\alpha ,n}\) extends to a bounded linear operator on \(\mathcal {L}^p_{\alpha ,n}\), and we have

$$\begin{aligned} \Vert \mathcal {F}^m_{\alpha ,n}f \Vert _{q, \alpha ,n}\leqslant |b |^{(\alpha +2n+1)(\frac{2}{q}-1)} \left( \dfrac{(C_{\alpha +2n}p)^{\frac{1}{p}}}{(C_{\alpha +2n}q)^{\frac{1}{q}}}\right) ^{\alpha +2n+1}\Vert f \Vert _{p, \alpha ,n}. \end{aligned}$$
(24)

Proof

Let \(f \in \mathcal {L}^p_{\alpha ,n}\),

$$\begin{aligned} \Vert \mathcal {F}^m_{\alpha ,n}f \Vert _{q, \alpha ,n}&=\Vert \mathcal {F}^m_{\alpha +2n}\left( \mathcal {M}^{-1}\right) f \Vert _{q, \alpha +2n}\\&\leqslant |b |^{(\alpha +2n+1)(\frac{2}{q}-1)} \left( \dfrac{(C_{\alpha +2n}p)^{\frac{1}{p}}}{(C_{\alpha +2n}q)^{\frac{1}{q}}}\right) ^{\alpha +2n+1}\Vert \mathcal {M}^{-1}f \Vert _{p, \alpha +2n}\\&= |b |^{(\alpha +2n+1)(\frac{2}{q}-1)} \left( \dfrac{(C_{\alpha +2n}p)^{\frac{1}{p}}}{(C_{\alpha +2n}q)^{\frac{1}{q}}}\right) ^{\alpha +2n+1}\Vert f \Vert _{p, \alpha ,n}. \end{aligned}$$

Heisenberg inequality for \(\mathcal {F}^m_{\alpha ,n}\)

Theorem 18

(Heisenberg uncertainty principle) For every \(f \in \mathcal {L}^2_{\alpha ,n}\), we have

$$\begin{aligned} \Vert xf \Vert _{2, \alpha ,n}^2 \Vert x\mathcal {F}^m_{\alpha ,n}f \Vert _{2, \alpha ,n}^2 \geqslant b^2(\alpha +2n +1)^2 \Vert f \Vert _{2, \alpha ,n}^4. \end{aligned}$$
(25)

Proof

$$\begin{aligned} \Vert xf \Vert _{2, \alpha ,n}^2 \Vert x\mathcal {F}^m_{\alpha ,n}f \Vert _{2, \alpha ,n}^2&=\Vert x \mathcal {M}^{-1} f \Vert _{2, \alpha +2n}^2 \Vert x\mathcal {F}^m_{\alpha +2n}\left( \mathcal {M}^{-1}f\right) \Vert _{2, \alpha +2n}^2 \\&\geqslant b^2(\alpha +2n +1)^2 \Vert \mathcal {M}^{-1}f \Vert _{2, \alpha +2n}^4 \\&= b^2(\alpha +2n +1)^2 \Vert f \Vert _{2, \alpha ,n}^4. \end{aligned}$$

Hardy uncertainty principle for \(\mathcal {F}^m_{\alpha ,n}\)

In this section, we are mainly interested in proving analogues of Hardy’s theorem for the generalized canonical Fourier-Bessel transform \(\mathcal {F}^m_{\alpha ,n}\). For this, we will need the following Lemma.

Lemma 5.1

(See [19]) Let h be an entire function on \(\mathbb {C}\) satisfying the following estimates:

$$\begin{aligned} |h(z) |\leqslant ce^{a \vert z \vert ^2},\;\; z \in \mathbb {C} \qquad and \qquad |h(x) |\leqslant ce^{-a x^2}, ,\;\; x \in \mathbb {R}, \end{aligned}$$

for some positive constants a and c. Then, there is a constant \(\lambda\) such that \(h(z)= \lambda e^{-a z^2}\), \(z \in \mathbb {C}\).

Theorem 19

(Hardy theorem) Let \(\nu\) and \(\beta\) be positive constants. Suppose f is a measurable function on \([0,\infty [\) satisfying:

$$\begin{aligned} |f(y) |\leqslant Ce^{-\nu y^2}\qquad and \qquad |\mathcal {F}^m_{\alpha ,n}f(y) |\leqslant Ce^{-\beta y^2}, \end{aligned}$$

where C is a positive constant. Then, we have the following results:

  1. a.

    If \(\nu \beta =\dfrac{1}{4b^2}\), then \(f(y)= Ae^{-(\nu +\frac{i}{2}\frac{a}{b})y^2}\), where A is an arbitrary constant.

  2. b.

    If \(\nu \beta > \dfrac{1}{4b^2}\), then f = 0.

  3. c.

    If \(\nu \beta < \dfrac{1}{4b^2}\), then there are infinitely many linearly independent functions satisfying the above estimates.

Proof

  1. 1.

    Let us introduce the function

    $$\begin{aligned} h(y)=e^{-\frac{1}{2}\frac{d}{b}y^2}\mathcal {F}^m_{\alpha ,n}f(y)=\dfrac{c_{\alpha +2n}}{(ib)^{\alpha +2n+1}}\int _0^{\infty }j_{\alpha +2n}(\frac{xy}{b})e^{\frac{i}{2}\frac{a}{b}x^2}f(x)x^{2\alpha +2n+1}dx,\;\; y \in \mathbb {R}. \end{aligned}$$

    The hypothesis on \(\mathcal {F}^m_{\alpha ,n}f(y)\) can be used to show

    $$\begin{aligned} |h(y) |=|\mathcal {F}^m_{\alpha ,n}f(y) |\leqslant C_1 e^{-\beta y^2},\;\; y\in \mathbb {R}. \end{aligned}$$

    We deduce that h can be extended to the complex plane as an entire function satisfying

    $$\begin{aligned} |h(z) |\leqslant C_1 e^{\beta \vert z \vert ^2}. \end{aligned}$$

    Therefore, by Lemma 5.1, we get

    $$\begin{aligned} h(z)= \lambda e^{-\beta z^2},\;z \in \mathbb {C}\;\; or \;\; \mathcal {F}^m_{\alpha ,n}f(z)=\lambda e^{-(\beta -\frac{i}{2}\frac{d}{b} )z^2}, \; z \in \mathbb {C}. \end{aligned}$$

    Appealing to the inversion formula (19), we see that

    $$\begin{aligned} f(y)=\lambda \mathcal {F}^{m^{-1}}_{\alpha ,n}\left( e^{-(\beta -\frac{i}{2}\frac{d}{b} )x^2} \right) (y)=\lambda \dfrac{c_{\alpha +2n}e^{-\frac{i}{2}\frac{a}{b}y^2}}{(-ib)^{\alpha +2n+1}}\int _0^{\infty }e^{-\beta x^2}j_{\alpha +2n}(\frac{xy}{b})x^{2\alpha +2n+1}dx,\;\; x \in \mathbb {R}. \end{aligned}$$

    To complete the proof, we need to evaluate the last integral. Substitute the series for \(j_{\alpha +2n}(\frac{xy}{b})\) in the integral to get

    $$\begin{aligned} \int _0^{\infty }e^{-\beta x^2}&j_{\alpha +2n}(\frac{xy}{b})x^{2\alpha +2n+1}dx \\&=\Gamma (\alpha +2n+1)\sum \limits _{k=0}^{+\infty } \dfrac{(-1)^k (\frac{x}{2b})^{2n}}{k!\, \Gamma (k+\alpha +2n+1)}\int _0^{+\infty }e^{-\beta x^2}x^{2\alpha +4n+1}dx \\&=\dfrac{\Gamma (\alpha +2n+1)}{2\beta ^{\alpha +2n+1}}e^{-\nu x^2}. \end{aligned}$$

    Consequently, \(f(y)= Ae^{-(\nu +\frac{i}{2}\frac{a}{b})y^2}\).

  2. 2.

    Let us assume that \(\nu \beta >\dfrac{1}{4b^2}\) and define \(\nu '=\dfrac{1}{4b^2 \beta }\). Applying the first case, we get A = 0.

  3. 3.

    See [7].

Nash-type inequality

The classical Nash inequality in \(\mathbb {R}^n\) may be stated as

$$\begin{aligned} \Vert f \Vert ^{2+\frac{4}{n}}_{\mathcal {L}^2(\mathbb {R}^n)}\leqslant C_n \Vert f \Vert ^{\frac{4}{n}}_{\mathcal {L}^1(\mathbb {R}^n)}\Vert |x|\mathcal {F} (f) \Vert ^{2}_{\mathcal {L}^2(\mathbb {R}^n)}, \end{aligned}$$
(26)

for all functions \(f \in \mathcal {L}^1(\mathbb {R}^n)\cap \mathcal {L}^2(\mathbb {R}^n)\). This inequality was first introduced by Nash [16] to obtain regularity properties on the solutions to parabolic partial differential equations and the optimal constant \(C_n\) has been computed more recently in [5].

Now, we will show an analogue of inequality (26) for the generalized canonical Fourier-Bessel transform.

Theorem 20

Let \(k>0\), then there exists a constant \(C_{\alpha ,n,k}>0\) such that for all \(f \in \mathcal {L}^1_{\alpha ,n} \cap \mathcal {L}^2_{\alpha ,n}\)

$$\begin{aligned} \Vert f \Vert ^2_{2,\alpha ,n}\leqslant C_{\alpha ,n,k} \Vert f \Vert ^{\frac{2k}{\alpha +2n+k+1}}_{1,\alpha ,n}\Vert x^k \mathcal {F}^m_{\alpha ,n}f \Vert ^{\frac{2\alpha +4n+2}{\alpha +2n+k+1}}_{2,\alpha ,n}, \end{aligned}$$
(27)

where \(C_{\alpha ,n,k}=C_{\alpha +2n,k}=\dfrac{c_{\alpha +2n}^2}{(2\alpha +4n+2)b^{2\alpha +4n+2}}\left( \dfrac{2kb^{2\alpha +4n+2}}{c_{\alpha +2n}^2}\right) ^{\frac{2\alpha +4n+2}{2\alpha +4n+2k+2}}+\left( \dfrac{2kb^{2\alpha +4n+2}}{c_{\alpha +2n}^2}\right) ^{\frac{-2k}{2\alpha +4n+2k+2}}\).

Proof

Using Eqs. 10 and 13, we have

$$\begin{aligned} \Vert f \Vert ^2_{2,\alpha ,n}&=\Vert \mathcal {M}^{-1} f \Vert ^2_{2,\alpha +2n}\\&\leqslant C_{\alpha +2n,k} \Vert \mathcal {M}^{-1}f \Vert ^{\frac{2k}{\alpha +2n+k+1}}_{1,\alpha +2n}\Vert x^k \mathcal {F}^m_{\alpha +2n}\left( \mathcal {M}^{-1}f\right) \Vert ^{\frac{2\alpha +4n+2}{\alpha +2n+k+1}}_{2,\alpha +2n}\\&\leqslant C_{\alpha ,n,k} \Vert f \Vert ^{\frac{2k}{\alpha +2n+k+1}}_{1,\alpha ,n}\Vert x^k \mathcal {F}^m_{\alpha ,n}f \Vert ^{\frac{2\alpha +4n+2}{\alpha +2n+k+1}}_{2,\alpha ,n}. \end{aligned}$$

Carlson-type inequality

In 1934, Carlson [6] proved that the following inequality

$$\begin{aligned} \left( \int _0^{\infty }|f(x) |dx\right) ^4 \leqslant \pi ^2 \left( \int _0^{\infty }|f(x) |^2 dx\right) \left( \int _0^{\infty }x^2|f(x) |dx\right) \end{aligned}$$
(28)

holds and \(\pi ^2\) is the best constant. Several generalizations and applications in different branches of mathematics have been given during the years.

Now, we will show an analogue of inequality (28) for the generalized canonical Fourier-Bessel transform.

Theorem 21

Let \(k>0\), then there exists a constant \(\nu _{\alpha ,n,k}>0\) such that for all \(f \in \mathcal {L}^1_{\alpha ,n} \cap \mathcal {L}^2_{\alpha ,n}\)

$$\begin{aligned} \Vert f \Vert ^{2k+\alpha +2n+1}_{1,\alpha ,n}\leqslant \nu _{\alpha ,n,k} \Vert f \Vert ^{2k}_{2,\alpha ,n} \Vert x^{2k}f \Vert ^{\alpha +2n+1}_{1,\alpha ,n}, \end{aligned}$$
(29)

where \(\nu _{\alpha ,k}\) is given by

$$\begin{aligned} \nu _{\alpha ,n,k}=\nu _{\alpha +2n,k}=\left( \dfrac{1}{\sqrt{2\alpha +4n+2}}\right) ^{2k}\left[ \left( \dfrac{2k}{\alpha +2n+1}\right) ^{\alpha +2n+1}+\left( \dfrac{2k}{\alpha +2n+1}\right) ^{-2k}\right] . \end{aligned}$$

Proof

Using Eqs. 11 and 13, we have

$$\begin{aligned} \Vert f \Vert ^{2k+\alpha +2n+1}_{1,\alpha ,n}&=\Vert \mathcal {M}^{-1}f \Vert ^{2k+\alpha +2n+1}_{1,\alpha +2n}\\&\leqslant \nu _{\alpha +2n,k} \Vert \mathcal {M}^{-1}f \Vert ^{2k}_{2,\alpha +2n} \Vert x^{2k}\mathcal {M}^{-1}f \Vert ^{\alpha +2n+1}_{1,\alpha +2n}\\&=\nu _{\alpha ,n,k} \Vert f \Vert ^{2k}_{2,\alpha ,n} \Vert x^{2k}f \Vert ^{\alpha +2n+1}_{1,\alpha ,n}. \end{aligned}$$

Global uncertainty principle

Many variations and generalizations of the Heisenberg-Pauli-Weyl uncertainty in both classical and quantum analysis have been treated and many versions have been obtained for several generalized Fourier transforms. Therefore, one may obtain new uncertainty principles involving weighted \(\mathcal {L}^p\) and \(\mathcal {L}^q\)-norms \(1<p\leqslant p<\infty\) by replacing the weight \(\vert x |\) with other weight functions u and \(\tilde{u}\) and search for inequalities of the form [10]

$$\begin{aligned} C\Vert f \Vert ^2_{\mathcal {L}^2(\mathbb {R}^n)}\leqslant \Vert uf \Vert _{\mathcal {L}^p(\mathbb {R}^n)}\Vert \tilde{u} \mathcal {F}( f) \Vert _{\mathcal {L}^q(\mathbb {R}^n)}, \;\; f \in \mathcal {L}^2(\mathbb {R}^n). \end{aligned}$$
(30)

From the two previous sections, we deduce a variation on Heisenberg uncertainty inequality for the generalized canonical Fourier-Bessel transform, more precisely, we will show the following theorem.

Theorem 22

Let \(k>0\) for all \(f \in \mathcal {L}^1_{\alpha ,n} \cap \mathcal {L}^2_{\alpha ,n}\)

$$\begin{aligned} \Vert f \Vert ^1_{1,\alpha ,n}\Vert f \Vert ^2_{2,\alpha ,n}\leqslant (C_{\alpha ,n,k})^{\frac{k+\alpha +2n+1}{\alpha +2n+1}}(\nu _{\alpha ,n,k})^{\frac{1}{\alpha +2n+1}}\Vert x^{2k}f \Vert _{1,\alpha ,n} \Vert x^k \mathcal {F}^m_{\alpha ,n}f \Vert ^{2}_{2,\alpha ,n}. \end{aligned}$$
(31)

Proof

Using Eqs. 12 and 13

$$\begin{aligned} \Vert f \Vert _{1,\alpha ,n}\Vert f \Vert ^2_{2,\alpha ,n}&=\Vert \mathcal {M}^{-1} f \Vert ^1_{1,\alpha +2n}\Vert \mathcal {M}^{-1}f \Vert ^2_{2,\alpha +2n} \\&\leqslant (C_{\alpha +2n,k})^{\frac{k+\alpha +2n+1}{\alpha +2n+1}}(\nu _{\alpha +2n,k})^{\frac{1}{\alpha +2n+1}}\Vert x^{2k}\mathcal {M}^{-1}f \Vert _{1,\alpha +2n} \Vert x^k \mathcal {F}^m_{\alpha +2n}\left( \mathcal {M}^{-1} f \right) \Vert ^{2}_{2,\alpha +2n} \\&=(C_{\alpha ,n,k})^{\frac{k+\alpha +2n+1}{\alpha +2n+1}}(\nu _{\alpha ,n,k})^{\frac{1}{\alpha +2n+1}}\Vert x^{2k}f \Vert _{1,\alpha ,n} \Vert x^k \mathcal {F}^m_{\alpha ,n}f \Vert ^{2}_{2,\alpha ,n}. \end{aligned}$$