1 Introduction

The Hausdorff–Young inequality is a cornerstone in Fourier analysis, stating the boundedness of the Fourier transform

$$\begin{aligned} {\mathscr {F}} f (\zeta )=\int _{{\mathbb {R}}^n} f(x) e^{-i 2\pi \langle x, \zeta \rangle } dx, \end{aligned}$$

between \(L^p({\mathbb {R}}^n)\) and \(L^q({\mathbb {R}}^n)\) spaces, with q being the Hölder conjugate of p for \(1< p \leqslant 2 \leqslant q < \infty .\) The optimal constant in this inequality, known as the Babenko–Beckner constant, has been established for \(q \geqslant 2\) and is achieved for Gaussian functions, as evidenced by analytic methods and maximization strategies [2, 4, 16].

Extensions of the Quaternionic Fourier transform [15] signify the evolution of multichannel signal processing. The quaternion approach is shown to better maintain signal integrity than traditional component-wise methods. This transformation offers detailed understanding of signal space structures, enhancing areas such as color imaging, vector field visualization, and speech signal processing [9].

However, extending the sharp Hausdorff–Young inequality to the non-associative octonion algebra poses significant challenges. This paper explores this area, also examining the broader Cayley–Dickson algebras, which include nested algebras developed through the Cayley–Dickson process. This sequence, represented by

$$\begin{aligned} {{\mathcal {C}}}_0\subset {{\mathcal {C}}}_1 \subset {{\mathcal {C}}}_2 \subset {{\mathcal {C}}}_3\subset \cdots , \end{aligned}$$

begins with the algebras \({\mathbb {R}},\) \({\mathbb {C}},\) \({\mathbb {H}},\) \({\mathbb {O}},\) and progresses into more intricate structures, often exhibiting more challenging properties [7, 8].

We aim to further the comprehensive Fourier transform theory for \({\mathcal {C}}_m\) with \(m \geqslant 4,\) building upon the foundational work on real-valued functions by Snopek [22, 23]. Our extension includes functions valued in \({\mathcal {C}}_m\) [10].

The octonion Fourier transform was initially introduced by Snopek [23] for real-valued functions, and later expanded by Błaszczyk [5, 6] for octonion-valued functions as:

$$\begin{aligned} {\mathcal {F}}_{{\mathbb {O}}}{{f}}({\textbf{y}})=\int _{{\mathbb {R}}^3}f({\textbf{x}})e^{-2\pi e_1x_1y_1}e^{-2\pi e_2x_2y_2}e^{-2\pi e_4x_3y_3}d{\textbf{x}}. \end{aligned}$$
(1.1)

The properties of this transform have been systematically studied by Błaszczyk and Lian [5, 6, 15], with applications spanning various fields [13,14,15, 22].

In the context of Cayley–Dickson algebras, Fourier transforms encounter several challenges

  • The product of two algebra elements may have a magnitude that does not match the product of their individual magnitudes.

  • The non-applicability of interpolation theory in this algebraic setting limits conventional proof techniques.

  • The Plancherel theorem is not valid within these algebras.

  • The classical relationship between derivatives and multiplications is altered in this non-associative setting.

To address these complexities, we use an innovative method, constructing the Cayley–Dickson Fourier transform through sequential classical Fourier transforms, following the complex structures within the algebra.

Specifically, the Cayley–Dickson Fourier transform \({\mathcal {F}}_m{f},\) defined for functions in \(L^1({\mathbb {R}}^m,{\mathcal {C}}m),\) is expressed as

$$\begin{aligned} {\mathcal {F}}_m{f}({\textbf{y}}) =\int _{{\mathbb {R}}^m}f({\textbf{x}})e^{-2\pi e_1x_1y_1}e^{-2\pi e_2x_2y_2}\cdots e^{-2\pi e_{2^{m-1}}x_my_m}d{\textbf{x}}, \end{aligned}$$

using a left-to-right multiplication rule for the integrand. This transformation is constructed through a series of classical Fourier transforms \({\mathcal {F}}_{{\mathbb {C}}_{e_{2^{t-1}}}},\) aligned with the complex structure indicated by \(e_{2^{t-1}}.\) This composite method is represented as a successive application of \({\mathcal {F}}_{{\mathbb {C}}_{e_{2^{i-1}}}}\) for each i from 1 to m.

Additionally, our approach utilizes a theorem about the norm-preserving extension of linear operators. Specifically, for any \(\sigma \)-finite measurable spaces \((X, \Gamma _X, \mu )\) and \((Y, \Gamma _Y, \nu ),\) a bounded linear operator

$$\begin{aligned} T: L^p(X, {\mathbb {C}}) \longrightarrow L^q(Y, {\mathbb {C}}) \end{aligned}$$

where q is the conjugate exponent of p and \(p\leqslant q,\) can be extended to

$$\begin{aligned} T: L^p(X, \ell ^2({\mathbb {C}})) \longrightarrow L^q(Y, \ell ^2({\mathbb {C}})) \end{aligned}$$

with an unchanged norm, utilizing the natural inclusion \({\mathbb {C}}\subset \ell ^2({\mathbb {C}}).\)

In \({{\mathcal {C}}}_m,\) we establish the Hausdorff–Young inequality for functions in \( L^p({\mathbb {R}}^m,{\mathcal {C}}_m)\) with \(1<p<2,\) expressed as

$$\begin{aligned} \Vert {\mathcal {F}}{f}\Vert _q\leqslant \left( p^{\frac{1}{p}} q^{-\frac{1}{q}}\right) ^{m/2}\Vert f\Vert _p. \end{aligned}$$

Notably, for Gaussian functions, this inequality is exact and achievable, showing a distinct jump between the cases of \(m=1\) and \(m>1.\)

Our study also highlights practical implications, especially concerning the Beckner–Hirschman entropic inequality

$$\begin{aligned} S(\vert f\vert ^2)+S(\vert {\mathcal {F}}_m{f}\vert ^2)\geqslant m(1-\ln 2), \end{aligned}$$
(1.2)

for functions \(f\in L^2({\mathbb {R}}^m,{\mathcal {C}}_m)\) with \(\Vert f\Vert _2=1.\) Here, S denotes the von Neumann entropy.

2 Preliminaries

In this section, we will provide an overview of the Cayley–Dickson algebras and the Fourier transform in the setting of the Cayley–Dickson algebras. For the recent development related to the Cayley–Dickson algebras, we refer to [1, 12, 13, 18,19,20,21].

Convention: In this paper, we will adopt the convention of left-to-right multiplication, due to the non-associativity of \({\mathcal {C}}_m\) for \(m\geqslant 3,\) unless explicitly stated otherwise. Thus, for any \(x_j\in {\mathcal {C}}_m,\ j=1,2,\ldots ,n,\) we have

$$\begin{aligned} x_1x_2x_3x_4x_5\cdot \cdot \cdot x_{n-1}x_n=(\cdot \cdot \cdot ((((x_1x_2)x_3)x_4)x_5)\cdot \cdot \cdot x_{n-1})x_n. \end{aligned}$$

2.1 Cayley–Dickson Process

The Cayley–Dickson algebras consist of a set of \(2^m\)-dimensional real algebras. These algebras include all finite-dimensional division algebras over the reals that are alternative, such as the real numbers \({\mathbb {R}},\) complex numbers \({\mathbb {C}},\) quaternions \({\mathbb {H}},\) and octonions \({\mathbb {O}}.\) However, it is worth noting that Cayley–Dickson algebras with dimensions greater than 8 are not alternative and have zero divisors.

The definition of the Cayley–Dickson algebra \({\mathcal {C}}_m,\) where \(m\geqslant 1\) and \({\mathcal {C}}_0={\mathbb {R}},\) can be found in [3, 7, 8]. As a real linear space, \({\mathcal {C}}_{m-1}\) has a dimension of \(2^{m-1}.\) Its standard orthogonal basis consists of \(2^{m-1}\) elements, denoted as

$$\begin{aligned} {e_0, e_1, \ldots , e_{2^{m-1}-1}}, \end{aligned}$$
(2.1)

where \(e_0=1.\) The multiplication rule for the basis is given by

$$\begin{aligned} e_ke_l=-\delta _{kl}+\gamma _{kls}e_s \end{aligned}$$
(2.2)

for all \(k,l,s=1,2, \ldots ,2^{m-1}-1,\) where the coefficients \(\gamma _{kls}\) are totally anti-symmetric with respect to the interchange of k and l,  and any two of k, l, s uniquely determine the third, provided that k, l, s are not equal to each other.

We can represent any arbitrary element x in \({\mathcal {C}}_{m-1}\) using an orthonormal basis as:

$$\begin{aligned} x=x_0+\sum _{k=1}^{2^{m-1}-1}x_ke_k, \end{aligned}$$
(2.3)

where \(x_k\) belongs to the set of real numbers. The conjugation of x is defined as:

$$\begin{aligned} \overline{x}=x_0-\sum _{k=1}^{2^{m-1}-1}x_ke_k. \end{aligned}$$
(2.4)

Using the anti-symmetry property of the coefficients, i.e., \(\gamma _{kls}=-\gamma _{lks},\) it can be shown that conjugation is an involution. Therefore,

$$\begin{aligned} \overline{\overline{x}}=x \quad \text {and} \quad \overline{xy}=\overline{y}\ \overline{x}. \end{aligned}$$
(2.5)

The expressions for the real part \(\Re {\mathfrak {e}}(x)\) and imaginary part \(\Im (x)\) of x are given by:

$$\begin{aligned} \Re {\mathfrak {e}}(x)=\frac{x+\overline{x}}{2},\quad \Im (x)=x-\Re {\mathfrak {e}}(x). \end{aligned}$$
(2.6)

We can define the Cayley–Dickson inner product \(\langle x,y\rangle ,\) the real inner product \(\langle x,y\rangle _{{\mathbb {R}}},\) and the norm \(\vert x\vert \) for any \(x, y\in {\mathcal {C}}_{m-1}\) as follows:

$$\begin{aligned} \langle x,y\rangle= & {} x\overline{y}, \end{aligned}$$
(2.7)
$$\begin{aligned} \langle x,y\rangle _{{\mathbb {R}}}= & {} \Re {\mathfrak {e}} \langle x,y\rangle =\sum _{k=0}^{2^{m-1}-1}x_ky_k, \end{aligned}$$
(2.8)
$$\begin{aligned} \vert x\vert= & {} \langle x,x\rangle ^{{1}/{2}}=\Bigg (\sum _{k=0}^{2^{m-1}-1}x_k^2\Bigg )^{{1}/{2}}. \end{aligned}$$
(2.9)

Now we consider the relation between \({\mathcal {C}}_m\) and \({\mathcal {C}}_{m-1}.\) We take the orthonormal basis of \({\mathcal {C}}_{m-1}\) in (2.1) and let \(e_{2^{m-1}}\) is an imaginary unit in \({\mathcal {C}}_{m}\) that anti-commutes with the basis elements of \(e_{2^{m-1}},\) i.e.,

$$\begin{aligned} e_ke_{2^{m-1}}=-e_{2^{m-1}} e_k \end{aligned}$$
(2.10)

for all \( k=1,2,\ldots ,2^{m-1}-1.\) We define

$$\begin{aligned} e_{k+2^{m-1}}= e_ke_{2^{m-1}}. \end{aligned}$$

Then we have a standard orthonormal basis of \({\mathcal {C}}_m,\) given by

$$\begin{aligned} e_0, e_1, \ldots , e_{2^{m-1}-1}, e_{2^{m-1}}, \ldots , e_{2^m-1}. \end{aligned}$$

Using the Cayley–Dickson construction, we can define the multiplication law for \({\mathcal {C}}_m\) in terms of \({\mathcal {C}}_{m-1}\) and an additional imaginary unit \(e_{2^{m-1}}.\) Specifically, for any \(x,y,z,w\in {\mathcal {C}}_{m-1},\) we have:

$$\begin{aligned} (x+ye_{2^{m-1}})(z+we_{2^{m-1}})=(xz-\overline{w}y)+(wx+y\overline{z} )e_{2^{m-1}}. \end{aligned}$$
(2.11)

Alternatively, we can express this as:

$$\begin{aligned} x(we_{2^{m-1}}){} & {} =(wx)e_{2^{m-1}}, \quad (ye_{2^{m-1}})z=(y\overline{z})e_{2^{m-1}}, \quad (ye_{2^{m-1}})(we_{2^{m-1}})\\{} & {} =-\overline{w}y. \end{aligned}$$

The multiplication table provided by the Cayley–Dickson construction is shown below:

$$\begin{aligned} (e_{2^{m-1}}e_k)e_l= & {} e_{2^{m-1}}(e_le_k), \end{aligned}$$
(2.12)
$$\begin{aligned} e_k(e_le_{2^{m-1}})= & {} (e_le_k)e_{2^{m-1}}, \end{aligned}$$
(2.13)
$$\begin{aligned} e_{2^{m-1}}(e_{2^{m-1}}e_k)= & {} -e_k, \end{aligned}$$
(2.14)
$$\begin{aligned} (e_le_{2^{m-1}})e_{2^{m-1}}= & {} -e_l,\end{aligned}$$
(2.15)
$$\begin{aligned} (e_{2^{m-1}}e_k)(e_le_{2^{m-1}})= & {} -e_le_k. \end{aligned}$$
(2.16)

The Cayley–Dickson construction allows us to express \({{\mathcal {C}}}_m\) in terms of \({{\mathcal {C}}}_{m-1}\) and \(e_{2^{m-1}}.\) With this decomposition, we can compute the inner product, norm, and real inner product in \({{\mathcal {C}}}_m.\) More precisely, let \(u, v\in {{\mathcal {C}}}_m\) be written as

$$\begin{aligned} u=x+ye_{2^{m-1}}, \quad v=z+we_{2^{m-1}}, \end{aligned}$$

where \(x,y,z,w\in {\mathcal {C}}_{m-1}.\) We can define

$$\begin{aligned} \overline{u}:=\overline{x}-ye_{2^{m-1}} \end{aligned}$$
(2.17)

and obtain

$$\begin{aligned} \langle u,v\rangle= & {} u\overline{v}= x\overline{z}+\overline{w}y+(yz-wx)e_{2^{m-1}}, \end{aligned}$$
(2.18)
$$\begin{aligned} \vert u\vert= & {} \langle u,u\rangle ^{\frac{1}{2}}=(\vert x\vert ^2+\vert y\vert ^2)^{\frac{1}{2}}, \end{aligned}$$
(2.19)
$$\begin{aligned} \langle u,v\rangle _{{\mathbb {R}}}= & {} \langle x,z\rangle _{{\mathbb {R}}}+\langle y,w\rangle _{{\mathbb {R}}}=\sum _{k=0}^{2^{m-1}-1}(x_kz_k+y_kw_k). \end{aligned}$$
(2.20)

We can observe that the norm of any \(u,v\in {\mathcal {C}}_m\) satisfies the triangle inequality

$$\begin{aligned} \vert u+v\vert \leqslant \vert u\vert +\vert v\vert . \end{aligned}$$
(2.21)

Additionally, for every non-zero \(u\in {\mathcal {C}}_m,\) there exists an inverse given by:

$$\begin{aligned} u^{-1}:=\frac{\overline{u}}{\vert u\vert }. \end{aligned}$$
(2.22)

2.2 Multiplicativity of Absolute Value

In this subsection, we explore the relationship between multiplication and absolute value in Cayley–Dickson algebras.

If \(m\geqslant 4,\) for any \(x,y\in {\mathcal {C}}_m,\) the value of \(\vert xy\vert \) can be greater than, equal to, or less than \(\vert x\vert \vert y\vert .\)

To illustrate this, consider the example where \(x=e_1+e_{10}\) and \(y=e_5+e_{14}.\) In this case, \(xy=0,\) and so \(\vert xy\vert =0\) which is less than \(\vert x\vert \vert y\vert =2.\)

On the other hand, suppose \(x=e_1-e_{10}\) and \(y=e_0+e_1+e_4-e_{15}.\) Then \(\vert xy\vert =2\sqrt{3},\) which is greater than \(\vert x\vert \vert y\vert =2\sqrt{2}.\)

Finally, if both x and y are real, then \(|xy|=|x||y|.\)

We define the set \(\Gamma _m\) as the collection of all complex planes generated by the imaginary unit \(e_{2^{j-1}}\) for j ranging from 1 to m. In other words,

$$\begin{aligned} \Gamma _m=\bigcup _{j=1}^m {\mathbb {C}}_{e_{2^{j-1}}}, \end{aligned}$$

where

$$\begin{aligned} {\mathbb {C}}_{e_{2^{j-1}}}={\mathbb {R}}+{\mathbb {R}}e_{2^{j-1}}. \end{aligned}$$

Lemma 2.1

Let m be a positive integer. For any \(x\in {\mathcal {C}}_m\) and \(y\in \Gamma _m,\) we have

$$\begin{aligned} \vert xy\vert =\vert x\vert \vert y\vert . \end{aligned}$$

Proof

This result can be derived through the Cayley–Dickson construction (2.11) and the process of induction. More details can be found in [10, Lemma 3.2.]. \(\square \)

2.3 Vector-Valued Function Spaces

When considering function spaces, we can treat the Cayley–Dickson algebras \({\mathcal {C}}_m\) as \({\mathbb {R}}^{2^m},\) which leads to all function spaces being vector-valued. In this context, we will focus on two function spaces:

  • \(L^p({\mathbb {R}}^m,{\mathcal {C}}_m)\) for \(1\leqslant p<\infty ,\)

  • \({\mathcal {S}}({\mathbb {R}}^m,{\mathcal {C}}_m),\) also known as the Schwartz space.

For any function \(f:{\mathbb {R}}^m\rightarrow {\mathcal {C}}_m,\) there exists a standard basis of \({\mathcal {C}}_m\) which allows us to express f as

$$\begin{aligned} \sum _{j=0}^{2^m-1}f_j({\textbf{x}})e_j, \end{aligned}$$

where \(f_j:{\mathbb {R}}^m\rightarrow {\mathbb {R}}\) are real-valued functions. It is important to note that \(f\in L^p({\mathbb {R}}^m,{\mathcal {C}}_m)\) if and only if \(f_j\in L^p({\mathbb {R}}^m,{\mathbb {R}})\) for all \(j=0,1,\ldots ,2^{m}-1.\) Similar results hold for other vector-valued spaces.

2.4 Cayley–Dickson Fourier Transforms

The Cayley–Dickson Fourier Transform is briefly introduced along with its properties in this subsection. For more detailed information, we refer to the paper [10].

To define the Cayley–Dickson Fourier Transform of \(f\in L^1({\mathbb {R}}^m,{\mathcal {C}}_m),\) we denote the function as \({\mathcal {F}}f\) and express it as follows:

$$\begin{aligned} {\mathcal {F}}f({\textbf{y}}) = {\mathcal {F}}_m{f}({\textbf{y}}) :=\int _{{\mathbb {R}}^m}f({\textbf{x}})e^{-2\pi e_1x_1y_1}e^{-2\pi e_2x_2y_2}\cdots e^{-2\pi e_{2^{m-1}}x_my_m}d{\textbf{x}}.\nonumber \\ \end{aligned}$$
(2.23)

Here, \({\textbf{x}}\) and \({\textbf{y}}\) are m-dimensional vectors, \({\mathcal {C}}_m\) is identified with the \(2^m\)-dimensional real space, and \(e_i\) is the i-th unit vector.

Proposition 2.2

For any \(f\in L^1({\mathbb {R}}^m,{\mathcal {C}}_m),\) \({\mathcal {F}}{f}\) is uniformly continuous on \({\mathbb {R}}^m,\) and

$$\begin{aligned} \Vert {\mathcal {F}}{f}\Vert _{L^{\infty }({\mathbb {R}}^m,{\mathcal {C}}_m)}\leqslant \Vert f\Vert _{L^1({\mathbb {R}}^m,{\mathcal {C}}_m)}. \end{aligned}$$
(2.24)

Proposition 2.3

(Parseval) For any \(f\in L^2({\mathbb {R}}^m,{\mathcal {C}}_m),\) we have

$$\begin{aligned} \Vert {\mathcal {F}}{f}\Vert _{L^2({\mathbb {R}}^m,{\mathcal {C}}_m)}=\Vert f\Vert _{L^2({\mathbb {R}}^m,{\mathcal {C}}_m)}. \end{aligned}$$
(2.25)

We will now prove that the Cayley–Dickson-Fourier transform preserves the Schwartz space \({\mathcal {S}}({\mathbb {R}}^m,{\mathcal {C}}_m).\) To do this, we need to introduce a critical involution given by

$$\begin{aligned} \tau _{\alpha ,\beta }({\textbf{y}})=\big (y_1,(-1)^{\alpha _1+\beta _1}y_2, \ldots ,(-1)^{\sum _{l=1}^{m-1}(\alpha _l+\beta _l)}y_m\big ), \end{aligned}$$

where \(\alpha \) and \(\beta \) are multi-indices in \({\mathbb {N}}^m.\)

Proposition 2.4

[10] Let \(\alpha \) and \(\beta \) be multi-indices in \({\mathbb {N}}^m.\) Then,  the Cayley–Dickson Fourier Transform of \(\partial ^{\alpha }({\textbf{x}}^{\beta }f),\) evaluated at \({\textbf{y}},\) can be expressed as follows : 

$$\begin{aligned} {\mathcal {F}}{\{\partial ^{\alpha }({\textbf{x}}^{\beta }f)}\} ({\textbf{y}})=C{\textbf{y}}^{\alpha }\partial ^{\beta } {\mathcal {F}}{f}(\tau _{\alpha ,\beta }({\textbf{y}}))e_{2^{m-1}}^{-\beta _m}\cdots e_1^{-\beta _1}e_{2^{m-1}}^{\alpha _m}\cdots e_1^{\alpha _1}, \end{aligned}$$
(2.26)

where

$$\begin{aligned} C=(-1)^{\vert \beta \vert }(2\pi )^{\vert \alpha \vert -\vert \beta \vert }. \end{aligned}$$

3 The Proof of Sharp Hausdorff–Young Inequalities

In order to establish our main theorem, it is necessary to extend an operator that is associated with vector-valued functions, as presented in [17, Theorem 5.5.1.].

Definition 3.1

Let \(1\leqslant p<\infty ,\) and \((X, \Gamma _X, \mu )\) be a \(\sigma \)-finite measure space. Let \(f_j\) be complex-valued functions in \(L^p(X,{\mathbb {C}}).\) We define f as the sequence \(\{f_j\}_{j=1}^\infty ,\) where \(f_j\in L^p(X,{\mathbb {C}})\) for all j. We say that \(f\in L^p(X, \ell ^2({\mathbb {C}}))\) if

$$\begin{aligned} |f|_{\ell ^2({\mathbb {C}})}=\Bigg (\sum _{j=1}^\infty |f_j|^2\Bigg )^{1/2} \end{aligned}$$

belongs to \(L^p(X,{\mathbb {C}}).\) We denote the norm of f in \(L^p(X, \ell ^2({\mathbb {C}}))\)

The space \(L^p(X, \ell ^2({\mathbb {C}}))\) is a Banach space.

Theorem 3.2

[17, Theorem 5.5.1.] Suppose \(1\leqslant p\leqslant q<\infty \) and let \((X, \Gamma _X, \mu )\) and \((Y, \Gamma _Y, \nu )\) be two \(\sigma \)-finite measure spaces. Let T be a bounded linear operator from \(L^p(X,{\mathbb {C}})\) to \(L^q(Y,{\mathbb {C}})\) with norm N. Then T has a norm-preserving extension,  also denoted by T,  from to , where we use the canonical embedding . In other words,  for any , we have

(3.1)

The Cayley–Dickson algebra \({{\mathcal {C}}}_m\) can be viewed as a complex linear space consisting of the direct sum of

$$\begin{aligned} \bigoplus _{j=0}^{2^{m-1}-1}e_j{\mathbb {C}}_{e_{2^{m-1}}}, \end{aligned}$$

which in turn can be seen as \({\mathbb {C}}^{2^{m-1}}\) in a certain way.

Lemma 3.3

The Cayley–Dickson algebras,  as a real linear space,  can be represented by a direct sum of orthogonal complex planes. Specifically,  there exists an isometric isomorphism

$$\begin{aligned} I:{\mathcal {C}}_m \longrightarrow \bigoplus _{j=0}^{2^{m-1}-1}e_j{\mathbb {C}}_{e_{2^{m-1}}} \end{aligned}$$

given by

$$\begin{aligned} I(x)=I\left( \sum _{j=0}^{2^m-1}x_je_j\right) = \sum _{j=0}^{2^{m-1}-1}e_j(x_j+x_{j+2^{m-1}}e_{2^{m-1}}). \end{aligned}$$
(3.2)

Proof

To prove this, it is sufficient to show that I is an isomorphism and isometric. It is clear that I is an isomorphism. To see that I is isometric, we have

$$\begin{aligned} \vert I(x)\vert ^2&=\sum _{j=0}^{2^{m-1}-1}(x_j^2+x_{j+2^{m-1}}^2)&=\sum _{j=0}^{2^m-1}x_j^2=\vert x\vert ^2. \end{aligned}$$

The proof is complete. \(\square \)

We present a generalization of Beckner’s result [4] to the case of Cayley–Dickson algebras \({\mathcal {C}}_m.\) Our main theorem is as follows.

Theorem 3.4

(Hausdorff–Young) Let \(f\in L^p({\mathbb {R}}^m,{\mathcal {C}}_m)\) with \(1<p<2.\) Then,  we have

$$\begin{aligned} \Vert {\mathcal {F}}{f}\Vert _q\leqslant A_p^m\Vert f\Vert _p, \end{aligned}$$
(3.3)

where

$$\begin{aligned} A_p=\left( p^{\frac{1}{p}} q^{-\frac{1}{q}}\right) ^{1/2} \end{aligned}$$

is sharp and can be attained if f is a Gaussian function

$$\begin{aligned} f({\textbf{x}})=ae^{-\pi (\sum _{j=1}^mb_jx_j^2+2c_jx_j)}, \end{aligned}$$
(3.4)

where \(b_j\) is positive for all j,  \((c_1, c_2, \ldots , c_m)\in {\mathbb {C}}_{e_1}\times {\mathbb {R}}^{m-1},\) and

$$\begin{aligned} a\in {\left\{ \begin{array}{ll} {\mathbb {C}}_{e_1}, &{} m=1,\\ {\mathbb {R}}, &{} m\geqslant 2. \end{array}\right. } \end{aligned}$$

Remark 3.5

The inequality of Hausdorff–Young holds for \(p=1\) and \(p=2,\) where (3.3) reduces to (2.24) and (2.25), respectively. However, the extremizers differ from the \(1<p<2\) case, where there exist only a few functions that satisfy equality in (3.3). In the cases of \(p=1\) and \(p=2,\) there are numerous functions that satisfy equality, especially in \(L^2({\mathbb {R}}^m,{\mathcal {C}}_m),\) where all functions satisfy equality as demonstrated in Theorem 2.3.

Let us return to the proof of Theorem 3.4.

Proof

Let us assume \(f\in {\mathcal {S}}({\mathbb {R}}^m,{\mathcal {C}}_m)\) and use an approximation process and Theorem 2.4 to show that it suffices to prove (3.3) only for functions in the Schwartz space.

To begin, we note that the constant \(A_p^m\) is sharp. We represent \({\mathcal {F}}_{m-1}\) as the composition of the classic Fourier transform over the complex plane \({\mathbb {C}}_{ e_{2^{t-1}}}.\) More precisely, we have

$$\begin{aligned} {\mathcal {F}}_{m-1 }={\mathcal {F}}_{{\mathbb {C}}_{e_{2^{m-2}}}}\circ \cdots \circ {\mathcal {F}}_{{\mathbb {C}}_{e_1}}, \end{aligned}$$

where \({\mathcal {F}}_{{\mathbb {C}}{e_{2^{i-1}}}}\) for \(i=1, \ldots , m-1\) represents the classic Fourier transform over the complex plane \({{\mathbb {C}}_{e_{2^{i-1}}}}.\) We then express \({\mathcal {F}}_{m-1}f\) in real-valued measurable components, as given by

$$\begin{aligned} {\mathcal {F}}_{m-1} {f} =\sum _{j=0}^{2^m-1}g_je_j=\sum _{j=0}^{2^{m-1}-1}e_j(g_j+g_{j+2^{m-1}}e_{2^{m-1}}), \end{aligned}$$
(3.5)

where \(g_j\) are the Fourier coefficients of f. We define auxiliary functions

$$\begin{aligned} h_j:=g_j+g_{j+2^{m-1}}e_{2^{m-1}} \end{aligned}$$

and observe that the associator

$$\begin{aligned}{}[e_j,h_j,e^{-2\pi e_{2^{m-1}}x_my_m}] \end{aligned}$$

is zero, which implies that

$$\begin{aligned} {\mathcal {F}}{f}={\mathcal {F}}_{{\mathbb {C}}_{e_{2^{m-1}}}}\circ {\mathcal {F}}_{m-1}{f}=\sum _{j=0}^{2^{m-1}-1}e_j{\mathcal {F}}_{{\mathbb {C}}_{e_{2^{m-1}}}}{{h_j}}. \end{aligned}$$
(3.6)

Lemma 3.3 implies that the complex planes \(e_j{\mathbb {C}}_{e_{2^{m-1}}}\) are mutually orthogonal. Using this, we obtain the expressions

$$\begin{aligned} \vert {\mathcal {F}}_{m-1}{f}\vert ^2=\sum _{j=0}^{2^{m-1}-1}\vert h_j\vert ^2, \end{aligned}$$
(3.7)

and

$$\begin{aligned} \vert {\mathcal {F}}{f} \vert ^2=\sum _{j=0}^{2^{m-1}-1}\vert {\mathcal {F}}_{{\mathbb {C}}_{e{2^{m-1}}}}{{h_j}}\vert ^2, \end{aligned}$$
(3.8)

where \(h_j\) denotes the j-th component of f under the isometric isomorphism I from Lemma 3.3.

Using induction, we will prove our theorem by assuming that inequality (3.9) holds for \(t=1\) and \(t=m-1\) for any \(\varphi \in {\mathcal {S}}({\mathbb {R}}^m,{\mathcal {C}}_m).\) It states that

$$\begin{aligned} \Vert {\mathcal {F}}_t{ \varphi }(\cdot ,x_{t+1},\ldots ,x_m)\Vert _q\leqslant A_p^t\Vert \varphi (\cdot ,x_{t+1},\ldots , x_m)\Vert _p. \end{aligned}$$
(3.9)

We should note that the \(t=1\) case corresponds to the classical result, while the \(t=m-1\) case serves as our induction hypothesis.

Then we prove that (3.9) also holds for \(t=m.\) We denote \({\textbf{x}}'=(x_1,\ldots ,x_{m-1}).\) By (3.7), \(f({\textbf{x}})\) is \(L^p\) integrable for almost every \(x_m\in {\mathbb {R}}\) and induction (3.9), we have

$$\begin{aligned} \Bigg (\int _{{\mathbb {R}}^{m-1}}\!\Bigg (\sum _{j=0}^{2^m\!-\!1}\vert h_j({\textbf{y}}',\!x_m)\vert ^2\Bigg )^{q/2}d{\textbf{y}}'\Bigg )^{1/q}= & {} \Bigg (\int _{{\mathbb {R}}^{m-1}}\vert {\mathcal {F}}_{m-1}{f} ({\textbf{y}}',x_m)\vert ^qd{\textbf{y}}'\Bigg )^{1/q}\nonumber \\\leqslant & {} A_p^{m-1} \Bigg (\int _{{\mathbb {R}}^{m-1}}\vert f({\textbf{x}})\vert ^pd\mathbf {x'}\Bigg )^{1/p}.\nonumber \\ \end{aligned}$$
(3.10)

Integrate on both sides of (3.10) with respect to \(x_m,\) we obtain

$$\begin{aligned} \int _{{\mathbb {R}}}\Bigg (\int _{{\mathbb {R}}^{m-1}}\Bigg (\sum _{j=0}^{2^m-1}\vert h_j({\textbf{y}}',x_m)\vert ^2\Bigg )^{q/2}d{\textbf{y}}'\Bigg )^{p/q}dx_m\leqslant A_p^{pm-p}\Vert f\Vert _p^p. \end{aligned}$$

Then we invoke the Minkowski inequality to the left side of above integral inequality to get

$$\begin{aligned}{} & {} \Bigg (\int _{{\mathbb {R}}^{m-1}}\Bigg (\int _{{\mathbb {R}}} \Bigg (\sum _{j=0}^{2^m-1}\vert h_j({\textbf{y}}',x_m)\vert ^2 \Bigg )^{p/2}dx_m\Bigg )^{q/p}d{\textbf{y}}'\Bigg )^{p/q}\nonumber \\{} & {} \quad \leqslant \int _{{\mathbb {R}}}\Bigg (\int _{{\mathbb {R}}^{m-1}}\Bigg (\sum _{j=0}^{2^m-1}\vert h_j({\textbf{y}}',x_m)\vert ^2\Bigg )^{q/2}d{\textbf{y}}'\Bigg )^{p/q}dx_m<\infty . \end{aligned}$$
(3.11)

This show that each \(h_j\) is \(L^p\) integrable with respect to \(x_m\) for almost every \({\textbf{y}}'=(y_1,\ldots ,y_{m-1})\in {\mathbb {R}}^{m-1}.\)

Next, we claim that the quantity on the left side of (3.11) is greater than or equal to

$$\begin{aligned} A_p^{-p}\Bigg (\int _{{\mathbb {R}}^m}\Bigg (\sum _{j=0}^{2^m-1} \vert {\mathcal {F}}_{{\mathbb {C}}_{e_{2^{m-1}}}}{h_j }({\textbf{y}})\vert ^2\Bigg )^{\frac{q}{2}}d{\textbf{y}}\Bigg )^{\frac{p}{q}}. \end{aligned}$$
(3.12)

Indeed, we note that the functions \(h_j,\) which are \({\mathbb {C}}_{2^{m-1}}\)-valued and operated on by the classic Fourier Transform \({\mathcal {F}}_{{\mathbb {C}}_{e_{2^{m-1}}}},\) satisfy the conditions of Lemma 3.2 with respect to \(x_m.\) By the induction step (3.9), the norm N in Lemma 3.2 can be taken to be \(A_p.\) Also, since \(1<p<2<q,\) we can apply Theorem 2.4 and Tonelli’s theorem to complete the proof of our claim.

Furthermore, we can see that (3.12) is exactly equal to \(A_p^{-p}\Vert {\mathcal {F}}{\{f}\}\Vert _q^p\) by using (3.8).

Now we come to show that the sharp constant \(A_p^m\) can be attained by the Gaussian function given in (3.4). Additionally, we can verify that this condition indeed leads to equality in (3.3).

To do so, we refer to two well-known classical results:

$$\begin{aligned}{} & {} \Vert g\Vert _{L^p({\mathbb {R}},{\mathbb {C}})}=\vert a\vert e^{\pi b^{-1}(\Re {\mathfrak {e}}(c))^2}(pb)^{-\frac{1}{2p}}, \end{aligned}$$
(3.13)
$$\begin{aligned}{} & {} \Vert {\mathcal {F}}_{{\mathbb {C}}}{g}\Vert _{L^q({\mathbb {R}},{\mathbb {C}})} =\Vert ab^{-\frac{1}{2}}e^{-\pi b^{-1}(y+e_1c)^2}\Vert _{L^q({\mathbb {R}}, {\mathbb {C}})}\nonumber \\{} & {} \quad =\vert a\vert b^{-\frac{1}{2}}e^{\pi b^{-1}(\Re {\mathfrak {e}}(c))^2}(q^{-1}b)^{\frac{1}{2q}}, \end{aligned}$$
(3.14)

where

$$\begin{aligned} g(x)=ae^{-\pi bx^2+2\pi cx} \end{aligned}$$

with \(a\in {\mathbb {C}},b>0,\) and \(c\in {\mathbb {C}}.\)

Using (3.13)–(3.14), (3.4), and Lemma 2.1, we can compute the following expressions:

$$\begin{aligned}{} & {} \Vert f\Vert _{L^p({\mathbb {R}}^m,{\mathcal {C}}_m)}=\vert a\vert e^{\pi \sum _{j=1}^mb_j^{-1}(\Re {\mathfrak {e}}(c_j))^2}\prod _{j=1}^m(pb_j)^{-\frac{1}{2p}}, \end{aligned}$$
(3.15)
$$\begin{aligned}{} & {} \Vert {\mathcal {F}}{f}\Vert _{L^q({\mathbb {R}},{\mathcal {C}}_m)}\nonumber \\{} & {} \quad =\left\| a\left( \prod _{j=1}^mb_j^{-\frac{1}{2}}\right) e^{-\pi b_1^{-1}(y_1+e_1c_1)^2}\cdots e^{-\pi b_m^{-1}(y_m+{e_{2^{m-1}}}c_m)^2}\right\| _{L^q({\mathbb {R}},{\mathcal {C}}_m)}\nonumber \\{} & {} \quad =\vert a\vert \left( \prod _{j=1}^mb_j^{-\frac{1}{2}}e^{\pi b_j^{-1}(\Re {\mathfrak {e}}(c_j))^2}(q^{-1}b_j)^{\frac{1}{2q}}\right) . \end{aligned}$$
(3.16)

It is clear that (3.15)–(3.16) make (3.3) become an equality. This completes the proof of our theorem. \(\square \)

Remark 3.6

We do not know if all \(L^p\) functions that satisfy (3.4) are those that attain the optimal constant \(A_p^m.\) If \(m\geqslant 2,\) we can prove by induction that (3.10) is an equality if and only if f is a Gaussian function of the form

$$\begin{aligned} f({\textbf{x}}) = a(x_m) e^{-\pi \left( \sum _{j=1}^{m-1} b_j(x_m) x_j^2 + 2c_j(x_m) x_j\right) } \end{aligned}$$
(3.17)

where \(a(x_m)\in {\mathbb {C}}{e_1},\) \(b_j(x_m)>0,\) \(c_1(x_m)\in {\mathbb {C}}{e_1},\) and \(c_j(x_m)\in {\mathbb {R}}\) for \(j\geqslant 2.\) However, it is challenging to demonstrate that \(c_j(x_m)\) is independent of \(x_m.\)

Using the Hausdorff–Young inequality in Theorem 3.4, we can derive the following direct implication, known as the sharp Beckner–Hirschman entropic inequality.

Theorem 3.7

(Beckner–Hirschman) Let \(f\in L^2({\mathbb {R}}^m,{\mathcal {C}}_m)\) and \(\Vert f\Vert _2=1.\) Then we have

$$\begin{aligned} S(\vert f\vert ^2)+S(\vert {\mathcal {F}}{f}\vert ^2)\geqslant m(1-\ln 2) \end{aligned}$$
(3.18)

whenever the left hand side has meaning,  where

$$\begin{aligned} S(\vert f\vert )=-\int _{{\mathbb {R}}^m}\vert f({\textbf{x}})\vert \ln \vert f({\textbf{x}})\vert d{\textbf{x}} \end{aligned}$$

is the Shannon entropy of \(\vert f\vert .\)

Proof

The result follows by differentiating (3.3) with respect to p at \(p=2.\) \(\square \)

This theorem is the generalization of Hirschman’s result [11] in the setting of Cayley–Dickson algebras.

4 Concluding Remarks

We have proved the sharp Hausdorff–Young inequality for Fourier transforms over the Cayley–Dickson algebra \({{\mathcal {C}}}_m\) for any positive integer m. This result is attained by the Gaussian function given by

$$\begin{aligned} f({\textbf{x}})=ae^{-\pi (\sum _{j=1}^mb_jx_j^2+2c_jx_j)}, \end{aligned}$$

where \(b_j\) is positive for all j,  \((c_1, c_2, \ldots , c_m)\in {\mathbb {C}}_{e_1}\times {\mathbb {R}}^{m-1},\) and

$$\begin{aligned} a\in {\left\{ \begin{array}{ll} {\mathbb {C}}_{e_1}, &{} m=1,\\ {\mathbb {R}}, &{} m\geqslant 2. \end{array}\right. } \end{aligned}$$

In [16], Lieb showed that when \(m=1,\) the aforementioned functions are the sole extremizers of the inequality. This result remains valid for any \(m\in {{\mathbb {N}}},\) as long as the extremizers are even functions. Nevertheless, it is currently unknown whether these functions remain the only extremizers even when \(m=2\) in the context of quaternions.