We first state the following result, which describes the Hausdorff–Young inequality related to the WLCT.
Theorem 5.1
(WLCT Hausdorff–Young)
For any \(1\leq r\leq 2\) such that \(\frac{1}{r} + \frac{1}{s} =1\). Suppose that ϕ in \(L^{s}(\mathbb{R})\) and f in \(L^{r}(\mathbb{R})\), then we have
$$\begin{aligned} \bigl\Vert G_{\phi }^{B} f(w,v) \bigr\Vert _{L^{s}(\mathbb{R})} \leq \Vert \phi \Vert _{L^{s}( \mathbb{R})} \Vert f \Vert _{L^{r}(\mathbb{R})}. \end{aligned}$$
(33)
Proof
We assume that \(\| \phi \|_{L^{s}(\mathbb{R})} = 1\). An application of the identity (10) together with the identity (16) will lead to
$$\begin{aligned} \bigl\Vert G_{\phi }^{B} f(w,v) \bigr\Vert _{L^{\infty }(\mathbb{R})} & = \bigl\Vert \mathcal{C}_{B} \{ f T_{v} \bar{\phi } \} \bigr\Vert _{L^{\infty }(\mathbb{R})} \\ &\leq \Vert f T_{v} \bar{\phi } \Vert _{L^{1}(\mathbb{R})} \\ &\leq \Vert f \Vert _{L^{1}(\mathbb{R})} \Vert \phi \Vert _{L^{\infty }(\mathbb{R})} \\ &= \Vert f \Vert _{L^{1}(\mathbb{R})}. \end{aligned}$$
(34)
For \(r=2\), we obtain
$$\begin{aligned} \bigl\Vert G_{\phi }^{B} f(w,v) \bigr\Vert _{L^{2}(\mathbb{R})} = \Vert \phi \Vert _{L^{2}( \mathbb{R})} \Vert f \Vert _{L^{2}(\mathbb{R})} \leq \Vert f \Vert _{L^{2}( \mathbb{R})}, \end{aligned}$$
(35)
and applying Riesz–Thorin interpolation theorem yields
$$\begin{aligned} \bigl\Vert G_{\phi }^{B} f(w,v) \bigr\Vert _{L^{s}(\mathbb{R})} \leq \Vert f \Vert _{L^{r}( \mathbb{R})}. \end{aligned}$$
(36)
Let \(\phi \in L^{r}(\mathbb{R})\) be a window function and putting \(\psi = \frac{\phi }{\|\phi \|_{L^{s}(\mathbb{R})}}\), then we immediately get
$$\begin{aligned} G_{\psi }^{B} f =\frac{1}{ \Vert \phi \Vert _{L^{s}(\mathbb{R})}} G_{\phi }^{B} f. \end{aligned}$$
(37)
From (36), we conclude that
$$\begin{aligned} \bigl\Vert G_{\psi }^{B} f(w,v) \bigr\Vert _{L^{s}(\mathbb{R})} \leq \Vert f \Vert _{L^{r}( \mathbb{R})}. \end{aligned}$$
(38)
This implies that
$$\begin{aligned} \bigl\Vert G_{\phi }^{B} f(w,v) \bigr\Vert _{L^{s}(\mathbb{R})} \leq \Vert \phi \Vert _{L^{s}( \mathbb{R})} \Vert f \Vert _{L^{r}(\mathbb{R})}, \end{aligned}$$
(39)
and the proof is complete. □
We derive the following theorem, which is little different from those proposed in the paper [5].
Theorem 5.2
Let ϕ be a window function belonging to \(L^{2}(\mathbb{R})\) such that \(\|\phi \|_{L^{2}(\mathbb{R})} =1\). Let \(f\in L^{2}(\mathbb{R})\) be a function with \(\|f\|_{L^{2}(\mathbb{R})}=1\). If
$$\begin{aligned} \int \int _{V} \bigl\vert G_{\phi }^{B} f(w,v) \bigr\vert ^{2} \,dw \,dv \geq 1-\epsilon, \end{aligned}$$
(40)
then for every \(\epsilon \geq 0\) one has
$$\begin{aligned} \mu (V) \geq \sqrt{2\pi b} (1-\epsilon ). \end{aligned}$$
(41)
Here \(V \subseteq \mathbb{R} \times \mathbb{R}\) is a measurable subset, and \(\mu (V)\) is the Lebesgue measure of V.
Proof
Applying the Cauchy–Schwarz inequality results in
$$\begin{aligned} \bigl\vert G_{\phi }^{B} f(w,v) \bigr\vert &= \biggl\vert \frac{1}{\sqrt{2\pi b }} \int _{ \mathbb{R}} f(x) \overline{\phi (x -v)} e^{\frac{i}{2} (\frac{a}{b}x^{2}- \frac{2}{b} xw + \frac{d}{b} w^{2} -\frac{\pi }{2})} \,dx \biggr\vert \\ & \leq \frac{1}{\sqrt{2\pi b }} \biggl( \int _{\mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \,dx \biggr)^{\frac{1}{2}} \biggl( \int _{\mathbb{R}} \bigl\vert \phi (x-v) \bigr\vert ^{2} \,dx \biggr)^{\frac{1}{2}} \\ & = \frac{1}{\sqrt{2\pi b }} \Vert f \Vert _{L^{2}(\mathbb{R})} \Vert \phi \Vert _{L^{2}( \mathbb{R})}, \end{aligned}$$
(42)
which implies that
$$\begin{aligned} 1-\epsilon \leq \int \int _{V} \bigl\vert G_{\phi }^{B} f(w,v) \bigr\vert ^{2} \,dw \,dv \leq \bigl\Vert G_{\phi }^{B} f \bigr\Vert _{L^{\infty }(\mathbb{R})}^{2} \mu (V) \leq \frac{1}{\sqrt{2\pi b }} \mu (V), \end{aligned}$$
(43)
This is the desired result. □
Theorem 5.3
Let \(\phi,\psi \) be two window functions, and \(f \in L^{2}(\mathbb{R})\), then for all \(r \in [1,\infty )\), we have
$$\begin{aligned} &\biggl( \int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert G_{\phi }^{B} f(w,v) G_{\psi }^{B} g(w,v) \bigr\vert ^{r} \,dw \,dv \biggr)^{\frac{1}{r}} \\ &\quad \leq \biggl(\frac{1}{2\pi b} \biggr)^{\frac{r-1}{r}} \Vert f \Vert _{L^{2}( \mathbb{R})} \Vert g \Vert _{L^{2}(\mathbb{R})} \Vert \phi \Vert _{L^{2}(\mathbb{R})} \Vert \psi \Vert _{L^{2}(\mathbb{R})}. \end{aligned}$$
(44)
Proof
According to the Cauchy–Schwarz inequality, we obtain
$$\begin{aligned} &\int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert G_{\phi }^{B} f(w,v) G_{ \psi }^{B} g(w,v) \bigr\vert \,dw \,dv \\ &\quad\leq \biggl( \int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert G_{\phi }^{B} f(w,v) \bigr\vert ^{2} \,dw \,dv \biggr)^{\frac{1}{2}} \biggl( \int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert G_{\psi }^{B} g(w,v) \bigr\vert ^{2} \,dw \,dv \biggr)^{ \frac{1}{2}}. \end{aligned}$$
(45)
By virtue of (20), we see that
$$\begin{aligned} \int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert G_{\phi }^{B} f(w,v) G_{ \psi }^{B} g(w,v) \bigr\vert \,dw \,dv \leq \Vert f \Vert _{L^{2}(\mathbb{R})} \Vert g \Vert _{L^{2}( \mathbb{R})} \Vert \phi \Vert _{L^{2}(\mathbb{R})} \Vert \psi \Vert _{L^{2}( \mathbb{R})}. \end{aligned}$$
(46)
Thus,
$$\begin{aligned} & \biggl( \int _{\mathbb{R}} \int _{\mathbb{R}} |G_{\phi }^{B} f(w,v) \bigl\vert G_{\psi }^{B} g(w,v) \bigr\vert ^{r} \,dw \,dv \biggr)^{\frac{1}{r}} \\ &\quad\leq \bigl\Vert G_{\phi }^{B} f G_{\psi }^{B} g \bigr\Vert ^{\frac{r-1}{r}}_{L^{ \infty }(\mathbb{R}\times \mathbb{R} )} \biggl( \int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert G_{\phi }^{B} f(w,v) G_{\psi }^{B} g(w,v) \bigr\vert \,dw \,dv \biggr)^{\frac{1}{r}} \\ &\quad\leq \biggl(\frac{1}{2\pi b} \Vert f \Vert _{L^{2}(\mathbb{R})} \Vert \phi \Vert _{L^{2}( \mathbb{R})} \Vert g \Vert _{L^{2}(\mathbb{R})} \Vert \psi \Vert _{L^{2}(\mathbb{R})} \biggr)^{\frac{r-1}{r}} \bigl( \Vert f \Vert _{L^{2}(\mathbb{R})} \Vert g \Vert _{L^{2}( \mathbb{R})} \Vert \phi \Vert _{L^{2}(\mathbb{R})} \Vert \psi \Vert _{L^{2}( \mathbb{R})} \bigr)^{\frac{1}{r}} \\ &\quad= \biggl(\frac{1}{2\pi b} \biggr)^{\frac{r-1}{r}} \Vert f \Vert _{L^{2}( \mathbb{R})} \Vert g \Vert _{L^{2}(\mathbb{R})} \Vert \phi \Vert _{L^{2}(\mathbb{R})} \Vert \psi \Vert _{L^{2}(\mathbb{R})}, \end{aligned}$$
(47)
which completes the proof of Theorem 5.3 as desired. □
As an easy consequence of Theorem 5.3 mentioned above, we get the following result.
Corollary 5.4
Let \(\phi, \psi \) be two window functions. Then for every \(f \in L^{2}(\mathbb{R})\) with \(r\in [2,\infty )\), one has
$$\begin{aligned} \biggl( \int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert G_{\phi }^{B} f(w,v) \bigr\vert ^{r} \,dw \,dv \biggr)^{\frac{1}{r}} \leq \biggl( \frac{1}{2\pi b} \biggr)^{\frac{\frac{r}{2}-1}{r}} \Vert f \Vert _{L^{2}(\mathbb{R})} \Vert \phi \Vert _{L^{2}( \mathbb{R})}. \end{aligned}$$
(48)
Proof
For \(r=\infty \), we have
$$\begin{aligned} \bigl\vert G_{\phi }^{B} f(w,v) \bigr\vert \leq \frac{1}{2\pi b} \Vert f \Vert _{L^{2}(\mathbb{R})} \Vert \phi \Vert _{L^{2}(\mathbb{R})}. \end{aligned}$$
(49)
For all \(r \in [1, \infty )\), it holds
$$\begin{aligned} \biggl( \int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert G_{\phi }^{B} f(w,v) \bigr\vert ^{2r} \,dw \,dv \biggr)^{\frac{1}{r}} \leq \biggl( \frac{1}{2\pi b} \biggr)^{\frac{r-1}{r}} \Vert f \Vert ^{2}_{L^{2}(\mathbb{R})} \Vert \phi \Vert ^{2}_{L^{2}( \mathbb{R})}. \end{aligned}$$
(50)
Now putting \(s=2r\in [2,\infty )\) yields
$$\begin{aligned} \biggl( \int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert G_{\phi }^{B} f(w,v) \bigr\vert ^{s} \,dw \,dv \biggr)^{\frac{2}{s}} \leq \biggl( \frac{1}{2\pi b} \biggr)^{\frac{s-2}{s}} \Vert f \Vert ^{2}_{L^{2}(\mathbb{R})} \Vert \phi \Vert ^{2}_{L^{2}( \mathbb{R})}. \end{aligned}$$
(51)
Hence,
$$\begin{aligned} \biggl( \int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert G_{\phi }^{B} f(w,v) \bigr\vert ^{s} \,dw \,dv \biggr)^{\frac{1}{s}} \leq \biggl( \frac{1}{2\pi b} \biggr)^{\frac{\frac{s}{2}-1}{s}} \Vert f \Vert _{L^{2}(\mathbb{R})} \Vert \phi \Vert _{L^{2}( \mathbb{R})}. \end{aligned}$$
(52)
We thus finish the proof. □
Below, we obtain an immediate generalization of Theorem 5.2 in the following form.
Theorem 5.5
With the notations of Theorem 5.2. For all \(r>2\), the following inequality holds
$$\begin{aligned} \mu (V) \geq (1-\epsilon )^{\frac{r}{(r-2)}} (2\pi b). \end{aligned}$$
(53)
Proof
Applying the Hölder inequality, we easily obtain
$$\begin{aligned} 1-\epsilon&\leq \int \int _{V} \bigl\vert G_{\phi }^{B} f(w,v) \bigr\vert ^{2} \,dw \,dv \\ &\leq \biggl( \int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert G_{\phi }^{B} f(w,v) \bigr\vert ^{2 .\frac{r}{2}} \,dw \,dv \biggr)^{\frac{2}{r}} \biggl( \int _{\mathbb{R}} \int _{\mathbb{R}} \chi _{V}(w, v)^{\frac{r}{r-2}} \,dw \,dv \biggr)^{ \frac{r-2}{r}}, \end{aligned}$$
(54)
where \(\chi _{V}\) denotes the indicator function of the set V. Substituting relation (48) into the first term in the right-hand side of the equation (54), we see that
$$\begin{aligned} 1-\epsilon &\leq \biggl( \biggl(\frac{1}{2\pi b} \biggr)^{ \frac{\frac{r}{2}-1}{r}} \Vert f \Vert _{L^{2}(\mathbb{R})} { \Vert \phi \Vert _{L^{2}(\mathbb{R})}} \biggr)^{2} \bigl(\mu (V) \bigr)^{ \frac{r-2}{r}}. \end{aligned}$$
(55)
For all \(r>2\), we get
$$\begin{aligned} 1-\epsilon &\leq \biggl(\frac{1}{2\pi b} \biggr)^{\frac{r-2}{r}} \bigl(\mu (V) \bigr)^{\frac{r-2}{r}}. \end{aligned}$$
(56)
After simplifying the above relation, we get
$$\begin{aligned} \mu (V) \geq (1-\epsilon )^{\frac{r}{(r-2)}} (2\pi b). \end{aligned}$$
(57)
This proves the claim. □
Definition 5.1
A function \(f \in L^{2}(\mathbb{R})\) is said to be the ϵ-concentrated on a measurable set \(X \subseteq \mathbb{R}\) if for any \(\epsilon > 0\), it holds
$$\begin{aligned} \biggl( \int _{\mathbb{R}\setminus X} \bigl\vert f(x) \bigr\vert ^{r} \,dx \biggr)^{1/r} \leq \epsilon \Vert f \Vert _{L^{r}(\mathbb{R})},\quad 0< r\leq \infty. \end{aligned}$$
(58)
Below, based on Definition 5.1 above, we immediately obtain the alternative form of Theorem 5.5.
Theorem 5.6
Let ϕ be a complex window function, and \(f \in L^{2}(\mathbb{R})\). If \(G_{\phi }^{B} f\) is the ϵ-concentrated on \(L^{2}\)-norm on measurable set X of \(\mathbb{R}\), then for all \(r>2\), one has
$$\begin{aligned} \mu (V) \geq \bigl(1-\epsilon ^{2}\bigr)^{\frac{r}{(r-2)}} (2\pi b). \end{aligned}$$
(59)
Proof
For simplicity, we first assume that \(\|f \|^{2}_{L^{2}(\mathbb{R})} = \|\phi \|^{2}_{L^{2}(\mathbb{R})} =1\). Invoking (58), we get
$$\begin{aligned} \int _{\mathbb{R}\setminus X} \int _{\mathbb{R}\setminus X} \bigl\vert G_{\phi }^{B} f( \omega,u) \bigr\vert ^{2} \,d\omega \,du \leq \epsilon ^{2} \int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert G_{\phi }^{B} f( \omega,u) \bigr\vert ^{2} \,d\omega \,du. \end{aligned}$$
(60)
An application of relation (8) together with relation (16) will lead to
$$\begin{aligned} \int _{\mathbb{R}\setminus X} \int _{\mathbb{R}\setminus X} \bigl\vert G_{\phi }^{B} f( \omega,u) \bigr\vert ^{2} \,d\omega \,du &\leq \epsilon ^{2} \int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert \mathcal{C}_{B} \{ f T_{u} \bar{\phi } \} \bigr\vert ^{2} \,d\omega \,du \\ &= \epsilon ^{2} \Vert f \Vert ^{2}_{L^{2}(\mathbb{R})} \Vert \phi \Vert ^{2}_{L^{2}( \mathbb{R})} \\ &= \epsilon ^{2}. \end{aligned}$$
Hence,
$$\begin{aligned} \int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert G_{\phi }^{B} f( \omega,u) \bigr\vert ^{2} \,d\omega \,du \leq \int _{X} \int _{X} \bigl\vert G_{\phi }^{A} f( \omega,u) \bigr\vert ^{2} \,d\omega \,du + \epsilon ^{2}. \end{aligned}$$
(61)
Applying equation (20) and the Hölder inequality, we obtain
$$\begin{aligned} 1-\epsilon ^{2} &\leq \int _{X} \int _{X} \bigl\vert G_{\phi }^{B} f( \omega,u) \bigr\vert ^{2} \,d\omega \,du \\ &\leq \biggl( \int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert G_{\phi }^{B} f( \omega,u) \bigr\vert ^{2 \frac{r}{2}} \,d\omega \,du \biggr)^{\frac{2}{r}} \bigl( \mu (X) \bigr)^{\frac{r-2}{r}} \\ &\leq \biggl(\frac{1}{2\pi b} \biggr)^{\frac{r-2}{r}} \bigl(\mu (X) \bigr)^{\frac{r-2}{r}}, \end{aligned}$$
which completes the proof of the theorem. □
Next, we present a lemma, which describes the basic concept of Nazarovs’s uncertainty principle for the Fourier transform (see [18]).
Lemma 5.7
Suppose that \(X_{1}\) and \(X_{2}\) are two finite, measurable subsets of \(\mathbb{R}\). Then for every \(f \in L^{2}(\mathbb{R})\) there exists a constant \(C>0\) such that
$$\begin{aligned} \int _{\mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \,dx \leq C e^{C\mu (X_{1}) \mu (X_{2})} \biggl( \int _{\mathbb{R}\setminus X_{1}} \bigl\vert f(x) \bigr\vert ^{2} \,dx + \int _{ \mathbb{R}\setminus X_{2}} \bigl\vert F\{f\}(w) \bigr\vert ^{2} \,dw \biggr). \end{aligned}$$
(62)
We are ready to obtain a straightforward generalization of Nazarovs’s uncertainty principle in the framework of the WLCT.
Theorem 5.8
With the notations of Lemma 5.7above, if \(\phi \in L^{2}(\mathbb{R})\), then
$$\begin{aligned} & \Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \,dx \\ &\quad \leq C e^{C\mu (X_{1}) \mu (X_{2})} \\ &\qquad{}\times \biggl( \Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}\setminus X_{1}} \bigl\vert f(x) \bigr\vert ^{2} \,dx + 2\pi \int _{\mathbb{R}} \int _{\mathbb{R}\setminus b X_{2}} \bigl\vert G_{ \phi }^{B} f(w,v) \bigr\vert ^{2} \,dw \,dv \biggr). \end{aligned}$$
(63)
Proof
Replacing \(f(x)\) by \(e^{\frac{ia}{2b}x^{2}} f T_{v} \bar{\phi }(x)\) on both sides of (62), we obtain
$$\begin{aligned} &\int _{\mathbb{R}} \bigl\vert e^{\frac{ia}{2b}x^{2}}f T_{v} \bar{\phi }(x) \bigr\vert ^{2} \,dx \\ &\quad \leq C e^{C\mu (X_{1}) \mu (X_{2})} \\ &\qquad {}\times \biggl( \int _{\mathbb{R}\setminus X_{1}} \bigl\vert e^{\frac{ia}{2b}x^{2}} f T_{v} \bar{\phi }(x) \bigr\vert ^{2} \,dx + \int _{\mathbb{R}\setminus X_{2}} \bigl\vert F \bigl\{ e^{\frac{ia}{2b}x^{2}}f T_{v} \bar{\phi }\bigr\} (w) \bigr\vert ^{2} \,dw \biggr). \end{aligned}$$
(64)
Hence,
$$\begin{aligned} &\Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \,dx \\ &\quad \leq C e^{C\mu (X_{1}) \mu (X_{2})} \\ &\qquad{}\times \biggl( \Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}\setminus X_{1}} \bigl\vert f(x) \bigr\vert ^{2} \,dx + \int _{\mathbb{R}} \int _{\mathbb{R}\setminus X_{2}} \bigl\vert F \bigl\{ e^{\frac{ia}{2b}x^{2}} f T_{v} \bar{\phi } \bigr\} (w) \bigr\vert ^{2} \,dw \,dv \biggr). \end{aligned}$$
(65)
This further leads to
$$\begin{aligned} &\Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \,dx \\ &\quad \leq C e^{C\mu (X_{1}) \mu (X_{2})} \\ &\qquad{}\times \biggl( \Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}\setminus X_{1}} \bigl\vert f(x) \bigr\vert ^{2} \,dx + \frac{1}{b} \int _{\mathbb{R}} \int _{\mathbb{R} \setminus b X_{2}} \biggl\vert F\bigl\{ e^{\frac{ia}{2b}x^{2}} f T_{v} \bar{\phi } \bigr\} { \biggl(\frac{w}{b} \biggr)} \biggr\vert ^{2} \,dw \,dv \biggr). \end{aligned}$$
(66)
Applying relation (13), we see that
$$\begin{aligned} &\Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \,dx \\ &\quad \leq C e^{C\mu (X_{1}) \mu (X_{2})} \\ &\qquad{}\times \biggl( \Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}\setminus X_{1}} \bigl\vert f(x) \bigr\vert ^{2} \,dx \\ &\qquad{}+ \frac{1}{b} \int _{\mathbb{R}} \int _{\mathbb{R} \setminus b X_{2}} \bigl\vert \sqrt{2\pi b} e^{i\frac{\pi }{4}} e^{-i \frac{d}{2b}w^{2}} G_{\phi }^{B} f(w,v) \bigr\vert ^{2} \,dw \,dv \biggr), \end{aligned}$$
(67)
and the required result follows. □
The following results concern some consequences of Nazarov’s uncertainty principles described by equation (62).
Corollary 5.9
Using the notations as in Lemma 5.7, we have
$$\begin{aligned} \int _{\mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \,dx \leq C e^{C\mu (X_{1}) \mu (X_{2})} \biggl( \int _{\mathbb{R}\setminus X_{1}} \bigl\vert f(x) \bigr\vert ^{2} \,dx + 2\pi \int _{ \mathbb{R}\setminus bX_{2}} \bigl\vert \mathcal{C}_{B}\{ f \} (w) \bigr\vert ^{2} \,dw \biggr), \end{aligned}$$
(68)
which is Nazarovs’s uncertainty principle in the context of the LCT.
Proof
Including \(f(x)\) as \(e^{\frac{ia}{2b}x^{2}}f(x)\) on both sides of (62) and then implementing (5) will lead to the desired result. □
The above corollary is consistent with the result studied in [19], where, in this case, the LCT is a particular case of the offset linear canonical transform.
Corollary 5.10
Under the assumptions of Lemma 5.7, if \(\phi \in L^{2}(\mathbb{R})\), then one has
$$\begin{aligned} &\Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \,dx \\ &\quad \leq C e^{C\mu (X_{1}) \mu (X_{2})} \\ &\qquad {}\times\biggl( \Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}\setminus X_{1}} \bigl\vert f(x) \bigr\vert ^{2} \,dx + 2\pi \int _{\mathbb{R}} \int _{\mathbb{R}\setminus X_{2}} \bigl\vert G_{\phi } f(w,v) \bigr\vert ^{2} \,dw \,dv \biggr), \end{aligned}$$
(69)
which is Nazarovs’s uncertainty principle associated with the WFT defined by (14).
Proof
By (15), the proof is straightforward. □
Theorem 5.11
(WLCT local uncertainty principle)
Let \(\phi \in L^{2}(\mathbb{R})\) be a window function. If \(0 <\alpha <\frac{1}{2}\), then there exists some constant \(C_{\alpha }\) such that for any \(f\in L^{2}(\mathbb{R})\) and \(X\subseteq \mathbb{R}\) measurable, one has
$$\begin{aligned} \int _{\mathbb{R}} \int _{bX} \bigl\vert G_{\phi }^{B} f(w,v) \bigr\vert ^{2} \,dw \,dv \leq \frac{C_{\alpha } \Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} }{2\pi } \bigl[\mu (X) \bigr]^{2 \alpha } \int _{\mathbb{R}} \vert x \vert ^{2\alpha } \bigl\vert f(x) \bigr\vert ^{2} \,dx, \end{aligned}$$
(70)
and for \(\alpha > \frac{1}{2}\), it holds
$$\begin{aligned} 2\pi \int _{\mathbb{R}} \int _{bX} \bigl\vert G_{\phi }^{B} f(w,v) \bigr\vert ^{2} \,dw \,dv \leq{}& C_{\alpha } \mu (X) \Vert \phi \Vert _{L^{2}(\mathbb{R})}^{ \frac{2\alpha -1}{\alpha }} \biggl( \int _{\mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \,dx \biggr)^{\frac{2\alpha -1}{2\alpha }} \\ & {}\times \Vert \phi \Vert _{L^{2}(\mathbb{R})}^{\frac{1}{\alpha }} \biggl( \int _{ \mathbb{R}} \bigl\vert \vert x \vert ^{\alpha } f(x) \bigr\vert ^{2} \,dx \biggr)^{ \frac{1}{2\alpha }}, \end{aligned}$$
(71)
Proof
Let a set \(X\subseteq \mathbb{R}\) with finite measure. The local uncertainty principle for the Fourier transform is expressed as (see [20])
$$\begin{aligned} \int _{X} \bigl\vert F\{f\}(w) \bigr\vert ^{2} \,dw \leq C_{\alpha } \bigl[\mu (X)\bigr]^{2\alpha } \bigl\Vert \vert x \vert ^{ \alpha } f \bigr\Vert ^{2}_{L^{2}(\mathbb{R})}. \end{aligned}$$
(72)
In fact, by inserting \(f(x)\) as \(f T_{v} \bar{\phi }(x)\) on both sides of the relation (72) above, we get
$$\begin{aligned} \int _{X} \bigl\vert F\{f T_{v} \bar{\phi }\}(w) \bigr\vert ^{2} \,dw \leq C_{\alpha } \bigl[ \mu (X) \bigr]^{2\alpha } \bigl\Vert \vert x \vert ^{\alpha } f T_{v} \bar{\phi } \bigr\Vert ^{2}_{L^{2}( \mathbb{R})}. \end{aligned}$$
(73)
Furthermore,
$$\begin{aligned} &\int _{\mathbb{R}} \int _{X} \bigl\vert F\bigl\{ e^{\frac{ia}{2b}x^{2}}f T_{v} \bar{\phi }\bigr\} (w) \bigr\vert ^{2} \,dw \,dv \\ &\quad \leq C_{\alpha } \bigl[\mu (X)\bigr]^{2\alpha } \int _{ \mathbb{R}} \int _{\mathbb{R}} \bigl\vert \vert x \vert ^{\alpha } e^{\frac{ia}{2b}x^{2}} f(x) \overline{\phi (x -v)} \bigr\vert ^{2} \,dx \,dv. \end{aligned}$$
(74)
Setting \(w = \frac{w}{b}\), we obtain
$$\begin{aligned} &\frac{1}{b} \int _{\mathbb{R}} \int _{bX} \biggl\vert F\bigl\{ e^{\frac{ia}{2b}x^{2}} f T_{v} \bar{\phi }\bigr\} \biggl(\frac{w}{b}\biggr) \biggr\vert ^{2} \,dw \,dv \\ &\quad \leq C_{\alpha } \bigl[\mu (X) \bigr]^{2 \alpha } \int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert \vert x \vert ^{\alpha } f(x) \overline{\phi (x -v)} \bigr\vert ^{2} \,dx \,dv. \end{aligned}$$
(75)
It follows from the expression (13) that
$$\begin{aligned} 2\pi \int _{\mathbb{R}} \int _{bX} \bigl\vert G_{\phi }^{B} f(w,v) \bigr\vert ^{2} \,dw \,dv \leq C_{\alpha } \bigl[\mu (X) \bigr]^{2\alpha } \Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}} \vert x \vert ^{2\alpha } \bigl\vert f(x) \bigr\vert ^{2} \,dx, \end{aligned}$$
(76)
which gives the proof of (70). Based on the local uncertainty principle for the Fourier transform
$$\begin{aligned} \int _{X} \bigl\vert F\{f\}(w) \bigr\vert ^{2} \,dw \leq C_{\alpha } \mu (X) \Vert f \Vert _{L^{2}( \mathbb{R})}^{2-\frac{1}{\alpha }} \bigl\Vert \vert x \vert ^{\alpha } f \bigr\Vert _{L^{2}( \mathbb{R})}^{\frac{1}{\alpha }}, \end{aligned}$$
(77)
we apply the same arguments to get equation (71). □
Recently, Kubo et al. [21] have proposed the logarithmic Sobolev-type uncertainty principle for the Fourier transform. Below, we search this uncertainty principle in the setting of the WLCT. For this purpose, we present the following.
Definition 5.2
For \(1 \leq r < \infty \) and \(s>0\) define the weighted Lebesgue space as
$$\begin{aligned} \mathcal{W}_{s}^{r}(\mathbb{R}) = \bigl\{ f\in L^{r}(\mathbb{R}): \langle x\rangle ^{s} \in L^{r}(\mathbb{R}) \bigr\} , \end{aligned}$$
(78)
where \(\langle x\rangle = (1 +x^{2})^{\frac{1}{2}}\) is the weight function.
We have the following.
Theorem 5.12
Let \(\phi \in L^{2}(\mathbb{R})\) be a window function. For every \(f\in \mathcal{S}(\mathbb{R})\cap \mathcal{W}_{1}^{2}(\mathbb{R})\), we have
$$\begin{aligned} &\Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \ln \biggl(\frac{1+ \vert x \vert ^{2}}{2} \biggr) \,dx + 2\pi \int _{\mathbb{R}} \int _{ \mathbb{R}} \ln \vert w \vert \bigl\vert G_{\phi }^{B} f(w,v) \bigr\vert ^{2} \,dw \,dv \\ &\quad \geq \biggl( \frac{\Gamma ^{\prime } ( 1/2 )}{\Gamma ( 1/2 )} + 2 \pi \ln b \biggr) \Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \,dx. \end{aligned}$$
(79)
In this case, \(\mathcal{S}(\mathbb{R})\) is the Sobolev space on \(\mathbb{R}\) defined by
$$\begin{aligned} \mathcal{S}(\mathbb{R}) = \bigl\{ f\in L^{2}(\mathbb{R}): Df \in L^{2}( \mathbb{R}) \bigr\} , \end{aligned}$$
where D stands for the differential operator, and Γ indicates the Gamma function.
Proof
From the logarithmic Sobolev-type uncertainty principle for the Fourier transform, we obtain that (see [21])
$$\begin{aligned} \int _{\mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \ln \biggl(\frac{1+ \vert x \vert ^{2}}{2} \biggr) \,dx + \int _{\mathbb{R}} \ln \vert w \vert \bigl\vert F\{f\}(w) \bigr\vert ^{2} \,dw \geq \biggl( \frac{\Gamma ^{\prime } ( 1/2 )}{\Gamma ( 1/2 )} \biggr) \int _{\mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \,dx. \end{aligned}$$
(80)
Now replacing \(f(x)\) by \(e^{\frac{ia}{2b}x^{2}} f T_{v} \bar{\phi }(x)\) on both sides of identity (80) yields
$$\begin{aligned} &\int _{\mathbb{R}} \bigl\vert e^{\frac{ia}{2b}x^{2}} f T_{v} \bar{\phi }(x) \bigr\vert ^{2} \ln \biggl(\frac{1+ \vert x \vert ^{2}}{2} \biggr) \,dx + \int _{\mathbb{R}} \ln \vert w \vert \bigl\vert F\bigl\{ e^{\frac{ia}{2b}x^{2}} f T_{v} \bar{\phi } \bigr\} (w) \bigr\vert ^{2} \,dw \\ &\quad\geq \biggl( \frac{\Gamma ^{\prime } ( 1/2 )}{\Gamma ( 1/2 )} \biggr) \int _{\mathbb{R}} \bigl\vert e^{\frac{ia}{2b}x^{2}}f T_{v} \bar{\phi }(x) \bigr\vert ^{2} \,dx. \end{aligned}$$
(81)
Hence,
$$\begin{aligned} &\int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert f T_{v} \bar{\phi }(x) \bigr\vert ^{2} \ln \biggl(\frac{1+ \vert x \vert ^{2}}{2} \biggr) \,dx \,dv + \int _{\mathbb{R}} \int _{\mathbb{R}} \ln \vert w \vert \bigl\vert F\bigl\{ e^{\frac{ia}{2b}x^{2}}f T_{v} \bar{\phi } \bigr\} (w) \bigr\vert ^{2} \,dw \,dv \\ &\quad \geq \biggl( \frac{\Gamma ^{\prime } ( 1/2 )}{\Gamma ( 1/2 )} \biggr) \int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert f T_{v} \bar{\phi }(x) \bigr\vert ^{2} \,dx\,dv. \end{aligned}$$
(82)
This implies that
$$\begin{aligned} &\Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \ln \biggl(\frac{1+ \vert x \vert ^{2}}{2} \biggr) \,dx + \frac{1}{b} \int _{\mathbb{R}} \int _{\mathbb{R}} \ln \biggl\vert \frac{w}{b} \biggr\vert \biggl\vert F\bigl\{ e^{ \frac{ia}{2b}x^{2}}f T_{v} \bar{\phi } \bigr\} \biggl(\frac{w}{b}\biggr) \biggr\vert ^{2} \,dw \,dv \\ &\quad \geq \biggl( \frac{\Gamma ^{\prime } ( 1/2 )}{\Gamma ( 1/2 )} \biggr) \Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \,dx. \end{aligned}$$
(83)
Applying (13) gives
$$\begin{aligned} &\Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \ln \biggl(\frac{1+ \vert x \vert ^{2}}{2} \biggr) \,dx + 2\pi \int _{\mathbb{R}} \int _{ \mathbb{R}} \ln \biggl\vert \frac{w}{b} \biggr\vert \bigl\vert G_{\phi }^{B} f(w,v) \bigr\vert ^{2} \,dw \,dv \\ & \quad\geq \biggl( \frac{\Gamma ^{\prime } ( 1/2 )}{\Gamma ( 1/2 )} \biggr) \Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \,dx. \end{aligned}$$
(84)
According to (20), we deduce that
$$\begin{aligned} &\Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \ln \biggl(\frac{1+ \vert x \vert ^{2}}{2} \biggr) \,dx + 2\pi \int _{\mathbb{R}} \int _{ \mathbb{R}} \ln \vert w \vert \bigl\vert G_{\phi }^{B} f(w,v) \bigr\vert ^{2} \,dw \,dv \\ &\quad \geq \biggl( \frac{\Gamma ^{\prime } ( 1/2 )}{\Gamma ( 1/2 )} + 2 \pi \ln b \biggr) \Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \,dx, \end{aligned}$$
and the proof is complete. □
As an immediate consequence of the above theorem, we obtain the following (see [7]).
Corollary 5.13
Let \(\phi \in L^{2}(\mathbb{R})\) be a window function. For every \(f\in \mathcal{S}(\mathbb{R})\), we have
$$\begin{aligned} &\Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}} \ln \vert x \vert \bigl\vert f(x) \bigr\vert ^{2} \,dx + \int _{\mathbb{R}} \int _{\mathbb{R}} \ln \vert w \vert \bigl\vert G_{\phi }^{B} f(w,v) \bigr\vert ^{2} \,dw \,dv \\ & \quad\geq (D + \ln b ) \Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{ \mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \,dx, \end{aligned}$$
(85)
where \(D = \Phi (\frac{1}{2})-\ln \pi, \Phi (x) = \frac{d}{dx} \ln [ \Gamma (x)]\).