In this work, we establish \(L^p\) local uncertainty principle for the Hankel-Stockwell transform and we deduce \(L^p\) version of Heisenberg-Pauli-Weyl uncertainty principle. Next, By combining these principles and the techniques of Donoho-Stark we present uncertainty principles of concentration type in the \(L^p\) theory, when \(1\)<\(p\leqslant2\). Finally, Pitt’s inequality and Beckner’s uncertainty principle are proved for this transform.
In harmonic analysis, uncertainty principles play an important role. It states that a non-zero function and its Fourier transform cannot be simultaneously sharply concentrated. many of them have already been studied from several points of view for the Fourier transform, Heisenberg-Pauli-Weyl inequality [1] and local uncertainty inequality [2]. As a classical uncertainty principle, the Heisenberg uncertainty principle has been extended to transforms such as the spherical mean transforms [3,4], the Dunkl transform [5] and so forth.
The Hankel transform \(\mathcal{H}_\alpha\) is defined for every integrable function \(f\) on \(\mathbb{R}_+=[0,+\infty[\) with respect to the measure \(d\nu_\alpha\), by
\begin{equation*} \mathcal{H}_\alpha(f)(\lambda)=\int_0^{+\infty}f(x)j_\alpha(\lambda x)d\nu_\alpha(x),\end{equation*} where \(d\nu_\alpha\) is the measure defined on \(\mathbb{R}_+ \) by \[d\nu_\alpha(x)=\frac{x^{2\alpha+1}}{2^{\alpha}\Gamma(\alpha+1)}dx,\] and \(j_\alpha\) is the modified Bessel function given in the next section.The Hankel transform is found as a very useful mathematical tool in many fields of physics, signal processing and other [6,7]. Also, many uncertainty principles related to this transform \(\mathcal{H}_\alpha\) have been proved [8,9,10].
Time-frequency analysis plays an important role in harmonic analysis, in particular in signal theory. With the development of time-frequency analysis, the study of uncertainty principles have gained considerable attention and have been extended to a wide class of integral transforms such as Weinstein transforms [11,12], Dunkl transforms [13], Hankel-Stockwell transforms [14] and so on.
Based on the ideas of Faris [15] and Price [2,16], we show a general form of the local uncertainty principles for the Hankel-Stockwell transform and we deduce \(L^p\) version of Heisenberg-Pauli-Weyl uncertainty principle. We shall use also the Heisenberg uncertainty principle, the properties of the Hankel-Stockwell transform and the techniques of Donoho-Stark [17,18], we show a continuous-time principle for the \(L^p\) theory, when \(1 < p \leqslant 2\). Finally, Pitt's inequality and Beckner's uncertainty principle are proved for this transform.
This work is organized as follows; in Section 2 we recall some harmonic analysis results related to the Hankel transform. In Section 3, we present some elements of harmonic analysis related to the Hankel-Stockwell transform. In Section 4, we introduce some uncertainty principles for this transform.
