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The present study introduces the two-sided and right-sided Quaternion Hyperbolic Fourier Transforms (QHFTs) for analyzing two-dimensional quaternion-valued signals defined in an open rectangle of the Euclidean plane endowed with a hyperbolic measure. The different forms of these transforms are defined by replacing the Euclidean plane waves with the corresponding hyperbolic plane waves in one dimension, giving the hyperbolic counterpart of the corresponding Euclidean Quaternion Fourier Transforms. Using hyperbolic geometry tools, we study the main operational and mapping properties of the QHFTs, such as linearity, shift, modulation, dilation, symmetry, inversion, and derivatives. Emphasis is placed on novel hyperbolic derivative and hyperbolic primitive concepts, which lead to the differentiation and integration properties of the QHFTs. We further prove the Riemann–Lebesgue Lemma and Parseval’s identity for the two-sided QHFT. Besides, we establish the Logarithmic, Heisenberg–Weyl, Donoho–Stark, and Benedicks’ uncertainty principles associated with the two-sided QHFT by invoking hyperbolic counterparts of the convolution, Pitt’s inequality, and the Poisson summation formula. This work is motivated by the potential applications of the QHFTs and the analysis of the corresponding hyperbolic quaternionic signals.

Weighted (Lp,Lq) inequalities are studied for a variety of integral transforms of Fourier type. In particular, weighted norm inequalities for the Fourier, Hankel, and Jacobi transforms are derived from Calderón-type rearrangement estimates. The obtained results keep their novelty even in the simplest cases of the studied transforms, the cosine and sine Fourier transforms. Sharpness of the conditions on weights is discussed.

The aim of this paper is to prove a generalization of uncertainty principles for the continuous wavelet transform connected with the Riemann–Liouville operator in L p -norm. More precisely, we establish the Heisenberg–Pauli–Weyl uncertainty principle, Donoho–Stark’s uncertainty principles and local Cowling-Price’s type inequalities. Finally, Pitt’s inequality and Beckner’s uncertainty principle are proved for this transform.

The linear canonical wavelet transform is a nontrivial generalization of the classical wavelet transform in the context of the linear canonical transform. In this article, we first present a direct interaction between the linear canonical transform and Fourier transform to obtain the generalization of the uncertainty principles related to the linear canonical transform. We develop these principles for constructing some uncertainty principles concerning the linear canonical wavelet transform.

In this paper, we prove the sharp Pitt’s inequality for a generalized Clifford-Fourier transform which is given by a similar operator exponential as the classical Fourier transform but containing generators of Lie superalgebra. As an application, the Beckner’s logarithmic uncertainty principle for the Clifford-Fourier transform is established.

In this paper, we prove the sharp Pitt’s inequality for the Weinstein transform. As an application, the Beckner’s logarithmic uncertainty principle for the Weinstein transform is established.

In this work, we introduce the quaternion linear canonical S-transform, which is a generalization of the linear canonical S-transform using quaternion. We investigate its properties and seek the different types of uncertainty principles related to this transformation. The obtained results can be looked as an extension of the uncertainty principles for the quaternion linear canonical transform and the quaternion windowed linear canonical transform.

A novel representation is developed as a measure for multilinear fractional embedding. Corresponding extensions are given for the Bourgain-Brezis-Mironescu theorem and Pitt＇s inequal- ity. New results are obtained for diagonal trace restriction on submanifolds as an application of the Hardy-Littlewood-Sobolev inequality. Smoothing estimates are used to provide new structural un- derstanding for density functional theory, the Coulomb interaction energy and quantum mechanics of phase space. Intriguing connections are drawn that illustrate interplay among classical inequalities in Fourier analysis.

In the present paper, we prove weighted inequalities for the Dunkl transform (which generalizes the Fourier transform) when the weights belong to the well-known class B . As application, we obtain the Pitt’s inequality for power weights.

The aim of this paper is to define and study the Stockwell transform associated with the Riemann–Liouville operator and we present some interesting harmonic analysis results. We establish the Pitt’s and Beckner logarithmic inequalities related to this transform.