Amri Besma

University of Tunis El Manar | FST · Mathematics and Computer Science Department

We consider the Bessel–Kingman hypergroup ([0,+∞[,∗) and we denote by dμα,α⩾-12,the measure defined on [0,+∞[ by dμα(x)=x2+12αΓ(α+1)dx.We define the Gabor multiplier Gu,v(σ) associated with two square integrable functions u, v on [0,+∞[ with respect to the measure dμα called window functions and σ is a measurable function on [0,+∞[ called a signal....

For every real number p > 0, we define the p-dispersion ρ p,να (f) of a measurable function f on [0,+∞[×R, where ν α is some positive measure. We prove that for every orthonormal basis (ϕ m,n ) (m,n)∈N 2 of L ² (dν α ), the sequences (ρ p,να (ϕ m,n )) , ( (m,n)∈ N ² (ρ p,να (Fα (ϕ m,n ))) (m,n)∈ N2can not be simultaneously bounded, where F α is som...

We define and study the windowed Fourier transform, called also the Gabor transform, associated with singular partial differential operators defined on the half plane ]0,+∞[×R . We prove a Plancherel theorem and an inversion formula that we use to establish the classical Heisenberg uncertainty principle. Next, we study this transform on subsets of...

We define and study the windowed Fourier transform, called also the Gabor transform, associated with singular partial differential operators defined on the half plane ]0,+∞[×R . We prove a Plancherel theorem and an inversion formula that we use to establish the classical Heisenberg uncertainty principle. Next, we study this transform on subsets of...

We define the Hardy–Littlewood operator Hα associated with the Riemann–Liouville transform defined on the half plane [0,+∞[×R. We study the boundedness of this operator on the Lebesgue space Lp(dνα), where dνα is the measure defined on [0,+∞[×R by dνα(r,x)=r2α+1dr2αΓ(α+1)⊗dx2π.We prove that for every p∈]1,+∞], the Hardy–Littlewood operator Hα is of...

We define wavelets and wavelet transforms associated with spherical mean operator. We establish a Plancherel theorem, orthogonality property and inversion formula for the wavelet transform. Next, we define the Toeplitz operators \(\mathfrak {T}_{\varphi ,\psi }(\sigma )\) associated with two wavelets \(\varphi ,\psi \) and with symbol \(\sigma .\)...

We study the Gauss and Poisson semigroups connected with the Riemann-Liouville operator defined on the half plane. Next, we establish a principle of maximum for the singular partial differential operator Later, we define the Littlewood-Paley g-function and using the principle of maximum, we prove that for every p ε ]1;+∞[there exists a positive con...

We prove Hausdorff-Young inequality for the Fourier transform connected with Riemann-Liouville operator. We use this inequality to establish the uncertainty principle in terms of entropy. Next, we show that we can derive the Heisenberg-Pauli-Weyl inequality for the precedent Fourier transform.

We introduce the translation operators and convolution product connected with some singular partial differential operators defined on the half plane [0,+∞[×ℜ. We investigate Calderon's reproducing formula associated with the convolution * involving finite Borel measures, leading to results on L p -norm for functions on half plane.

First, we establish the Stein–Weiss inequality for the B-Riesz potential generated by the Riemann–Liouville operator. Next, we prove the Pitt's and Beckner logarithmic inequalities related to the connected Fourier transform.

First, we study the Gauss and Poisson semigroups connected with the Riemann-Liouville operator. Next, we dene and study the Littlewood-Paley g-function associated with the Riemann-Liouville operator for which we prove the Lp-boundedness for p ∈ ]1, 2].