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Ahmed A. Abdelhakim

Publications and source records attributed to Ahmed A. Abdelhakim.

13 recordsLinked to original sources
arXiv2026

A Littlewood-Paley approach to the Mittag-Leffler function in the frequency space and applications to nonlocal problems

Let $0<α<2$, $β>0$ and $α/2<|s|\leq 1$. In a previous work, we obtained all possible values of the Lebesgue exponent $p=p(γ)$ for which the Fourier transform of $ E_{α,β}(e^{\dot{\imath}πs} |\cdot|^γ )$ is an $L^{p}(\mathbb{R}^d)$ function, when $γ>(d-1)/2$. We recover the more interesting lower regularity case $0<γ\leq (d-1)/2$, using tools from the Littlewood-Paley theory. This question arises in the analysis of certain space-time fractional diffusion and Schrödinger problems and has been solved for the particular cases $α\in (0,1)$, $β=α,1$, and $s=-1/2,1$ via asymptotic analysis of Fox $H$-functions. The Littlewood-Paley theory provides a simpler proof that allows considering all values of $β,γ>0$ and $s\in (-1,1]\setminus [-α/2,α/2]$. This enabled us to prove various key estimates for a general class of nonlocal space-time problems.

math.CA↗
arXiv2026

On the asymptotic behaviour of the Fourier transform of the Mittag-Leffler function

Let $α\in (0,2)$ and let $β>0$. Fix $-π<φ\leq π$ such that $|φ|>απ/2$. We obtain asymptotic upper bounds on the Fourier transform of the radially symmetric tempered distribution \begin{equation*} \mathbb{R}^n\ni x\mapsto E_{α,β}(e^{\dot{\imath} φ} |x|^σ), \end{equation*} for $σ>(n-1)/2$, where $E_{α,β}$ is the two-parameter Mittag-Leffler function. As an application, we obtain some values of the Lebesgue exponent $p=p(σ)$, $σ>(n-1)/2$, for which the Fourier transform is in $L^{p}(\mathbb{R}^{n})$. Such values cannot be obtained via the well-known $L^{p}(\mathbb{R}^{n})$ properties of $E_{α,β}$ and the Hausdorff-Young inequality, when $σ\leq n/2$.

math.CA↗
arXiv2024

A characterization of a nonlinear integral triangle inequality

Let $(E,\|.\|)$ be a Banach space and let $(Ω,μ)$ be a Lebesgue measure space. We characterize, for all $p>0$, measurable functions $u:Ω\rightarrow \mathbb{R}$ for which \begin{equation*} \left\| \int_Ω f\,dμ\right\|^{p}\,\leq\,\int_Ω u \| f \|^{p}\,dμ.\tag{I} \end{equation*} We characterize $u$ for the reverse of (I) as well. The discrete counterpart of this problem is also solved.

math.FA↗
arXiv2024

A critical analysis of the conformable derivative

We prove that conformable ``fractional" differentiability of a function $f:[0,\infty[\,\longrightarrow \mathbb{R}$ is nothing else than the classical differentiability. More precisely, the conformable $α$-derivative of $f$ at some point $x>0$, where $0<α<1$, is the pointwise product $x^{1-α}f^{\prime}(x)$. This proves the lack of significance of recent studies of the conformable derivatives. The results imply that interpreting fractional derivatives in the conformable sense alters fractional differential problems into differential problems with the usual integer-order derivatives that no longer describe the original fractional physical phenomena. A general fractional viscoelasticity model is analysed to illustrate this state of affairs. We also test the modelling efficiency of the conformable derivative using a fractional model of viscoelastic deformation of tight sandstone, and a fractional world population growth model.

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arXiv2020

Optimal critical exponent $L^{p}$ inequalities of Hardy type on the sphere via Xiao's method

First, we correct the proof presented in [Abimbola Abolarinwa, Kamilu Rauf, Songting Yin, Sharp $L^{p}$ Hardy type and uncertainty principle inequalities on the sphere, Journal of Mathematical Inequalities, 13, 4 (2019), 1011 - 1022] and obtain a correct sharp version of an $L^{p}$ Hardy inequality on the sphere $\mathbb{S}^{n}$ for all $2\leq p<n$. Secondly, we prove sharp critical exponent $L^{n}$ inequalities on the sphere $\mathbb{S}^{n}$ in $\mathbb{R}^{n+1}$, $n\geq 2$. The singularity in this problem is the geodesic distance from an arbitrary point on the sphere.

math.FA↗
arXiv2020

Sharp subcritical and critical $L^{p}$ Hardy inequalities on the sphere

We prove sharp inequalities of Hardy type for functions in the Sobolev space $W^{1,p}$ on the unit sphere $\mathbb{S}^{n-1}$ in $\mathbb{R}^{n}$. We achieve this in both the subcritical and critical cases. The method we use to show optimality takes into account all the constants involved in our inequalities.

math.FA↗
arXiv2018

Precise interpretation of the conformable fractional derivative

Let $α\in\,]0,1[$. We prove that the existence of the conformable fractional derivative $T_αf$ of a function $f:[0,\infty[\,\longrightarrow \mathbb{R}$ introduced by Khalil et al. in [R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math. 264 (2014) 65-70] is equivalent to classical differentiability. Precisely the fractional $α$-derivative of $f$ is the pointwise product $T_αf(x)=x^{1-α}f^{\prime}(x)$, $x>0$. This simplifies the recent results concerning conformable fractional calculus.

math.CA↗
arXiv2018

On the boundary terms in Hardy's inequalities for $W^{1,p}$ functions

With the help of a radially invariant vector field, we derive inequalities of the Hardy kind, with no boundary terms, for $W^{1,p}$ functions on bounded star domains. Our results are not obtainable from the classical inequalities for $W^{1,p}_{0}$ functions. Unlike in $W^{1,p}_{0}$, our inequalities admit maximizers that we describe explicitly.

math.AP↗
arXiv2017

Limiting case Hardy inequalities on the sphere

We give sharp limiting case Hardy inequalities on the sphere $\mathbb{S}^{2}$ and show that their optimal constants are unattainable by any $f\in H^{1}\left(\mathbb{S}^{2}\right)\setminus\{0\}$. The singularity of the problem is related to the geodesic distance from a point on the sphere.

math.AP↗
arXiv2015

A convexity of functions on convex metric spaces of Takahashi and applications

We quickly review and make some comments on the concept of convexity in metric spaces due to Takahashi. Then we introduce a concept of convex structure based convexity to functions on these spaces and refer to it as $W-$convexity. $W-$convex functions generalize convex functions on linear spaces. We discuss illustrative examples of (strict) $ W-$convex functions and dedicate the major part of this paper to proving a variety of properties that make them fit in very well with the classical theory of convex analysis. Finally, we apply some of our results to the metric projection problem and fixed point theory.

math.FA↗
arXiv2015

On the summablility of truncated double Fourier series

We estimate the truncated double trigonometric series $\sum_{n=0}^{N}\sum_{m=0}^{M}a_{mn} {e}^{2π\imath \left(m x+n y\right)}$, $a_{mn} \in\mathbb{C}$, in Lebesgue spaces with mixed norms in terms of the $p^{th}-q^{th}$ power finite double sums of its coefficients. We obtain these estimates for all possible values of the exponents involved then we provide examples of matrices in ${\mathbb{C}}^{M\times N}$ that maximize some of them up to a constant independent of $M$ and $N$.

math.FA↗
arXiv2015

$L^p$-$L^q$ boundedness of integral operators with oscillatory kernels: Linear versus quadratic phases

Let $\,T^{j,k}_{N}:L^{p}(B)\, \rightarrow\,L^{q}([0,1])\,$ be the oscillatory integral operators defined by $\;\displaystyle T^{j,k}_{N}f(s):=\int_{B} \,f(x)\,e^{\imath N{|x|}^{j}s^{k}}\,dx, \quad (j,k)\in\{1,2\}^{2},\,$ where $\,B\,$ is the unit ball in ${\mathbb{R}}^{n}\,$ and $\,N\,>>1.$ We compare the asymptotic behaviour as $\,N\rightarrow +\infty\,$ of the operator norms $\,\parallel T^{j,k}_{N} \parallel_ {L^{p}(B)\rightarrow L^{q}([0,1])}\,$ for all $\,p,\,q\in [1,+\infty].\,$ We prove that, except for the dimension $n=1,\,$ this asymptotic behaviour depends on the linearity or quadraticity of the phase in $s$ only. We are led to this problem by an observation on inhomogeneous Strichartz estimates for the Schrödinger equation.

math.AP↗