arXiv subjects

Solange Mukeshimana

Publications and source records attributed to Solange Mukeshimana.

3 recordsLinked to original sources

arXiv2026

Sparse Bounds for Rough Fourier Integral Operators

We prove pointwise bounds for rough Fourier integral operators by the $L^p$ Hardy-Littlewood maximal function. We assume the Fourier integral operators have amplitudes in $L^\infty S^m_ρ$ and phases $φ$ such that $φ(x,ξ) - x\cdotξ\in L^\infty Φ^1$, and assume a non-degeneracy condition on the matrix $\partial^2_ξφ(x,ξ)$. The pointwise bound holds when \begin{equation*} m < -\fracρ{2}(n-1) - \fracρ{p} - \frac{n}{p}(1-ρ), \end{equation*} which is known to a be sharp condition on $m$ when $ρ=1$, modulo the end-point. Making use of this pointwise bound and known $L^p$ boundedness results when the phase satisfies an additional non-degeneracy condition, we go on to prove sparse form bounds for these operators.

math.CA↗

arXiv2026

Some More Sparse Bounds for Rough and Smooth Pseudodifferential Operators

Beltran \& Cladek~\cite{BC} use $L^r$ to $L^s$ bounds to prove sparse form bounds for pseudodifferential operators with Hörmander symbols in $S^m_{ρ,δ}$ up to, but not including, the sharp end-point in decay $m$. We further develop their technique, obtaining pointwise sparse bounds for rough pseudodifferential operators that are merely measurable in their spatial variables and an alternative proof of their results which avoids proving geometrically decaying sparse bounds. We also provide sufficient conditions for sparse form bounds to hold and use these to reprove know sparse bounds for pseudodifferential operators with symbols in $S^0_{1,δ}$ for $δ< 1$.

math.CA↗

arXiv2023

On Chaplygin models in f(G) gravity

The current work treats cosmological perturbation in a mixture of standard matter, Chaplygin gas as well as Gauss-bonnet fluids using a 1+3 covariant approach in the context of modified $f(G)$ gravity. We define the gradient variables to obtain linear perturbation equations. After scalar and redshift transformations, we consider both an original Chaplygin and generalized Chaplygin gas models under Gauss-bonnet gravity. For pedagogical purposes, the consideration of polynomial $f(G)$ gravity model was used to solve the perturbation equations for short- and long- wavelength modes and investigate the late time evolution. The numerical solutions were obtained. The results show that the energy overdensity perturbations decay with an increase in redshift. The treatment recovers GR results under limiting cases.

gr-qc↗