arXiv subjects

Magali Folch-Gabayet

Publications and source records attributed to Magali Folch-Gabayet.

2 recordsLinked to original sources

arXiv2024

Estimates for strongly singular operators along curves

For a proper function $f$ on the plane, we study the operator \[ Tf(x,y) = \lim_{\varepsilon\to 0} \int_\varepsilon^1 f(x-t,y-t^k) \frac{e^{2πi γ(t)}}{ψ(t)} dt, \] where $k\ge1$ and $ψ$ and $γ$ are functions defined near the origin such that $ψ(t)\to 0$ and $|γ(t)|\to\infty$ as $t\to 0$. We give sufficient regularity and growth conditions on $ψ$ and $γ$ for its multiplier to be a bounded function, and thus for the operator to be bounded on $L^2(\mathbb R^2)$. We consider an extension to $L^p(\mathbb R^2)$, for certain $p's$.

math.CA↗

arXiv2019

Weak-type (1,1) estimates for strongly singular operators

Let $ψ$ be a positive function defined near the origin such that $\lim_{t\to 0^{+}}ψ(t)=0$. We consider the operator \begin{equation*} T_θf(x) = \lim_{\varepsilon\to 0^+} \int_\varepsilon^1 e^{iγ(t)}f(x-t) \frac{dt}{t^θψ(t)^{1-θ}}, \end{equation*} where $γ$ is a real function with $\lim_{t\to 0^+}|γ(t)| = \infty$ and $0 \le θ\le 1$. Assuming certain regularity and growth conditions on $ψ$ and $γ$, we show that $T_1$ is of weak type $(1,1)$.

math.CA↗