arXiv subjects

Bassam Shayya

Publications and source records attributed to Bassam Shayya.

5 recordsLinked to original sources

arXiv2022

Mizohata-Takeuchi estimates in the plane

Suppose $S$ is a smooth compact hypersurface in $\Bbb R^n$ and $σ$ is an appropriate measure on $S$. If $Ef= \hat{fdσ}$ is the extension operator associated with $(S,σ)$, then the Mizohata-Takeuchi conjecture asserts that $\int |Ef(x)|^2 w(x) dx \leq C (\sup_T w(T)) \| f \|_{L^2(σ)}^2$ for all functions $f \in L^2(σ)$ and weights $w : \Bbb R^n \to [0,\infty)$, where the $\sup$ is taken over all tubes $T$ in $\Bbb R^n$ of cross-section 1, and $w(T)= \int_T w(x) dx$. This paper investigates how far we can go in proving the Mizohata-Takeuchi conjecture in $\Bbb R^2$ if we only take the decay properties of $\hatσ$ into consideration. As a consequence of our results, we obtain new estimates for a class of convex curves that include exponentially flat ones such as $(t,e^{-1/t^m})$, $0 \leq t \leq c_m$, $m \in \Bbb N$.

math.CA↗

arXiv2022

A family of fractal Fourier restriction estimates with implications on the Kakeya problem

In a recent paper [Ann. of Math. 189 (2019), 837--861], Du and Zhang proved a fractal Fourier restriction estimate and used it to establish the sharp $L^2$ estimate on the Schrödinger maximal function in $\Bbb R^n$, $n \geq 2$. In this paper, we show that the Du-Zhang estimate is the endpoint of a family of fractal restriction estimates such that each member of the family (other than the original) implies a sharp Kakeya result in $\Bbb R^n$ that is closely related to the polynomial Wolff axioms. We also prove that all the estimates of our family are true in $\Bbb R^2$.

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arXiv2022

Improved weighted restriction estimates in $\Bbb R^3$

Suppose $0 < α\leq n$, $H: \Bbb R^n \to [0,1]$ is a Lebesgue measurable function, and $A_α(H)$ is the infimum of all numbers $C$ for which the inequality $\int_B H(x) dx \leq C R^α$ holds for all balls $B \subset \Bbb R^n$ of radius $R \geq 1$. After Guth introduced polynomial partitioning to Fourier restriction theory, weighted restriction estimates of the form $\| Ef \|_{L^p(B,Hdx)} \leq C R^εA_α(H)^{1/p} \| f \|_{L^q(σ)}$ have been studied and proved in several papers, leading to new results about the decay properties of spherical means of Fourier transforms of measures and, in some cases, to progress on Falconer's distance set conjecture in geometric measure theory. This paper improves on the known estimates when $E$ is the extension operator associated with the unit paraboloid ${\mathcal P} \subset \Bbb R^3$, reaching the full possible range of $p,q$ exponents (up to the sharp line) for $p \geq 3 + (α-2)/(α+1)$ and $2 < α\leq 3$.

math.CA↗

arXiv2020

Fourier restriction in low fractal dimensions

Let $S \subset \Bbb R^n$ be a smooth compact hypersurface with a strictly positive second fundamental form, $E$ be the Fourier extension operator on $S$, and $X$ be a Lebesgue measurable subset of $\Bbb R^n$. If $X$ contains a ball of each radius, then the problem of determining the range of exponents $(p,q)$ for which the estimate $\| Ef \|_{L^q(X)} \leq C \| f \|_{L^p(S)}$ holds is equivalent to the restriction conjecture. In this paper, we study the estimate under the following assumption on the set $X$: there is a number $0 < α\leq n$ such that $|X \cap B_R| \leq c \, R^α$ for all balls $B_R$ in $\Bbb R^n$ of radius $R \geq 1$. On the left-hand side of this estimate, we are integrating the function $|Ef(x)|^q$ against the measure $χ_X dx$. Our approach consists of replacing the characteristic function $χ_X$ of $X$ by an appropriate weight function $H$, and studying the resulting estimate in three different regimes: small values of $α$, intermediate values of $α$, and large values of $α$. In the first regime, we establish the estimate by using already available methods. In the second regime, we prove a weighted Hölder-type inequality that holds for general non-negative Lebesgue measurable functions on $\Bbb R^n$, and combine it with the result from the first regime. In the third regime, we borrow a recent fractal Fourier restriction theorem of Du and Zhang and combine it with the result from the second regime. In the opposite direction, the results of this paper improve on the Du-Zhang theorem in the range $0 < α< n/2$.

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arXiv2017

Weighted restriction estimates using polynomial partitioning

We use the polynomial partitioning method of Guth to prove weighted Fourier restriction estimates in $\Bbb R^3$ with exponents $p$ that range between $3$ and $3.25$, depending on the weight. As a corollary to our main theorem, we obtain new (non-weighted) local and global restriction estimates for compact $C^\infty$ surfaces $S \subset \Bbb R^3$ with strictly positive second fundamental form. For example, we establish the global restriction estimate $\| Ef \|_{L^p(\Bbb R^3)} \leq C \, \| f \|_{L^q(S)}$ in the full conjectured range of exponents for $p > 3.25$ (up to the sharp line), and the global restriction estimate $\| Ef \|_{L^p(Ω)} \leq C \, \| f \|_{L^2(S)}$ for $p>3$ and certain sets $Ω\subset \Bbb R^3$ of infinite Lebesgue measure. As a corollary to our main theorem, we also obtain new results on the decay of spherical means of Fourier transforms of positive compactly supported measures on $\Bbb R^3$ with finite $α$-dimensional energies.

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