Bassam Shayya

An affine restriction estimate in ℝ

We prove that the Fourier transform of an L 4/3 function can be restricted to any compact convex C 2 surface of revolution in R 3 .

Weighted restriction estimates using polynomial partitioning

We use the polynomial partitioning method of Guth [J. Amer. Math. Soc. 29 (2016) 371–413] to prove weighted Fourier restriction estimates in R3 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∞ surfaces S⊂R3 with strictly positive second fundamental form. For example, we establish the global restriction estimate ∥Ef∥Lp(R3)≲∥f∥Lq(S) in the full conjectured range of exponents for p>3.25 (up to the sharp line), and the global restriction estimate ∥Ef∥Lp(Ω)≲∥f∥L2(S) for p>3 and certain sets Ω⊂R3 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 R3 with finite α-dimensional energies.

Nonisotropic strongly singular integral operators

We consider a class of strongly singular integral operators which include those studied by Wainger, and Fefferman and Stein, and extend the results concerning the L p boundedness of these operators to the nonisotropic setting. We also describe a geometric property of the underlying space which helps us show that our results are sharp.

The WAT conjecture on the torus

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Spherical Means Associated with Non-isotropic Dilation Groups

Let {δt}t>0 be a non-isotropic dilation group on Rn. Let τ: Rn → [0,∞) be a continuous function that vanishes only at the origin and satisfies τ(δtx) = tτ(x), t > 0, x ∈ Rn. In this paper we obtain two-sided inequalities for spherical means of the form % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A!

Adjoint restriction estimates and scaling on spheres

\int_{S^{n-1}}\tau(r_1\omega_1,\cdots,r_n\omega_n)^{-\alpha}d\sigma (\omega),

Fourier restriction to convex surfaces of revolution in R3

where α is a positive constant, and r1,…, rn are positive parameters.

Singular integrals with nonhomogeneous oscillatory kernels

We test the restriction conjecture, in its adjoint form, against a class of measures Φδdσ on the sphere S n-1 . The densities Φδ are smoothed out characteristic functions of δ a 2 x δ a 3 × ... × δ a n rectangular caps on S n-1 , where a 2 , a 3 ,...,a n are fixed nonnegative numbers.