Fourier restriction in low fractal dimensions
نویسندگان
چکیده
Let $S \subset \Bbb R^n$ be a smooth compact hypersurface with strictly positive second fundamental form, $E$ the Fourier extension operator on $S$, and $X$ Lebesgue measurable subset of $\Bbb R^n$. If contains ball each radius, then problem determining range exponents $(p,q)$ for which estimate $\| Ef \|_{L^q(X)} \leq C \| f \|_{L^p(S)}$ holds is equivalent to restriction conjecture. In this paper, we study under following assumption set $X$: there number $0 < \alpha n$ such that $|X \cap B_R| c \, R^\alpha$ all balls $B_R$ in radius $R \geq 1$. On left-hand side estimate, are integrating function $|Ef(x)|^q$ against measure $\chi_X dx$. Our approach consists replacing characteristic $\chi_X$ by an appropriate weight $H$, studying resulting three different regimes: small values $\alpha$, intermediate large $\alpha$. first regime, establish using already available methods. prove weighted H\"{o}lder-type inequality general non-negative functions R^n$, combine it result from regime. third borrow recent fractal theorem Du Zhang opposite direction, results paper improve Du-Zhang n/2$.
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ژورنال
عنوان ژورنال: Proceedings of the Edinburgh Mathematical Society
سال: 2021
ISSN: ['1464-3839', '0013-0915']
DOI: https://doi.org/10.1017/s0013091521000201