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• W2051615888 abstract "In this paper we study the L p − L r boundedness of the extension operators associated with paraboloids in $${{mathbb F}_{q}^{d}}$$ , where $${mathbb{F}_{q}}$$ is a finite field of q elements. In even dimensions d ≥ 4, we estimate the number of additive quadruples in the subset E of the paraboloids, that is the number of quadruples $${(x,y,z,w) in E^4}$$ with x + y = z+w. As a result, in higher even dimensions, we obtain the sharp range of exponents p for which the extension operator is bounded, independently of q, from L p to L 4 in the case when −1 is a square number in $${mathbb{F}_{q}}$$ . Using the sharp L p −L 4 result, we improve upon the range of exponents r, for which the L 2 − L r estimate holds, obtained by Mockenhaupt and Tao (Duke Math 121:35–74, 2004) in even dimensions d ≥ 4. In addition, assuming that −1 is not a square number in $${mathbb{F}_{q}}$$ , we extend their work done in three dimension to specific odd dimensions d ≥ 7. The discrete Fourier analytic machinery and Gauss sum estimates make an important role in the proof." @default.
• W2051615888 created "2016-06-24" @default.
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• W2051615888 date "2009-07-28" @default.
• W2051615888 modified "2023-10-12" @default.
• W2051615888 title "Extension theorems for paraboloids in the finite field setting" @default.
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• W2051615888 doi "https://doi.org/10.1007/s00209-009-0580-1" @default.
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