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Let $$\\gamma(t)=\\left (t,t^2,\\cdots,t^{n-1},\\phi(t)\\right).$$ In this paper, we show that $$\\left(\\int_I \\big|\\hat f(\\gamma(t))\\big|^q \\big|\\phi^{(n)}(t)\\big|^{\\frac{2}{n(n+1)}} dt\\right)^{1/q}\\le C\\|f\\|_{L^p(\\mathbb R^n)}$$ holds in the range $$1\\le p<\\frac{n^2+n+2}{n^2+n},\\quad 1\\le q<\\frac{2}{n^2+n}p'.$$ This generalizes an affine restriction theorem of Sj\\\"olin (1974) for $n=2$. 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