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Let $$\phi $$ϕ be a smooth function on a compact interval I. Let $$\begin{aligned} \gamma (t)=\left( t,t^2,\ldots ,t^{n-1},\phi (t)\right) . \end{aligned}$$γ(t)=t,t2,…,tn-1,ϕ(t).In this paper, we show that $$\begin{aligned} \left( \int _I… Click to show full abstract
Let $$\phi $$ϕ be a smooth function on a compact interval I. Let $$\begin{aligned} \gamma (t)=\left( t,t^2,\ldots ,t^{n-1},\phi (t)\right) . \end{aligned}$$γ(t)=t,t2,…,tn-1,ϕ(t).In this paper, we show that $$\begin{aligned} \left( \int _I \big |\hat{f}(\gamma (t))\big |^q \big |\phi ^{(n)}(t)\big |^{\frac{2}{n(n+1)}} \mathrm{{d}}t\right) ^{1/q}\le C\Vert f\Vert _{L^p(\mathbb R^n)} \end{aligned}$$∫I|f^(γ(t))|q|ϕ(n)(t)|2n(n+1)dt1/q≤C‖f‖Lp(Rn)holds in the range $$\begin{aligned} 1\le p<\frac{n^2+n+2}{n^2+n},\quad 1\le q<\frac{2}{n^2+n}p'. \end{aligned}$$1≤p
Journal Title: Journal of Fourier Analysis and Applications
Year Published: 2017
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