Weighted restriction theorems for space curves |
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Authors: | Jong-Guk Bak Jungjin Lee |
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Institution: | a Pohang University of Science and Technology, Pohang, Republic of Korea b Department of Mathematics, Seoul National University, Seoul, Republic of Korea |
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Abstract: | Consider a nondegenerate Cn curve γ(t) in Rn, n?2, such as the curve γ0(t)=(t,t2,…,tn), t∈I, where I is an interval in R. We first prove a weighted Fourier restriction theorem for such curves, with a weight in a Wiener amalgam space, for the full range of exponents p, q, when I is a finite interval. Next, we obtain a generalization of this result to some related oscillatory integral operators. In particular, our results suggest that this is a quite general phenomenon which occurs, for instance, when the associated oscillatory integral operator acts on functions f with a fixed compact support. Finally, we prove an analogue, for the Fourier extension operator (i.e. the adjoint of the Fourier restriction operator), of the two-weight norm inequality of B. Muckenhoupt for the Fourier transform. Here I may be either finite or infinite. These results extend two results of J. Lakey on the plane to higher dimensions. |
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Keywords: | Fourier restriction theorem Oscillatory integral operator Amalgam space Weighted norm inequality |
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