Abstract. In this paper we consider adjoint restriction estimates for space curves with respect to general measures and obtain optimal estimates when the curves satisfy a finite type condition. The argument here is new in that it doesn't rely on the offspring curve method, which has been extensively used in the previous works. Our work was inspired by the recent argument due to Bourgain and Guth which was used to deduce linear restriction estimates from multilinear estimates for hypersurfaces.

Abstract. We consider Strichartz estimates for the wave equation with respect to general measures which satisfy certain growth conditions. In R 3+1 we obtain the sharp estimate and in higher dimensions improve the previous results.

Let φ be a smooth function on a compact interval I. Let γ(t) = t, t 2 , · · · , t n−1 , φ(t). In this paper, we show that I ˆ f (γ(t)) q φ (n) (t) 2 n(n+1) dt 1/q ≤ Cf L p (R n) holds in the range 1 ≤ p < n 2 + n + 2 n 2 + n , 1 ≤ q < 2 n 2 + n p. This generalizes an affine restriction theorem of Sjölin [22] for n = 2. Our proof relies on ideas of Sjölin [22] and Drury [11], and more recently Bak-Oberlin-Seeger [3] and Stovall [24], as well as a variation bound for smooth functions.

<p style='text-indent:20px;'>We prove new <inline-formula><tex-math id="M1">\begin{document}$ L^p $\end{document}</tex-math></inline-formula>–<inline-formula><tex-math id="M2">\begin{document}$ L^q $\end{document}</tex-math></inline-formula> estimates for averages over dilates of the circle with respect to fractal measures, which unify different types of maximal estimates for the circular average. Our results are consequences of <inline-formula><tex-math id="M3">\begin{document}$ L^p $\end{document}</tex-math></inline-formula>–<inline-formula><tex-math id="M4">\begin{document}$ L^q $\end{document}</tex-math></inline-formula> smoothing estimates for the wave operator relative to fractal measures. We also discuss similar results concerning the spherical averages.</p>