Let \(\Gamma \) be a compact patch of a well-curved \(C^{n+1}\) curve in \(\mathbb {R}^n\) with induced Lebesgue measure \(\textrm{d} \lambda \) , and let \(g \mapsto \widehat{g \,\textrm{d}\lambda }\) be the Fourier extension operator for \(\Gamma \) . Then we have, for arbitrary non-negative weights w, \(\begin{aligned} \int _{B_R} |\widehat{g \,\textrm{d}\lambda }|^2w \le C_{n,a} R^{a} \sup _S \left( \int _S w\right) \int _\Gamma |g|^2 \, \textrm{d} \lambda \end{aligned}\) for any \(a> \frac{n-3}{2} + \frac{2}{n} - \frac{2}{n^2(n+1)}\) , where the \(\sup \) is over all 1-neighbourhoods S of hyperplanes whose normals are parallel to the tangent at some point of \(\Gamma \) . This represents partial progress on the Mizohata–Takeuchi conjecture for curves in dimensions \(n \ge 3\) , improving upon the exponent \(a=n-1\) which can be obtained as a consequence of the Agmon–Hörmander trace inequality. Our main tool in establishing this inequality will be a weighted formulation of refined decoupling for well-curved curves. We also discuss the sharpness of the exponents we obtain in this and in auxiliary results, and further explore this in the context of axiomatic decoupling for curves.
