Different perspectives on the Mizohata-Takeuchi conjecture

Ferrante, Michele (2024). Different perspectives on the Mizohata-Takeuchi conjecture. University of Birmingham. Ph.D.

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Abstract

The Mizohata–Takeuchi conjecture states that one should be able to control the $L^2$ weighted norm of the Fourier extension operator associated with a measure $\mu$ supported on a smooth hypersurface in $\R^n$, with the $L^\infty$ norm of the X-ray transform of the weight, i.e.
$$\int_{\mathbb{R}^n} |\widehat{gd\mu}|^2\, w \lesssim \norm{L^\infty}{Xw} \int|g|^2d\mu, $$
where
$$\widehat{gd\mu}(x)=\int e^{ix\cdot\xi}g(\xi)d\mu(\xi)$$
and the X-ray transform is a function of lines defined as
$$Xw(\ell)=\int_\ell w .$$
We will discuss some possible generalization of the Mizohata–Takeuchi conjecture. In particular, we will see some results for measures not supported on smooth surfaces, we will discuss the possibility of introducing a general Mizohata–Takeuchi conjecture for measures satisfying only dimensionality hypotheses and we will face the Mizohata–Takeuchi problem in the setting of locally compact abelian group.

Type of Work:

Thesis (Doctorates > Ph.D.)

Award Type:

Doctorates > Ph.D.

Supervisor(s):