Let \(L \in \mathbb{R}^d_+\) and \(\mathbb{T}^d_L=[0,2\pi L_1]\times \cdots \times [0,2\pi L_d]\) be the scaled \(d\)-torus . If \(f\in L^p(\mathbb{T}^d_L)\) with \(p\in [1,\infty ]\) such that \(\supp \hat{f}\subseteq J\), where \(J\) is a \(d\)-dimensional rectangle with side lengths \(b_1,\ldots ,b_d\), and \(S\subseteq \mathbb{R}^d\) is a \((\gamma,a)\)-thick set with \(a=(a_1, \ldots , a_d)\) such that \(0< a_i\le 2\pi L_i\) for every \(i=1,\ldots ,d\), then
\[ \lVert f\rVert_{L^p(\mathbb{T}^d_L)}\le \biggl(\frac{c^d}{\gamma}\biggr)^{ca\cdot b + \frac{6d+1}{p}}\lVert f\rVert_{L^p(S\cap \mathbb{T}^d_L)}, \]where \(c\) is a numerical constant. [1, Theorem 2.1]
Remarks
- The authors used the same technique as Kovrijkine in his UP in the Euclidean space .
- The proof uses a Bernstein inequality. Alexander and I attempted to proof this result by avoiding it.
See also Link to heading
References Link to heading
- M. Egidi and I. Veselić,
Scale-free Unique Continuation Estimates and Logvinenko–Sereda Theorems on the Torus,
Annales Henri Poincaré, vol. 21, no. 12, pp. 3757–3790, 2020. doi:10.1007/s00023-020-00957-7