Let \(d\ge 1\), \(Q\) be a parallelepiped with sides \(b_1, b_2, \ldots, b_d\) parallel to the coordinate axes, \(p \in [1,\infty]\), \(\gamma \in (0,1]\), \(a\in \mathbb{R}_+^d\) and \(S \subset \mathbb{R}^d\) a \((\gamma, a)\)-thick set . Then two universal constants \(c_1, c_2>0\) exists such that
\begin{equation*} \lVert f \rVert_{L^p(\mathbb{R}^d)} \le \Bigl(\frac{c_1^d}{\gamma}\Bigr)^{c_2(a \cdot b + d)}\lVert f\rVert_{L^p(S)} \end{equation*}for every \(f\in L^p(\mathbb{R}^d)\) with \(\supp \hat{f} \subset Q\) [1, Theorem 1].
Remarks
- This result improves classical Logvinenko-Sereda theorem , since the constant has only polynomial dependence on \(\gamma\).
- This Result is qualitatively optimal.
- Note, that \(\hat{f}\) are tempered distributions in general.
Proof
\(p\neq \infty\) Link to heading
The proof is similar to the one of the one-dimensional case .
1. Simplification of the problem. Link to heading
By transforming the argument of \(f\) we may consider cubes with sides of length 1. In this case the support of the Fourier transform scales. See [1] for details.
2. Apply complex lemma on thick lines to get an estimate where some supremum is involved. Link to heading
According to (0x669e0e75) there is a point \(x\in B\) such that \(\lvert f(x)\rvert\ge \lVert f\rVert_{L^p(B)}\lvert B\rvert^{-1/p}\).
3. Estimate the supremum on good cubes Link to heading
4. Combine the estimates of step 2 and 3 to obtain the result. Link to heading
See also Link to heading
References Link to heading
- O. Kovrijkine,
Some estimates of Fourier transforms,
Ph.D. thesis, United States -- California: California Institute of Technology, 2000.