Multiscale version of the Logvinenko-Sereda Theorem

这个项目的目的是证明Logvinenko-Sereda定理的一个多尺度版本。经典的Logvinenko-Sereda定理属于调和分析或傅立叶分析的领域，它断言在实线上的L^p$函数对粗子集的限制与整个轴上的函数具有相似的L^p$-范数，只要它的傅立叶变换在紧区间上得到支持。只有区间的大小进入估计，而不是它的位置。限制集的厚度也在常数中。Kovrijkine将结果推广到傅里叶变换在有限个相同长度的区间的并集中有支持的情况。同样，只有间隔的数量和大小进入估计，而不是它们的位置。最近关于特征函数和Schrödinger算子的谱投影的无标度唯一延拓估计或不确定性原理表明了Logvinenko-Sereda定理的多标度版本的有效性。这里的函数是在长度为$L$的区间上考虑的，其中$L$的范围大于正实数。虽然预期的估计乍一看似乎比整个轴的情况更简单，但人们有额外的任务来有效地控制估计对额外大小参数$L$的依赖性。理想情况下，人们希望显示估计阈值在$L$中是一致的。虽然推测界是调和分析的结果，但它立即引发了反问题理论中的结果，特别是在适当的稀疏性假设下。因此，它可以看作是压缩感知和稀疏恢复的连续体。此外，推测估计在Schrödinger算符的谱理论和热方程的控制理论中也有应用。这一估计的多维扩展将具有更广泛的相关性。

The aim of this project is to prove a multiscale version of the Logvinenko-Sereda Theorem.The classical Logvinenko-Sereda Theorem belongs to the realm of Harmonic or Fourier Analysis and asserts that the restriction of an $L^p$ function on the real lineto a thick subset has comparable $L^p$-norm to the function on the whole axis, provided that its Fourier transform is supported on a compact interval. Only the size of the interval enters in the estimate, not its position. The thicknessof the restriction set enters in the constant as well. Kovrijkine extended the result to the case where the Fourier transform has support in a union of a finite number of intervals of the same length. Again, only the number and the size of the intervals enters the estimate, not their position.Recent scale-free unique continuation estimates or uncertainty principles for eigenfunctions and spectral projections of Schrödinger operator suggest the validity of a multiscale version of the Logvinenko-Sereda Theorem.Here the functions are considered on intervals of length $L$, where $L$ ranges over the positive reals.While the intended estimate appears at first sight simpler than in the case of the whole axis, one has the additional taskto control effectively the dependence of the estimate on the additional size parameter $L$. Ideally, one would like to show that the estimateshold uniformly in $L$. While the conjectured bound is a Harmonic Analysis result, it immediately triggers consequencesin the theory of Inverse Problems, in particular under appropriate sparsity assumptions. Thus it can be seen as an continuum relative of compressed sensing and sparse recovery. Moreover, the conjectured estimateshave applications in the spectral theory of Schrödinger operators and in the control theory of the heat equation.A multidimensional extension of this estimate will have even wider relevance.