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Heisenberg's inequality in $L^p$

Miquel Saucedo, Sergey Tikhonov

TL;DR

This work extends the classical Heisenberg-type uncertainty principle to weighted Lebesgue spaces with broken power weights, treating both non-symmetric and symmetric variants. It develops a local uncertainty principle and delivers sharp, complete characterizations of when these weighted inequalities hold, together with explicit optimal bounds expressed as $H_{A,B}$ and its exponents. The results are complemented by an analysis of special Hadamard operators, showing which bounds carry over to operators sharing Fourier-like properties, and an appendix detailing weighted Fourier inequalities that underpin the main theorems. Overall, the paper clarifies how decay and concentration interact at the origin and at infinity, generalizing the classical HUP to a versatile, finely balanced weighted framework with precise regimes of validity and sharpness.

Abstract

In this paper, we obtain non-symmetric and symmetric versions of the classical Heisenberg-Pauli-Weyl uncertainty principle in Lebesgue spaces with power weights.

Heisenberg's inequality in $L^p$

TL;DR

This work extends the classical Heisenberg-type uncertainty principle to weighted Lebesgue spaces with broken power weights, treating both non-symmetric and symmetric variants. It develops a local uncertainty principle and delivers sharp, complete characterizations of when these weighted inequalities hold, together with explicit optimal bounds expressed as and its exponents. The results are complemented by an analysis of special Hadamard operators, showing which bounds carry over to operators sharing Fourier-like properties, and an appendix detailing weighted Fourier inequalities that underpin the main theorems. Overall, the paper clarifies how decay and concentration interact at the origin and at infinity, generalizing the classical HUP to a versatile, finely balanced weighted framework with precise regimes of validity and sharpness.

Abstract

In this paper, we obtain non-symmetric and symmetric versions of the classical Heisenberg-Pauli-Weyl uncertainty principle in Lebesgue spaces with power weights.
Paper Structure (15 sections, 16 theorems, 159 equations)

This paper contains 15 sections, 16 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Non-symmetric HUP
  3. Symmetric HUP
  4. Broken power weights
  5. Structure of the paper
  6. Local uncertainty principle
  7. Non-symmetric uncertainty inequalities
  8. Main result
  9. Proof of $\lesssim$ part in Theorem \ref{['theorem:GH']}
  10. Proof of $\gtrsim$ part of Theorem \ref{['theorem:GH']}
  11. Symmetric uncertainty inequalities
  12. Proof of sufficiency in Theorem \ref{['theorem:broken']}
  13. Proof of necessity in Theorem \ref{['theorem:broken']}
  14. Heisenberg type uncertainty for special Hadamard operators
  15. Appendix. Weighted Fourier inequalities

Key Result

Theorem 1.1

Let $1 \leq p,q \leq \infty$ and $\alpha,\beta> 0$. Then, inequality eq:uncerxiao holds if and only if one of the following conditions holds: and one of the following conditions holds:

Theorems & Definitions (33)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 3.1
  • Lemma 3.2
  • ...and 23 more