Knowlpedia

Notation

Japanese bracket

The smooth positive weight (1 + norm-squared)^(1/2) used to combine local and asymptotic estimates.

Core idea

For x∈Rdx\in\mathbb R^d, the Japanese bracket is the notation

⟨x⟩=(1+∣x∣2)1/2.\langle x\rangle=(1+|x|^2)^{1/2}.

It is smooth and positive everywhere, comparable to 1+∣x∣1+|x|, and comparable to ∣x∣|x| for large ∣x∣|x|.

Use in weighted estimates

A bound such as ∣f(x)∣≤C⟨x⟩−N|f(x)|\le C\langle x\rangle^{-N} describes polynomial decay without a singularity at the origin. Symbol estimates are often written ∣Dαa(x)∣≤Cα⟨x⟩m−∣α∣|D^\alpha a(x)|\le C_\alpha\langle x\rangle^{m-|\alpha|}.

Basic inequality

There is an absolute constant CC such that ⟨x+y⟩≤C⟨x⟩⟨y⟩\langle x+y\rangle\le C\langle x\rangle\langle y\rangle. This permits weighted convolution and translation estimates.

References
  1. Lars Hörmander, The Analysis of Linear Partial Differential Operators I, Springer, 2003. DOI record.