📂Lebesgue Spaces

Weighted Lp Spaces

Definition¹

A function space defined as follows is called a weighted $L^{p}$ space or specifically a $w$-weighted $L^{p}$ space.

$$ L_{w}^{p}(a,b):= \left\{ f : \mathbb{R}\to \mathbb{C}\ \big|\ \int_{a}^{b} \left| f(x) \right|^{p}w(x)dx <\infty \right\} $$

Here, $w:\mathbb{R}\to[0,\infty)$ is called a weight function.

Description

It is one of the spaces that generalizes the $L^{p}$ space. When it is $w(x)=1$, $L_{w}^{p}=L^{p}$ holds. The norm of the weighted $L^{p}$ space is defined as follows for $1\le p <\infty$.

$$ \left\| f\right\|_{p,w}=\left\| f\right\|_{L_{w}^{p}(a,b)}=\left( \int_{a}^{b}\left| f(x) \right|^{p}w(x)dx \right)^{\frac{1}{p}} $$

It is obvious that the above value is finite by the definition of the $L_{w}^{p}$ space. Especially, when it is $p=2$, the inner product can be defined as follows.

$$ \langle f,g \rangle_{L_{w}^{2}(a,b)}=\int_{a}^{b}f(x)\overline{g(x)}w(x)dx,\quad f,g \in L_{w}^{2}(\mathbb{R}) $$

Similar to the $L^{2}$ space, it becomes a Hilbert space.