First the standard $L2$ space : $$L^2(\mathbb{R}) = \Big \{ f : \| f \|_2 = \left( \int_{\mathbb{R}} | f(x) |^2 dx \right)^{1/2} < \infty \Big \}.$$
Let $s \geqslant 1/2$. Define a weighted $L2$ space as follows :
$$L^2_{s} := \{ f \in L^2(\mathbb{R}) : \| (2+|x|)^s f(x) \|_2 < \infty \}.$$
There is also another Banach space $B$ (in the literature that I have seen it either doesn't have a special name, or is just called a "Besov" space) :
$$B := \{ f \in L^2(\mathbb{R}) : \sum_{n=0} ^{\infty} \sqrt{2^n} \| \mathbb{1}_{\Omega_n} (x) f(x) \|_2 <\infty \}.$$
Here $\mathbb{1}_{\Omega_n}(x)$ is the indicator function onto the sets $\Omega_n := \{ x \in \mathbb{R} : 2^{n-1} \leqslant |x| < 2^n \}$, $n \geqslant 1$, and $\Omega_0 := \{x \in \mathbb{R} : |x| < 1 \}$. Note $\{ \Omega_n \}$ is a partition of $\mathbb{R}$.
Then one can show that the following inclusions hold (I can include a proof if requested) :
$$L^2_s \subsetneq B \subsetneq L^2_{1/2}, \quad \forall s >1/2.$$
Now let us define another type of weighted $L2$ space involving logarithms. For $s,p \geqslant 1/2$,
$$L^2_{s,p} := \{ f \in L^2(\mathbb{R}) : \| (2+|x|)^s \left(\log(2+|x|)\right)^p f(x) \|_2 < \infty \}.$$
Then one can also show that the following inclusions hold :
$$L^2_s \subsetneq L^2 _{1/2,p} \subsetneq B \subsetneq L^2_{1/2}, \quad \forall s,p >1/2.$$
We also have trivially
$$L^2_s \subsetneq L^2 _{1/2,p} \subsetneq L^2_{1/2,1/2} \subsetneq L^2_{1/2}, \quad \forall s,p >1/2.$$
My question is : what is the relationship between $L^2_{1/2,1/2}$ and $B$ ? Is one included in the other ? Thanks for any tips or references
Let $a_n:= \sqrt{\int_{\Omega_n}f^2}$. A function $f$ belongs to $B$ if and only if $\sum_{n\geqslant 0}2^{n/2}a_n$ is finite, and to $L^2_{1/2,1/2}$ if and only if $\sum_{n\geqslant 0} 2^nna_n^2$ is finite.