Different Besov-Norm Definitions

Question

first some notation: $\langle x\rangle=\sqrt{1+x^2}$, $P_{j}$ is the Littlewood Paley Projector and $P_{\leq0}$ corresponds to the small frequencies.

I have a the following definition of the Besov norm:

$$||u||_{B_{p,q}^s}= ||P_{\leq0}(f)||+(\sum^\infty_{j=1}(2^{js}||P_j(f)||_{p})^{q})^{1/q}$$ and $$||u||_{B_{p,\infty}^s}= ||P_{\leq0}(f)||+\sup_{j\geq1}2^{js}||P_j(f)||_{p}$$ I was told they were equivalent to the following but I'm not sure (especially about whether I want to allow negative $j$ and whether I have to use $\langle\rangle$ or not) and I'm failing to prove the equivalence of the Norms: $$||u||_{B_{p,q}^s}= (\sum^\infty_{j=-\infty}(\langle2^{j}\rangle^s||P_j(f)||_{p})^{q})^{1/q}$$ and $$||u||_{B_{p,\infty}^s}= \sup_{j\in\mathbb{Z}}\langle2^j\rangle^{s}||P_j(f)||_{p}$$

My question is: Did I get the second definitons right and how can I prove equivalence?

Answer 1

In this paper http://arxiv.org/abs/1007.3418 one can find definitions for both inhomogenous and homogenous besov norms. The second definitions are the ones for the inhomogenous besov norms if you take away the ⟨⟩ brackets.

Furthermore it is obvious (since all norms on $\mathbb{R}^n$ are equivalent) that: $$||P_{\leq0}(f)||+(\sum^\infty_{j=1}(2^{js}||P_j(f)||_{p})^{q})^{1/q} \sim (\sum^\infty_{j=0}(2^{js}||P_j(f)||_{p})^{q})^{1/q}$$

IF by definition $P_{\leq0} = P_{0}$. This is also explained in the referenced paper.