Consider the interpolation space $Z=(X,Y)_{\theta,p}$. In the case $Y\subseteq X$ do we have that, for all $a>0$ the following norm: $$N_a:x\mapsto\left(\int_{0}^{a} \vert t^{-\theta}k(t,x)\vert^p \frac{dt}{t}\right)^{1/p} $$
is equivalent to the norm of $Z$: $$\Vert x\Vert_Z=\left(\int_{0}^{+\infty} \vert t^{-\theta}k(t,x)\vert^p \frac{dt}{t}\right)^{1/p} \quad ?$$
In the case of $Y\subseteq X$, only the behavior near $t=0$ of $t^{-\theta}k(t,x)$ plays a role in the definition of $Z$. Because $\vert k(t,x)\vert\leq \Vert x\Vert_X$.
In other words I'm asking if we could replace the half line $(0,+\infty)$ by any interval $(0,a)$.
$$ \theta\in(0,1), p\in(1,+\infty), \quad \text{and } k(t,x)=\inf_{x=b+c\in X+Y}\big({\Vert b\Vert+t\Vert c\Vert}\big). $$ Thank you.
