Consider the interpolation space $Z=(X,Y)_{\theta,p}$, in the case $Y\subseteq X$ do we have that the following norm: $x\longrightarrow\left(\int_{0}^{a} \vert t^{-\theta}k(t,x)\vert^p \frac{dt}{t}\right)^{1/p} $ for all $a>0$

is equivalent to the norm of Z: $$\Vert x\Vert_Z=\left(\int_{0}^{+\infty} \vert t^{-\theta}k(t,x)\vert^p \frac{dt}{t}\right)^{1/p} $$

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 $(0,a)$.

$$ \theta\in(0,1), p\in(1,+\infty), \quad \text{and } k(t,x)=\inf_{x=a+b\in X+Y}({\Vert a\Vert+t\Vert b\Vert}). $$ Thank you.