# Interpolation theory

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.