$K(t,x)=\inf_{x=a+b,\ a\in X,\ b\in Y}\{\|a\|_X+t\|b\|_Y\}$ and $\int_0^\infty \frac{|K(t,x)|^p}{t}dt<\infty$ implies $x=0$?

Question

Let $X,Y$ be two Banach spaces with respective norms $\|\cdot\|_X$ and $\|\cdot\|_Y$. Suppose that $X$ and $Y$ are subsets of a vector space $Z$. Define $K(t,x)$ for $t\in (0,\infty)$ and $x\in X+Y$ by $$K(t,x)=\inf_{x=a+b,\ a\in X,\ b\in Y}\{\|a\|_X+t\|b\|_Y\}$$

Assume that (for fixed $x$ and $p\in [1,\infty)$) $$\int_0^\infty \frac{|K(t,x)|^p}{t}dt<\infty$$

Can we conclude that $x=0$?

Thank you

Answer 1

Observe that $K(t,x) $ is a nondecreasing function of $t$. If $K(t_0,x)>0$ for some $t_0$, then $$\int_{t_0}^\infty \frac{K(t,x)^p}{t}dt \ge K(t_0,x)^p \int_{t_0}^\infty \frac{1}{t} dt=\infty $$ Thus, $K(t,x)=0$ for all $t$. On the other hand, $K(1,x)$ bounds the norm of $x$ via the triangle inequality.