If $X_n$ converges to $X$ in $L^p$, do we have $X_n^p$ converges to $X^p$ in $L^1$?
We can prove that it is true when p=1,2 easily. I am curious whether this is true for all $p>0$.
If $X_n$ converges to $X$ in $L^p$, do we have $X_n^p$ converges to $X^p$ in $L^1$?
We can prove that it is true when p=1,2 easily. I am curious whether this is true for all $p>0$.
Note that $|x^p - y^p| = \left|\int_y^x p t^{p-1}\ dt \right| \le p (|x|+|y|)^{p-1} |x - y|$ Thus if $1/p + 1/q = 1$ (where $1 < p,q < \infty$) $\eqalign{\|X^p - Y^p\|_1 &\le \int p (|X| + |Y|)^{p-1} |X - Y| \cr &\le p \|(|X| + |Y|)^{p-1}\|_q \|X - Y\|_p \cr &= p \left(\int (|X| + |Y|)^p\right)^{1/q} \|X - Y\|_p \cr &= p \||X|+|Y|\|_p^{p/q} \|X - Y\|_p \cr &\le p (\|X\|_p + \|Y\|_p)^{p/q} \|X - Y\|_p\cr}$ If $X_n \to X$ in $L^p$, $\|X_n\|_p$ is bounded, and so we get $\|X_n^p - X^p\|_1 \to 0$.
It's not correct for $0 < p < 1$. Take $X_n: [0,1] \to \mathbb{R}$ as $X_n(x) = n \cdot I_{[0,1/n)}(x)$, where $I_A$ = indicator function of a set $A$. Then $\|X_n\|_p = n^{1-1/p}$ and therefore $X_n \to 0$ in $L^p$ for $p < 1$, but $X_n$ does not converge in $L^1$.