My question is about operators on $L^p$ spaces

Question

I am not sure how to proceed. This is the question: let $1<p<+\infty$ and let $q$ be the conjugate exponent. Show that if $f\in L^q((0,\infty))$ then $Tf(x)=x^{-1/p}\int_0^x f(t)dt\in C_0((0,+\infty))$ and furthermore $\|Tf\|_\infty \leq C\|f\|_q.$

Answer 1

First of all, notice that $T: L^q(\mathbb R^+_0) \to C(\mathbb R^+_0)$ is well defined by Holder's inequality since $$|Tf(x)| = \left|x^{-1/p}\int_0^x f(t)~dt\right| \le x^{-1/p} \|\boldsymbol 1_{(0, x)}\|_p \|f\|_q = \|f\|_q. $$ Moreover, for $y \in (0, +\infty)$ and $(x_n)_n \subset (0, \infty)$ satisfying $x_n \to y$, we have $\boldsymbol 1_{(0,x_n)}(x) f(x) \to f(x)\boldsymbol 1_{(0,y)}(x)$ a.e. By dominated convergence theorem, we conclude that $$\int_0^{x_n}f(t)~dt \to \int_0^y f(t)~dt.$$ Since the function $x \mapsto x^{-1/p}$ is continuous on $(0, +\infty)$, we deduce that $Tf(x)$ is also continuous as a product of continuous functions.