Let $t_0 > 0$, $p \in (1,+\infty)$ and define $K:(0,t_0)\times (0,t_0) \rightarrow \mathbb{R}$ by $K(x,y) := \frac{1}{x+y}$. Is it true that for all $f \in L^p\bigl((0,t_0);\mathbb{R}\bigr)$ the mapping $$ y \mapsto \int_{(0,t_0)} K(x,y)f(x) \ dx = \int_{(0,t_0)} \frac{f(x)}{x+y} \ dx $$
is contained in $L^p\bigl((0,t_0);\mathbb{R}\bigr)$? I tried using Hölder's inequality as well as Jensen's inequality and Minkowski's integral inequality, but nothing worked.
Furthermore, if the statement above is true, is the operator $A: L^p\bigl((0,t_0);\mathbb{R}\bigr) \rightarrow L^p\bigl((0,t_0);\mathbb{R}\bigr)$ defined by $$ Af := \Bigl(y \mapsto \int_{(0,t_0)} \frac{f(x)}{x+y} \ dx\Bigr) $$
bounded? I think this would follow from the Closed Graph Theorem, but I am not so sure how.