There is an operator given:
$K: X \to Y$,
$(Kf)(x) = \int \limits_0^1 k(x, y)f(y) \mbox{d}y$.
Let's define: $X = L^P([0, 1]), Y = L^q([0, 1])$, where $\frac{1}{p}+\frac{1}{q} = 1, k \in L^q([0, 1]^2)$.
The task is to prove whether the operator $K$ is bounded and then to calculate it's norm.
My attempt $$||Kf||_q^q = \int \limits_{0}^{1} \bigg( \bigg|\int \limits_0^1 k(x,y)f(y) \mbox{d}y \bigg| \bigg)^q \mbox{d}x \le \int \limits_{0}^{1} \bigg( \int \limits_0^1 |k(x,y)f(y)| \mbox{d}y \bigg)^q \mbox{d}x.$$ Now I would say that a "trick" with adding the $p$-th power and afterwards the usage of Hölder's inequality. I would appreciate any hints or tips.
To show that the operator is bounded you can use Hölder's inequality and get
$\int_0^1 \left( \vert k(x,y) \vert \vert f(y)\vert dy \right)^q dx \leq \int_0^1 \left( \left( \int_0^1 \vert k(x,y) \vert^q dy \right)^{\frac{1}{q}} \Vert f \Vert_p \right)^q dx = \Vert f \Vert_p^q \int_0^1 \left( \left( \int_0^1 \vert k(x,y) \vert^q dy \right)^{\frac{1}{q}} \right)^q dx $. The latter integral exists as $k \in L^q([0,1]^2)$. This shows that $K$ is bounded.