# Show that $\lVert Tf\rVert_p \leq C \lVert f \rVert_p$ for $p \in [1, \infty)$

I want to show the following statement:

Let $$(X,\mathcal{A}, \mu)$$ be a measurable space, $$K: X \times X \rightarrow \mathbb{R}$$ measurable. For real-valued functions $$f$$, we define $$Tf(x)=\int_X K(x,y)f(y) \mu(dy)$$.

If there exists a constant $$C \in \mathbb{R}$$ such that

$$\int_X |K(x,y)| \mu(dy) \leq C$$ for every $$x$$ and

$$\int_X |K(x,y)| \mu(dx) \leq C$$ for every $$y$$,

Show that for $$1\leq p <\infty$$, $$\lVert Tf \rVert_p \leq C \lVert f \rVert_p$$.

My attempt:

\begin{align*} \|Tf\|_p &= \left(\int_{X} |Tf(x)|^p \mu(dx) \right)^{\frac{1}{p}} \\ &= \left(\int_{X} \left|\int_X K(x,y) f(y) \mu(dy) \right|^p \mu(dx) \right)^{\frac{1}{p}} \\ &\leq \left(\int_{X} \int_X |K(x,y)|^p |f(y)|^p \mu(dy) \mu(dx) \right)^{\frac{1}{p}} \\ &\leq \left(\int_{X} \int_X C^p |f(y)|^p \mu(dy) \mu(dx) \right)^{\frac{1}{p}} \\ &\leq C \left(\int \int_X |f(y)|^p \mu(dy) \mu(dx) \right)^{\frac{1}{p}}. \end{align*}

My problem is, that I don't know how to get rid of the inner integral $$\int_X \mu(dy)$$. Further, I didn't use both conditions that, i.e. $$\int |K(x,y)| \leq C$$ is bounded for all $$x$$ if $$y$$ is fixed, and for all $$y$$ if $$x$$ is fixed. So to me, it seems that this approach is the wrong one. I have a feeling that I somehow need to use the topic of "bounded linear functionals", to find a solution. But I couldn't think of another one.

Let $$q$$ be the conjugate of $$p$$. Notice \begin{aligned} \int_X\left|K(x,y)f(y)\right|\mu(dy) &=\int_X\left|K(x,y)\right|^{1/q}\left|K(x,y)\right|^{1/p}\left|f(y)\right|\mu(dy)\\ &\le \left(\int_X\left|K(x,y)\right|\mu(dy)\right)^{1/q}\left(\int_X\left|K(x,y)\right|\left|f(y)\right|^p\mu(dy)\right)^{1/p}. \end{aligned}
From here, you may write \begin{aligned} \int_X\left|Tf(x)\right|^p\mu(dx) &\le C^{p/q}\int_X\left(\int_X\left|K(x,y)\right|\left|f(y)\right|^p\mu(dy)\right)\mu(dx)\\ &\le C^{p/q}\int_X\left(\int_X\left|K(x,y)\right|\mu(dx)\right)\left|f(y)\right|^p\mu(dy)\\ &\le C^{p/q}\int_XC\left|f(y)\right|^p\mu(dy)\\ &=C^{p}\int_X\left|f(y)\right|^p\mu(dy), \end{aligned} which yields \begin{aligned} \left\|Tf(x)\right\|_p \le C\left\|f\right\|_p. \end{aligned}