Problem related to $L^p$ space Problem Let $k: \mathbb R^{d\times d} \to \mathbb R^d$ be a measurable function such that there is $c>0$ with $$\sup_{x \in \mathbb R^d}\int |k(x,y)|dy \leq c, \space \sup_{y \in \mathbb R^d}\int |k(x,y)|dx \leq c$$
Show that for $1<p<\infty$, the function $K:L^p(\mathbb R^d) \to L^p(\mathbb R^d)$ given by $$K(f)(x)=\int k(x,y)f(y)dy$$
is well defined and uniformly continuous.
I got stuck trying to show both properties. First I would like to prove that $K(f) \in L^p(\mathbb R^d)$ for each $f \in L^p(\mathbb R^d)$, so take a function $f$ from that space and $$\int |K(f)(x)|^pdx=\int |\int (k(x,y)fy)dy|^pdx$$
I thought of applying Hölder's inequality to get $$\int |\int (k(x,y)fy)dy|^pdx \leq \int(\int |k(x,y)|^qdy)^{\frac{p}{q}})(\int|f(y)|^pdy)dx$$$$=\int|f(y)|^pdy(\int(\int |k(x,y)|^qdy)^{\frac{p}{q}}dx)$$
I don't know where to go from there.
As for uniform continuity I am lost. I would appreciate hints and suggestions to prove the two properties of the exercise. Thanks in advance.

Use (Riesz-Thorin) interpolation. Show that $\|K\|_{L^1(\Bbb R^d) \to L^1(\Bbb R^d)} \le c$ using $\sup_y \int |k(x,y)|\, dx \le c$, and show that $\|K\|_{L^\infty(\Bbb R^d)\to L^\infty(\Bbb R^d)} \le c$ using $\sup_x \int |k(x,y)|\, dy \le c$. These imply $\|K\|_{L^p(\Bbb R^d) \to L^p(\Bbb R^d)} \le c$ for all $1 < p < \infty$ by interpolation. This also gives well-definedness. For all $f,g\in L^p(\Bbb R^d)$, $$\|K(f) - K(g)\|_{L^p(\Bbb R^d)} = \|K(f-g)\|_{L^p(\Bbb R^d)} \le c\|f - g\|_{L^p(\Bbb R^d)}$$ So $K$ is uniformly continuous.
