Extension of bounded linear operator: a conceptual question

Question

Let $Tf(x)=\int_{\mathbb R} K(x,y)f(y)dy$ where $K(x,y):\mathbb R\times \mathbb R\to \mathbb C$ is some measurable integration kernel. Suppose there are two dense subspaces of test functions $D_1$ and $D_2$ of $L^p(\mathbb R)$ such that for all $f\in D_i$, $i=1,2$, the representation given by $Tf$ is well defined. Also, assume $\|Tf\|_q\leq C\|f\|_p$ for all $f\in D_1$. Then is it true that for all $f\in D_2$, we also have $\|Tf\|_q\leq C\|f\|_p$?

Answer 1

These are considerations of measure theory.

Essentially:

Let $f_n$ in $D_1$ approximate some $f\in D_2$. Then $\|Tf_n\|_q≤ C \|f_n\|_p$, where the right-hand side is bounded so you can use some lemma from integration theory (for example Fatou) to see that $\|Tf\|_q ≤ \liminf_n C\|f_n\|_p=C\|f\|_p$.

To carry out the details:

Let $f_n\in D_1$ be such that $\|f_n -f \|_p\to0$ and $f_n\to f$ pointwise ae with $|f_n| ≤ |f|$. Then for every $x$ $K(x,y)f_n(y)$ converges to $K(x,y)f(y)$ pw ae and is dominated by $|K(x,y) f(y)|$ hence you get also that $$\int K(x,y) f_n(y)\, dy \to \int K(x,y) f(y)\,dy.$$ Now we have just said that $Tf_n\to Tf$ pointwise, while we know that $\|T f_n\|_q≤ C\|f_n\|_p$. Since $f_n$ is convergent in $L^p$ the norms on the right have $\liminf$ being $C\|f\|_q$, so Fatou gives you the inequality $\|Tf\|_q=\|g\|_q ≤ C\|f\|_p$.