How do we compute this specific $2$ variables integral?

Question

Let $X=\mathcal{C}_{0}(\mathbb{R}^{2})\subset \overline{\mathcal{C_{c}}(\mathbb{R}^{2})}$ the closure in norm $\|\cdot\|_{+\infty}$ of continous functions with compact support in $\mathbb{R}^{2}$. Given a sequence $(a_{n})\subset\mathbb{R}^{+}$, for every $n\in\mathbb{N}$ and $u\in X$ we set: $$T_{n}u=\int_{-a_{n}}^{a_{n}}u(x,nx)dx$$ once shown $T_{n}$ is a linear operator, how do we compute its norm? Initially one may observe that $$|x|\leq a_{n}\qquad\text{and}\qquad|nx|\leq a_{n}\implies |x|\leq \frac{a_{n}}{n}$$ But I'm not sure how to change the first integral: $$\int^{\frac{a_{n}}{n}}_{-\frac{a_{n}}{n}}\int_{-a_{n}}^{a_{n}}u(x,y)dxdy\qquad\text{or}\qquad\frac{1}{n}\int^{\frac{a_{n}}{n}}_{-\frac{a_{n}}{n}}\int_{-a_{n}}^{a_{n}}u(x,y)dxdy$$ the question is actually stupid but it got me confused. This is just the first part of an exercise, and giving the wrong answer would change the rest quite drastically.

Answer 1

The goal is to compute $||T_n|| := \sup_{||u||_\infty=1}\{ |T(u)|\}$ where $||u||_\infty$ is the sup norm on $C_0(\mathbb{R}^2)$

By definition of the sup norm we have $|u(x,nx)| \leq ||u||_\infty$ so we have $$|T_n(u)| = \left| \int_{-a_n}^{a_n} u(x, nx) dx \right| \leq \int_{-a_n}^{a_n} |u(x, nx)| dx \leq \int_{-a_n}^{a_n} ||u||_\infty dx = 2a_n ||u||_\infty$$

So $||T_n||\leq 2a_n$

The next step is to produce a function $u\in C_0(\mathbb{R}^2)$ that makes the above inequalities an equality. For this take any function which is 1 on the rectangle $K = [-a_n, a_n]\times [-na_n, na_n]\subset \mathbb{R}^2$, zero outside an open $O$ containing $K$ and bounded by $1$ everywhere.

Now $$|T_n(u_n)| = \left| \int_{-a_n}^{a_n} u_n(x, nx) dx \right| = \left| \int_{-a_n}^{a_n} 1 dx \right| = 2a_n $$