# Approximations of BMO functions

I couldn't think of a good title. There was a small detail when I was trying to prove something about the norm of BMO functions. This small detail is not directly related to the problem itself, but here it is:

Fix $$Q \in \mathbb{R}^n$$ a cube, and consider the following two classes of functions:

$$\mathcal{F} = \left\{ a(x) - m_Qa: \text{supp}(a) \subset Q, \lVert a \rVert_{L^2(Q)} \leq 1 \right\}$$

$$\mathcal{G} = \left\{ a(x) : \text{supp}(a) \subset Q, \int_{Q} a = 0, \lVert a \rVert_{L^2(Q)} \leq 1 \right\}$$

where $$m_Q a = \vert Q \vert^{-1} \int_Q a(x) dx$$

I want to show that， for any $$h \in L^2(Q)$$ $$\sup_{f \in \mathcal{F}} \ \left\vert \ \int_Q h(x) f(x) \ \right\vert = \sup_{g \in \mathcal{G}} \ \left\vert \ \int_Q h(x) g(x) \ \right\vert$$

In other words can I say that $$\sup_{f \in \mathcal{F}} \ f(x) = \sup_{g \in \mathcal{G}} g(x)$$

• @CalvinKhor You are right that the previous question I posted did not make too much sense and is not what I really wanted to ask. I changed my question. Dec 1, 2022 at 6:29
• Oh, I misunderstood a little. For the last sup question, what is $x$? A fixed point? Dec 1, 2022 at 6:42
• Please explain the acronym BMO. Dec 1, 2022 at 6:59
• @CalvinKhor Yes, but I feel that the last line is a slightly stronger statement. What really mattered is the integration with respect to $h(x)$ part. Dec 1, 2022 at 7:01
• @JeanMarie Bounded Mean Oscillation. Dec 1, 2022 at 7:02