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If [itex]f : X \to \mathbb{C}[/itex] is a measurable function, define the essential supremum of f to be:

[tex]||f||_{\infty} = \inf \{a \in [0,\infty ] : \mu (\{x : |f(x)| > a\}) = 0\}[/tex]

where [itex]\mu[/itex] is a measure, and we adopt the convention [itex]\inf \emptyset = \infty[/itex]. Note that

[tex]||f||_{\infty} \in \{a \in [0,\infty ] : \mu (\{x : |f(x)| > a\}) = 0\}[/tex]

If f has finite essential supremum, we say f is an [itex]L^{\infty}[/itex] function. The set of [itex]L^{\infty}[/itex] functions forms a Banach space and [itex]||.||_{\infty}[/itex] defines a norm on this space. So if f and g are [itex]L^{\infty}[/itex] functions, then so is f+g, and the following inequality holds:

[tex]||f+g||_{\infty} \leq ||f||_{\infty} + ||g||_{\infty}[/tex]

(Note: We will treat two functions as identical if the subset of the domain on which they differ has measure 0. All the terms defined above remain well-defined upon adopting this convention.)

Problem

When does equality hold in the above inequality?

Attempt

Define [itex]z : X \to C[/itex] where C is the complex circle by:

[tex]z(x) = \frac{|f(x)|}{f(x)} \mbox{ if } f(x) \neq 0;\ z(x) = 1\mbox{ if } f(x) = 0[/tex]

Then fz is a non-negative real-valued function, and

[tex]|fz| - |f| = |gz| - |g| = |(f+g)z| - |f+g| = 0[/tex]

hence

[tex]||fz||_{\infty} - ||f||_{\infty} = ||gz||_{\infty} - ||g||_{\infty} = ||(f+g)z||_{\infty} - ||f+g||_{\infty} = 0[/tex]

So assume w.l.o.g. that f is a non-negative real-valued function. Let A and B denote the essential suprema of f and g respectively. Right now my rough idea is that we get the desired equality iff for all a < A, for all b < B, and for all c > 0, the following holds:

[tex]\mu ( \{x : f(x) > a, |g(x)| > b, ||g(x)| - g(x)| < c|g(x)|\} ) > 0[/tex]

It basically says that equality holds iff there is a sizeable region of the domain where f is close to its maximum, |g| is close to its maximum, and g is close to being a positive real. Is this right? Is there a nicer way to put it?

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# Minkowski's Inequality

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