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In this example:
Barack Obama on food and climate change: 'We can still act and it won't be too late'.
what does “Obama on food and climate change” mean?
Barack Obama on food and climate change: XXX.
This structure simply implies that XXX concerns food and climate change, that XXX is an opinion about this subject. WordHippo defines "on" as
Dealing with the subject of
and provides synonyms as about, concerning, regarding, having to do with, touching and many others.
A on B: XXX is a formal structure you would expect to read in newspapers or formal written texts, when quoting an author for example. Certainly, it can be used in less formal situations ironically. Referring to my friend John, I could say:
Mr John Smith on the intricate ways of smuggling money from one's parents: XXX
Does it imply the person is a subject matter expert? Not necessarily.
I found other instances of this structure online:
Biden on climate crisis: 10 executive actions president-elect is taking on first day (Independent)
OR
Trump on climate change report: 'I don't believe it'
from BBC News
That doesn't make Mr Trump an expert on climate change.
See also another way of putting it in this title of an article from Reuters:
Trump, on climate, says he won't jeopardize U.S. wealth on 'dreams'
