Not explicitely, but it is implied that Cooper's diving into the black hole with the ship enables him to gather the missing data needed for "the equation" to be completed. Script:
DR MANN: Amelia, your father solved his equation before I even left.
BRAND: Then why wouldn’t he use it?!
DR MANN: The equation couldn’t
reconcile relativity with quantum mechanics. You need more -
COOPER: More what?!
DR MANN: More data. You need to see inside a black hole.
And the laws of nature prohibit a naked singularity.
Romilly): Is that true?
ROMILLY: If a black hole is an oyster, the
singularity is the pearl inside. Its gravity is so strong, it’s always
hidden in darkness, behind the horizon. That’s why we call it a black
COOPER: If we could look beyond the horizon
And then,in the end, all we get is this hint:
MURPH: People didn’t believe me, they thought I’d done it all myself ... (Taps watch.)
Where "it" is "solving the gravity problem" or "saving Earth". After all, it's just a macguffin for all we are concerned. More details on how the communication is achieved are already in this excellent answer, so I won't repeat them here.