@JochenWengenroth: my most sincere apologies for asking.

@OlivierBégassat: the choice of $\epsilon > 0$ will depend on $f$, as in the formulation of the question. If it belongs to the Lebesgue space with exponent $p+\epsilon$, then it will belong to all Lebesgue spaces with smaller exponent.

@G.Blaickner : thank you so much. The equivalence to the Poincare conjecture is honestly quite shocking since many people will intuitively take "combinatorial" for granted.

Could you please provide a few references for the claims in your first paragraph?

Explain the downvote. It is a reasonable question.

It's nice that you mention projections onto a one-parameter family of functions. That's how you can motivate the Fourier transform.