Hilbert transform is bounded
By Parseval's theorem
{ L^{2}} -> { L^{2}}.
less obvious fact
bounded { L^{1}} -> {L^{1,w}}.
By
#Marcinkiewicz's bounded { L^{p}} -> { L^{p}} : 1 < p < 2.
By Duality arguments
bounded : 2 < p < ∞.
really unbounded for p= 1 or ∞.
Hilbert transform is bounded
By Parseval's theorem
{ L^{2}} -> { L^{2}}.
less obvious fact
bounded { L^{1}} -> {L^{1,w}}.
By
#Marcinkiewicz's bounded { L^{p}} -> { L^{p}} : 1 < p < 2.
By Duality arguments
bounded : 2 < p < ∞.
really unbounded for p= 1 or ∞.