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$(Tf)(s)=\int_{0}^{1}k(s,t)f(t)dt$
I want to show that it is a continuous operator $T:L^p([0,1]->L^p([0,1])$
Proof: What I need to show is that $\exists C>0$ with $||Tf||_{L^\infty([0,1]^2)}\le||C||||f||_{L^\infty([0,1])}$, correct?
I take squares: $||Tf||^2_{L^\infty([0,1]^2)}=\mathrm{ess } \sup\int_{0}^{1}(\int_{0}^{1}k(s,t)f(t)dt)^2ds$ right?
Now I would like to use Hölder inequality, but I do not know how, the essential supremum looks confusing.