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Problem 3. Let a € (0, 1), ƒ € Co (R"), and u be its Newtonian potential. Show that D²u €

Ca(Rn) by using the following representation formula

+ [ Di¡jG(x−y) f (y) dy,

Diju(x) = |_ Dij¡G(x−y)(ƒ(y)—ƒ(x)) dy+f(x) [_D,G(x−y)v; døy +

where z No (I left the verification of the this formula as an exercise).

Fig: 1