Let ( R , m ) (R,\mathfrak {m}) be a Noetherian regular local ring of characteristic p > 0 p>0 and let I I be a nonzero ideal of R R . Let D ( − ) = Hom R ⁡ ( − , E ) D(-)= \operatorname {Hom}_R(-, E) be the Matlis dual functor, where E = E R ( R / m ) E = E_R(R/{\mathfrak {m}}) is the injective hull of the residue field R / m R/{\mathfrak {m}} . In this short note, we prove that if H I i ( R ) ≠ 0 {H}^i_I(R)\neq 0 , then Supp R ⁡ ( D ( H I i ( R ) ) ) = Spec ⁡ ( R ) \operatorname {Supp}_R(D({H}^i_{I}(R)))=\operatorname {Spec}(R) .