The author proposes a result about the Fourier transforms of spaces \(L^ p(G)\). Let \(G\) be a separable compact group. Fix a normed Haar measure \(dx\) on \(G\) such that \(\int_ G dx=1\). The natural Fourier transform \(\widehat f(\lambda)=\int_ G f(x)\lambda(x^{-1})dx\) of an absolutely integrable function \(f\in L^ 1(G)\) is a operator-valued function on the unitary dual \(\widehat G\) of \(G\). Suppose that there is a neighbourhood \(V\) of the identity element \(1\in G\) such that the restricted function \(f_ V:=f|_ V\) is \(p\)-summable, \(1\leq p\leq 2\), and that \(\widehat f(\lambda)\) is Hermitian non-negative, for every \(\lambda\in\widehat G\). Then the Fourier transform \(\widehat f(\lambda)\) is \(q\)-summable, where \(p^{-1}+q^{-1}=1\). The particular results with the assumption that \(f\) is either \(K\)-biinvariant for a symmetric pair \((G,K)\) or central were previously obtained by T. Kawazoe and H. Myazaki [Tokyo J. Math. 12, No. 1, 241-246 (1989; Zbl 0729.43009)].