The first author of this paper introduced the special atom space \(B^ p\) for \(p>\). In this paper, the authors first give several descriptions of \(B^ p\) for \(p>1\). Let f be a function which may be decomposed into a finite or countable linear combinations of characteristic functions of intervals in T, this is, \(f(x)=\sum \alpha_ n\chi_{I_ n}(x)\), where T is the unit circle in the plane. Let \(\| f\|_{B^ p}=\inf \sum^{\infty}_{n=1}\| \alpha_ n\chi_{I_ n}\|_ p\), where the infimum is taken over all possible decompositions of f and \(\| \cdot \|_ p\) is the \(L^ p\)-norm on T. Next, let \(\| f\|^*_{B^ p}=\inf \sum^{\infty}_{n=1}\| b_ n\|_ p\), where f a countable sum of special atoms \(b_ n\), and let \(\| f\|^ T_{B^ p}=\inf \sum^{\infty}_{n=1}\| t_ n\|_ p\), where the infimum is taken over all possible representations of f, \(f=\sum^{\infty}_{n=1}t_ n\), by triangular functions \(t_ n\). Also let \(\| f\|^ H_{B^ p}=\inf \sum \| c_{nk}\phi_{nk}\|_ p\), where f is a countable linear combination of Haar functions \(\phi_{nk}\). They show that \(B^ p\) is the set of \(f\in L^ p\) such that \(\| f\|_{B^ p}\), \(\| f\|^*_{B^ p}\), \(\| f\|^ T_{B^ p}\) and \(\| f\|^ H_{B^ p}\) are all finite, and these norms are equivalent to each other in \(B^ p\). The authors also give various consequences of \(B^ p\). Some of these results are the following:
(1) if \(<p<1<q<\infty\) and \(1/q=1/p-1\) then \(f\in B^ p\) if and only if \(F\in B^ p\), where F is the indefinite integral of f,
(2) if \(<p<1\), then \(L^ 1\) of the boundary of the unit disc is continuously embedded in \(B^ p,\)
(3) if \(f\in B^ p\), \(<p<\infty\) and \(\sigma_ n\) is the (C,1) means of the partial sums \(s_ n\) of the Fourier expansion of f then \(\sigma_ n\) tends to f in \(B^ p\).