A note on the dyadic Hardy spaces

Abstract. The usual \(L^p\)-norms are trivially invariant with respect to the multiplication by Walsh functions. The analogous question will be investigated in the dyadic Hardy space \({\bf H}\). We introduce an invariant subspace \({\bf H_*}\) of \({\bf H}\) in this sense and show some properties of \({\bf H_*}\). For example a function in \({\bf H_*}\) will be constructed such that its Walsh–Fourier series diverges in \(L^1\)-norm.