The convolution algebra H¹(R)

Abstract

H¹(R) is a Banach algebra which has better mapping properties under singular integrals than L¹(R) . We show that its approximate identity sequences are unbounded by constructing one unbounded approximate identity sequence {v_n}. We introduce a Banach algebra Q that properly lies between H¹ and L¹, and use it to show that c(1 + ln n) ≤ ||v_n||_H¹ ≤ Cn^1/2. We identify the maximal ideal space of H¹ and give the appropriate version of Wiener's Tauberian theorem.