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 {vn}. We introduce a Banach algebra Q that properly lies between H1 and L1, and use it to show that c(1 + ln n) ≤ ||vn|| ≤ Cn1/2. We identify the maximal ideal space of H1 and give the appropriate version of Wiener′s Tauberian theorem.
Highlights
The Hardy space H1(R) is known to have better mapping properties than L1(R); for example the Hilbert transform and other singular integral operators are unbounded on L1(R), but are bounded on H1(R)
We investigate whether the properties of H1(R) as a convolution algebra are better than those of L1(R), in particular for spectral synthesis
We give a bound for the rate of increase that depends on the behavior of the approximate identity sequences in a convolution algebra Q lying between H1(R) and L1(R), but the bounds are not sharp
Summary
The Hardy space H1(R) is known to have better mapping properties than L1(R); for example the Hilbert transform and other singular integral operators are unbounded on L1(R), but are bounded on H1(R). We give a bound for the rate of increase that depends on the behavior of the approximate identity sequences in a convolution algebra Q lying between H1(R) and L1(R), but the bounds are not sharp. We have since discovered an explanation for the above results (except for the bound on the growth of the norms) using essential ideals. We will discuss this in a paper to appear ([6])
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