Norm-Controlled Inversion in Measure Algebras

Abstract

We solve Nikolski's norm-controlled inversion problem for measure algebras of locally compact Abelian groups. For every δ>1/2 there is a constant C_M(δ), independent of the group G, such that |μ|_M(G)≤1 and inf_γ∈Ĝ|μ̂(γ)|≥δ imply that μ is invertible in M(G) and |μ⁻¹|_M(G)≤ C_M(δ). Together with Nikolski's negative results, this shows that 1/2 is the sharp universal threshold. The main step is a uniform inversion theorem for Fourier algebras A(K) of compact Abelian groups; its proof is based on a profile decomposition and a resolvent argument. We also obtain norm-controlled inversion for Toeplitz operators with non-vanishing Wiener symbols of winding number zero and give direct endpoint constructions showing failure of norm control at δ=1/2 in A(K) for every infinite compact Abelian group K, and in the unitization of L¹(G) for every nondiscrete locally compact Abelian group G.

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BibTeX RIS

BibTeX

@misc{ohrysko2026normcontrolled,
  title = {{Norm-Controlled Inversion in Measure Algebras}},
  author = {Przemysław Ohrysko},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.27714},
  eprint = {2609.27714},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.27714},
}

RIS

TY  - UNPB
TI  - Norm-Controlled Inversion in Measure Algebras
AU  - Przemysław Ohrysko
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.27714
UR  - https://arxiv.org/abs/2609.27714
AB  - We solve Nikolski's norm-controlled inversion problem for measure algebras of locally compact Abelian groups. For every δ>1/2 there is a constant C_M(δ), independent of the group G, such that |μ|_M(G)≤1 and inf_γ∈Ĝ|μ̂(γ)|≥δ imply that μ is invertible in M(G) and |μ^−1|_M(G)≤ C_M(δ). Together with Nikolski's negative results, this shows that 1/2 is the sharp universal threshold. The main step is a uniform inversion theorem for Fourier algebras A(K) of compact Abelian groups; its proof is based on a profile decomposition and a resolvent argument. We also obtain norm-controlled inversion for Toeplitz operators with non-vanishing Wiener symbols of winding number zero and give direct endpoint constructions showing failure of norm control at δ=1/2 in A(K) for every infinite compact Abelian group K, and in the unitization of L^1(G) for every nondiscrete locally compact Abelian group G.
ER  -

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