For spaces with weak doubling, the symbol norm bounds the Wiener-Hopf operator norm from below; with separated doubling, half the symbol norm bounds the Kuratowski measure of noncompactness.
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Lower estimates for the norm and the Kuratowski measure of noncompactness of Wiener-Hopf type operators
For spaces with weak doubling, the symbol norm bounds the Wiener-Hopf operator norm from below; with separated doubling, half the symbol norm bounds the Kuratowski measure of noncompactness.