TL;DR
This paper constructs Fourier multipliers that are weak (1,1) bounded but not weak (p,p) bounded for any p > 1, demonstrating the sharpness of this property.
Contribution
It introduces a class of Fourier multipliers with specific weak boundedness properties and proves the sharpness of these results.
Findings
Constructed Fourier multipliers with weak (1,1) boundedness
Showed these operators are not weak (p,p) bounded for p > 1
Proved the sharpness of the boundedness results
Abstract
We construct a class of Fourier multipliers whose associated operators are weak (1,1) bounded but fail to be weak (p, p) bounded for any 1 < p \leq \infty. Moreover, we show that this result is sharp.
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Taxonomy
TopicsAdvanced Harmonic Analysis Research · Holomorphic and Operator Theory · Advanced Banach Space Theory