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Some more sparse bounds for rough and smooth pseudodifferential operators
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics , First View
- Published online by Cambridge University Press:
- 17 April 2026, pp. 1-30
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Beltran & Cladek use
$L^r$ to 
$L^s$ bounds to prove sparse form bounds for pseudodifferential operators with Hörmander symbols in 
$S^m_{\rho,\delta}$ up to, but not including, the sharp end-point in decay 
$m$. We further develop their technique, obtaining pointwise sparse bounds for rough pseudodifferential operators that are merely measurable in their spatial variables and an alternative proof of their results, which avoids proving geometrically decaying sparse bounds. We also provide sufficient conditions for sparse form bounds to hold and use these to reprove known sparse bounds for pseudodifferential operators with symbols in 
$S^0_{1,\delta}$ for 
$\delta \lt 1$.
Improved A1−A∞ and Related Estimates for Commutators of Rough Singular Integrals
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 61 / Issue 4 / November 2018
- Published online by Cambridge University Press:
- 06 August 2018, pp. 1069-1086
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An A1−A∞ estimate, improving on a previous result for [b, TΩ] with
$\Omega \in L^{infty}({\open S}^{n - 1})$ and b∈BMO is obtained. A new result in terms of the A∞ constant and the one supremum Aq−A∞exp constant is also proved, providing a counterpart for commutators of the result obtained by Li. Both of the preceding results rely upon a sparse domination result in terms of bilinear forms, which is established using techniques from Lerner.
