Sharp uniform resolvent estimate outside the uniform boundedness range

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Let n≥2n\geq2, define the region R0\mathcal R_0 and points B,E,B′,E′,D,C,D′B,E,B',E',D,C,D' as in the source, and let z∈C∖[0,∞)z\in\mathbb C\setminus[0,\infty). Set

γp,q=max⁡{0,1−n+12(1p−1q),n+12−np,nq−n−12}.\gamma_{p,q}=\max\left\{0,1-\frac{n+1}{2}\left(\frac1p-\frac1q\right),\frac{n+1}{2}-\frac np,\frac nq-\frac{n-1}{2}\right\}.

Sharp resolvent conjecture. If (1p,1q)∈R0∖([B,E]∪[B′,E′]∪[D,C)∪[D′,C))(\frac1p,\frac1q)\in\mathcal R_0\setminus([B,E]\cup[B',E']\cup[D,C)\cup[D',C)), then

∥(−Δ−z)−1∥p→q≃p,q,n∣z∣−1+n2(1p−1q)+γp,qdist⁡(z,[0,∞))−γp,q.\|(-\Delta-z)^{-1}\|_{p\to q}\simeq_{p,q,n}|z|^{-1+\frac n2(\frac1p-\frac1q)+\gamma_{p,q}}\operatorname{dist}(z,[0,\infty))^{-\gamma_{p,q}}.

The conjecture concerns sharp LpL^p-to-LqL^q resolvent bounds outside the uniform boundedness range. The source does not state whether this precise claim has been resolved.

References

Primary source

Chuanwei Gao, Jingyue Li and Liang Wang, “A type of oscillatory integral operator and its applications”, arXiv:2201.01021 (2022).

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