Sharp uniform resolvent estimate outside the uniform boundedness range

Let n2n\geq2, define the region R0\mathcal R_0 and points B,E,B,E,D,C,DB,E,B',E',D,C,D' as in the source, and let zC[0,)z\in\mathbb C\setminus[0,\infty). Set

γp,q=max{0,1n+12(1p1q),n+12np,nqn12}.\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

(Δz)1pqp,q,nz1+n2(1p1q)+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.

Sources & referencesView supporting material

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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