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Sharp Resolvent Estimates Outside of the Uniform Boundedness Range

Abstract

In this paper we are concerned with resolvent estimates for the Laplacian \(\Delta \) in Euclidean spaces. Uniform resolvent estimates for \(\Delta \) were shown by Kenig et al. (Duke Math J 55(2):329–347, 1987) who established rather a complete description of the Lebesgue spaces allowing such estimates. However, the problem of obtaining sharp \(L^p\)\(L^q\) bounds depending on z has not been considered in a general framework which admits all possible pq. In this paper, we present a complete picture of sharp \(L^p\)\(L^q\) resolvent estimates, which may depend on z. We also obtain the sharp resolvent estimates for the fractional Laplacians and a new result for the Bochner–Riesz operators of negative index.

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Notes

  1. 1.

    The midpoint of the line segment \(AA'\) in Fig. 2.

  2. 2.

    Sharpness here refers to the optimal dependence of \(\Vert (-\Delta -z)^{-1}\Vert _{p\rightarrow q}\) on the spectral parameter z.

  3. 3.

    \({\mathbb {S}}^1:=\{z\in {\mathbb {C}}: |z|=1\}\).

  4. 4.

    \({\mathcal {P}}=[(1,0), E, B, B', E']{\setminus } ([E, B]\cup [E', B'] )\), \({\mathcal {T}}=[B, D, D', B']{\setminus } [B, B']\), and \({\mathcal {Q}}=[(0,0), D, B, E]{\setminus }([D, B]\cup [B, E])\). See Figs. 1 and 2.

  5. 5.

    Here we choose the branch of \(\sqrt{z}\), \(z\in {\mathbb {C}}{\setminus }[0,\infty )\), such that the imaginary part is positive. Note that \(\Xi _\delta =\{z\in {\mathbb {C}}{\setminus }[0,\infty ): (\mathop {\mathrm {Im}}z)^2\ge 4\delta ^2(\mathop {\mathrm {Re}}z+\delta ^2) \}\). In the complex plane this region excludes a neighborhood of the origin and a parabolic region opening to the right.

  6. 6.

    This is equivalent to saying that the matrix \(\partial _u^2\langle v, \partial _x\Phi \rangle \) is either positive or negative definite.

  7. 7.

    When \(d=2\) this is \(1/q<1/4\). When \(d\ge 3\) this is equivalent to saying that (1 / p, 1 / q) lies strictly below the line passing through the points \(P_*\) and \(P_\circ \). See Figs. 3 and 4.

  8. 8.

    For a proof of existence of such \(\phi \) we refer the reader to [30, Lemma 2.2]. Also, see [12, Lemma 2.1].

  9. 9.

    If \(\gamma _{p,q}>0\) and \(\mathop {\mathrm {Re}}z\ge |\mathop {\mathrm {Im}}z|\gg 1\), then the region is roughly determined by \(|\mathop {\mathrm {Im}}z|\gtrsim (\mathop {\mathrm {Re}}z)^{\frac{\gamma _{p,q}-\omega _{p,q}}{\gamma _{p,q}}}\). Likewise, if \(\gamma _{p,q}>0\) and \(0<\mathop {\mathrm {Re}}z< |\mathop {\mathrm {Im}}z|\), \(|\mathop {\mathrm {Im}}z|\gtrsim \ell ^{-1/\omega _{p,q}} \) for \(z\in {\mathcal {Z}}_{p,q}(\ell )\).

  10. 10.

    We recall from the introduction that \(\widetilde{{\mathcal {R}}}_{3,\pm }=\{(\frac{1}{p},\frac{1}{q}):\pm (\gamma _{p,q}-\omega _{p,q})<0\}\) and \(\widetilde{\mathcal R}_{3,0}=\{(\frac{1}{p},\frac{1}{q}):\gamma _{p,q}-\omega _{p,q}=0\}\).

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Acknowledgements

The authors were supported by NRF-2018R1A2B2006298. We would like to thank Ihyeok Seo and Younghun Hong for discussions on related issues, and the anonymous referees for various helpful comments.

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Correspondence to Yehyun Kwon.

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Kwon, Y., Lee, S. Sharp Resolvent Estimates Outside of the Uniform Boundedness Range. Commun. Math. Phys. (2019). https://doi.org/10.1007/s00220-019-03536-y

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