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A Carleson Problem for the Boussinesq Operator

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Abstract

In this paper, we show that the Boussinesq operator \({{\cal B}_t}f\) converges pointwise to its initial data f ∈ Hs(ℝ) as t → 0 provided \(s \ge {1 \over 4}\)—more precisely—on one hand, by constructing a counterexample in ℝ we discover that the optimal convergence index \({s_{c,1}} = {1 \over 4}\); on the other hand, we find that the Hausdorff dimension of the divergence set for \({{\cal B}_t}f\) is

$${\alpha _{1,{\cal B}}}\left( s \right) = \left\{ {\matrix{{1 - 2s,} \hfill & {{\rm{as}}\,\,{1 \over 4} \le s \le {1 \over 2};} \hfill\cr {1,} \hfill & {{\rm{as}}\,\,0 > s > {1 \over 4}.} \hfill\cr} } \right.$$

Moreover, a higher dimensional lift was also obtained for f being radial.

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Acknowledgements

We thank the referees for their time and comments.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Beijing Technology and Business University, Beijing, 100048, P. R. China

    Dan Li

  2. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, P. R. China

    Jun Feng Li

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  1. Dan Li
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  2. Jun Feng Li
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Correspondence to Jun Feng Li.

Additional information

The second author is supported by NSFC (Grant No. 12071052) and the Fundamental Research Funds for the Central Universities; The first author is supported by the Research Initiation Fund for Young Teachers of Beijing Technology and Business University (Grant No. QNJJ2021-02)

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Li, D., Li, J.F. A Carleson Problem for the Boussinesq Operator. Acta. Math. Sin.-English Ser. 39, 119–148 (2023). https://doi.org/10.1007/s10114-022-1221-4

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  • DOI: https://doi.org/10.1007/s10114-022-1221-4

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