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Fock-Sobolev Spaces of Fractional Order

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Abstract

For the full range of index \(0<p\le \infty \), real weight α and real Sobolev order s, two types of weighted Fock-Sobolev spaces over \(\mathbb C^{n}\), \(F^{p}_{\alpha , s}\) and \(\widetilde F^{p}_{\alpha ,s}\), are introduced through fractional differentiation and through fractional integration, respectively. We show that they are the same with equivalent norms and, furthermore, that they are identified with the weighted Fock space \(F^{p}_{\alpha -sp,0}\) for the full range of parameters. So, the study on the weighted Fock-Sobolev spaces is reduced to that on the weighted Fock spaces. We describe explicitly the reproducing kernels for the weighted Fock spaces and then establish the boundedness of integral operators induced by the reproducing kernels. We also identify dual spaces, obtain complex interpolation result and characterize Carleson measures.

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Authors and Affiliations

  1. Department of Mathematics, Pusan National University, Pusan, 609-735, Republic of Korea

    Hong Rae Cho

  2. Department of Mathematics, Korea University, Seoul, 136-713, Republic of Korea

    Boo Rim Choe & Hyungwoon Koo

Authors
  1. Hong Rae Cho
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  2. Boo Rim Choe
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  3. Hyungwoon Koo
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Correspondence to Boo Rim Choe.

Additional information

H. Cho was supported by NRF of Korea(2014R1A1A2056828) and B. Choe was supported by NRF of Korea(2013R1A1A2004736). Also, H. Koo was supported by NRF of Korea(2012R1A1A2000705) and NSFC(11271293).

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Cho, H.R., Choe, B.R. & Koo, H. Fock-Sobolev Spaces of Fractional Order. Potential Anal 43, 199–240 (2015). https://doi.org/10.1007/s11118-015-9468-3

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