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  • HONG RAE CHO (a1) and SOOHYUN PARK (a2)


Let $s\in \mathbb{R}$ and $0<p\leqslant \infty$ . The fractional Fock–Sobolev spaces $F_{\mathscr{R}}^{s,p}$ are introduced through the fractional radial derivatives $\mathscr{R}^{s/2}$ . We describe explicitly the reproducing kernels for the fractional Fock–Sobolev spaces $F_{\mathscr{R}}^{s,2}$ and then get the pointwise size estimate of the reproducing kernels. By using the estimate, we prove that the fractional Fock–Sobolev spaces $F_{\mathscr{R}}^{s,p}$ are identified with the weighted Fock spaces $F_{s}^{p}$ that do not involve derivatives. So, the study on the Fock–Sobolev spaces is reduced to that on the weighted Fock spaces.



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The author was supported by NRF of Korea (NRF-2016R1D1A1B03933740).



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[1] Aronszajn, N., Theory of reproducing kernels , Trans. Amer. Math. Soc. 68(3) (1950), 337404.
[2] Bongioanni, B. and Torrea, J. L., Sobolev spaces associated to the harmonic oscillator , Proc. Indian Acad. Sci. Math. Sci. 116 (2006), 337360.
[3] Cho, H. R., Choe, B. R. and Koo, H., Linear combinations of composition operators on the Fock–Sobolev spaces , Potential Anal. 41 (2014), 12231246.
[4] Cho, H. R., Choe, B. R. and Koo, H., Fock–Sobolev spaces of fractional order , Potential Anal. 43 (2015), 199240.
[5] Cho, H. R., Choi, H. and Lee, H.-W., Boundedness of the Segal–Bargmann Transform on Fractional Hermite–Sobolev Spaces , J. Funct. Spaces 2017 (2017), Article ID 9176914, 6 pages.
[6] Cho, H. R. and Zhu, K., Fock–Sobolev spaces and their Carleson measures , J. Funct. Anal. 263(8) (2012), 24832506.
[7] Choe, B. R. and Yang, J., Commutants of Toeplitz operators with radial symbols on the Fock–Sobolev space , J. Math. Anal. Appl. 415(2) (2014), 779790.
[8] Hall, B. and Lewkeeratiyutkul, W., Holomorphic Sobolev spaces and the generalised Segal–Bargmann transform , J. Funct. Anal. 217 (2004), 192220.
[9] Mengestie, T., Schatten class weighted composition operators on weighted Fock spaces , Arch. Math. (Basel) 101(4) (2013), 349360.
[10] Mengestie, T., Volterra type and weighted composition operators on weighted Fock spaces , Integral Equations Operator Theory 76(1) (2013), 8194.
[11] Mengestie, T., On trace ideal weighted composition operators on weighted Fock spaces , Arch. Math. (Basel) 105(5) (2015), 453459.
[12] Radha, R. and Thangavelu, S., Holomorphic Sobolev spaces, Hermite and special Hermite semigroups and a Paley–Wiener theorem for the windowed Fourier transform , J. Math. Anal. Appl. 354(2) (2009), 564574.
[13] Wang, X. F., Cao, G. F. and Xia, J., Toeplitz operators on Fock–Sobolev spaces with positive measure symbols , Sci. China Math. 57(7) (2014), 14431462.
[14] Zhu, K. H., Operator Theory in Function Spaces, Monographs and Textbooks in Pure and Applied Mathematics 139 , Marcel Dekker, Inc., New York, 1990, xii+258 pp.
[15] Zhu, K., Analysis on Fock spaces, Graduate Texts in Mathematics 263 , Springer, New York, 2012, x+344 pp.
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  • HONG RAE CHO (a1) and SOOHYUN PARK (a2)


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