The heat transform on the complex plane. (English) Zbl 07823203
Summary: The heat transform \(H_t\), for positive time \(t>0\), is the convolution on the complex plane with the heat kernel. In the field of analytic function spaces and related operator theory, \(H_t\) coincides with the Berezin transform for the Fock space \(F_t^2\) induced by the Gaussian measure \(e^{-|z|^{2}/t} d\,A(z)\). We study fixed-points of \(H_t\) and the limit behavior of \(H_t \, f\) as \(t\rightarrow 0^+\). Fixed-points of \(H_t\) are shown to be closely related to eigenfunctions of the Laplacian corresponding to certain special eigenvalues, while the limit behavior of \(H_t\,f\) as \(t\rightarrow 0^+\) depends on certain continuity and oscillation properties of \(f\).
The paper is expository, although it contains a few new results. In particular, the main results about fixed-points of \(H_t\) are known, but we present a completely new proof here.
The paper is expository, although it contains a few new results. In particular, the main results about fixed-points of \(H_t\) are known, but we present a completely new proof here.
MSC:
| 30H20 | Bergman spaces and Fock spaces |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 35P99 | Spectral theory and eigenvalue problems for partial differential equations |
References:
| [1] | P. Ahern, M. Flores, and W. Rudin, “An invariant volume-mean-value property”, J. Funct. Anal. 111:2 (1993), 380-397. Digital Object Identifier: 10.1006/jfan.1993.1018 Google Scholar: Lookup Link · Zbl 0771.32006 · doi:10.1006/jfan.1993.1018 |
| [2] | W. Bauer, “Mean oscillation and Hankel operators on the Segal-Bargmann space”, Integral Equations Operator Theory 52:1 (2005), 1-15. Digital Object Identifier: 10.1007/s00020-003-1272-6 Google Scholar: Lookup Link · Zbl 1067.47028 · doi:10.1007/s00020-003-1272-6 |
| [3] | W. Bauer and L. A. Coburn, “Heat flow, weighted Bergman spaces, and real analytic Lipschitz approximation”, J. Reine Angew. Math. 703 (2015), 225-246. Digital Object Identifier: 10.1515/crelle-2015-0016 Google Scholar: Lookup Link · Zbl 1326.32035 · doi:10.1515/crelle-2015-0016 |
| [4] | W. Bauer and L. A. Coburn, “Toeplitz operators with uniformly continuous symbols”, Integral Equations Operator Theory 83:1 (2015), 25-34. Digital Object Identifier: 10.1007/s00020-015-2235-4 Google Scholar: Lookup Link · Zbl 1351.32030 · doi:10.1007/s00020-015-2235-4 |
| [5] | W. Bauer and L. A. Coburn, “Uniformly continuous functions and quantization on the Fock space”, Bol. Soc. Mat. Mex. (3) 22:2 (2016), 669-677. Digital Object Identifier: 10.1007/s40590-016-0108-8 Google Scholar: Lookup Link · Zbl 1356.47034 · doi:10.1007/s40590-016-0108-8 |
| [6] | W. Bauer, B. R. Choe, and H. Koo, “Commuting Toeplitz operators with pluriharmonic symbols on the Fock space”, J. Funct. Anal. 268:10 (2015), 3017-3060. Digital Object Identifier: 10.1016/j.jfa.2015.03.003 Google Scholar: Lookup Link · Zbl 1327.47023 · doi:10.1016/j.jfa.2015.03.003 |
| [7] | W. Bauer, L. A. Coburn, and R. Hagger, “Toeplitz quantization on Fock space”, J. Funct. Anal. 274:12 (2018), 3531-3551. Digital Object Identifier: 10.1016/j.jfa.2018.01.001 Google Scholar: Lookup Link · Zbl 1441.47034 · doi:10.1016/j.jfa.2018.01.001 |
| [8] | F. A. Berezin, “Covariant and contravariant symbols of operators”, Izv. Akad. Nauk SSSR Ser. Mat. 36:5 (1972), 1134-1167. In Russian; translated in Math USSR-Izv. 6:5 (1972), 1117-1151. Digital Object Identifier: 10.1070/IM1972v006n05ABEH001913 Google Scholar: Lookup Link · Zbl 0259.47004 · doi:10.1070/IM1972v006n05ABEH001913 |
| [9] | F. A. Berezin, “Quantization”, Izv. Akad. Nauk SSSR Ser. Mat. 38:5 (1974), 1116-1175. In Russian; translated in Math USSR-Izv. 8:5 (1974), 1109-1165. Digital Object Identifier: 10.1070/IM1974v008n05ABEH002140 Google Scholar: Lookup Link · Zbl 0312.53049 · doi:10.1070/IM1974v008n05ABEH002140 |
| [10] | F. A. Berezin, “General concept of quantization”, Comm. Math. Phys. 40 (1975), 153-174. Digital Object Identifier: 10.1007/BF01609397 Google Scholar: Lookup Link · Zbl 1272.53082 · doi:10.1007/BF01609397 |
| [11] | F. A. Berezin, “Quantization in complex symmetric spaces”, Izv. Akad. Nauk SSSR Ser. Mat. 39:2 (1975), 363-402. In Russian; translated in Math USSR-Izv. 9:2 (1975), 341-379. Digital Object Identifier: 10.1070/IM1975v009n02ABEH001480 Google Scholar: Lookup Link · Zbl 0324.53049 · doi:10.1070/IM1975v009n02ABEH001480 |
| [12] | F. A. Berezin, “The connection between covariant and contravariant symbols of operators on classical complex symmetric spaces”, Dokl. Akad. Nauk SSSR 241:1 (1978), 15-17. In Russian; translated in Soviet Math. Dokl. 19 (1978), 786-789. · Zbl 0439.47038 |
| [13] | C. A. Berger and L. A. Coburn, “Heat flow and Berezin-Toeplitz estimates”, Amer. J. Math. 116:3 (1994), 563-590. Digital Object Identifier: 10.2307/2374991 Google Scholar: Lookup Link · Zbl 0839.46018 · doi:10.2307/2374991 |
| [14] | I. Casseli, “Fixed points of the Berezin transform of multidimensional polyanalytic Fock spaces”, J. Math. Anal. Appl. 481:2 (2020), art. id. 123479. Digital Object Identifier: 10.1016/j.jmaa.2019.123479 Google Scholar: Lookup Link · Zbl 1431.32001 · doi:10.1016/j.jmaa.2019.123479 |
| [15] | M. Engliš, “Functions invariant under the Berezin transform”, J. Funct. Anal. 121:1 (1994), 233-254. Digital Object Identifier: 10.1006/jfan.1994.1048 Google Scholar: Lookup Link · Zbl 0807.47017 · doi:10.1006/jfan.1994.1048 |
| [16] | K. Zhu, Operator theory in function spaces, 2nd ed., Mathematical Surveys and Monographs 138, Amer. Math. Soc., Providence, RI, 2007. Digital Object Identifier: 10.1090/surv/138 Google Scholar: Lookup Link · Zbl 1123.47001 · doi:10.1090/surv/138 |
| [17] | K. Zhu, Analysis on Fock spaces, Graduate Texts in Mathematics 263, Springer, 2012. Digital Object Identifier: 10.1007/978-1-4419-8801-0 Google Scholar: Lookup Link · Zbl 1262.30003 · doi:10.1007/978-1-4419-8801-0 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
