Sergeev, A. G. Estimates of integral representations in a future tube. (English. Russian original) Zbl 0598.32006 Russ. Math. Surv. 40, No. 4, 215-216 (1985); translation from Usp. Mat. Nauk 40, No. 4(244), 195-196 (1985). The author considers the future light cone \[ V^+ = \{y=(y_ 0,y_ 1,...,y_ n)\in {\mathbb{R}}^{n+1}: y_ 0>0, y^ 2_ 0>\sum^{n}_{j=1}y^ 2_ j\} \] and the associated tube domain \(\tau^+:={\mathbb{R}}^{n+1}+iV^+\). For holomorphic functions \(f: \tau\) \({}^+\to {\mathbb{C}}\) one has the Cauchy-Fantappié-Bochner, Jost-Lehmann- Dyson and the Bergman integral representations. These integral representations are considered respectively as Cauchy-Fantappié- Bochner, Jost-Lehmann-Dyson and Bergman transforms on the space \(L^{\infty}_{comp}\) of \(L^{\infty}({\mathbb{R}}^{n+1})\) functions with compact support. The author states local and global estimates for these transforms. Analogous estimates hold for solutions of the \({\bar \partial}\)-problem. Compare also the author’s article in Teor. Mat. Fiz. 54, No.1, 99-110 (1983; Zbl 0529.32001) and B. Jöricke in Math. Nachr. 112, 227-244 (1983; Zbl 0579.32006). Reviewer: R.Braun MSC: 32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.) 32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010) 32A22 Nevanlinna theory; growth estimates; other inequalities of several complex variables 44A20 Integral transforms of special functions 32A40 Boundary behavior of holomorphic functions of several complex variables 32D05 Domains of holomorphy 32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators 35N15 \(\overline\partial\)-Neumann problems and formal complexes in context of PDEs Keywords:special domains; Lie ball; future tube; integral representations; Cauchy- Fantappié-Bochner; Jost-Lehmann-Dyson; Bergman transforms; estimates; \({\bar \partial }\)-problem Citations:Zbl 0529.32001; Zbl 0579.32006 PDFBibTeX XMLCite \textit{A. G. Sergeev}, Russ. Math. Surv. 40, No. 4, 215--216 (1985; Zbl 0598.32006); translation from Usp. Mat. Nauk 40, No. 4(244), 195--196 (1985) Full Text: DOI