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On the behaviour of solutions of the \({\overline \partial}\)-equation on the boundary of the future tube. (English. Russian original) Zbl 0691.32007

Sov. Math., Dokl. 37, No. 1, 83-87 (1988); translation from Dokl. Akad. Nauk SSSR 298, No. 2, 294-298 (1988).
Let \(T^+=\{z\in \mathbb C^ 3: (y^ 0)^ 2>(y^ 1)^ 2+(y^ 2)^ 2, y^ 0>0\}\) be the future tube in complex 3-space \((z^ j=x^ j+y^ j,\, j=0,1,2)\). The author constructs a polynomially compact set \(X^+\) in the boundary of \(T^+\) and a sequence of uniformly bounded \({\overline\partial}\)-closed \((0,1)\) forms \(f_ n\) on a decreasing sequence of strictly pseudoconvex neighborhoods \(U_ n\) of \(X^+\), such that the supremum of any sequence \(v_ n\) of solutions of \({\overline \partial}_{v_ n}=f_ n\) tends to \(\infty\) as \(n\to \infty\). This result is related to the conjecture that there is no sup-norm estimate for solutions of \({\overline\partial}\) on \(T^+\) near the distinguished boundary \(\{z: y=0\}\).
The author’s construction is similar in spirit to an earlier construction of N. Sibony [Invent. Math. 62, 235–242 (1980; Zbl 0429.32026)].
Reviewer: R. M. Range

MSC:

32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)

Citations:

Zbl 0429.32026
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