Optimal Flat Functions in Carleman-Roumieu Ultraholomorphic Classes in Sectors.

Area of Science:

• Complex Analysis
• Harmonic Analysis
• Functional Analysis

Background:

• Carleman-Roumieu classes define sequences of functions with specific growth properties.
• Ultraholomorphic functions are generalizations of holomorphic functions with broader applicability.
• Nonquasianalytic weight sequences are key to constructing these function classes.

Purpose of the Study:

• To construct optimal flat functions within Carleman-Roumieu ultraholomorphic classes.
• To develop linear continuous extension operators for these classes.
• To explore explicit constructions for specific weight sequences, including q-Gevrey.

Main Methods:

• Construction of optimal flat functions using general strongly nonquasianalytic weight sequences.
• Development of a general procedure for linear continuous extension operators.
• Analysis of regular weight sequences in the sense of Dyn'kin.
• Application to specific examples like the q-Gevrey case.

Main Results:

• Successful construction of optimal flat functions in specified ultraholomorphic classes.
• Establishment of a general method for creating extension operators, which act as right inverses of the Borel map.
• Demonstration of explicit constructions for certain classes of weight sequences.

Conclusions:

• The study provides a framework for understanding and constructing flat functions in ultraholomorphic settings.
• The developed extension operators offer significant tools for further research in complex analysis and related fields.
• The explicit examples highlight the practical applicability of the theoretical constructions.