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On Calderón's conjecture for the bilinear Hilbert transform
1School of Mathematics, Georgia Institute of Technology, Atlanta GA 30332, USA.
Summary
This study demonstrates the boundedness of the bilinear Hilbert transform. It maps function spaces Lp and Lq to Lr under specific conditions for p, q, and r.
Area of Science:
- Harmonic Analysis
- Functional Analysis
- Operator Theory
Background:
- The Hilbert transform is a fundamental operator in harmonic analysis.
- Understanding the boundedness of bilinear operators is crucial for extending classical results.
- The behavior of the bilinear Hilbert transform on various function spaces is an active area of research.
Purpose of the Study:
- To establish the boundedness of the bilinear Hilbert transform.
- To determine the precise conditions on the underlying function spaces (Lp, Lq, Lr) for this boundedness.
Main Methods:
- The study employs techniques from harmonic analysis and functional analysis.
- Analysis involves mapping properties between Lebesgue spaces (Lp).
- The core of the method is verifying the operator's boundedness under specific parameter constraints.
Main Results:
- The bilinear Hilbert transform is shown to map from Lp x Lq into Lr.
- The conditions for boundedness are established as 1 < p, q <= infinity, 1/p + 1/q = 1/r, and 2/3 < r < infinity.
- This extends the understanding of multilinear operator theory.
Conclusions:
- The boundedness of the bilinear Hilbert transform is proven for a significant range of function spaces.
- The results contribute to the theory of multilinear operators and their mapping properties.
- This work provides a foundation for further investigations into related operators.