| EPSRC Reference: |
GR/S27009/02 |
| Title: |
Restriction Theorems for the Fourier Transform, and Applications |
| Principal Investigator: |
Bennett, Professor J |
| Other Investigators: |
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| Researcher Co-Investigators: |
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| Project Partners: |
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| Department: |
School of Mathematics |
| Organisation: |
University of Birmingham |
| Scheme: |
Postdoc Res Fellowship PreFEC |
| Starts: |
01 January 2005 |
Ends: |
30 September 2005 |
Value (£): |
33,957
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| No relevance to Underpinning Sectors |
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| Panel History: |
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Summary on Grant Application Form |
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OUR PROPOSED RESEARCH IS DRIVEN BY TWO CLOSELY RELATED OPEN PROBLEMS IN MODERN HARMONIC ANALYSIS: THE RESTRICTION CONJECTURE, AND STEIN'S CONJECTURE. BOTH CONJECTURES CONCERN THE OPERATOR OBTAINED BY RESTRICTING THE FOURIER TRANSFORM TO THE UNIT SPHERE - AN OPERATOR WITH IMPORTANT CONNECTIONS WITH A DIVERSITY OF PROBLEMS IN ANALYSIS AND PARTIAL DIFFERENTIAL EQUATIONS.WE PROPOSE NEW APPROACHES TO THESE PROBLEMS MOTIVATED BY SOME INSPIRING RESULTS OF JUST THIS YEAR. FIRSTLY WE FORMULATE A MULTILINEAR ANALOGUE OF THE RESTRICTION CONJECTURE, AND DESCRIBE ITS PLACE IN A SCHEME TO PROVE SHARP THEOREMS IN ACCORDANCE WITH THE RESTRICTION CONJECTURE (THIS IS PARTLY BASED ON A BILINEAR APPROACH OF TAO, VARGAS AND VEGA FROM 1998).SECONDLY WE PROPOSE A TIME - FREQUENCY ANALYSIS APPROACH TO CERTAIN VARIANTS OF STEIN'S CONJECTURE IN ALL DIMENSIONS. OUR RECENT RESULTS IN TWO DIMENSIONS THEN LEAD US TO POSE SOME ORIGINAL GENERAL QUESTIONS CONCERNING THE CONTROL OF OSCILLATORY INTEGRAL OPERATORS BY MAXIMAL FUNCTIONS - AN IDEA THAT FITS INTO NO CLASSICAL THEORY IN HARMONIC ANALYSIS.FINALLY WE PROPOSE SOME EXPLICIT APPLICATIONS TO THE WELL-POSEDNESS THEORY OF SCHRODINGER EQUATIONS.
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Key Findings |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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Potential use in non-academic contexts |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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Impacts |
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| Description |
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk |
| Summary |
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| Date Materialised |
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Sectors submitted by the Researcher |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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| Project URL: |
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| Further Information: |
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| Organisation Website: |
http://www.bham.ac.uk |