When does e−/τ/ maximize Fourier extension for a conic section?
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Abstract
In the past decade, much effort has gone into understanding maximizers for Fourier restriction and extension inequalities. Nearly all of the cases in which maximizers for inequalities involving the restriction or extension operator have been successfully identified can be seen as partial answers to the question in the title. In this survey, we focus on recent developments in sharp restriction theory relevant to this question. We present results in the algebraic case for spherical and hyperbolic extension inequalities. We also discuss the use of the Penrose transform leading to some negative answers in the case of the cone.
Abstract
In the past decade, much effort has gone into understanding maximizers for Fourier restriction and extension inequalities. Nearly all of the cases in which maximizers for inequalities involving the restriction or extension operator have been successfully identified can be seen as partial answers to the question in the title. In this survey, we focus on recent developments in sharp restriction theory relevant to this question. We present results in the algebraic case for spherical and hyperbolic extension inequalities. We also discuss the use of the Penrose transform leading to some negative answers in the case of the cone.
Chapters in this book
- Frontmatter I
- Contents V
- Algebraically integrable bodies and related properties of the Radon transform 1
- The covariogram problem 37
- The logarithmic Minkowski conjecture and the Lp-Minkowski problem 83
- Bellman functions and continuous time 119
- Volume product 163
- Inequalities for sections and projections of convex bodies 223
- Borderline estimates for weighted singular operators and concavity 257
- Extremal sections and projections of certain convex bodies: a survey 343
- When does e−/τ/ maximize Fourier extension for a conic section? 391
- Affine surface area 427
- Analysis and geometry near the unit ball: proofs, counterexamples, and open questions 445
- Index 469
Chapters in this book
- Frontmatter I
- Contents V
- Algebraically integrable bodies and related properties of the Radon transform 1
- The covariogram problem 37
- The logarithmic Minkowski conjecture and the Lp-Minkowski problem 83
- Bellman functions and continuous time 119
- Volume product 163
- Inequalities for sections and projections of convex bodies 223
- Borderline estimates for weighted singular operators and concavity 257
- Extremal sections and projections of certain convex bodies: a survey 343
- When does e−/τ/ maximize Fourier extension for a conic section? 391
- Affine surface area 427
- Analysis and geometry near the unit ball: proofs, counterexamples, and open questions 445
- Index 469
