# A Constrained Optimization Problem for the Fourier Transform: Quantitative Analysis

A Constrained Optimization Problem for the Fourier Transform: Quantitative Analysis JGeomAnal https://doi.org/10.1007/s12220-018-0038-y A Constrained Optimization Problem for the Fourier Transform: Quantitative Analysis Dominique Maldague Received: 15 March 2018 © Mathematica Josephina, Inc. 2018 Abstract Among functions f majorized by indicator functions 1 , which functions 1/p have maximal ratio  f  /|E | ? We establish a quantitative answer to this question for exponents q sufﬁciently close to even integers, building on previous work proving the existence of such maximizers. Keywords Harmonic analysis · Extremization · Fourier transform · Quantitative analysis Mathematics Subject Classiﬁcation 42A16 · 42B10 1 Introduction −2πix ·ξ Deﬁne the Fourier transform as F ( f )(ξ ) = f (ξ ) = e f (x )dx for a d 1 d function f : R → C. The Fourier transform is a contraction from L (R ) to ∞ d 2 d L (R ) and is unitary on L (R ). Interpolation gives the Hausdorff–Young inequal- 1 1 ity  f    f  , where p ∈ (1, 2),1 = + .In[1], Beckner proved the sharp q p p q Hausdorff–Young inequality f   C  f  , (1.1) q p 1/2 p −1/2q where C = p http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The Journal of Geometric Analysis Springer Journals

# A Constrained Optimization Problem for the Fourier Transform: Quantitative Analysis

, Volume OnlineFirst – May 29, 2018
43 pages

/lp/springer_journal/a-constrained-optimization-problem-for-the-fourier-transform-Dtlgb9FExU
Publisher
Springer US
Subject
Mathematics; Differential Geometry; Convex and Discrete Geometry; Fourier Analysis; Abstract Harmonic Analysis; Dynamical Systems and Ergodic Theory; Global Analysis and Analysis on Manifolds
ISSN
1050-6926
eISSN
1559-002X
D.O.I.
10.1007/s12220-018-0038-y
Publisher site
See Article on Publisher Site

### Abstract

JGeomAnal https://doi.org/10.1007/s12220-018-0038-y A Constrained Optimization Problem for the Fourier Transform: Quantitative Analysis Dominique Maldague Received: 15 March 2018 © Mathematica Josephina, Inc. 2018 Abstract Among functions f majorized by indicator functions 1 , which functions 1/p have maximal ratio  f  /|E | ? We establish a quantitative answer to this question for exponents q sufﬁciently close to even integers, building on previous work proving the existence of such maximizers. Keywords Harmonic analysis · Extremization · Fourier transform · Quantitative analysis Mathematics Subject Classiﬁcation 42A16 · 42B10 1 Introduction −2πix ·ξ Deﬁne the Fourier transform as F ( f )(ξ ) = f (ξ ) = e f (x )dx for a d 1 d function f : R → C. The Fourier transform is a contraction from L (R ) to ∞ d 2 d L (R ) and is unitary on L (R ). Interpolation gives the Hausdorff–Young inequal- 1 1 ity  f    f  , where p ∈ (1, 2),1 = + .In[1], Beckner proved the sharp q p p q Hausdorff–Young inequality f   C  f  , (1.1) q p 1/2 p −1/2q where C = p

### Journal

The Journal of Geometric AnalysisSpringer Journals

Published: May 29, 2018

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