Math 2.0

Harmonic analysis on Euclidean spaces Fourier transform Theorem

Babenko–Beckner inequality

Builds on Fourier transformLp spaceHausdorff–Young inequalityFourier transform of the Gaussian

Statement

Let n≥1n \ge 1, let 1<p≤21 < p \le 2, and let qq be the conjugate exponent, 1p+1q=1\frac{1}{p} + \frac{1}{q} = 1. Put Ap=(p1/p/q1/q)1/2A_p = \left( p^{1/p} / q^{1/q} \right)^{1/2}. Let f^(ξ)=∫ℝnf(x)e−2πix⋅ξ dx\hat f(\xi) = \int_{\R^n} f(x) e^{-2\pi i x \cdot \xi}\, dx be the Fourier transform. Then for every f∈L1(ℝn)∩Lp(ℝn)f \in L^1(\R^n) \cap L^p(\R^n), ∥f^∥Lq(ℝn)≤Apn ∥f∥Lp(ℝn).\norm{\hat f}_{L^q(\R^n)} \le A_p^n\, \norm{f}_{L^p(\R^n)}. The constant ApnA_p^n is the best possible: equality holds for f(x)=e−π∣x∣2f(x) = e^{-\pi \abs{x}^2}.

Remarks

The inequality sharpens the Hausdorff–Young inequality, whose constant is 11; for 1<p<21 < p < 2 one has Ap<1A_p < 1. K. I. Babenko proved it when qq is an even integer, and W. Beckner proved it for all pp in the range. Beckner's proof goes through a sharp hypercontractive estimate for the Hermite semigroup and a central limit argument.

The equality case can be checked directly with the Fourier transform of the Gaussian: for f(x)=e−π∣x∣2f(x) = e^{-\pi \abs{x}^2} one has f^=f\hat f = f and ∥f∥r=r−n/(2r)\norm{f}_r = r^{-n/(2r)} for every r>0r > 0. So ∥f^∥q/∥f∥p=(p1/p/q1/q)n/2=Apn\norm{\hat f}_q / \norm{f}_p = \left( p^{1/p} / q^{1/q} \right)^{n/2} = A_p^n.

Differentiating the inequality in pp at p=2p = 2, where both sides are equal by Plancherel, gives the entropic uncertainty principle.

Going deeper

For a student

This is a typical sharp inequality in analysis, with Gaussians as extremal functions. The same happens for the logarithmic Sobolev inequality and the hypercontractivity of the Ornstein–Uhlenbeck semigroup, to which it is closely related. Its derivative at p=2p = 2 gives the Hirschman–Beckner entropic uncertainty principle, which in turn implies the Heisenberg inequality.

Reading path

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Almost everywhere, concept, no evidence levelAlmost everywhereconceptFourier transform, concept, no evidence levelFourier transformconceptHilbert space, concept, no evidence levelHilbert spaceconceptIntegrable function, concept, no evidence levelIntegrable functionconceptIntegral of a non-negative function, concept, no evidence levelIntegral of a non-neg…conceptLebesgue integral, concept, no evidence levelLebesgue integralconceptLp space, concept, no evidence levelLp spaceconceptMeasurable function, concept, no evidence levelMeasurable functionconceptOperator norm, concept, no evidence levelOperator normconceptSequence, definition, no evidence levelSequencedefinitionSigma-finite measure, concept, no evidence levelSigma-finite measureconceptUnitary operator, concept, no evidence levelUnitary operatorconceptPlancherel theorem, statement, no evidence levelPlancherel theoremstatementMonotone convergence theorem, statement, no evidence levelMonotone convergence …statementRiesz–Thorin theorem, statement, no evidence levelRiesz–Thorin theoremstatementReal sequence, definition, no evidence levelReal sequencedefinitionHausdorff–Young inequality, statement, no evidence levelHausdorff–Young inequ…statementBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit inferior, definition, no evidence levelLimit inferiordefinitionFatou's lemma, statement, no evidence levelFatou's lemmastatementDominated convergence theorem, statement, no evidence levelDominated convergence…statementDifferentiation and multiplication under the Fourier transform, statement, no evidence levelDifferentiation and m…statementFourier transform of the Gaussian, statement, no evidence levelFourier transform of …statementBabenko–Beckner inequality, statement, no evidence levelBabenko–Beckner inequ…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Almost everywhere concept
  2. Fourier transform concept
  3. Hilbert space concept
  4. Integrable function concept
  5. Integral of a non-negative function concept
  6. Lebesgue integral concept
  7. Lp space concept
  8. Measurable function concept
  9. Operator norm concept
  10. Sequence definition
  11. Sigma-finite measure concept
  12. Unitary operator concept
  13. Plancherel theorem statement
  14. Monotone convergence theorem statement
  15. Riesz–Thorin theorem statement
  16. Real sequence definition
  17. Hausdorff–Young inequality statement
  18. Bounded sequence definition
  19. Limit inferior definition
  20. Fatou's lemma statement
  21. Dominated convergence theorem statement
  22. Differentiation and multiplication under the Fourier transform statement
  23. Fourier transform of the Gaussian statement
  24. Babenko–Beckner inequality statement

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Further reading

  1. Web https://en.wikipedia.org/wiki/Babenko%E2%80%93Beckner_inequality
About this page
TagFFM7Y61KRT
URIm2://FFM7Y61KRT
Kindstatement, theorem
WritingW0 stub
MSC42B10, 42A38
StatusThis copy of the site has no build data. Run m2 status FFM7Y61KRT.
AboutFourier transform
UsesFourier transform, Lp space, Hausdorff–Young inequality, Fourier transform of the Gaussian
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