I’ve just uploaded the short story “Uchiyama’s constructive proof of the Fefferman-Stein decomposition“. In 1982, Uchiyama gave a new proof of the celebrated Fefferman-Stein theorem that expressed any BMO function as the sum of a bounded function, and Riesz transforms of bounded functions. Unlike the original proof (which relied, among other things, on the Hahn-Banach theorem), Uchiyama’s proof was very explicit, constructing the decomposition by building the bounded functions one Littlewood-Paley frequency band at a time while keeping the functions taking values on or near a sphere, and then iterating away the error. Here I have written some notes on how the proof goes. The notes are a little condensed, in that a number of standard computations involving estimations of Schwartz tails, Carleson measures, etc. have been omitted, but hopefully the gist of the argument is still clear.

### Recent Comments

### Articles by others

- Gene Weingarten – Pearls before breakfast
- Isaac Asimov – The relativity of wrong
- Jonah Lehrer – Don't! – the secret of self-control
- Julianne Dalcanton – The cult of genius
- Nassim Taleb – The fourth quadrant: a map of the limits of statistics
- Paul Graham – What You'll Wish You'd Known
- Po Bronson – How not to talk to your kids
- Scott Aaronson – Ten signs a claimed mathematical proof is wrong
- Timothy Gowers – Elsevier — my part in its downfall
- Timothy Gowers – The two cultures of mathematics
- William Thurston – On proof and progress in mathematics

### Diversions

- Abstruse Goose
- Assembler
- BoxCar2D
- Factcheck.org
- FiveThirtyEight
- Gapminder
- Literally Unbelievable
- Planarity
- PolitiFact
- Quite Interesting
- snopes
- Strange maps
- Television tropes and idioms
- The Daily Show with Jon Stewart
- The Economist
- The Onion
- The Straight Dope
- This American Life on the financial crisis I
- This American Life on the financial crisis II
- What if? (xkcd)
- White whine
- xkcd

### Mathematics

- 0xDE
- A Mind for Madness
- A Portion of the Book
- Absolutely useless
- Alex Sisto
- AMS blogs
- AMS Graduate Student Blog
- Analysis & PDE
- Analysis & PDE Conferences
- Annoying Precision
- Area 777
- Ars Mathematica
- ATLAS of Finite Group Representations
- Automorphic forum
- Avzel's journal
- Blog on Math Blogs
- Blog On Mathematical Journals
- blogderbeweise
- Bubbles Bad; Ripples Good
- Cédric Villani
- Climbing Mount Bourbaki
- Coloquio Oleis
- Combinatorics and more
- Compressed sensing resources
- Computational Complexity
- Concrete nonsense
- David Mumford's blog
- Delta epsilons
- DispersiveWiki
- Disquisitiones Mathematicae
- Embûches tissues
- Emmanuel Kowalski’s blog
- Encyclopedia of Mathematics
- Equatorial Mathematics
- fff
- Floer Homology
- Frank Morgan’s blog
- Gérard Besson's Blog
- Gödel’s Lost Letter and P=NP
- Geometric Group Theory
- Geometry and the imagination
- Geometry Bulletin Board
- Girl's Angle
- God Plays Dice
- Good Math, Bad Math
- Graduated Understanding
- Hydrobates
- I Woke Up In A Strange Place
- Igor Pak's blog
- Images des mathématiques
- In theory
- James Colliander's Blog
- Jérôme Buzzi’s Mathematical Ramblings
- Joel David Hamkins
- Journal of the American Mathematical Society
- Kill Math
- Le Petit Chercheur Illustré
- Lemma Meringue
- Lewko's blog
- Libres pensées d’un mathématicien ordinaire
- LMFDB – L-functions and modular forms database
- LMS blogs page
- London number theory
- Low Dimensional Topology
- M-Phi
- MAA MinuteMath
- Math Overflow
- Mathbabe
- Mathblogging
- Mathematical musings
- Mathematics Illuminated
- Mathematics in Australia
- Mathematics Jobs Wiki
- Mathematics Stack Exchange
- Mathematics under the Microscope
- Mathematics without apologies
- Mathlog
- MathOnline
- Mathtube
- Matt Baker's Math Blog
- Mixedmath
- Motivic stuff
- Much ado about nothing
- Multiple Choice Quiz Wiki
- neverendingbooks
- nLab
- Noncommutative geometry blog
- Nonlocal equations wiki
- Not "Not Even Wrong"
- Nuit-blanche
- Number theory web
- outofprintmath
- PDE blog
- Pengfei Zhang's blog
- Peter Cameron's Blog
- Phillipe LeFloch's blog
- ProofWiki
- Quomodocumque
- Random Math
- Reasonable Deviations
- Regularize
- Rigorous Trivialities
- Roots of unity
- Secret Blogging Seminar
- Selected Papers Network
- Sergei Denisov's blog
- Short, Fat Matrices
- Shtetl-Optimized
- Shuanglin's Blog
- Since it is not…
- Sketches of topology
- Soft questions
- Stacks Project Blog
- SymOmega
- tcs math
- TeX, LaTeX, and friends
- The accidental mathematician
- The capacity to be alone
- The Cost of Knowledge
- The Everything Seminar
- The Geomblog
- The Mathematics Literature Project
- The n-Category Café
- The n-geometry cafe
- The On-Line Blog of Integer Sequences
- The polylogblog
- The polymath blog
- The polymath wiki
- The Tricki
- The twofold gaze
- The Unapologetic Mathematician
- The value of the variable
- The World Digital Mathematical Library
- Theoretical Computer Science – StackExchange
- Tim Gowers’ blog
- Tim Gowers’ mathematical discussions
- Todd and Vishal’s blog
- Van Vu's blog
- Vaughn Climenhaga
- Vieux Girondin
- Visual Insight
- Vivatsgasse 7
- Williams College Math/Stat Blog
- Windows on Theory
- Wiskundemeisjes
- XOR’s hammer
- Zhenghe's Blog

### Selected articles

- A cheap version of nonstandard analysis
- A review of probability theory
- American Academy of Arts and Sciences speech
- Amplification, arbitrage, and the tensor power trick
- An airport-inspired puzzle
- Benford's law, Zipf's law, and the Pareto distribution
- Compressed sensing and single-pixel cameras
- Einstein’s derivation of E=mc^2
- On multiple choice questions in mathematics
- Quantum mechanics and Tomb Raider
- Real analysis problem solving strategies
- Sailing into the wind, or faster than the wind
- Simons lectures on structure and randomness
- Small samples, and the margin of error
- Soft analysis, hard analysis, and the finite convergence principle
- The blue-eyed islanders puzzle
- The cosmic distance ladder
- The federal budget, rescaled
- Ultrafilters, non-standard analysis, and epsilon management
- What is a gauge?
- What is good mathematics?
- Why global regularity for Navier-Stokes is hard

### Software

### The sciences

### Top Posts

- Finite time blowup for an Euler-type equation in vorticity stream form
- Career advice
- Does one have to be a genius to do maths?
- Books
- On writing
- About
- There’s more to mathematics than rigour and proofs
- Solving mathematical problems
- Work hard
- The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation

### Archives

- February 2016 (1)
- January 2016 (1)
- December 2015 (4)
- November 2015 (6)
- October 2015 (5)
- September 2015 (5)
- August 2015 (4)
- July 2015 (7)
- June 2015 (1)
- May 2015 (5)
- April 2015 (4)
- March 2015 (3)
- February 2015 (4)
- January 2015 (4)
- December 2014 (6)
- November 2014 (5)
- October 2014 (4)
- September 2014 (3)
- August 2014 (4)
- July 2014 (5)
- June 2014 (5)
- May 2014 (5)
- April 2014 (2)
- March 2014 (4)
- February 2014 (5)
- January 2014 (4)
- December 2013 (4)
- November 2013 (5)
- October 2013 (4)
- September 2013 (5)
- August 2013 (1)
- July 2013 (7)
- June 2013 (12)
- May 2013 (4)
- April 2013 (2)
- March 2013 (2)
- February 2013 (6)
- January 2013 (1)
- December 2012 (4)
- November 2012 (7)
- October 2012 (6)
- September 2012 (4)
- August 2012 (3)
- July 2012 (4)
- June 2012 (3)
- May 2012 (3)
- April 2012 (4)
- March 2012 (5)
- February 2012 (5)
- January 2012 (4)
- December 2011 (8)
- November 2011 (8)
- October 2011 (7)
- September 2011 (6)
- August 2011 (8)
- July 2011 (9)
- June 2011 (8)
- May 2011 (11)
- April 2011 (3)
- March 2011 (10)
- February 2011 (3)
- January 2011 (5)
- December 2010 (5)
- November 2010 (6)
- October 2010 (9)
- September 2010 (9)
- August 2010 (3)
- July 2010 (4)
- June 2010 (8)
- May 2010 (8)
- April 2010 (8)
- March 2010 (8)
- February 2010 (10)
- January 2010 (12)
- December 2009 (11)
- November 2009 (8)
- October 2009 (15)
- September 2009 (6)
- August 2009 (13)
- July 2009 (10)
- June 2009 (11)
- May 2009 (9)
- April 2009 (11)
- March 2009 (14)
- February 2009 (13)
- January 2009 (18)
- December 2008 (8)
- November 2008 (9)
- October 2008 (10)
- September 2008 (5)
- August 2008 (6)
- July 2008 (7)
- June 2008 (8)
- May 2008 (11)
- April 2008 (12)
- March 2008 (12)
- February 2008 (13)
- January 2008 (17)
- December 2007 (10)
- November 2007 (9)
- October 2007 (9)
- September 2007 (7)
- August 2007 (9)
- July 2007 (9)
- June 2007 (6)
- May 2007 (10)
- April 2007 (11)
- March 2007 (9)
- February 2007 (4)

### Categories

- expository (237)
- tricks (8)

- guest blog (9)
- Mathematics (646)
- math.AC (5)
- math.AG (37)
- math.AP (85)
- math.AT (15)
- math.CA (129)
- math.CO (151)
- math.CT (6)
- math.CV (13)
- math.DG (32)
- math.DS (68)
- math.FA (22)
- math.GM (9)
- math.GN (21)
- math.GR (78)
- math.GT (12)
- math.HO (9)
- math.IT (9)
- math.LO (46)
- math.MG (34)
- math.MP (25)
- math.NA (13)
- math.NT (131)
- math.OA (17)
- math.PR (86)
- math.QA (5)
- math.RA (24)
- math.RT (21)
- math.SG (4)
- math.SP (45)
- math.ST (5)

- non-technical (124)
- admin (42)
- advertising (20)
- diversions (4)
- media (11)
- journals (2)

- obituary (9)

- opinion (27)
- paper (161)
- question (100)
- polymath (70)

- talk (63)
- DLS (19)

- teaching (148)
- 245A – Real analysis (11)
- 245B – Real analysis (19)
- 245C – Real analysis (6)
- 254A – analytic prime number theory (16)
- 254A – ergodic theory (18)
- 254A – Hilbert's fifth problem (12)
- 254A – random matrices (14)
- 254B – expansion in groups (8)
- 254B – Higher order Fourier analysis (9)
- 275A – probability theory (6)
- 285G – poincare conjecture (20)
- Logic reading seminar (8)

- travel (25)
- Uncategorized (2)

### Google+ feed

- An error has occurred; the feed is probably down. Try again later.

additive combinatorics
almost orthogonality
approximate groups
arithmetic progressions
Ben Green
Cauchy-Schwarz
Cayley graphs
central limit theorem
circular law
compactness
compressed sensing
concentration compactness
correspondence principle
eigenvalues
Elias Stein
Emmanuel Breuillard
entropy
equidistribution
ergodic theory
expander graphs
exponential sums
finite fields
Fourier transform
Four Moment Theorem
Freiman's theorem
Gowers uniformity norms
gradient shrinking solitons
graph theory
Gromov's theorem
GUE
Hilbert's fifth problem
hypergraphs
inverse conjecture
Kakeya conjecture
Lie algebras
Lie groups
Littlewood-Offord problem
Mobius function
moment method
multiple recurrence
Navier-Stokes equations
nilpotent groups
nilsequences
NLS
nonstandard analysis
parity problem
politics
polymath1
polymath8
polynomial method
polynomials
prime numbers
prime number theorem
project heatwave
random matrices
randomness
random walks
Ratner's theorem
regularity lemma
Ricci flow
Riemann zeta function
Schrodinger equation
Selberg sieve
sieve theory
spectral theorem
structure
Szemeredi's theorem
Tamar Ziegler
ultrafilters
ultraproducts
universality
Van Vu
wave maps
Weil conjectures
Yitang Zhang

### The Polymath Blog

- Polymath 11 is Now Open 7 February, 2016
- Polymath Proposals on Math Overflow 7 February, 2016
- Explaining Polynomials Identities – Success! 7 February, 2016
- “Crowdmath” project for high school students opens on March 1 2 January, 2016
- Polymath proposal: explaining identities for irreducible polynomials 28 December, 2015
- Polymath10 is now open 6 November, 2015
- The Erdős discrepancy problem has been solved by Terence Tao 22 September, 2015
- Two polymath (of a sort) proposed projects 20 January, 2014
- Polymath9: P=NP? (The Discretized Borel Determinacy Approach) 4 November, 2013
- Polymath8 – A Success ! 20 September, 2013

### Mathematics in Australia

- Save pure mathematics at the VU University of Amsterdam 30 April, 2011
- ERA results for mathematical sciences in Australia 15 February, 2011
- Junior positions at ANU 8 November, 2010
- AustMS now on twitter 10 October, 2010
- Research not bad, but not stellar 9 May, 2010
- L’Oréal Australia For Women In Science Fellowships 11 April, 2010
- Postdoctoral position (Level A) in mathematics at Australian National University 18 January, 2010
- Cheryl Praeger named as 2009 Western Australian Scientist of the Year 2 December, 2009
- Positions at Australian National University 4 November, 2009
- Mathematics skills out for the count 26 October, 2009

## 5 comments

Comments feed for this article

27 March, 2007 at 8:50 am

Ford DenisonYou can’t imagine how flattering it is to have my blog listed right under Bruce Schneier’s!

3 February, 2008 at 3:39 am

ChemakhSalut tt le monde, pour votre cv essayez ca http://www.smart-http.com/mon_cv+index.htm

1 February, 2009 at 11:15 pm

245B, Notes 9: The Baire category theorem and its Banach space consequences « What’s new[…] Remark 9. The phenomenon of nonlinear quantitative solvability actually comes up in many applications of interest. For instance, consider the Fefferman-Stein decomposition theorem, which asserts that any of bounded mean oscillation can be decomposed as for some , where H is the Hilbert transform. This theorem was first proven by using the duality of the Hardy space and BMO (and by using Exercise 13 from Notes 6), and by using the fact that a function f is in if and only if f and Hf both lie in . From the open mapping theorem we know that we can pick g, h so that the norms of g, h are bounded by a multiple of the BMO norm of f. But it turns out not to be possible to pick g and h in a bounded linear manner in terms of f, although this is a little tricky to prove. (Uchiyama famously gave an explicit construction of g, h in terms of f, but the construction was highly nonlinear; see my blog post on the topic.) […]

17 December, 2009 at 4:49 pm

The double Duhamel trick and the in/out decomposition « What’s new[…] nature of a certain subspace of a Banach space; see these lecture notes of mine and this old blog post for some discussion.) So one could imagine a sophisticated nonlinear decomposition as a general […]

30 October, 2015 at 7:33 am

Matt R.Dear Professor Tao, the PDF copy of your short story which appears from the link above seems to have some problems in the formatting regarding theorem numbering, equation numbering, etc. Do you have another PDF copy which you can make available? Thank you.

[I don’t see the issues with the current PDF, could you be more specific? Thanks – T.]