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Inequalities in Harmonic Analysis A modern panorama on classical ideas
William BecknerThe University of Texas at Austin
Nanjing May 2012
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Purpose: Development of models to rigorgously describe many-body
interactions and behavior of dynamical phenomena has suggested novelmultilinear embedding estimates and forms that characterize fractionalsmoothness. This framework increases understanding for genuinelyn-dimensional aspects of Fourier analysis.
Goals: To have an understanding of the tools we use from rst principles,and to gain insight for the balance between weighted inequalities thatconnect size estimates for a function and its Fourier transform.
Eli Stein (1967)we shall begin by studying the fractional powers of the Laplacian
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CLASSICAL INEQUALITIES
HardyLittlewoodSobolev Inequality
f | x| f , 0 < < n
| x|
f Lq(R n ) A f L p(R n )1q =
n +
1 p 1
HausdorffYoung Inequality
(F f )( x) = f ( x) = e2 ixy f ( y) dyF f L p (R n ) A f L p(R n)
1
p +
1
p = 1 , 1 p 2
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Sobolev Embedding
R n | f |2 d c R n ( / 42)/ 2 f p
dx2/ p
= n1 p
12
0 , 1 < p 2
Uncertainty & Pitts Inequality
R n | f |2 dx
2
B R n | x| | f |2 dx R n | | | f |
2 dx
B 4n
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Paradigms & Principles1. Characterization of smoothness
2. Rigorous description of many-body interactions
3. Establish sharp embedding estimates
4. Expand working framework fora. Fourier transformb. convolutionc. Riesz potentialsd. Stein-Weiss integrals (Hardy-Littlewood-Sobolev inequalities)
e. weights & symmetrizationf. analysis on Lie groups and manifolds with negative curvature
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5. Gain new insighta. uncertaintyb. restriction phenomenac. geometric symmetry
6. Effort for optimal constantsa. new features for exact model calculationsb. encoded geometric informationc. precise lower-order effects
7. Symmetry determines structure
8. Multilinear analysis
understanding for genuinelyn-dimensional aspects of Fourier analysis
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O BJECTS OF STUDY
= ( / 42)/ 2 , > 0 , 0 < < 1 and 1 p < n/
R n R n | f ( x) f ( y)| p| x y|n+ p dx dy R n R n |( f )( x) ( f )( y)| p| x y|n+ p dx dy R n R n |( f )( x) ( f )( y)| p| x y|n+ p dx dy
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1. A RONSZAJN -S MITH FORMULAS
Classical Formula: 0 < < 2
R n R n | f ( x) f ( y)|2| x y|n+ dx dy = D R n | | | f ( )|2 d Frank-Lieb-Seiringer: 0 < < min (2, n); g = | x| f , 0 < < n
D
R n
| | |
f ( )|2 d =
R n R n
|g( x) g( y)|2
| x y|n+ | x| | y|
dx dy
+ ( ,, n) R n | x| | f ( x)|2 dx
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Beckner: 0 < < 2, < n; g = | x|(n )/ 2( / 42 )( )/ 4 f
C R n | | |
f ( )|2 d R n | x| | f ( x)|2 dx+
C D R n R n |g( x) g( y)|2| x y|n+ | x| | y| (n )/ 2dx dy
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2. M ULTILINEAR FRACTIONAL EMBEDDING(APRES GROSS & PITAEVSKI )
Pitts inequality: n = mn , = k , 0 < k < n,(m 1) < / n < m
R n| x| | f ( x, , x)|2 dx C R n R n|( ) k / 42) k / 4 f |2dx1 dxmHardy-Littlewood-Sobolev inequality: mn = 2n/ q
R n| f ( x, , x)|q dx
2/ q
F R n R n
|( k / 42 ) k / 4 f |2dx1 dxm
Similar results on S n
Key insight on multilinear products
F ( x) =R mn
gk ( x yk ) H ( y) dy ; H L p(R mn ) F Lq(R n)
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3. R ESTRICTION TO SUBMANIFOLD & UNCERTAINTY
classical uncertainty principle
c R n | f |2 dx R n ( / 42)/ 4 | x|/ 2 f ( x) 2 dxRestriction to k -dimensional linear sub-variety
d R k |R f |2 dx R n ( / 42)/ 4 | x|/ 2 f ( x) 2 dxwith n = k , n k > > 0
d = (/ 2)(/ 2)
k + 4
k 4
2
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4. T RIANGLE INEQUALITY ESTIMATES
R n R n |g( y x) f ( x) h( x y) f ( y)| p dx dy
R n
|g( y)| | h( y)| p
dy
R n
| f ( x)| p dx
Proof: p 1
R n R n |g( y) f ( x) h( y) f ( y)| p dx 1/ p p dy R n |g( y)| f p | h( y)| f p p dy=
R n|g( y)| | h( y)|
pdy
R n| f ( x)| p dx
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This proves for 0 < < 1 and 1 p < n/
R n R n | f ( x) f ( y)| p| x y|n+ p dx dy D p, R n | x| p | f ( x)| p dx D p, = R n 1 | x| p | x | n p dx
for = ( n p )/ p and S n 1
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5. SURFACE CONVOLUTION (APRES KLAINERMAN &M ACHEDON )
S
1|w y|
1| y|
d
w R m and S = smooth submanifold in R n
(g f 1 f m)(w) , g L1(R n) , f k Ln/ k (R n)
= k = n(m 1) , 0 < k < n
Replace f k s by Riesz potentials; constrain multivariable integration to
hyperbolic surface
|w| R n R n | xk |2 | xm |2 w xk | xk | k dx1 dxn
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O BJECTIVE : MULTILINEAR EMBEDDING ESTIMATES
xk | f |r d
q
dw d p / (rq )
c p f ; { }
p f ; { } = R n R nm
n= 1
k / 42 k / 2 f
pdx1 . . . dxm
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6. A PRES BOURGAIN -B REZIS -M IRONESCU THEOREM
TheoremFor f S (R n) , 0 < < 1 and 1 p < n/ ( + )
R n R n |( f )( x) ( f )( y)| p| x y|n+ p dx dy c R n | f |q dx p/ q ,(1)
q = pn
n p( + )
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Proof: (1) Set g = f and apply Symmetrization Lemma
R n R n|g( x) g( y)| p
| x y|n+ p dx dy R n R n|g ( x) g ( y)| p
| x y|n+ p dx dy
(2) Apply triangle inequality estimate
R n R n|g ( x) g ( y)| p
| x y|n+ p dx dy D p, R n | x| p
|g ( x)| p
dx
g non-negative & radial decreasing
g ( x) c| x| n/ q , q = pn/ (n p )
(3) R n | x| p |g ( x)| pdx c R n |g ( x)|qdx p/ q = c R n | f |qdx p/ q
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(4) R n | f |q dx p/ q c R n | f |q dx p/ qsince
1| x|n f Lq (R n) c f L
q (R n )
for q = np/ (n p( + ))
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Tools Symmetrization Lemma
R
n R
n
| f ( x) f ( y)| p
| x y|n+ p dx dy
R n R n | f ( x) f ( y)| p| x y|n+ p dx dyfor p 1 and 0 < < 1
f = radial equimeasurable decreasing rearrangment of | f |
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7. H AUSDORFF -Y OUNG INEQUALITY FOR FRACTIONALD ERIVATIVES
Aronszajn-Smith
R n R n | f ( x) f ( y)|2| x y|n+ 2 dx dy = D R n | |2 | f ( )|2 d
R n R n| f ( x) f ( y)| p
| x y|n+ p dx dy R n | |
| f ( )| p
d
Theorem0 < < 1 , 1 < p < , 1/ p + 1/ p = 1
R n R n | f ( x) f ( y)| p
| x y|n+ p dx dy c R n | | | f ( )|
pd
p/ p
1 < p 2
cR n
| | | f ( )| p
d p/ p
2 p <
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9. N EW OBJECTS OF STUDY
R n R n K ( x y) f ( x)( f )( y) f ( y)( f )( x) p dx dy R n R n K ( x y) f ( x)g( y) f ( y)g( x) p dx dy
R n R n
K ( y) f ( x + y) + f ( x y) 2 f ( x) p dx dy
role of convolution
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10. A NALYSIS ON LIE GROUPS
n-dimensional Euclidean space
manifold with non-positive sectional curvature
homogeneous under action of non-unimodular Lie group
hyperbolic space H n Ls = H + s(s n + 1)1, s (n 1)/ 2
potentials fundamental solutions
F Lq (H n)2
Aq
H n
F ( LsF ) d , q > 2
Question: When can you compute optimal values for Aq?
Model embedding structure from Euclidean framework.
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REFERENCES
W. Beckner, Weighted inequalities and Stein-Weiss potentials ,Forum Math. 20 (2008), 587606.W. Beckner, Pitts inequality with sharp convolution estimates ,Proc. Amer. Math. Soc. 136 (2008), 18711885.W. Beckner, Pitts inequality and the fractional Laplacian: sharp error estimates , Forum Math. 24 (2012), 177209.W. Beckner, Multilinear embedding estimates for the fractional Laplacian ,Math. Res. Lett. 19 (2012), 115.W. Beckner, Multilinear embedding convolution estimates on smoothsubmanifolds , arXiv: 1204.5684
W. Beckner, Embedding estimates and fractional smoothness ,(in preparation)W. Beckner, Analysis on Lie groups embedding potentials ,(in preparation)
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