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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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