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Inversion and extension of the finite Hilbert transform Guillermo P . Curbera Universidad de Sevilla September 11, 2019 Workshop on Banach spaces and Banach lattices ICMAT Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform

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Inversion and extension of the finite Hilbert transform

Guillermo P . Curbera

Universidad de Sevilla

September 11, 2019 Workshop on Banach spaces and Banach lattices ICMAT

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 1 / 28

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Authorship

Joint work with: Susumu Okada University of Tasmania Australia Werner J. Ricker Katholische Universität Eichstätt Germany

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 2 / 28

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The finite Hilbert transform

1

The finite Hilbert transform

2

Inversion of the FHT

3

Extension of the FHT

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 3 / 28

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The finite Hilbert transform

The airfoil equation

“The study of an ideal flow past a thin airfoil” lead in aerodynamics to the airfoil equation: pAEq p.v.1 π ż 1

´1

fpxq x ´ t dx “ gptq, a.e. t P p´1, 1q.

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 4 / 28

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The finite Hilbert transform

The airfoil equation

“The study of an ideal flow past a thin airfoil” lead in aerodynamics to the airfoil equation: pAEq p.v.1 π ż 1

´1

fpxq x ´ t dx “ gptq, a.e. t P p´1, 1q. Studied by:

Birnbaum 1920’s; von Kármán 1930’s; Söhngen 1940’s; Tricomi 1950’s. Tricomi “Integral Equations” (1957) for the spaces Lpp´1, 1q.

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 4 / 28

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The finite Hilbert transform

The airfoil equation

“The study of an ideal flow past a thin airfoil” lead in aerodynamics to the airfoil equation: pAEq p.v.1 π ż 1

´1

fpxq x ´ t dx “ gptq, a.e. t P p´1, 1q. Studied by:

Birnbaum 1920’s; von Kármán 1930’s; Söhngen 1940’s; Tricomi 1950’s. Tricomi “Integral Equations” (1957) for the spaces Lpp´1, 1q. Nowadays is used in Tomography (image reconstruction).

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 4 / 28

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The finite Hilbert transform

The finite Hilbert transform FHT

The finite Hilbert transform is defined, for f P L1p´1, 1q, by the principal value integral: Tfptq :“ p.v.1 π ż 1

´1

fpxq x ´ t dx, t P p´1, 1q.

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 5 / 28

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The finite Hilbert transform

The finite Hilbert transform FHT

The finite Hilbert transform is defined, for f P L1p´1, 1q, by the principal value integral: Tfptq :“ p.v.1 π ż 1

´1

fpxq x ´ t dx, t P p´1, 1q. The setting of the Lp-spaces is not the most adequate for studying the FHT, because:

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 5 / 28

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The finite Hilbert transform

The finite Hilbert transform FHT

The finite Hilbert transform is defined, for f P L1p´1, 1q, by the principal value integral: Tfptq :“ p.v.1 π ż 1

´1

fpxq x ´ t dx, t P p´1, 1q. The setting of the Lp-spaces is not the most adequate for studying the FHT, because:

T : X Ñ X is injective ð ñ L2,8p´1, 1q Ę X. T : X Ñ X has non-dense range ð ñ X Ď L2,1p´1, 1q.

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 5 / 28

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The finite Hilbert transform

Rearrangement invariant space (r.i.s.)

Function space X on p´1, 1q such that:

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 6 / 28

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The finite Hilbert transform

Rearrangement invariant space (r.i.s.)

Function space X on p´1, 1q such that: X consists of measurable functions, X Ď L0p´1, 1q. X has a complete norm } ¨ }X. X in an ideal of measurable functions: |g| ď |f| a.e. & f P X ù ñ g P X & }g}X ď }f}X. X is rearrangement invariant: mptx : |gpxq| ą λu “ mptx : |fpxq| ą λu, for all λ ą 0 and f P X ù ñ g P X and }g}X “ }f}X.

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 6 / 28

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The finite Hilbert transform

Rearrangement invariant space (r.i.s.)

Function space X on p´1, 1q such that: X consists of measurable functions, X Ď L0p´1, 1q. X has a complete norm } ¨ }X. X in an ideal of measurable functions: |g| ď |f| a.e. & f P X ù ñ g P X & }g}X ď }f}X. X is rearrangement invariant: mptx : |gpxq| ą λu “ mptx : |fpxq| ą λu, for all λ ą 0 and f P X ù ñ g P X and }g}X “ }f}X. Examples: Lp spaces, weak-Lp spaces, Orlicz spaces, Lorentz Lp,q spaces, Lorentz Λφ spaces, Marcinkiewicz spaces,.....

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 6 / 28

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The finite Hilbert transform

Boundedness of T on r.i.s.

Theorem (M. Riesz): For H the Hilbert transform on R H : LppRq Ñ LppRq ð ñ 1 ă p ă 8.

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 7 / 28

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The finite Hilbert transform

Boundedness of T on r.i.s.

Theorem (M. Riesz): For H the Hilbert transform on R H : LppRq Ñ LppRq ð ñ 1 ă p ă 8. Theorem (Boyd): For X r.i.s. on R H : X Ñ X ð ñ 0 ă αX ď αX ă 1, where the Boyd indices of X (with E1{t the dilation operator f ÞÑ fp¨{tq): 0 ď αX :“ lim

tÑ0`

log }E1{t} log t ď αX :“ lim

tÑ8

log }E1{t} log t ď 1.

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 7 / 28

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The finite Hilbert transform

Boundedness of T on r.i.s.

Theorem (M. Riesz): For H the Hilbert transform on R H : LppRq Ñ LppRq ð ñ 1 ă p ă 8. Theorem (Boyd): For X r.i.s. on R H : X Ñ X ð ñ 0 ă αX ď αX ă 1, where the Boyd indices of X (with E1{t the dilation operator f ÞÑ fp¨{tq): 0 ď αX :“ lim

tÑ0`

log }E1{t} log t ď αX :“ lim

tÑ8

log }E1{t} log t ď 1. Theorem: For X r.i.s. on p´1, 1q T : X Ñ X ð ñ 0 ă αX ď αX ă 1

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 7 / 28

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Inversion of the FHT

1

The finite Hilbert transform

2

Inversion of the FHT

3

Extension of the FHT

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 8 / 28

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Inversion of the FHT

Inversion of the FHT

Tricomi gave inversion formulae for Lpp´1, 1q in two cases:

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 9 / 28

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Inversion of the FHT

Inversion of the FHT

Tricomi gave inversion formulae for Lpp´1, 1q in two cases:

When 1 ă p ă 2. When 2 ă p ă 8. Not for p “ 2.

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 9 / 28

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Inversion of the FHT

Inversion of the FHT

Tricomi gave inversion formulae for Lpp´1, 1q in two cases:

When 1 ă p ă 2. When 2 ă p ă 8. Not for p “ 2.

We give inversion formulae for r.i.s. X on p´1, 1q in two cases :

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 9 / 28

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Inversion of the FHT

Inversion of the FHT

Tricomi gave inversion formulae for Lpp´1, 1q in two cases:

When 1 ă p ă 2. When 2 ă p ă 8. Not for p “ 2.

We give inversion formulae for r.i.s. X on p´1, 1q in two cases :

When 1{2 ă αX ď αX ă 1. When 0 ă αX ď αX ă 1{2. Not when 1{2 P rαX, αXs. For example, X=L2,q for 1 ď q ď 8.

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 9 / 28

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Inversion of the FHT

The case 0 ă αX ď αX ă 1{2

Theorem (C., Okada & Ricker, 2018) Let X be a r.i.s. on p´1, 1q satisfying 0 ă αX ď αX ă 1{2.

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 10 / 28

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Inversion of the FHT

The case 0 ă αX ď αX ă 1{2

Theorem (C., Okada & Ricker, 2018) Let X be a r.i.s. on p´1, 1q satisfying 0 ă αX ď αX ă 1{2. (a) T : X Ñ X is injective. (b) q T : X Ñ X and satisfies q TT “ I, for q Tpfqpxq :“ ´ a 1 ´ x2 T ˆ fptq ? 1 ´ t2 ˙ pxq.

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 10 / 28

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Inversion of the FHT

The case 0 ă αX ď αX ă 1{2

Theorem (C., Okada & Ricker, 2018) Let X be a r.i.s. on p´1, 1q satisfying 0 ă αX ď αX ă 1{2. (a) T : X Ñ X is injective. (b) q T : X Ñ X and satisfies q TT “ I, for q Tpfqpxq :“ ´ a 1 ´ x2 T ˆ fptq ? 1 ´ t2 ˙ pxq. (c) The range of T is RpTq “ " f P X : ż 1

´1

fpxq ? 1 ´ x2 dx “ 0 * . (d) q T is an isomorphism from RpTq onto X. (e) X “ " f P X : ż 1

´1

fpxq ? 1 ´ x2 dx “ 0 * ‘ B 1 F .

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 10 / 28

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Inversion of the FHT

The case 1{2 ă αX ď αX ă 1

Theorem (C., Okada & Ricker, 2018) Let X be a r.i.s. on p´1, 1q satisfying 1{2 ă αX ď αX ă 1.

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 11 / 28

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Inversion of the FHT

The case 1{2 ă αX ď αX ă 1

Theorem (C., Okada & Ricker, 2018) Let X be a r.i.s. on p´1, 1q satisfying 1{2 ă αX ď αX ă 1. (a) T : X Ñ X is surjective and KerpTq “ x1{ ? 1 ´ x2y. (b) p T : X Ñ X and satisfies T p T “ I, for p Tpfqpxq :“ ´1 ? 1 ´ x2 T ´a 1 ´ t2 ¨ fptq ¯ pxq.

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 11 / 28

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Inversion of the FHT

The case 1{2 ă αX ď αX ă 1

Theorem (C., Okada & Ricker, 2018) Let X be a r.i.s. on p´1, 1q satisfying 1{2 ă αX ď αX ă 1. (a) T : X Ñ X is surjective and KerpTq “ x1{ ? 1 ´ x2y. (b) p T : X Ñ X and satisfies T p T “ I, for p Tpfqpxq :“ ´1 ? 1 ´ x2 T ´a 1 ´ t2 ¨ fptq ¯ pxq. (c) The range of p T is Rpp Tq “ " f P X : ż 1

´1

fpxqdx “ 0 * . (d) p T is an isomorphism onto its range. (e) X “ " f P X : ż 1

´1

fpxqdx “ 0 * ‘ B 1 ? 1 ´ x2 F .

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 11 / 28

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Inversion of the FHT

Solution to the airfoil equation Tf “ g

Theorem (C., Okada & Ricker, 2018) Let X be a r.i.s. on p´1, 1q.

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 12 / 28

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Inversion of the FHT

Solution to the airfoil equation Tf “ g

Theorem (C., Okada & Ricker, 2018) Let X be a r.i.s. on p´1, 1q. 1{2 ă αX ď αX ă 1: given g P X, all solutions f P X of the airfoil equation (AE) are given by fpxq “ 1 ? 1 ´ x2 T ´a 1 ´ t2 gptq ¯ pxq ` λ ? 1 ´ x2 , λ P C.

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 12 / 28

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Inversion of the FHT

Solution to the airfoil equation Tf “ g

Theorem (C., Okada & Ricker, 2018) Let X be a r.i.s. on p´1, 1q. 1{2 ă αX ď αX ă 1: given g P X, all solutions f P X of the airfoil equation (AE) are given by fpxq “ 1 ? 1 ´ x2 T ´a 1 ´ t2 gptq ¯ pxq ` λ ? 1 ´ x2 , λ P C. 0 ă αX ď αX ă 1{2: given g P X satisfying ż 1

´1

gpxq ? 1 ´ x2 dx “ 0, the unique solution f P X of the airfoil equation (AE) is fpxq :“ ´ a 1 ´ x2 T ˆ gptq ? 1 ´ t2 ˙ pxq.

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 12 / 28

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Extension of the FHT

1

The finite Hilbert transform

2

Inversion of the FHT

3

Extension of the FHT

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 13 / 28

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Extension of the FHT

Extension of the FHT on Lp

Theorem (Okada, Ricker & Sánchez-Pérez, 2008) For 1 ă p ă 8 with p ­“ 2, the finite Hilbert transform T : Lpp´1, 1q Ñ Lpp´1, 1q cannot be extended to a larger Banach function space.

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 14 / 28

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Extension of the FHT

Extension of the FHT on Lp

Theorem (Okada, Ricker & Sánchez-Pérez, 2008) For 1 ă p ă 8 with p ­“ 2, the finite Hilbert transform T : Lpp´1, 1q Ñ Lpp´1, 1q cannot be extended to a larger Banach function space. Lp T

✲ Lp ❄

Lp Ĺ Z T

✟✟✟✟✟✟✟ ✟ ✯

The proof is based on Tricomi’s decomposition of Lp in terms of T.

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 14 / 28

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Extension of the FHT

The extension problem for L2p´1, 1q

Facts:

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 15 / 28

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Extension of the FHT

The extension problem for L2p´1, 1q

Facts: T : L2p´1, 1q Ñ L2p´1, 1q is bounded. The range TpL2q Ă L2 is proper and dense. There is no close description of functions belonging to TpL2q. Okada (1991) gave a (rather complicate) characterization of when g P TpL2q. Consequently, there is no inversion formula for T on L2p´1, 1q.

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 15 / 28

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Extension of the FHT

The extension problem for L2p´1, 1q

Facts: T : L2p´1, 1q Ñ L2p´1, 1q is bounded. The range TpL2q Ă L2 is proper and dense. There is no close description of functions belonging to TpL2q. Okada (1991) gave a (rather complicate) characterization of when g P TpL2q. Consequently, there is no inversion formula for T on L2p´1, 1q. Question (1991): Is it possible to extend T : L2p´1, 1q Ñ L2p´1, 1q to a larger space: T : Z Ñ L2p´1, 1q?

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 15 / 28

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Extension of the FHT

Solution: for all r.i.s.!

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 16 / 28

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Extension of the FHT

Solution: for all r.i.s.!

Theorem (C., Okada & Ricker, 2019) Let X be a r.i.s. on p´1, 1q with 0 ă αX ď αX ă 1. The finite Hilbert transform T : X Ñ X cannot be extended to any genuinely larger Banach function space

  • ver p´1, 1q.

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 16 / 28

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Extension of the FHT

Strategy of the proof

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 17 / 28

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Extension of the FHT

Strategy of the proof

(1) Construct the function space rT, Xs :“ ! f P L1p´1, 1q : Tphq P X, for all |h| ď |f| ) ,

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 17 / 28

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Extension of the FHT

Strategy of the proof

(1) Construct the function space rT, Xs :“ ! f P L1p´1, 1q : Tphq P X, for all |h| ď |f| ) , which is the optimal lattice domain for T: X T

✲ X ❄

rT, Xs T

✟✟✟✟✟✟ ✟ ✯

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 17 / 28

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Extension of the FHT

Strategy of the proof

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 18 / 28

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Extension of the FHT

Strategy of the proof

Showing that rT, Xs is a Banach function space (with the Fatou property), for the norm }f}rT,Xs :“ sup ! }Tphq}X : |h| ď |f| ) , is non-trivial as T is not a positive operator.

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 18 / 28

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Extension of the FHT

Strategy of the proof

Showing that rT, Xs is a Banach function space (with the Fatou property), for the norm }f}rT,Xs :“ sup ! }Tphq}X : |h| ď |f| ) , is non-trivial as T is not a positive operator. For the proof we use a theorem of Talagrand concerning factorization of L0-valued measures.

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 18 / 28

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Extension of the FHT

Strategy of the proof

Showing that rT, Xs is a Banach function space (with the Fatou property), for the norm }f}rT,Xs :“ sup ! }Tphq}X : |h| ď |f| ) , is non-trivial as T is not a positive operator. For the proof we use a theorem of Talagrand concerning factorization of L0-valued measures. Since T : X Ñ X, we always have X Ď rT, Xs.

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 18 / 28

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Extension of the FHT

Strategy of the proof

Showing that rT, Xs is a Banach function space (with the Fatou property), for the norm }f}rT,Xs :“ sup ! }Tphq}X : |h| ď |f| ) , is non-trivial as T is not a positive operator. For the proof we use a theorem of Talagrand concerning factorization of L0-valued measures. Since T : X Ñ X, we always have X Ď rT, Xs. Thus, it suffices to prove that rT, Xs Ď X.

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 18 / 28

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Extension of the FHT

Strategy of the proof

(2) Proving rT, Xs Ď X is equivalent to showing, for some constant C ą 0 and all simple functions φ, that: p˚q C}φ}X ď sup

|θ|“1

}Tpθφq}X,

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 19 / 28

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SLIDE 47

Extension of the FHT

Strategy of the proof

(2) Proving rT, Xs Ď X is equivalent to showing, for some constant C ą 0 and all simple functions φ, that: p˚q C}φ}X ď sup

|θ|“1

}Tpθφq}X, which is showing, for φ :“ řN

n“1 anχAn, that

C › › › ›

N

ÿ

n“1

anχAn › › › ›

X

ď sup

|θ|“1

› › › ›T ˆ θ ¨

N

ÿ

n“1

anχAn ˙› › › ›

X

for some C ą 0 independent of φ.

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 19 / 28

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Extension of the FHT

Strategy for proving p˚q

(3) Consider the probability space Λ “ t´1, 1uN.

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 20 / 28

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Extension of the FHT

Strategy for proving p˚q

(3) Consider the probability space Λ “ t´1, 1uN. Define the function F by σ “ pσnqN

1 P Λ ÞÝ

Ñ Fpσq :“ › › › ›T ˆ

N

ÿ

n“1

σnanχAn ˙› › › ›

X

.

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 20 / 28

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SLIDE 50

Extension of the FHT

Strategy for proving p˚q

(3) Consider the probability space Λ “ t´1, 1uN. Define the function F by σ “ pσnqN

1 P Λ ÞÝ

Ñ Fpσq :“ › › › ›T ˆ

N

ÿ

n“1

σnanχAn ˙› › › ›

X

. Since Λ is a probability space }F}L1pΛq ď }F}L8pΛq .

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 20 / 28

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SLIDE 51

Extension of the FHT

Strategy for proving p˚q

(3) Consider the probability space Λ “ t´1, 1uN. Define the function F by σ “ pσnqN

1 P Λ ÞÝ

Ñ Fpσq :“ › › › ›T ˆ

N

ÿ

n“1

σnanχAn ˙› › › ›

X

. Since Λ is a probability space }F}L1pΛq ď }F}L8pΛq . The claim follows if we show that p˚˚q C}φ}X ď }F}L1pΛq ď }F}L8pΛq ď sup

|θ|“1

}Tpθφq}X.

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 20 / 28

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SLIDE 52

Extension of the FHT

Proof of p˚˚q : }F}L8pΛq ď sup|θ|“1 }Tpθφq}X

(4) Since σn “ ˘1, }F}L8pΛq “ sup

σPΛ

› › › ›T ˆ

N

ÿ

n“1

σnanχAn ˙› › › ›

X

“ sup

σPΛ

#› › › ›T ˆ θ

N

ÿ

n“1

anχAn ˙› › › ›

X

: θ “

N

ÿ

n“1

σnχAn + ď sup

|θ|“1

› ›Tpθφq › ›

X.

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 21 / 28

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SLIDE 53

Extension of the FHT

Proof of p˚˚q : C}φ}X ď }F}L1pΛq

(5) Fubini’s theorem and duality yields }F}L1pΛq “ ż

Λ

› › › ›

N

ÿ

n“1

σnanT pχAnq › › › ›

X

dσ “ ż

Λ

ˆ sup

}g}X1“1

ż 1

´1

|gptq| ˇ ˇ ˇ ˇ

N

ÿ

n“1

σnanT pχAnq ptq ˇ ˇ ˇ ˇ dt ˙ dσ ě sup

}g}X1“1

ż 1

´1

|gptq| ˆ ż

Λ

ˇ ˇ ˇ ˇ

N

ÿ

n“1

σnanT pχAnq ptq ˇ ˇ ˇ ˇ dσ ˙ dt.

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 22 / 28

slide-54
SLIDE 54

Extension of the FHT

Proof of p˚˚q : C}φ}X ď }F}L1pΛq

(5) Let Pn be the coordinate projections σ P Λ ÞÑ Pnpσq :“ σn P t´1, 1u.

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 23 / 28

slide-55
SLIDE 55

Extension of the FHT

Proof of p˚˚q : C}φ}X ď }F}L1pΛq

(5) Let Pn be the coordinate projections σ P Λ ÞÑ Pnpσq :“ σn P t´1, 1u. For t P p´1, 1q fixed, via Khintchine inequality, we have ż

Λ

ˇ ˇ ˇ ˇ

N

ÿ

n“1

σn anTpχAnqptq ˇ ˇ ˇ ˇ dσ ě 1 ? 2 ˆ

N

ÿ

n“1

|an|2 |TpχAnqptq|2 ˙1{2 .

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 23 / 28

slide-56
SLIDE 56

Extension of the FHT

Proof of p˚˚q : C}φ}X ď }F}L1pΛq

(5) Let Pn be the coordinate projections σ P Λ ÞÑ Pnpσq :“ σn P t´1, 1u. For t P p´1, 1q fixed, via Khintchine inequality, we have ż

Λ

ˇ ˇ ˇ ˇ

N

ÿ

n“1

σn anTpχAnqptq ˇ ˇ ˇ ˇ dσ ě 1 ? 2 ˆ

N

ÿ

n“1

|an|2 |TpχAnqptq|2 ˙1{2 . Consequently, via duality again, }F}L1pΛq ě 1 ? 2 › › › › ˆ

N

ÿ

n“1

|an|2 |T pχAnq|2 ˙1{2› › › ›

X

.

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 23 / 28

slide-57
SLIDE 57

Extension of the FHT

Proof of p˚˚q : C}φ}X ď }F}L1pΛq

(6) From the Stein-Weiss formula for the distribution function of TpχAq, it follows that mptt P A : |TpχAqptq| ą λuq “ 2mpAq eπλ ` 1.

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 24 / 28

slide-58
SLIDE 58

Extension of the FHT

Proof of p˚˚q : C}φ}X ď }F}L1pΛq

(6) From the Stein-Weiss formula for the distribution function of TpχAq, it follows that mptt P A : |TpχAqptq| ą λuq “ 2mpAq eπλ ` 1. We set λ “ 1, and find disjoint sets A1

n Ď An with

mpA1

nq “ δmpAnq,

for some 0 ă δ ă 1, with |TpχAnq| ą 1 on A1

n.

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 24 / 28

slide-59
SLIDE 59

Extension of the FHT

Proof of p˚˚q : C}φ}X ď }F}L1pΛq

(6) From the Stein-Weiss formula for the distribution function of TpχAq, it follows that mptt P A : |TpχAqptq| ą λuq “ 2mpAq eπλ ` 1. We set λ “ 1, and find disjoint sets A1

n Ď An with

mpA1

nq “ δmpAnq,

for some 0 ă δ ă 1, with |TpχAnq| ą 1 on A1

n.

So : ˆ

N

ÿ

n“1

|an|2 |T pχAnq|2 ˙1{2 ě

N

ÿ

n“1

|an|χA1

n. Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 24 / 28

slide-60
SLIDE 60

Extension of the FHT

Proof of p˚˚q : C}φ}X ď }F}L1pΛq

(7) Consequently, }F}L1pΛq ě 1 ? 2 › › › › ˆ

N

ÿ

n“1

|an|2 |T pχAnq|2 ˙1{2› › › ›

X

. ě 1 ? 2 › › › ›

N

ÿ

n“1

|an|χA1

n

› › › ›

X

ě 1 ? 2 1 }Eδ} › › › ›

N

ÿ

n“1

|an|χAn › › › ›

X

“ C}φ}X. Q.E.D.

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 25 / 28

slide-61
SLIDE 61

Extension of the FHT

Consequence

Corollary (C., Okada & Ricker, 2019) Let X be a r.i.s. on p´1, 1q satisfying 0 ă αX ď αX ă 1. Given f P L1p´1, 1q, the following conditions are equivalent. (a) f P X. (b) TpfχAq P X for every A P B. (c) Tpfθq P X for every θ P L8 with |θ| “ 1 a.e. (d) Tphq P X for every h P L0 with |h| ď |f| a.e.

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 26 / 28

slide-62
SLIDE 62

Extension of the FHT

References

By G. P . Curbera, S. Okada and W. J. Ricker: Inversion and extension of the finite Hilbert transform on (-1,1), Annali di Matematica Pura ed Applicata, to appear. Extension and integral representation of the finite Hilbert transform in rearrangement invariant spaces, Quaestiones Mathematicae, to appear. Special issue dedicated to the memory of Joe Diestel. Non-extendability of the finite Hilbert transform, preprint. The fine spectrum of the finite Hilbert transform, in preparation.

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 27 / 28

slide-63
SLIDE 63

Extension of the FHT

Thank you for your attention

Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 28 / 28