![Page 1: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/1.jpg)
Some inequalities in the interface betweenharmonic analysis and differential equations
Mateus SousaUniversidad de Buenos Aires
Seminario de Ecuaciones Diferenciales y Analisis Numerico2018
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 2: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/2.jpg)
This presentation is based on the work
Extremizers for Fourier restriction on hyperboloids, (with E. Carneiro, andD. Oliveira e Silva), to appear at Ann. Henri Poincare, arXiv:1708.03826(2017).
But some events may be changed for dramatization effects.
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 3: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/3.jpg)
This presentation is based on the work
Extremizers for Fourier restriction on hyperboloids, (with E. Carneiro, andD. Oliveira e Silva), to appear at Ann. Henri Poincare, arXiv:1708.03826(2017).
But some events may be changed for dramatization effects.
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 4: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/4.jpg)
Our main actor
Given a subset S ⊂ Rd , and a measure σ supported in S , we define formally theFourier extension operator T , associated to the pair (S , σ), as
Tf (ξ) =
∫Sf (x)e ix ·ξdσ(x).
In this talk we will learn sharp inequalities related to those operators.
By that I mean: given a functional inequality associated to T , we will discusswhat are the best constants involved in the inequality, which functions areextremizers in case they exist etc.
Remark: When S = Rd and σ is the Lebesgue measure Tf = f , i.e, the morefamiliar Fourier transform of f . It will be useful today, but not the focus.
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 5: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/5.jpg)
Our main actor
Given a subset S ⊂ Rd , and a measure σ supported in S , we define formally theFourier extension operator T , associated to the pair (S , σ), as
Tf (ξ) =
∫Sf (x)e ix ·ξdσ(x).
In this talk we will learn sharp inequalities related to those operators.
By that I mean: given a functional inequality associated to T , we will discusswhat are the best constants involved in the inequality, which functions areextremizers in case they exist etc.
Remark: When S = Rd and σ is the Lebesgue measure Tf = f , i.e, the morefamiliar Fourier transform of f . It will be useful today, but not the focus.
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 6: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/6.jpg)
Our main actor
Given a subset S ⊂ Rd , and a measure σ supported in S , we define formally theFourier extension operator T , associated to the pair (S , σ), as
Tf (ξ) =
∫Sf (x)e ix ·ξdσ(x).
In this talk we will learn sharp inequalities related to those operators.
By that I mean: given a functional inequality associated to T , we will discusswhat are the best constants involved in the inequality, which functions areextremizers in case they exist etc.
Remark: When S = Rd and σ is the Lebesgue measure Tf = f , i.e, the morefamiliar Fourier transform of f .
It will be useful today, but not the focus.
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 7: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/7.jpg)
Our main actor
Given a subset S ⊂ Rd , and a measure σ supported in S , we define formally theFourier extension operator T , associated to the pair (S , σ), as
Tf (ξ) =
∫Sf (x)e ix ·ξdσ(x).
In this talk we will learn sharp inequalities related to those operators.
By that I mean: given a functional inequality associated to T , we will discusswhat are the best constants involved in the inequality, which functions areextremizers in case they exist etc.
Remark: When S = Rd and σ is the Lebesgue measure Tf = f , i.e, the morefamiliar Fourier transform of f . It will be useful today, but not the focus.
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 8: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/8.jpg)
Why these operators in a PDE seminar?
Again, we wish to study operators of the form
Tf (ξ) =
∫Sf (x)e ix ·ξdσ(x),
where S ⊂ Rd , and σ is a measure supported in S .
One of the oldest cases of interest is when S = P = {(x , |x |2); x ∈ Rd} anddσ(x , t) = δ(t − |x |2)dxdt, i.e, the paraboloid with projection measure. In thiscase, by identifying f : P → C with a function in Rd , one has
Tf (ξ, τ) =
∫Rd
f (x)e ix ·ξ+i |x |2τdx .
When f ∈ L2(Rd), one can see that u(ξ, τ) = Tf (ξ, τ) is obviously the solutionto the following Schrodinger equation
i∂tu = ∆xu, (x , t) ∈ Rd × R;
u(x , 0) = (2π)d f (x)
(If this last part was not so obvious to you, look at the board)
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 9: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/9.jpg)
Why these operators in a PDE seminar?
Again, we wish to study operators of the form
Tf (ξ) =
∫Sf (x)e ix ·ξdσ(x),
where S ⊂ Rd , and σ is a measure supported in S .
One of the oldest cases of interest is when S = P = {(x , |x |2); x ∈ Rd} anddσ(x , t) = δ(t − |x |2)dxdt, i.e, the paraboloid with projection measure. In thiscase, by identifying f : P → C with a function in Rd , one has
Tf (ξ, τ) =
∫Rd
f (x)e ix ·ξ+i |x |2τdx .
When f ∈ L2(Rd), one can see that u(ξ, τ) = Tf (ξ, τ) is obviously the solutionto the following Schrodinger equation
i∂tu = ∆xu, (x , t) ∈ Rd × R;
u(x , 0) = (2π)d f (x)
(If this last part was not so obvious to you, look at the board)
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 10: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/10.jpg)
Why these operators in a PDE seminar?
Again, we wish to study operators of the form
Tf (ξ) =
∫Sf (x)e ix ·ξdσ(x),
where S ⊂ Rd , and σ is a measure supported in S .
One of the oldest cases of interest is when S = P = {(x , |x |2); x ∈ Rd} anddσ(x , t) = δ(t − |x |2)dxdt, i.e, the paraboloid with projection measure. In thiscase, by identifying f : P → C with a function in Rd , one has
Tf (ξ, τ) =
∫Rd
f (x)e ix ·ξ+i |x |2τdx .
When f ∈ L2(Rd), one can see that u(ξ, τ) = Tf (ξ, τ) is obviously the solutionto the following Schrodinger equation
i∂tu = ∆xu, (x , t) ∈ Rd × R;
u(x , 0) = (2π)d f (x)
(If this last part was not so obvious to you, look at the board)
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 11: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/11.jpg)
Why these operators in a PDE seminar?
Again, we wish to study operators of the form
Tf (ξ) =
∫Sf (x)e ix ·ξdσ(x),
where S ⊂ Rd , and σ is a measure supported in S .
One of the oldest cases of interest is when S = P = {(x , |x |2); x ∈ Rd} anddσ(x , t) = δ(t − |x |2)dxdt, i.e, the paraboloid with projection measure. In thiscase, by identifying f : P → C with a function in Rd , one has
Tf (ξ, τ) =
∫Rd
f (x)e ix ·ξ+i |x |2τdx .
When f ∈ L2(Rd), one can see that u(ξ, τ) = Tf (ξ, τ) is obviously the solutionto the following Schrodinger equation
i∂tu = ∆xu, (x , t) ∈ Rd × R;
u(x , 0) = (2π)d f (x)
(If this last part was not so obvious to you, look at the board)
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 12: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/12.jpg)
Other cases
This is not an isolated case:
The cone C = {(x , |x |) : x ∈ Rd} with the measuredσ(x , t) = δ(t − |x |)dxdtt : generates solutions to the wave equation.
The sphere with its surface measure generations solutions to Helmholtz’sequation.
The twisted paraboloid P = {(x , sign(x)x2) : x ∈ R} with projectionmeasure generates solutions to the Benjamin-Ono equation.
The cubic surface A = {(x , x3) : x ∈ R} with projection measure generatessolutions to Airy’s equation (KdV for some).
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 13: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/13.jpg)
Other cases
This is not an isolated case:
The cone C = {(x , |x |) : x ∈ Rd} with the measuredσ(x , t) = δ(t − |x |)dxdtt : generates solutions to the wave equation.
The sphere with its surface measure generations solutions to Helmholtz’sequation.
The twisted paraboloid P = {(x , sign(x)x2) : x ∈ R} with projectionmeasure generates solutions to the Benjamin-Ono equation.
The cubic surface A = {(x , x3) : x ∈ R} with projection measure generatessolutions to Airy’s equation (KdV for some).
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 14: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/14.jpg)
Other cases
This is not an isolated case:
The cone C = {(x , |x |) : x ∈ Rd} with the measuredσ(x , t) = δ(t − |x |)dxdtt : generates solutions to the wave equation.
The sphere with its surface measure generations solutions to Helmholtz’sequation.
The twisted paraboloid P = {(x , sign(x)x2) : x ∈ R} with projectionmeasure generates solutions to the Benjamin-Ono equation.
The cubic surface A = {(x , x3) : x ∈ R} with projection measure generatessolutions to Airy’s equation (KdV for some).
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 15: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/15.jpg)
Other cases
This is not an isolated case:
The cone C = {(x , |x |) : x ∈ Rd} with the measuredσ(x , t) = δ(t − |x |)dxdtt : generates solutions to the wave equation.
The sphere with its surface measure generations solutions to Helmholtz’sequation.
The twisted paraboloid P = {(x , sign(x)x2) : x ∈ R} with projectionmeasure generates solutions to the Benjamin-Ono equation.
The cubic surface A = {(x , x3) : x ∈ R} with projection measure generatessolutions to Airy’s equation (KdV for some).
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 16: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/16.jpg)
Other cases
This is not an isolated case:
The cone C = {(x , |x |) : x ∈ Rd} with the measuredσ(x , t) = δ(t − |x |)dxdtt : generates solutions to the wave equation.
The sphere with its surface measure generations solutions to Helmholtz’sequation.
The twisted paraboloid P = {(x , sign(x)x2) : x ∈ R} with projectionmeasure generates solutions to the Benjamin-Ono equation.
The cubic surface A = {(x , x3) : x ∈ R} with projection measure generatessolutions to Airy’s equation (KdV for some).
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 17: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/17.jpg)
PDE x Restriction estimate
There is a sort of duality: an estimate for the the operator T gives you anspace-time control of the solution.
For the Schrodinger/Paraboloid case,one has the following inequality:
‖Tf ‖Lp(Rd+1) . ‖f ‖L2(Rd ),
when p = 2 + 4d . Or
‖u‖Lp(Rd×R) . ‖u(·, 0)‖L2(Rd ).
The same paradigm is true for the other equations: a L2(σ)→ Lp(Rd+1) isequivalent to a Hs
x → Lpx ,t estimate. The cone case for instance corresponds tothis:
‖u‖Lp(Rd×R) . ‖u(·, 0)‖H1/2(Rd ) + ‖∂tu(·, 0)‖H−1/2(Rd )
where u is a solution to ∂ttu = ∆xu and p = 2 + 4d−1 .
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 18: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/18.jpg)
PDE x Restriction estimate
There is a sort of duality: an estimate for the the operator T gives you anspace-time control of the solution.
For the Schrodinger/Paraboloid case,one has the following inequality:
‖Tf ‖Lp(Rd+1) . ‖f ‖L2(Rd ),
when p = 2 + 4d .
Or
‖u‖Lp(Rd×R) . ‖u(·, 0)‖L2(Rd ).
The same paradigm is true for the other equations: a L2(σ)→ Lp(Rd+1) isequivalent to a Hs
x → Lpx ,t estimate. The cone case for instance corresponds tothis:
‖u‖Lp(Rd×R) . ‖u(·, 0)‖H1/2(Rd ) + ‖∂tu(·, 0)‖H−1/2(Rd )
where u is a solution to ∂ttu = ∆xu and p = 2 + 4d−1 .
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 19: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/19.jpg)
PDE x Restriction estimate
There is a sort of duality: an estimate for the the operator T gives you anspace-time control of the solution.
For the Schrodinger/Paraboloid case,one has the following inequality:
‖Tf ‖Lp(Rd+1) . ‖f ‖L2(Rd ),
when p = 2 + 4d . Or
‖u‖Lp(Rd×R) . ‖u(·, 0)‖L2(Rd ).
The same paradigm is true for the other equations: a L2(σ)→ Lp(Rd+1) isequivalent to a Hs
x → Lpx ,t estimate. The cone case for instance corresponds tothis:
‖u‖Lp(Rd×R) . ‖u(·, 0)‖H1/2(Rd ) + ‖∂tu(·, 0)‖H−1/2(Rd )
where u is a solution to ∂ttu = ∆xu and p = 2 + 4d−1 .
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 20: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/20.jpg)
PDE x Restriction estimate
There is a sort of duality: an estimate for the the operator T gives you anspace-time control of the solution.
For the Schrodinger/Paraboloid case,one has the following inequality:
‖Tf ‖Lp(Rd+1) . ‖f ‖L2(Rd ),
when p = 2 + 4d . Or
‖u‖Lp(Rd×R) . ‖u(·, 0)‖L2(Rd ).
The same paradigm is true for the other equations: a L2(σ)→ Lp(Rd+1) isequivalent to a Hs
x → Lpx ,t estimate.
The cone case for instance corresponds tothis:
‖u‖Lp(Rd×R) . ‖u(·, 0)‖H1/2(Rd ) + ‖∂tu(·, 0)‖H−1/2(Rd )
where u is a solution to ∂ttu = ∆xu and p = 2 + 4d−1 .
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 21: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/21.jpg)
PDE x Restriction estimate
There is a sort of duality: an estimate for the the operator T gives you anspace-time control of the solution.
For the Schrodinger/Paraboloid case,one has the following inequality:
‖Tf ‖Lp(Rd+1) . ‖f ‖L2(Rd ),
when p = 2 + 4d . Or
‖u‖Lp(Rd×R) . ‖u(·, 0)‖L2(Rd ).
The same paradigm is true for the other equations: a L2(σ)→ Lp(Rd+1) isequivalent to a Hs
x → Lpx ,t estimate. The cone case for instance corresponds tothis:
‖u‖Lp(Rd×R) . ‖u(·, 0)‖H1/2(Rd ) + ‖∂tu(·, 0)‖H−1/2(Rd )
where u is a solution to ∂ttu = ∆xu and p = 2 + 4d−1 .
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 22: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/22.jpg)
Back to restriction
Consider the hyperboloid Hd ⊂ Rd+1 defined by
Hd ={
(y , y ′) ∈ Rd × R : y ′ =√
1 + |y |2},
equipped with the measure
dσ(y , y ′) = δ(y ′ −
√1 + |y |2
) dy dy ′√1 + |y |2
.
The Fourier extension operator on Hd is defined by
Tf (x , t) =
∫Hd
f (y)e i(x ,t)·ydσ(y)
=
∫Rd
e ix ·ye it√
1+|y |2f (y ,√
1 + |y |2)dy√
1 + |y |2,
where (x , t) ∈ Rd × R, and f : Hd → C is in a appropriate dense class.
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 23: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/23.jpg)
Back to restriction
Consider the hyperboloid Hd ⊂ Rd+1 defined by
Hd ={
(y , y ′) ∈ Rd × R : y ′ =√
1 + |y |2},
equipped with the measure
dσ(y , y ′) = δ(y ′ −
√1 + |y |2
) dy dy ′√1 + |y |2
.
The Fourier extension operator on Hd is defined by
Tf (x , t) =
∫Hd
f (y)e i(x ,t)·ydσ(y)
=
∫Rd
e ix ·ye it√
1+|y |2f (y ,√
1 + |y |2)dy√
1 + |y |2,
where (x , t) ∈ Rd × R, and f : Hd → C is in a appropriate dense class.
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 24: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/24.jpg)
Hyperbolic setting
The classical work of Strichartz establishes that
‖Tf ‖Lp(Rd+1) ≤ Hd ,p ‖f ‖L2(dσ) ,
with a finite constant Hd ,p (independent of f ), in the following admissible range{6 ≤ p <∞, if d = 1;
2 + 4d ≤ p ≤ 2 + 4
d−1 , if d ≥ 2.
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 25: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/25.jpg)
Hyperbolic setting
-3 -2 -1 1 2 3
1
2
3
4
5
Figure: The paraboloid y = 1 + |x|22 osculates the hyperboloid y =
√1 + |x |2 at its
vertex. The cone y = |x | approximates the same hyperboloid at infinity.
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 26: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/26.jpg)
Klein–Gordon equation
The restriction operator T is intimately connected with the Klein-Gordonequation
∂2t u = ∆xu − u, (x , t) ∈ Rd × R;
u(x , 0) = u0(x), ∂tu(x , 0) = u1(x).
trough the following operator, the Klein–Gordon propagator,
e it√
1−∆g(x) :=1
(2π)d
∫Rd
g(ξ) e ix ·ξ e it√
1+|ξ|2dξ.
One can see that solutions to the equation can be written as
u(·, t) =1
2
(e it√
1−∆u0(·)− ie it√
1−∆(√
1−∆)−1u1(·))
+
1
2
(e−it
√1−∆u0(·) + ie−it
√1−∆(
√1−∆)−1u1(·)
),
and for f (x) = g(x)√
1 + |x |2 (or f = (1−∆)1/2g)
Tf (x , t) = (2π)d e it√
1−∆g(x),
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 27: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/27.jpg)
Klein-Gordon equation
This relation implies that the restriction estimate
‖Tf ‖Lp(Rd+1) ≤ Hd ,p ‖f ‖L2(dσ)
amounts to the following inequality
‖e it√
1−∆g‖Lpx,t(Rd×R) ≤ (2π)−d Hd ,p ‖g‖H
12 (Rd )
,
and although these are equivalent, each inequality has its advantages.
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 28: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/28.jpg)
Klein-Gordon equation
This relation implies that the restriction estimate
‖Tf ‖Lp(Rd+1) ≤ Hd ,p ‖f ‖L2(dσ)
amounts to the following inequality
‖e it√
1−∆g‖Lpx,t(Rd×R) ≤ (2π)−d Hd ,p ‖g‖H
12 (Rd )
,
and although these are equivalent, each inequality has its advantages.
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 29: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/29.jpg)
Problems to come
We are interested in sharp instances of the extension inequality on thehyperboloid.
More precisely, given a pair (d , p) in the admissible range, we studyextremizers and extremizing sequences for the restriction inequality , and areinterested in the value of the optimal constant
Hd ,p := sup06=f ∈L2(Hd )
‖Tf ‖Lp(Rd+1)
‖f ‖L2(dσ).
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 30: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/30.jpg)
Problems to come
We are interested in sharp instances of the extension inequality on thehyperboloid. More precisely, given a pair (d , p) in the admissible range, we studyextremizers and extremizing sequences for the restriction inequality , and areinterested in the value of the optimal constant
Hd ,p := sup06=f ∈L2(Hd )
‖Tf ‖Lp(Rd+1)
‖f ‖L2(dσ).
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 31: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/31.jpg)
Brief history
This problem has been considered in different surfaces:
Paraboloid: Gaussians are extremizers for even exponent cases and there areextremizers for all dimensions (Kunze, Foschi, Hundertmark-Zharnitsky,Carneiro, Shao).
Cone: Exponentials are extremizers for even exponent cases and there areextremizers for all dimensions(Foschi, Bez-Rogers, Ramos).
Sphere: Constants are the only real extremizers for the 2-sphere endpoint ofTomas-Stein and there are extremizers for the circle. Every even exponentcase in the mixed norm setting (L2 → Lprad(L2
ang )) has constants as uniqueextremizers and the same is true after some big exponent. (Foschi,Christ-Shao, Shao, Foschi-Oliveira e Silva, Carneiro-Oliveira e Silva-S.).
Hyperboloid: All the even exponent endpoints don’t have extremizers. Inlow dimensions there are extremizers for non-endpoint cases (Quilodran,Carneiro-Oliveira e Silva-S).
To see a picture of the history of sharp Fourier restriction theory, one can checkthe recent survey ”Some recent progress on in sharp Fourier restriction theory”by D. Foschi and D. Oliveira e Silva.
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 32: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/32.jpg)
Brief history
This problem has been considered in different surfaces:
Paraboloid: Gaussians are extremizers for even exponent cases and there areextremizers for all dimensions (Kunze, Foschi, Hundertmark-Zharnitsky,Carneiro, Shao).
Cone: Exponentials are extremizers for even exponent cases and there areextremizers for all dimensions(Foschi, Bez-Rogers, Ramos).
Sphere: Constants are the only real extremizers for the 2-sphere endpoint ofTomas-Stein and there are extremizers for the circle. Every even exponentcase in the mixed norm setting (L2 → Lprad(L2
ang )) has constants as uniqueextremizers and the same is true after some big exponent. (Foschi,Christ-Shao, Shao, Foschi-Oliveira e Silva, Carneiro-Oliveira e Silva-S.).
Hyperboloid: All the even exponent endpoints don’t have extremizers. Inlow dimensions there are extremizers for non-endpoint cases (Quilodran,Carneiro-Oliveira e Silva-S).
To see a picture of the history of sharp Fourier restriction theory, one can checkthe recent survey ”Some recent progress on in sharp Fourier restriction theory”by D. Foschi and D. Oliveira e Silva.
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 33: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/33.jpg)
Brief history
This problem has been considered in different surfaces:
Paraboloid: Gaussians are extremizers for even exponent cases and there areextremizers for all dimensions (Kunze, Foschi, Hundertmark-Zharnitsky,Carneiro, Shao).
Cone: Exponentials are extremizers for even exponent cases and there areextremizers for all dimensions(Foschi, Bez-Rogers, Ramos).
Sphere: Constants are the only real extremizers for the 2-sphere endpoint ofTomas-Stein and there are extremizers for the circle. Every even exponentcase in the mixed norm setting (L2 → Lprad(L2
ang )) has constants as uniqueextremizers and the same is true after some big exponent. (Foschi,Christ-Shao, Shao, Foschi-Oliveira e Silva, Carneiro-Oliveira e Silva-S.).
Hyperboloid: All the even exponent endpoints don’t have extremizers. Inlow dimensions there are extremizers for non-endpoint cases (Quilodran,Carneiro-Oliveira e Silva-S).
To see a picture of the history of sharp Fourier restriction theory, one can checkthe recent survey ”Some recent progress on in sharp Fourier restriction theory”by D. Foschi and D. Oliveira e Silva.
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 34: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/34.jpg)
Brief history
This problem has been considered in different surfaces:
Paraboloid: Gaussians are extremizers for even exponent cases and there areextremizers for all dimensions (Kunze, Foschi, Hundertmark-Zharnitsky,Carneiro, Shao).
Cone: Exponentials are extremizers for even exponent cases and there areextremizers for all dimensions(Foschi, Bez-Rogers, Ramos).
Sphere: Constants are the only real extremizers for the 2-sphere endpoint ofTomas-Stein and there are extremizers for the circle. Every even exponentcase in the mixed norm setting (L2 → Lprad(L2
ang )) has constants as uniqueextremizers and the same is true after some big exponent. (Foschi,Christ-Shao, Shao, Foschi-Oliveira e Silva, Carneiro-Oliveira e Silva-S.).
Hyperboloid: All the even exponent endpoints don’t have extremizers. Inlow dimensions there are extremizers for non-endpoint cases (Quilodran,Carneiro-Oliveira e Silva-S).
To see a picture of the history of sharp Fourier restriction theory, one can checkthe recent survey ”Some recent progress on in sharp Fourier restriction theory”by D. Foschi and D. Oliveira e Silva.
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 35: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/35.jpg)
Brief history
This problem has been considered in different surfaces:
Paraboloid: Gaussians are extremizers for even exponent cases and there areextremizers for all dimensions (Kunze, Foschi, Hundertmark-Zharnitsky,Carneiro, Shao).
Cone: Exponentials are extremizers for even exponent cases and there areextremizers for all dimensions(Foschi, Bez-Rogers, Ramos).
Sphere: Constants are the only real extremizers for the 2-sphere endpoint ofTomas-Stein and there are extremizers for the circle. Every even exponentcase in the mixed norm setting (L2 → Lprad(L2
ang )) has constants as uniqueextremizers and the same is true after some big exponent. (Foschi,Christ-Shao, Shao, Foschi-Oliveira e Silva, Carneiro-Oliveira e Silva-S.).
Hyperboloid: All the even exponent endpoints don’t have extremizers. Inlow dimensions there are extremizers for non-endpoint cases (Quilodran,Carneiro-Oliveira e Silva-S).
To see a picture of the history of sharp Fourier restriction theory, one can checkthe recent survey ”Some recent progress on in sharp Fourier restriction theory”by D. Foschi and D. Oliveira e Silva.
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 36: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/36.jpg)
Brief history
This problem has been considered in different surfaces:
Paraboloid: Gaussians are extremizers for even exponent cases and there areextremizers for all dimensions (Kunze, Foschi, Hundertmark-Zharnitsky,Carneiro, Shao).
Cone: Exponentials are extremizers for even exponent cases and there areextremizers for all dimensions(Foschi, Bez-Rogers, Ramos).
Sphere: Constants are the only real extremizers for the 2-sphere endpoint ofTomas-Stein and there are extremizers for the circle. Every even exponentcase in the mixed norm setting (L2 → Lprad(L2
ang )) has constants as uniqueextremizers and the same is true after some big exponent. (Foschi,Christ-Shao, Shao, Foschi-Oliveira e Silva, Carneiro-Oliveira e Silva-S.).
Hyperboloid: All the even exponent endpoints don’t have extremizers. Inlow dimensions there are extremizers for non-endpoint cases (Quilodran,Carneiro-Oliveira e Silva-S).
To see a picture of the history of sharp Fourier restriction theory, one can checkthe recent survey ”Some recent progress on in sharp Fourier restriction theory”by D. Foschi and D. Oliveira e Silva.
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 37: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/37.jpg)
What about the hyperboloid?
Theorem (Quilodran (2015))
Let (d , p) ∈ {(2, 4), (2, 6), (3, 4)}. Then
H2,4 = 234π, H2,6 = (2π)
56 , and H3,4 = (2π)
54 ,
and there are no extremizers for the restriction inequality in these cases.
Oberve that these cases are the only ones where d ≥ 2 and p is an even integer.In his paper, Quilodran left the question of what happens with the rest the eveninteger cases, which are many, since for d = 1 every integer bigger than 6 is anadmissible exponent.
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 38: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/38.jpg)
What about the hyperboloid?
Theorem (Quilodran (2015))
Let (d , p) ∈ {(2, 4), (2, 6), (3, 4)}. Then
H2,4 = 234π, H2,6 = (2π)
56 , and H3,4 = (2π)
54 ,
and there are no extremizers for the restriction inequality in these cases.
Oberve that these cases are the only ones where d ≥ 2 and p is an even integer.
In his paper, Quilodran left the question of what happens with the rest the eveninteger cases, which are many, since for d = 1 every integer bigger than 6 is anadmissible exponent.
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 39: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/39.jpg)
What about the hyperboloid?
Theorem (Quilodran (2015))
Let (d , p) ∈ {(2, 4), (2, 6), (3, 4)}. Then
H2,4 = 234π, H2,6 = (2π)
56 , and H3,4 = (2π)
54 ,
and there are no extremizers for the restriction inequality in these cases.
Oberve that these cases are the only ones where d ≥ 2 and p is an even integer.In his paper, Quilodran left the question of what happens with the rest the eveninteger cases, which are many, since for d = 1 every integer bigger than 6 is anadmissible exponent.
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 40: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/40.jpg)
Our contribution
Theorem 1 (Carneiro, Oliveira e Silva, and S. (2017))
The value of the optimal constant in the case (d , p) = (1, 6) is
H1,6 = 3−1
12 (2π)12 .
Moreover, extremizers do not exist in this case.
Theorem 2 (Carneiro, Oliveira e Silva, and S. (2017))
Extremizers for the restriction inequality do exist in the following cases:
(a) d = 1 and 6 < p <∞.
(b) d = 2 and 4 < p < 6.
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 41: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/41.jpg)
Our contribution
Theorem 1 (Carneiro, Oliveira e Silva, and S. (2017))
The value of the optimal constant in the case (d , p) = (1, 6) is
H1,6 = 3−1
12 (2π)12 .
Moreover, extremizers do not exist in this case.
Theorem 2 (Carneiro, Oliveira e Silva, and S. (2017))
Extremizers for the restriction inequality do exist in the following cases:
(a) d = 1 and 6 < p <∞.
(b) d = 2 and 4 < p < 6.
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 42: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/42.jpg)
Strategy: the case of Theorem 1
From Plancherel’s Theorem it follows that
‖Tf ‖3L6(R2) = ‖(f σ)3‖L2(R2) = ‖(f σ ∗ f σ ∗ f σ) ‖L2(R2) = 2π‖f σ ∗ f σ ∗ f σ‖L2(R2),
which in particular implies that
H31,6 = 2π sup
06=f ∈L2(H1)
‖f σ ∗ f σ ∗ f σ‖L2(R2)
‖f ‖3L2(H1)
.
We are thus led to studying convolution measure σ ∗ σ ∗ σ in order to obtainsharp bounds. In this case, we have access to the machinery of δ-calculusintroduced by Foschi.
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 43: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/43.jpg)
Strategy: the case of Theorem 1
From Plancherel’s Theorem it follows that
‖Tf ‖3L6(R2) = ‖(f σ)3‖L2(R2) = ‖(f σ ∗ f σ ∗ f σ) ‖L2(R2) = 2π‖f σ ∗ f σ ∗ f σ‖L2(R2),
which in particular implies that
H31,6 = 2π sup
06=f ∈L2(H1)
‖f σ ∗ f σ ∗ f σ‖L2(R2)
‖f ‖3L2(H1)
.
We are thus led to studying convolution measure σ ∗ σ ∗ σ in order to obtainsharp bounds. In this case, we have access to the machinery of δ-calculusintroduced by Foschi.
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 44: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/44.jpg)
Strategy: the case of Theorem 1
Step 1: After applying the convolution equivalence, apply Cauchy-Schwartz.This is gonna produce a bound in terms of the σ ∗ σ ∗ σ.
Step 2: Study σ ∗ σ ∗ σ and localize it maximum. This is gonna provide asharp constant candidante.
Step 3: Produce an extremizing sequence that converges to the candidate.
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 45: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/45.jpg)
Strategy: the case of Theorem 1
Step 1: After applying the convolution equivalence, apply Cauchy-Schwartz.This is gonna produce a bound in terms of the σ ∗ σ ∗ σ.
Step 2: Study σ ∗ σ ∗ σ and localize it maximum. This is gonna provide asharp constant candidante.
Step 3: Produce an extremizing sequence that converges to the candidate.
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 46: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/46.jpg)
Strategy: the case of Theorem 1
Step 1: After applying the convolution equivalence, apply Cauchy-Schwartz.This is gonna produce a bound in terms of the σ ∗ σ ∗ σ.
Step 2: Study σ ∗ σ ∗ σ and localize it maximum. This is gonna provide asharp constant candidante.
Step 3: Produce an extremizing sequence that converges to the candidate.
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 47: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/47.jpg)
Strategy: the case of Theorem 2
To produce extremizers, we consider extremizing sequences and show that theyconverge after suitable modifications. Recent work of Fanelli-Vega-Viscigliasuggest that in non endpoint cases concentration-compactness arguments canlead to the existence of extremizers.
The heart of the matter lies in the construction of a special cap, i.e. a set thatcontains a positive universal proportion of the total mass in an extremizingsequence, possibly after applying the symmetries of the problem. This rules outthe possibility of “mass concentration at infinity”, and is the missing part in aobservation of Quilodran, that had originally outlined the proof of a dichotomyfor extremizing sequences.
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 48: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/48.jpg)
Strategy: the case of Theorem 2
To produce extremizers, we consider extremizing sequences and show that theyconverge after suitable modifications. Recent work of Fanelli-Vega-Viscigliasuggest that in non endpoint cases concentration-compactness arguments canlead to the existence of extremizers.
The heart of the matter lies in the construction of a special cap, i.e. a set thatcontains a positive universal proportion of the total mass in an extremizingsequence, possibly after applying the symmetries of the problem. This rules outthe possibility of “mass concentration at infinity”, and is the missing part in aobservation of Quilodran, that had originally outlined the proof of a dichotomyfor extremizing sequences.
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 49: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/49.jpg)
1D setting
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 50: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/50.jpg)
2D setting
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 51: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/51.jpg)
Strategy: the case of Theorem 2
Step 1: Prove a bilinear estimate that show how caps interact.
Step 2: Use the bilinear inequality to prove a improved linear inequality.
Step 3: Use the improved inequality to prove that there is a special cap.Nonendpointness will produce a extremizer.
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 52: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/52.jpg)
Strategy: the case of Theorem 2
Step 1: Prove a bilinear estimate that show how caps interact.
Step 2: Use the bilinear inequality to prove a improved linear inequality.
Step 3: Use the improved inequality to prove that there is a special cap.Nonendpointness will produce a extremizer.
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 53: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/53.jpg)
Strategy: the case of Theorem 2
Step 1: Prove a bilinear estimate that show how caps interact.
Step 2: Use the bilinear inequality to prove a improved linear inequality.
Step 3: Use the improved inequality to prove that there is a special cap.Nonendpointness will produce a extremizer.
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE
![Page 54: Some inequalities in the interface between harmonic ...cms.dm.uba.ar/actividades/seminarios/sedan/MateusSeminar.pdf · d 1. Mateus Sousa Universidad de Buenos Aires Harmonic Analysis](https://reader035.vdocuments.us/reader035/viewer/2022070920/5fb91ab522224d2bdf359e31/html5/thumbnails/54.jpg)
Thank you!
¡Gracias!
Mateus Sousa Universidad de Buenos Aires Harmonic Analysis meets PDE