The principle of concentration-compactness and an application. Alexis Drouot September 3rd 2015
Plan.
Plan. ◮ The principle of concentration compactness.
Plan. ◮ The principle of concentration compactness. ◮ An application to the study of radial extremizing sequences for the Radon transform.
Context.
Context. ◮ Let T bounded from X to Y : | Tf | Y ≤ A | f | X for some minimal constant A .
Context. ◮ Let T bounded from X to Y : | Tf | Y ≤ A | f | X for some minimal constant A . ◮ A natural question is: are there functions realizing the equality?
Context. ◮ Let T bounded from X to Y : | Tf | Y ≤ A | f | X for some minimal constant A . ◮ A natural question is: are there functions realizing the equality? ◮ A natural approach is: take a sequence with | f n | X = 1 and | Tf n | Y → A and show that f n converges in X .
Context. ◮ Let T bounded from X to Y : | Tf | Y ≤ A | f | X for some minimal constant A . ◮ A natural question is: are there functions realizing the equality? ◮ A natural approach is: take a sequence with | f n | X = 1 and | Tf n | Y → A and show that f n converges in X . ◮ Problem: many interesting operators arise from physics and have many symmetries.
Context. ◮ Let T bounded from X to Y : | Tf | Y ≤ A | f | X for some minimal constant A . ◮ A natural question is: are there functions realizing the equality? ◮ A natural approach is: take a sequence with | f n | X = 1 and | Tf n | Y → A and show that f n converges in X . ◮ Problem: many interesting operators arise from physics and have many symmetries. ◮ Non compact groups of symmetries are hard to fight.
The concentration compactness principle.
The concentration compactness principle. Lemma Let φ n ≥ 0 on R d with | φ n | 1 = 1 . Then there exists a subsequence of φ n , still noted φ n with one of the following:
The concentration compactness principle. Lemma Let φ n ≥ 0 on R d with | φ n | 1 = 1 . Then there exists a subsequence of φ n , still noted φ n with one of the following: ◮ (tightness) There exists y n ∈ R d such that uniformly in n, � lim φ n = 1 . R →∞ B ( y n , R ) ◮ (vanishing) For all R, � n →∞ sup lim φ n = 0 . y ∈ R d B ( y , R ) ◮ (dichotomy) There exist 0 < α < 1 and φ n ≥ φ 1 n , φ 2 n ≥ 0 with d ( supp ( φ 1 n ) , supp ( φ 2 n )) → ∞ and | φ n − φ 1 n − φ 2 | φ 1 | φ 2 n | 1 → 0 , n | 1 → α, n | 1 → 1 − α.
Qualitative description. Let φ n satisfying the assumptions of the Lemma. Then one of the following happens:
Qualitative description. Let φ n satisfying the assumptions of the Lemma. Then one of the following happens: ◮ (tightness) There exists y n ∈ R d such that uniformly in n , � lim φ n = 1 . R →∞ B ( y n , R )
Qualitative description. Let φ n satisfying the assumptions of the Lemma. Then one of the following happens: ◮ (tightness) There exists y n ∈ R d such that uniformly in n , � lim φ n = 1 . R →∞ B ( y n , R ) φ n is mostly supported on a ball of radius R whose center y n moves around.
Qualitative description. Let φ n satisfying the assumptions of the Lemma. Then one of the following happens:
Qualitative description. Let φ n satisfying the assumptions of the Lemma. Then one of the following happens: ◮ (vanishing) For all R , � n →∞ sup lim φ n = 0 . y ∈ R d B ( y , R )
Qualitative description. Let φ n satisfying the assumptions of the Lemma. Then one of the following happens: ◮ (vanishing) For all R , � n →∞ sup lim φ n = 0 . y ∈ R d B ( y , R ) φ n is not really concentrated anywhere and somehow dissipates.
Qualitative description. Let φ n satisfying the assumptions of the Lemma. Then one of the following happens:
Qualitative description. Let φ n satisfying the assumptions of the Lemma. Then one of the following happens: ◮ (dichotomy) There exist 0 < α < 1 and φ n ≥ φ 1 n , φ 2 n ≥ 0 with d ( supp ( φ 1 n ) , supp ( φ 2 n )) → ∞ and | φ n − φ 1 n − φ 2 | φ 1 | φ 2 n | 1 → 0 , n | 1 → α, n | 1 → 1 − α.
Qualitative description. Let φ n satisfying the assumptions of the Lemma. Then one of the following happens: ◮ (dichotomy) There exist 0 < α < 1 and φ n ≥ φ 1 n , φ 2 n ≥ 0 with d ( supp ( φ 1 n ) , supp ( φ 2 n )) → ∞ and | φ n − φ 1 n − φ 2 | φ 1 | φ 2 n | 1 → 0 , n | 1 → α, n | 1 → 1 − α. φ n splits into two parts that get further and further from each other.
The Radon transform for radial functions in dimension 3
The Radon transform for radial functions in dimension 3 ◮ The Radon transform for radial functions takes the form � ∞ T f ( r ) = f ( u ) udu . r
The Radon transform for radial functions in dimension 3 ◮ The Radon transform for radial functions takes the form � ∞ T f ( r ) = f ( u ) udu . r ◮ T is continuous L p ( R + , u 2 du ) → L 4 ( R + , dr ), p = 4 / 3.
The Radon transform for radial functions in dimension 3 ◮ The Radon transform for radial functions takes the form � ∞ T f ( r ) = f ( u ) udu . r ◮ T is continuous L p ( R + , u 2 du ) → L 4 ( R + , dr ), p = 4 / 3. ◮ Goal: Prove that (radial) extremizing sequences for T converge modulo the group of dilations.
The Radon transform for radial functions in dimension 3 ◮ The Radon transform for radial functions takes the form � ∞ T f ( r ) = f ( u ) udu . r ◮ T is continuous L p ( R + , u 2 du ) → L 4 ( R + , dr ), p = 4 / 3. ◮ Goal: Prove that (radial) extremizing sequences for T converge modulo the group of dilations. ◮ Remark: this is a very weak form of a result of Christ: extremizing sequences for the Radon transform converge modulo the group of affine maps.
The Radon transform for radial functions in dimension 3 ◮ The Radon transform for radial functions takes the form � ∞ T f ( r ) = f ( u ) udu . r ◮ T is continuous L p ( R + , u 2 du ) → L 4 ( R + , dr ), p = 4 / 3. ◮ Goal: Prove that (radial) extremizing sequences for T converge modulo the group of dilations. ◮ Remark: this is a very weak form of a result of Christ: extremizing sequences for the Radon transform converge modulo the group of affine maps. But the goal is to apply the concentration compactness principle in a simple setting.
The case of f ∗ n .
The case of f ∗ n . ◮ Fix now 0 ≤ f n with | f n | p = 1 and |T f n | 4 → |T | .
The case of f ∗ n . ◮ Fix now 0 ≤ f n with | f n | p = 1 and |T f n | 4 → |T | . ◮ If f ∗ is the nonincreasing rearrangement of f then |T f ∗ | 4 ≥ |T f | 4 .
The case of f ∗ n . ◮ Fix now 0 ≤ f n with | f n | p = 1 and |T f n | 4 → |T | . ◮ If f ∗ is the nonincreasing rearrangement of f then |T f ∗ | 4 ≥ |T f | 4 . ◮ Thus f ∗ n is a nonincreasing extremizing sequence.
The case of f ∗ n . ◮ Fix now 0 ≤ f n with | f n | p = 1 and |T f n | 4 → |T | . ◮ If f ∗ is the nonincreasing rearrangement of f then |T f ∗ | 4 ≥ |T f | 4 . ◮ Thus f ∗ n is a nonincreasing extremizing sequence. ◮ Using some refined weak form inequalities there exists a rescaling of f ∗ n (still called f ∗ n ) so that f ∗ n converges weakly to a non-zero function .
The case of f ∗ n . ◮ Fix now 0 ≤ f n with | f n | p = 1 and |T f n | 4 → |T | . ◮ If f ∗ is the nonincreasing rearrangement of f then |T f ∗ | 4 ≥ |T f | 4 . ◮ Thus f ∗ n is a nonincreasing extremizing sequence. ◮ Using some refined weak form inequalities there exists a rescaling of f ∗ n (still called f ∗ n ) so that f ∗ n converges weakly to a non-zero function . ◮ Using Lieb’s lemma f ∗ n converges in L p .
What about f n ?
What about f n ? ◮ Rescale f ∗ n so that it converges and use the same rescaling for f n .
What about f n ? ◮ Rescale f ∗ n so that it converges and use the same rescaling for f n . ◮ f ∗ n and f n have the same distribution function.
What about f n ? ◮ Rescale f ∗ n so that it converges and use the same rescaling for f n . ◮ f ∗ n and f n have the same distribution function. ◮ Since f ∗ n converges there exists a set E n with f n ≥ 1 E n and | E n | ∼ 1.
What about f n ? ◮ Rescale f ∗ n so that it converges and use the same rescaling for f n . ◮ f ∗ n and f n have the same distribution function. ◮ Since f ∗ n converges there exists a set E n with f n ≥ 1 E n and | E n | ∼ 1. ◮ So vanishing does not occur!
The set E n does not move away from 0.
The set E n does not move away from 0. ◮ Assume that E n has a large part F n with d ( F n , 0) → ∞ .
The set E n does not move away from 0. ◮ Assume that E n has a large part F n with d ( F n , 0) → ∞ . ◮ There are not so many planes that have a big intersection with F n .
The set E n does not move away from 0. ◮ Assume that E n has a large part F n with d ( F n , 0) → ∞ . ◮ There are not so many planes that have a big intersection with F n . ◮ Hence T f n cannot be so big.
The set E n does not move away from 0. ◮ Assume that E n has a large part F n with d ( F n , 0) → ∞ . ◮ There are not so many planes that have a big intersection with F n . ◮ Hence T f n cannot be so big. ◮ Then f n cannot be an extremizing sequence.
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