Generalized BMO spaces on Riemannian manifolds G. Mauceri 1 S. Meda - PowerPoint PPT Presentation

Generalized BMO spaces on Riemannian manifolds G. Mauceri 1 S. Meda 2 M. Vallarino 3 1 Dipartimento di Matematica Università di Genova 2 Dipartimento di Matematica Università di Milano Bicocca 3 Dipartimento di Matematica Politecnico di Torino XXXIII Convegno Nazionale di Analisi Armonica Alba, 17-20 Giugno 2013

The classical H 1 − BMO theory ( X , d ,µ ) doubling metric measure space     H 1 ( X ) = � � � < ∞ � �  f = λ j a j : a j CW-atoms , � λ j  j j a is a CW-atom iff s uppa ⊂ B ◮ � a � 2 ≤ µ ( B ) − 1 / 2 ◮ � a d µ = 0 . ◮

The classical H 1 − BMO theory ( X , d ,µ ) doubling metric measure space     H 1 ( X ) = � � � < ∞ � �  f = λ j a j : a j CW-atoms , � λ j  j j a is a CW-atom iff s uppa ⊂ B ◮ � a � 2 ≤ µ ( B ) − 1 / 2 ◮ � a d µ = 0 . ◮ � � � � � f � H 1 = inf � λ j � , f = λ j a j . j j

The classical H 1 − BMO theory BMO ( X ) is the space of functions f such that � 1 / 2 � 1 � | f ( x ) − c | 2 d µ ( x ) � f � BMO = sup inf µ ( B ) c ∈ C B B

The classical H 1 − BMO theory BMO ( X ) is the space of functions f such that � 1 / 2 � 1 � | f ( x ) − c | 2 d µ ( x ) � f � BMO = sup inf µ ( B ) c ∈ C B B µ ( B ) − 1 / 2 � π B ( f ) � L 2 ( B ) = sup B π B : ⊥ projection on the orthogonal of constants in L 2 ( B ) .

The classical H 1 − BMO theory H 1 ( X ) ∗ = BMO ( X ) ◮ CZO ′ s : H 1 ( X ) → L 1 ( X ) , L ∞ ( X ) → BMO ( X ) ◮ [ H 1 ( X ) , BMO ( X )] θ = L p ( X ) , θ = 1 − 1 / p ◮

The classical H 1 − BMO theory H 1 ( X ) ∗ = BMO ( X ) ◮ CZO ′ s : H 1 ( X ) → L 1 ( X ) , L ∞ ( X ) → BMO ( X ) ◮ [ H 1 ( X ) , BMO ( X )] θ = L p ( X ) , θ = 1 − 1 / p ◮ Y Banach space, T linear operator. If ◮ � Ta � Y ≤ C ∀ a CW-atoms then � T � H 1 →Y ≤ C . [Meda, Sjögren,Vallarino]

Riemannian manifolds ( M , g ) noncompact, complete Riemannian manifold � d µ ( x ) = det g ( x ) d x Riemannian measure L = − divgrad Laplace-Beltrami operator

Riemannian manifolds ( M , g ) noncompact, complete Riemannian manifold � d µ ( x ) = det g ( x ) d x Riemannian measure L = − divgrad Laplace-Beltrami operator Natural singular integrals on M ◮ Riesz transforms: ∇ L − 1 / 2 , ∇ k L − k / 2 ◮ spectral multipliers: m ( L ) , L iu , u ∈ R

Riemannian manifolds ( M , g ) noncompact, complete Riemannian manifold � d µ ( x ) = det g ( x ) d x Riemannian measure L = − divgrad Laplace-Beltrami operator Natural singular integrals on M ◮ Riesz transforms: ∇ L − 1 / 2 , ∇ k L − k / 2 ◮ spectral multipliers: m ( L ) , L iu , u ∈ R Develop an analogue of the H 1 − BMO theory on M providing end-point estimates for ∇ k L − k / 2 , m ( L ) .

Previous results ( M , g ,µ ) doubling ◮ [Russ], [Marias, Russ] ◮ [Auscher, McIntosh, Russ]

Previous results ( M , g ,µ ) doubling ◮ [Russ], [Marias, Russ] ◮ [Auscher, McIntosh, Russ] ( M , g ,µ ) non-doubling: local Hardy spaces ◮ [Taylor] ◮ [Carbonaro, McIntosh, Morris] Endpoint estimates for s. i. operator that have only local singularities. But ∇ k L − k / 2 , L iu are singular also at ∞ .

Previous results ( M , g ,µ ) doubling ◮ [Russ], [Marias, Russ] ◮ [Auscher, McIntosh, Russ] ( M , g ,µ ) non-doubling: local Hardy spaces ◮ [Taylor] ◮ [Carbonaro, McIntosh, Morris] Endpoint estimates for s. i. operator that have only local singularities. But ∇ k L − k / 2 , L iu are singular also at ∞ . ( M , g ,µ ) non-doubling: global Hardy spaces ◮ [GM, Meda, Vallarino]

A class of nondoubling manifolds We assume M has bounded geometry ◮ inj M > 0 , Ric M bounded below M has spectral gap b = inf σ 2 ( L ) > 0 ◮ ( M , g ,µ ) is locally doubling but not globally doubling.

A class of nondoubling manifolds We assume M has bounded geometry ◮ inj M > 0 , Ric M bounded below M has spectral gap b = inf σ 2 ( L ) > 0 ◮ ( M , g ,µ ) is locally doubling but not globally doubling. Examples ◮ noncompact semisimple Lie groups with finite centre and any invariant metric ◮ noncompact symmetric spaces with Killing metric ◮ Damek-Ricci spaces ◮ Cartan-Hadamard manifolds with spectral gap.

The space H 1 A function a ∈ A s (atoms at scale s ) if (1) s uppa ⊂ B , r B < s (2) � a � 2 ≤ µ ( B ) − 1 / 2 � (3) a d µ = 0 . � � H 1 � < ∞ � � s ( M ) = f = � j λ j a , a j ∈ A s , � � λ j j     � : f = � � � � � f � H 1 s = inf � λ j λ j a , a j ∈ A s  j j  [Russ], [Carbonaro, M, Meda]

Properties of H 1 Main properties [Carbonaro, M, Meda] H 1 s 1 ( M ) = H 1 s 2 ( M ) , � f � H 1 s 1 ≈ � f � H 1 ◮ s 2 s 0 = 1 Henceforth H 1 ( M ) = H 1 s 0 ( M ) , 2 inj M [ H 1 ( M ) , L 2 ( M )] θ = L p ( M ) , θ = 2 ( 1 − 1 / p ) ◮

Properties of H 1 Main properties [Carbonaro, M, Meda] H 1 s 1 ( M ) = H 1 s 2 ( M ) , � f � H 1 s 1 ≈ � f � H 1 ◮ s 2 s 0 = 1 Henceforth H 1 ( M ) = H 1 s 0 ( M ) , 2 inj M [ H 1 ( M ) , L 2 ( M )] θ = L p ( M ) , θ = 2 ( 1 − 1 / p ) ◮ However L iu and ∇ L − 1 / 2 do not map H 1 ( M ) → L 1 ( M ) More cancellation is needed

Special atoms and the space X 1 ( M ) Quasiharmonic functions q ( M ) = { u ∈ C ∞ ( M ) : Lu = const in M } � � q 2 ( B ) = u ∈ L 2 ( B ) : Lu = const in B q 2 ( B ) = � � u : Lu = const in a ngbhd of B q ( M ) ⊂ q 2 ( B ) ⊂ q 2 ( B )

Special atoms and the space X 1 ( M ) Quasiharmonic functions q ( M ) = { u ∈ C ∞ ( M ) : Lu = const in M } � � q 2 ( B ) = u ∈ L 2 ( B ) : Lu = const in B q 2 ( B ) = � � u : Lu = const in a ngbhd of B q ( M ) ⊂ q 2 ( B ) ⊂ q 2 ( B ) A function A ∈ SA s (special atoms at scale s ) if (1) s uppA ⊂ B , r B < s (2) � A � 2 ≤ µ ( B ) − 1 / 2 ∀ u ∈ q 2 ( B ) . � (3) Au d µ = 0

Special atoms and the space X 1 ( M ) Quasiharmonic functions q ( M ) = { u ∈ C ∞ ( M ) : Lu = const in M } � � q 2 ( B ) = u ∈ L 2 ( B ) : Lu = const in B q 2 ( B ) = � � u : Lu = const in a ngbhd of B q ( M ) ⊂ q 2 ( B ) ⊂ q 2 ( B ) A function A ∈ SA s (special atoms at scale s ) if (1) s uppA ⊂ B , r B < s (2) � A � 2 ≤ µ ( B ) − 1 / 2 ∀ u ∈ q 2 ( B ) . � (3) Au d µ = 0 If B has no “holes" and ∂ B is smooth then (3) is equivalent to � (3’) Au d µ = 0 ∀ u ∈ q ( M ) . ∀ u ∈ q 2 ( B ) � (3”) Au d µ = 0 True if s ≤ s 0 = 1 2 inj M .

The space X 1 � � � < ∞ X 1 � � s = f = � j λ j A , A j ∈ SA s , � � λ j j     � : f = � � � � � f � X 1 s = inf � λ j λ j A , A j ∈ SA s  j j 

The space X 1 � � � < ∞ X 1 � � s = f = � j λ j A , A j ∈ SA s , � � λ j j     � : f = � � � � � f � X 1 s = inf � λ j λ j A , A j ∈ SA s  j j  Main properties [M, Meda, Vallarino] U = L ( σ I + L ) − 1 : H 1 → X 1 is an isomorphism for all s > 0 ◮ s and σ large enough (hard work!).

The space X 1 � � � < ∞ X 1 � � s = f = � j λ j A , A j ∈ SA s , � � λ j j     � : f = � � � � � f � X 1 s = inf � λ j λ j A , A j ∈ SA s  j j  Main properties [M, Meda, Vallarino] U = L ( σ I + L ) − 1 : H 1 → X 1 is an isomorphism for all s > 0 ◮ s and σ large enough (hard work!). X 1 s 1 = X 1 s 2 , � f � X 1 s 1 ≈ � f � X 1 ◮ s 2 Henceforth X 1 = X 1 s 0 = 1 s 0 , 2 inj M

The space X 1 � � � < ∞ X 1 � � s = f = � j λ j A , A j ∈ SA s , � � λ j j     � : f = � � � � � f � X 1 s = inf � λ j λ j A , A j ∈ SA s  j j  Main properties [M, Meda, Vallarino] U = L ( σ I + L ) − 1 : H 1 → X 1 is an isomorphism for all s > 0 ◮ s and σ large enough (hard work!). X 1 s 1 = X 1 s 2 , � f � X 1 s 1 ≈ � f � X 1 ◮ s 2 Henceforth X 1 = X 1 s 0 = 1 s 0 , 2 inj M [ X 1 , L 2 ( M )] θ = L p ( M ) , θ = 2 ( 1 − 1 / p ) ◮

The space X 1 � � � < ∞ X 1 � � s = f = � j λ j A , A j ∈ SA s , � � λ j j     � : f = � � � � � f � X 1 s = inf � λ j λ j A , A j ∈ SA s  j j  Main properties [M, Meda, Vallarino] U = L ( σ I + L ) − 1 : H 1 → X 1 is an isomorphism for all s > 0 ◮ s and σ large enough (hard work!). X 1 s 1 = X 1 s 2 , � f � X 1 s 1 ≈ � f � X 1 ◮ s 2 Henceforth X 1 = X 1 s 0 = 1 s 0 , 2 inj M [ X 1 , L 2 ( M )] θ = L p ( M ) , θ = 2 ( 1 − 1 / p ) ◮ ◮ If T = ∇ L − 1 / 2 , L iu then � TA � 1 ≤ C ∀ A ∈ SA s .

The space X 1 fin ( M ) QUESTION: Suppose Y Banach, T linear s. t. (UBA) � T A � Y ≤ C ∀ A ∈ SA s . Does T : X 1 ( M ) → Y boundedly? NOT OBVIOUS!

The space X 1 fin ( M ) QUESTION: Suppose Y Banach, T linear s. t. (UBA) � T A � Y ≤ C ∀ A ∈ SA s . Does T : X 1 ( M ) → Y boundedly? NOT OBVIOUS! X 1 ∋ f = � � � � λ j A j , � f � X 1 ≈ � λ j � j j