Background New results Off-diagonal estimates and weighted elliptic operators Cristian Rios University of Calgary Workshop on Harmonic Analysis, Partial Differential Equations and Geometric Measure Theory ICMAT - January 2015 Joint work with David Cruz-Uribe and Chema Martell Rios Off-diagonal estimates and Weighted elliptic operators 1 / 31
Background New results Background Main motivators and instigators Weighted elliptic operators Extended Calderón-Zygmund theory Operators defined by sesquilinear forms Weighted Sobolev Spaces Gaffney estimates Kato for weighted ellipticity New results Off Diagonal estimates The functional calculus Riesz transform bounds Square function estimates Kato estimates Unweighted Kato estimates Rios Off-diagonal estimates and Weighted elliptic operators 2 / 31
Background New results Some of the main motivations Auscher, Hofmann, Lacey, McIntosh, Tchamitchian, " The solution of the Kato square root problem for second order elliptic operators in R n ", Ann.Math. 2002. Rios Off-diagonal estimates and Weighted elliptic operators 2 / 31
Background New results Some of the main motivations Auscher, Hofmann, Lacey, McIntosh, Tchamitchian, " The solution of the Kato square root problem for second order elliptic operators in R n ", Ann.Math. 2002. Auscher, " On necessary and sufficient conditions for L p -estimates of Riesz transforms .... ", Mem.Amer.Math.Soc. 186 (2007) Rios Off-diagonal estimates and Weighted elliptic operators 2 / 31
Background New results Some of the main motivations Auscher, Hofmann, Lacey, McIntosh, Tchamitchian, " The solution of the Kato square root problem for second order elliptic operators in R n ", Ann.Math. 2002. Auscher, " On necessary and sufficient conditions for L p -estimates of Riesz transforms .... ", Mem.Amer.Math.Soc. 186 (2007) Auscher and Martell, " Weighted norm inequalities, off diagonal estimates and elliptic operators I,II, III, IV ", Adv.Math 2007, J.Evol.Eq. 2007, JFA 2006, Math Z 2008. Rios Off-diagonal estimates and Weighted elliptic operators 2 / 31
Background New results Some of the main motivations Auscher, Hofmann, Lacey, McIntosh, Tchamitchian, " The solution of the Kato square root problem for second order elliptic operators in R n ", Ann.Math. 2002. Auscher, " On necessary and sufficient conditions for L p -estimates of Riesz transforms .... ", Mem.Amer.Math.Soc. 186 (2007) Auscher and Martell, " Weighted norm inequalities, off diagonal estimates and elliptic operators I,II, III, IV ", Adv.Math 2007, J.Evol.Eq. 2007, JFA 2006, Math Z 2008. Cruz-Uribe, R. " The Kato problem for operators with weighted ellipticity ", TAMS (to appear) Rios Off-diagonal estimates and Weighted elliptic operators 2 / 31
Background New results Auscher and Martell " Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part III:Harmonic Analysis of elliptic operators ," JFA 241 (2006) 703-746. A ∈ E ( λ , Λ ) . L = − div A ( x ) ∇ , Functional calculus for L , and weighted f.c. � � � � � ∇ L − 1/2 f Riesz transform estimates (Auscher) � p ∼ � f � p , p − < p < q + . � � � � L p ( u ) � � f � L p ( u ) , � ∇ L − 1/2 f RT weighted estimates � p − r w < p < q + / ( s w ) � . � � √ 1, np − Reverse inequalities for L . max < p < p + , n + p − � � � � p � �∇ f � p � L 1/2 f � Square function estimates. Commutators with bmo functions � [ ϕ ( L ) , b ] � p � � b � BMO � f � p (more). Rios Off-diagonal estimates and Weighted elliptic operators 3 / 31
Background New results Weights, A p and reverse Hölder classes A weight is any nonnegative locally integrable function u in R n . The A p class p > 1 � � p − 1 � � 1 B u − p − 1 dx [ u ] A p = sup − B u ( x ) dx − < ∞ . B The A 1 class � u ( x ) − 1 − B u ( x ) dx < ∞ . [ u ] A 1 = sup esssup B x ∈ B The RH s class s > 1 � � − 1 � � 1/ s � � B u ( x ) s dx [ u ] RH s = sup − B u ( x ) dx − < ∞ . B The RH ∞ class � � − 1 � [ u ] RH ∞ = sup esssup u ( x ) − B u ( x ) dx < ∞ . B x ∈ B Rios Off-diagonal estimates and Weighted elliptic operators 4 / 31
Background New results Some well known properties of A p weights A 1 ⊂ A p ⊂ A q for 1 ≤ p ≤ q < ∞ . RH ∞ ⊂ RH q ⊂ RH p for 1 < p ≤ q ≤ ∞ . � � A ∞ = A p = RH s . 1 ≤ p < ∞ 1 < s ≤ ∞ A p is left open u ∈ A p , p > 1 = ⇒ ∃ ε > 0 : u ∈ A p − ε . RH s is right open u ∈ RH s , s < ∞ = ⇒ ∃ ε > 0 : u ∈ RH s + ε . 1 ⇒ w − p − 1 ∈ A p � , p � = p / ( p − 1 ) . 1 < p < ∞ , u ∈ A p ⇐ If w ∈ A ∞ then dw is doubling. Rios Off-diagonal estimates and Weighted elliptic operators 5 / 31
Background New results Extended Calderón-Zygmund theory Some notation : Given an Euclidean ball B = B r ( x ) ⊂ R n denote by C 1 ( B ) = 4 B C j ( B ) = 2 j + 1 B /2 j B , j ≥ 2. Rios Off-diagonal estimates and Weighted elliptic operators 6 / 31
Background New results Extended Calderón-Zygmund theory Theorem 1 (Auscher and Martell (II-III)) Given w ∈ A 2 with doubling order D, 1 ≤ p 0 < q 0 ≤ ∞ , T : L q 0 ( w ) − → L q 0 ( w ) (bounded) sublinear, {A r } r > 0 linear from L ∞ c into L q 0 ( w ) . Suppose that ∀ B = B r , f ∈ L ∞ c with support ( f ) ⊂ B and j ≥ 2, � � 1/ p 0 � � 1/ p 0 � � C j ( B ) | T ( I − A r ) f | p 0 dw B | f | p 0 dw − ≤ g ( j ) − , and for j ≥ 1, � � 1/ q 0 � � 1/ p 0 � � C j ( B ) |A r f | q 0 dw B | f | p 0 dw − ≤ g ( j ) − , where ∑ g ( j ) < ∞ . Then for all p 0 < p < q 0 , there is a constant C such that for all f ∈ L ∞ c , � Tf � L p ( w ) ≤ C � f � L p ( w ) . Rios Off-diagonal estimates and Weighted elliptic operators 7 / 31
Background New results Extended Calderón-Zygmund theory Theorem 2 (Auscher and Martell (II-III)) Given w ∈ A 2 , 1 ≤ p 0 < q 0 ≤ ∞ , T sublinear and bounded on L p 0 ( w ) , {A r } r > 0 linear and bounded from D ⊂ L p 0 ( w ) into L p 0 ( w ) , and S linear from D into measurable functions on R n . Suppose that ∀ f ∈ D , B = B r , � � 1/ p 0 � � 1/ p 0 � � ≤ ∑ B | T ( I − A r ) f | p 0 dw 2 j + 1 B | Sf | p 0 dw − g ( j ) − , j ≥ 1 � � 1/ q 0 � � 1/ p 0 � � ≤ ∑ B | T A r f | q 0 dw 2 j + 1 B | Tf | p 0 dw − g ( j ) − , j ≥ 1 where ∑ g ( j ) < ∞ . Then for all p 0 < p < q 0 , and weights v ∈ A p / p 0 ( w ) � RH ( q 0 / p ) � ( w ) , there is a constant C such that for all f ∈ D , � Tf � L p ( v dw ) ≤ C � Sf � L p ( v dw ) . Rios Off-diagonal estimates and Weighted elliptic operators 8 / 31
Background New results Operators given by sesquilinear forms Let a be a sesquilinear form with dense domain D ( a ) ⊂ H in a Hilbert space H such that Re a ( u , u ) ≥ 0, (accretive) | a ( u , v ) | ≤ M � u � a � v � a , with � f � a = ( Re a ( f , f ) + � f , f � H ) 1/2 , (continuous). ( D ( a ) , �·� a ) is complete (closed), Rios Off-diagonal estimates and Weighted elliptic operators 9 / 31
Background New results Operators given by sesquilinear forms Let a be a sesquilinear form with dense domain D ( a ) ⊂ H in a Hilbert space H such that Re a ( u , u ) ≥ 0, (accretive) | a ( u , v ) | ≤ M � u � a � v � a , with � f � a = ( Re a ( f , f ) + � f , f � H ) 1/2 , (continuous). ( D ( a ) , �·� a ) is complete (closed), then there exists an associated operator L a such that a ( u , v ) = � L a u , v � H , ∀ u ∈ D ( L a ) , v ∈ D ( a ) , with D ( L a ) dense in H . Rios Off-diagonal estimates and Weighted elliptic operators 9 / 31
Background New results Operators given by sectorial sesquilinear forms Let L a be the operator associated to a densely defined, accretive, continuous, closed sesquilinear form in a Hilbert space H . If for some 0 ≤ ϑ < π 2 , | Im a ( u , u ) | ≤ tan ( ϑ ) Re a ( u , u ) (sectorial of angle ϑ ) then L a is sectorial of angle ϑ + π 4 , i.e.: (i) σ ( L a ) ⊂ Σ ϑ + π 4 , � � < ∞ for all ω � > ϑ + π � z R ( z , L a ) � op | z ∈ C \ Σ ω � (ii) sup 4 , R ( z , L a ) = ( z − L a ) − 1 . Rios Off-diagonal estimates and Weighted elliptic operators 10 / 31
Background New results Operators given by sectorial sesquilinear forms Let L a be the operator associated to a densely defined, accretive, continuous, closed sesquilinear form in a Hilbert space H . If for some 0 ≤ ϑ < π 2 , | Im a ( u , u ) | ≤ tan ( ϑ ) Re a ( u , u ) (sectorial of angle ϑ ) then L a is sectorial of angle ϑ + π 4 , i.e.: (i) σ ( L a ) ⊂ Σ ϑ + π 4 , � � < ∞ for all ω � > ϑ + π � z R ( z , L a ) � op | z ∈ C \ Σ ω � (ii) sup 4 , R ( z , L a ) = ( z − L a ) − 1 . A consequence of (ii) is that L a has a bounded holomorphic calculus in H . (iii) If ϕ is a bounded holomorphic function in Σ ω � then � ϕ ( L a ) � op ≤ � ϕ � ∞ . � � � e − tL a u � In particular, H ≤ � u � H for all t > 0. Rios Off-diagonal estimates and Weighted elliptic operators 10 / 31
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