Banach algebras of convolution type operators with PSO data on - PowerPoint PPT Presentation

Spaces PSO Banach algebra A p , w Central algebra Localization Local spectra Main results Banach algebras of convolution type operators with PSO data on weighted Lebesgue spaces Yuri Karlovich Universidad Autónoma del Estado de Morelos, Cuernavaca, México IWOTA 2017, TU Chemnitz, Chemnitz, Germany, August 14-18, 2017

Spaces PSO Banach algebra A p , w Central algebra Localization Local spectra Main results Spaces Weighted Lebesgue spaces A measurable function w : R → [ 0 , ∞ ] is called a weight if the preimage w − 1 ( { 0 , ∞} ) has measure zero. For 1 < p < ∞ , a weight w belongs to the Muckenhoupt class A p ( R ) if � 1 � 1 / p � 1 � 1 / q � � w p ( x ) dx w − q ( x ) dx c p , w := sup < ∞ , | I | | I | I I I where 1 / p + 1 / q = 1, and supremum is taken over all intervals I ⊂ R of finite length | I | . Given 1 < p < ∞ and w ∈ A p ( R ) , we consider the weighted Lebesgue space L p ( R , w ) equipped with the norm � � � 1 / p | f ( x ) | p w p ( x ) dx � f � L p ( R , w ) := . R Let B ( X ) denote the Banach algebra of all bounded linear operators acting on a Banach space X .

Spaces PSO Banach algebra A p , w Central algebra Localization Local spectra Main results SO data The space SO ⋄ For a continuous function f : R → C and a set I ⊂ R , let osc ( f , I ) = sup {| f ( t ) − f ( s ) | : t , s ∈ I } . Given λ ∈ ˙ R := R ∪ {∞} , we denote by SO λ the C ∗ -algebra of functions slowly oscillating at λ , � � f ∈ C b ( ˙ SO ∞ := R \ {∞} ) : x → + ∞ osc ( f , [ − 2 x , − x ] ∪ [ x , 2 x ]) = 0 lim , � � f ∈ C b ( ˙ SO λ := R \ { λ } ) : lim x → 0 osc ( f , λ + ([ − 2 x , − x ] ∪ [ x , 2 x ])) = 0 for λ ∈ R , where C b ( ˙ R \ { λ } ) = C ( ˙ R \ { λ } ) ∩ L ∞ ( R ) . Let SO ⋄ be the minimal C ∗ -subalgebra of L ∞ ( R ) that contains all the C ∗ -algebras SO λ with λ ∈ ˙ R , and therefore contains C ( ˙ R ) . Let C n ( R ) be the set of all n times continuously differentiable functions a : R → C . Let F : L 2 ( R ) → L 2 ( R ) denote the Fourier � transform, f ( t ) e itx dt , x ∈ R . ( F f )( x ) := R

Spaces PSO Banach algebra A p , w Central algebra Localization Local spectra Main results Convolution operators from the viewpoint of pseudodifferential operators Fourier multipliers on L p ( R , w ) A function a ∈ L ∞ ( R ) is called a Fourier multiplier on L p ( R , w ) if the convolution operator W 0 ( a ) := F − 1 a F maps the dense subset L 2 ( R ) ∩ L p ( R , w ) of L p ( R , w ) into itself and extends to a bounded linear operator on L p ( R , w ) . Let M p , w stand for the Banach algebra of all Fourier multipliers on L p ( R , w ) equipped with the norm � a � M p , w := � W 0 ( a ) � B ( L p ( R , w )) . To define slowly oscillating functions in M p , w , we need such fact. Theorem (Yu. K./Juan Loreto-Hernández) If a ∈ C 3 ( R \ { 0 } ) and � D γ a � L ∞ ( R ) < ∞ for all γ = 0 , 1 , 2 , 3 , where ( Da )( x ) = xa ′ ( x ) for x ∈ R , then the convolution operator W 0 ( a ) = F − 1 a F is bounded on every weighted Lebesgue space L p ( R , w ) with 1 < p < ∞ and w ∈ A p ( R ) , and � � � W 0 ( a ) � B ( L p ( R , w )) ≤ c p , w max � D γ a � L ∞ ( R ) : γ = 0 , 1 , 2 , 3 < ∞ , where the constant c p , w ∈ ( 0 , ∞ ) depends only on p and w.

Spaces PSO Banach algebra A p , w Central algebra Localization Local spectra Main results Piecewise slowly oscillating functions Banach algebras SO ⋄ p , w and PSO ⋄ p , w For λ ∈ ˙ R , consider the commutative Banach algebras � � x → λ ( D γ SO 3 a ∈ SO λ ∩ C 3 ( R \ { λ } ) : lim λ := λ a )( x ) = 0 , γ = 1 , 2 , 3 � � � D γ with norm � a � SO 3 λ := max λ a � L ∞ ( R ) : γ = 0 , 1 , 2 , 3 , where ( D λ a )( x ) = ( x − λ ) a ′ ( x ) for λ ∈ R and ( D λ a )( x ) = xa ′ ( x ) if λ = ∞ . Then SO 3 λ ⊂ M p , w . Let SO λ, p , w denote the closure of SO 3 λ in M p , w , and let SO ⋄ p , w be the Banach subalgebra of M p , w generated by all algebras SO λ, p , w ( λ ∈ ˙ R ). By the continuous embedding M p , w ⊂ M 2 = L ∞ ( R ) , we get SO ⋄ p , w ⊂ SO ⋄ . Let V ( R ) be the Banach algebra of all functions a : R → C with finite total variation V ( a ) and the norm � a � V = � a � ∞ + V ( a ) . By Stechkin’s inequality, V ( R ) ⊂ M p , w . Let PSO ⋄ p , w := alg ( SO ⋄ p , w , PC p , w ) be the Banach subalgebra of M p , w generated by the algebras SO ⋄ p , w and PC p , w , where PC p , w is the closure in M p , w of the set PC ∩ V ( R ) of all piecewise continuous functions in V ( R ) .

Spaces PSO Banach algebra A p , w Central algebra Localization Local spectra Main results Piecewise slowly oscillating functions Maximal ideal spaces of SO ⋄ p , w and M ( PSO ⋄ p , w ) Identifying the points λ ∈ ˙ R with the evaluation functionals δ λ on R for f ∈ C ( ˙ ˙ R ) , δ λ ( f ) = f ( λ ) , we infer that the maximal ideal space M ( SO ⋄ ) of SO ⋄ has the form M ( SO ⋄ ) = � R M λ ( SO ⋄ ) λ ∈ ˙ � � where M λ ( SO ⋄ ) = ξ ∈ M ( SO ⋄ ) : ξ | C ( ˙ R ) = δ λ are the fibers of M ( SO ⋄ ) over λ ∈ ˙ R . One can show that for every λ ∈ ˙ R , M λ ( SO ⋄ ) = M λ ( SO λ ) = M ∞ ( SO ∞ ) = ( clos SO ∗ ∞ R ) \ R , ∞ R is the weak-star closure of R in SO ∗ where clos SO ∗ ∞ . Theorem If p ∈ ( 1 , ∞ ) and w ∈ A p ( R ) , then the maximal ideal spaces of p , w and SO ⋄ coincide as sets, M ( SO ⋄ SO ⋄ p , w ) = M ( SO ⋄ ) . Lemma If p ∈ ( 1 , ∞ ) and w ∈ A p ( R ) , then the maximal ideal space M ( PSO ⋄ p , w ) of PSO ⋄ p , w can be identified with M ( SO ⋄ ) × { 0 , 1 } .

Spaces PSO Banach algebra A p , w Central algebra Localization Local spectra Main results Compactness Compactness Let p ∈ ( 1 , ∞ ) , w ∈ A p ( R ) , and let K p , w be the ideal of compact operators in the Banach algebra B p , w = B ( L p ( R , w )) . Let us study the Banach subalgebra A p , w of B p , w generated by all operators aI ( a ∈ PSO ⋄ ) and W 0 ( b ) ( b ∈ PSO ⋄ p , w ). Theorem (Yu. K./Iván Loreto-Hernández) If either a ∈ PSO ⋄ and b ∈ SO ⋄ p , w , or a ∈ SO ⋄ and b ∈ PSO ⋄ p , w , or a ∈ alg ( C ( R ) , SO ∞ ) and b ∈ alg ( C p , w ( R ) , SO ∞ , p , w ) , then [ aI , W 0 ( b )] ∈ K p , w . It suffices to prove the compactness of [ aI , W 0 ( b )] on the space L 2 ( R ) . Let a ∈ SO ⋄ ⊂ VMO and b ∈ SO 3 ∞ . Then W 0 ( b ) is a bounded on L 2 ( R ) Calderón-Zygmund operator. Then � � � [ aI , W 0 ( b )] � B ( L 2 ( R )) ≤ C � a � BMO for every a ∈ BMO ( R ) . On the other hand, VMO is the closure in BMO ( R ) of the set C ( ˙ R ) , and hence [ aI , W 0 ( b )] ∈ K ( L 2 ( R )) .

Spaces PSO Banach algebra A p , w Central algebra Localization Local spectra Main results Central subalgebra A Banach algebra Z p , w Consider the Banach algebra � � aI , W 0 ( b ) : a ∈ SO ⋄ , b ∈ SO ⋄ Z p , w := alg ⊂ B p , w , p , w which is generated by all the operators aW 0 ( b ) ∈ B p , w with a ∈ SO ⋄ and b ∈ SO ⋄ p , w . Then K p , w ⊂ Z p , w ⊂ A p , w ⊂ B p , w . Consider the quotient Banach algebra A π p , w := A p , w / K p , w . Then Z π p , w := Z p , w / K p , w is a central subalgebra of A π p , w . Take the following commutative Banach subalgebras of Z π p , w : � ( aI ) π : a ∈ SO ⋄ � � � ( W 0 ( b )) π : b ∈ SO ⋄ A π , A π 1 := 2 := , p , w where A π := A + K p , w for all A ∈ B p , w , and their maximal ideals � � ( aI ) π : a ∈ SO ⋄ , ξ ( a ) = 0 I π ⊂ A π 1 ,ξ := and 1 � � (1) ( W 0 ( b )) π : b ∈ SO ⋄ I π ⊂ A π 2 ,η := p , w , η ( b ) = 0 2 . Lemma 2 = { cI π : c ∈ C } . If 1 < p < ∞ , w ∈ A p ( R ) , then A π 1 ∩ A π

Spaces PSO Banach algebra A p , w Central algebra Localization Local spectra Main results Central subalgebra The maximal ideal space of Z π p , w Theorem If p ∈ ( 1 , ∞ ) and w ∈ A p ( R ) , then the maximal ideal space � � Z π of the Banach algebra Z π M p , w is homeomorphic to the p , w � � � � � set � M λ ( SO ⋄ ) × M ∞ ( SO ⋄ ) M ∞ ( SO ⋄ ) × M τ ( SO ⋄ ) Ω := ∪ λ ∈ R τ ∈ R � � M ∞ ( SO ⋄ ) × M ∞ ( SO ⋄ ) ∪ equipped with topology induced by the product topology of M ( SO ⋄ ) × M ( SO ⋄ ) , and the Gelfand transform p , w → C (Ω) , A π �→ A ( · , · ) is defined on the generators Γ Z : Z π A π = [ aW 0 ( b )] π ( a ∈ SO ⋄ , b ∈ SO ⋄ p , w ) of the algebra Z π p , w by A ( ξ, η ) = a ( ξ ) b ( η ) for all ( ξ, η ) ∈ Ω . An operator A ∈ Z p , w is Fredholm on the space L p ( R , w ) if and only if the Gelfand transform of the coset A π is invertible, that is, if A ( ξ, η ) � = 0 for all ( ξ, η ) ∈ Ω .