Fourier Analysis for vector-measures OSCAR BLASCO Universidad - PowerPoint PPT Presentation

Introduction Fourier transform and the Riemann-Lebesgue lemma Convolution for vector measures Invariance under homeomorphisms References Fourier Analysis for vector-measures OSCAR BLASCO Universidad Valencia Integration, Vector Measures and Related Topics Bedlewo 15-21 June 2014 Oscar Blasco Fourier Analysis for vector-measures

Introduction Fourier transform and the Riemann-Lebesgue lemma Convolution for vector measures Invariance under homeomorphisms References Notation Throughout X is a complex Banach space, G be a compact abelian group, B ( G ) for the Borel σ -algebra of G , m G for the Haar measure of the group, L p ( G ) the space of � G | f | p dm G < ∞ . mesurable functions such that Oscar Blasco Fourier Analysis for vector-measures

Introduction Fourier transform and the Riemann-Lebesgue lemma Convolution for vector measures Invariance under homeomorphisms References Notation Throughout X is a complex Banach space, G be a compact abelian group, B ( G ) for the Borel σ -algebra of G , m G for the Haar measure of the group, L p ( G ) the space of � G | f | p dm G < ∞ . mesurable functions such that ( M ( G , X ) , �·� ) stands for the space of regular vector measures normed with the semivariation and M ac ( G , X ) for those such that ν << m G . Oscar Blasco Fourier Analysis for vector-measures

Introduction Fourier transform and the Riemann-Lebesgue lemma Convolution for vector measures Invariance under homeomorphisms References Notation Throughout X is a complex Banach space, G be a compact abelian group, B ( G ) for the Borel σ -algebra of G , m G for the Haar measure of the group, L p ( G ) the space of � G | f | p dm G < ∞ . mesurable functions such that ( M ( G , X ) , �·� ) stands for the space of regular vector measures normed with the semivariation and M ac ( G , X ) for those such that ν << m G . M ( G , X ) coincides with W C ( C ( G ) , X ), i.e. we identify ν with a weakly compact � operator T ν : C ( G ) → X and denote T ν ( φ ) = G φ d ν . Moreover � T ν � = � ν � . Oscar Blasco Fourier Analysis for vector-measures

Introduction Fourier transform and the Riemann-Lebesgue lemma Convolution for vector measures Invariance under homeomorphisms References Notation Throughout X is a complex Banach space, G be a compact abelian group, B ( G ) for the Borel σ -algebra of G , m G for the Haar measure of the group, L p ( G ) the space of � G | f | p dm G < ∞ . mesurable functions such that ( M ( G , X ) , �·� ) stands for the space of regular vector measures normed with the semivariation and M ac ( G , X ) for those such that ν << m G . M ( G , X ) coincides with W C ( C ( G ) , X ), i.e. we identify ν with a weakly compact � operator T ν : C ( G ) → X and denote T ν ( φ ) = G φ d ν . Moreover � T ν � = � ν � . Let 1 < p ≤ ∞ . A measure ν is said to have bounded p -semivariation with respect to m G if �� � � � � � � : π partition , � ∑ � ∑ � ν � p , m G = sup � α A ν ( A ) � α A χ A � L p ′ ( G ) ≤ 1 . (1.1) � A ∈ π A ∈ π X The case p = ∞ corresponds to � ν ( A ) � ≤ Cm G ( A ) for A ∈ B ( G ) for some constant C and � ν � ∞ , λ is the infimum of such constants. Oscar Blasco Fourier Analysis for vector-measures

Introduction Fourier transform and the Riemann-Lebesgue lemma Convolution for vector measures Invariance under homeomorphisms References Notation Throughout X is a complex Banach space, G be a compact abelian group, B ( G ) for the Borel σ -algebra of G , m G for the Haar measure of the group, L p ( G ) the space of � G | f | p dm G < ∞ . mesurable functions such that ( M ( G , X ) , �·� ) stands for the space of regular vector measures normed with the semivariation and M ac ( G , X ) for those such that ν << m G . M ( G , X ) coincides with W C ( C ( G ) , X ), i.e. we identify ν with a weakly compact � operator T ν : C ( G ) → X and denote T ν ( φ ) = G φ d ν . Moreover � T ν � = � ν � . Let 1 < p ≤ ∞ . A measure ν is said to have bounded p -semivariation with respect to m G if �� � � � � � � : π partition , � ∑ � ∑ � ν � p , m G = sup � α A ν ( A ) � α A χ A � L p ′ ( G ) ≤ 1 . (1.1) � A ∈ π A ∈ π X The case p = ∞ corresponds to � ν ( A ) � ≤ Cm G ( A ) for A ∈ B ( G ) for some constant C and � ν � ∞ , λ is the infimum of such constants. We use the notation M p ( G , X ) and this space can be identify with L ( L p ′ ( G ) , X ) and � ν � p , m G = � T ν � L p ′ ( G ) , X . Oscar Blasco Fourier Analysis for vector-measures

Introduction Fourier transform and the Riemann-Lebesgue lemma Convolution for vector measures Invariance under homeomorphisms References Motivation (part 1) L ( ν ) for the space of functions integrable with respect to a vector measure ν . If f ∈ L 1 ( ν ) we denote � ν f ( A ) = A fd ν . Then ν f is a vector measure and � ν f � = � f � L 1 ( ν ) . We write I ν the integration � operator, i.e. I ν : L 1 ( ν ) → X is defined by I ν ( f ) = ν f ( G ) = G fd ν Oscar Blasco Fourier Analysis for vector-measures

Introduction Fourier transform and the Riemann-Lebesgue lemma Convolution for vector measures Invariance under homeomorphisms References Motivation (part 1) L ( ν ) for the space of functions integrable with respect to a vector measure ν . If f ∈ L 1 ( ν ) we denote � ν f ( A ) = A fd ν . Then ν f is a vector measure and � ν f � = � f � L 1 ( ν ) . We write I ν the integration � operator, i.e. I ν : L 1 ( ν ) → X is defined by I ν ( f ) = ν f ( G ) = G fd ν Delgado y Miana (2009) introduced the notion of ”norm integral translation invariant” vector measures, as those satisfying � I ν ( τ a φ ) � = � I ν ( φ ) � , φ ∈ simple function , a ∈ G (1.2) where τ a ( φ )( s ) = φ ( s − a ). Oscar Blasco Fourier Analysis for vector-measures

Introduction Fourier transform and the Riemann-Lebesgue lemma Convolution for vector measures Invariance under homeomorphisms References Motivation (part 1) L ( ν ) for the space of functions integrable with respect to a vector measure ν . If f ∈ L 1 ( ν ) we denote � ν f ( A ) = A fd ν . Then ν f is a vector measure and � ν f � = � f � L 1 ( ν ) . We write I ν the integration � operator, i.e. I ν : L 1 ( ν ) → X is defined by I ν ( f ) = ν f ( G ) = G fd ν Delgado y Miana (2009) introduced the notion of ”norm integral translation invariant” vector measures, as those satisfying � I ν ( τ a φ ) � = � I ν ( φ ) � , φ ∈ simple function , a ∈ G (1.2) where τ a ( φ )( s ) = φ ( s − a ). For any norm integral translation invariant measure ν such that ν << m G they showed that L 1 ( ν ) ⊂ L 1 ( G ). Oscar Blasco Fourier Analysis for vector-measures

Introduction Fourier transform and the Riemann-Lebesgue lemma Convolution for vector measures Invariance under homeomorphisms References Motivation (part 1) L ( ν ) for the space of functions integrable with respect to a vector measure ν . If f ∈ L 1 ( ν ) we denote � ν f ( A ) = A fd ν . Then ν f is a vector measure and � ν f � = � f � L 1 ( ν ) . We write I ν the integration � operator, i.e. I ν : L 1 ( ν ) → X is defined by I ν ( f ) = ν f ( G ) = G fd ν Delgado y Miana (2009) introduced the notion of ”norm integral translation invariant” vector measures, as those satisfying � I ν ( τ a φ ) � = � I ν ( φ ) � , φ ∈ simple function , a ∈ G (1.2) where τ a ( φ )( s ) = φ ( s − a ). For any norm integral translation invariant measure ν such that ν << m G they showed that L 1 ( ν ) ⊂ L 1 ( G ). Hence convolution and Fourier transform of functions in L 1 ( ν ) are well defined. Oscar Blasco Fourier Analysis for vector-measures

Introduction Fourier transform and the Riemann-Lebesgue lemma Convolution for vector measures Invariance under homeomorphisms References Motivation (part 1) L ( ν ) for the space of functions integrable with respect to a vector measure ν . If f ∈ L 1 ( ν ) we denote � ν f ( A ) = A fd ν . Then ν f is a vector measure and � ν f � = � f � L 1 ( ν ) . We write I ν the integration � operator, i.e. I ν : L 1 ( ν ) → X is defined by I ν ( f ) = ν f ( G ) = G fd ν Delgado y Miana (2009) introduced the notion of ”norm integral translation invariant” vector measures, as those satisfying � I ν ( τ a φ ) � = � I ν ( φ ) � , φ ∈ simple function , a ∈ G (1.2) where τ a ( φ )( s ) = φ ( s − a ). For any norm integral translation invariant measure ν such that ν << m G they showed that L 1 ( ν ) ⊂ L 1 ( G ). Hence convolution and Fourier transform of functions in L 1 ( ν ) are well defined. They showed that if f ∈ L 1 ( G ) and g ∈ L p ( ν ) then f ∗ g ∈ L p ( ν ) for 1 ≤ p < ∞ . Oscar Blasco Fourier Analysis for vector-measures