Approximation in the Zygmund Class Od Soler i Gibert Joint work - PowerPoint PPT Presentation

Approximation in the Zygmund Class Od´ ı Soler i Gibert Joint work with Artur Nicolau Universitat Aut` onoma de Barcelona New Developments in Complex Analysis and Function Theory, 02 July 2018 Approximation in the Zygmund Class CAFT 2018 1 / 21

Introduction A Short Motivation Consider the spaces of functions f : I 0 “ r 0 , 1 s Ñ R L p , for 1 ď p ă 8 , with ˙ 1 { p ˆż | f p t q| p dt � f � L p “ , I 0 L 8 , with � f � L 8 “ sup | f p t q| , t P I 0 and BMO , with ˆ 1 ˙ 1 { 2 ż | f p t q ´ f I | 2 dt � f � BMO “ sup , | I | I Ď I 0 I ş where f I “ I f p t q dt Approximation in the Zygmund Class CAFT 2018 2 / 21

Introduction It is known that L 8 Ĺ BMO Ĺ L p Ĺ L q Ĺ L 1 , for 1 ă p ă q ă 8 A singular integral operator (e.g. H the Hilbert Transform) is bounded from L p to L p if 1 ă p ă 8 , from BMO to BMO and from L 8 to BMO J. Garnett and P. Jones (1978) characterised L 8 for � ¨ � BMO Approximation in the Zygmund Class CAFT 2018 3 / 21

Introduction A Different Setting Consider the spaces of continuous functions f : I 0 Ñ R Lip α , for 0 ă α ď 1 , with | f p x q ´ f p y q| ď C | x ´ y | α , x , y P R , and the Zygmund class Λ ˚ , with | f p x ` h q ´ 2 f p x q ` f p x ´ h q| � f � ˚ “ sup ă 8 h x P I 0 h ą 0 Approximation in the Zygmund Class CAFT 2018 4 / 21

Introduction It can be seen that Lip 1 Ĺ Λ ˚ Ĺ Lip α Ĺ Lip β , for 0 ă α ă β ă 1 Singular integral operators are bounded from Lip α to Lip α for 0 ă α ă 1 , from Λ ˚ to Λ ˚ and from Lip 1 to Λ ˚ What could be a characterisation of Lip 1 for � ¨ � ˚ ? Related open problem (J. Anderson, J. Clunie, C. Pommerenke; 1974): what is the characterisation of H 8 for the Bloch space norm? Approximation in the Zygmund Class CAFT 2018 5 / 21

Introduction Our Concepts A function f : I 0 Ñ R belongs to the Zygmund class Λ ˚ if it is continuous and | f p x ` h q ´ 2 f p x q ` f p x ´ h q| � f � ˚ “ sup ă 8 , h x P I 0 h ą 0 BMO if it is locally integrable and ˆ 1 ˙ ż | f p t q ´ f I | 2 dt � f � BMO “ sup ă 8 , | I | I Ď I 0 I I p BMO q if it is continuous and f 1 P BMO (distributional) Note that I p BMO q Ĺ Λ ˚ Approximation in the Zygmund Class CAFT 2018 6 / 21

Introduction Our Concepts What is the characterisation of I p BMO q for � ¨ � ˚ ? Related problem : P. G. Ghatage and D. Zheng (1993) characterised BMOA for the Bloch space norm Approximation in the Zygmund Class CAFT 2018 7 / 21

Introduction Notation Given x P R , h ą 0 , denote ∆ 2 f p x , h q “ f p x ` h q ´ 2 f p x q ` f p x ´ h q h If I “ p x ´ h , x ` h q , we say ∆ 2 f p I q “ ∆ 2 f p x , h q Given f , g P Λ ˚ , consider dist p f , g q “ � f ´ g � ˚ , and given X Ď Λ ˚ , we say dist p f , X q “ inf g P X � f ´ g � ˚ Approximation in the Zygmund Class CAFT 2018 8 / 21

Introduction A Characterisation for I p BMO q Theorem (R. Strichartz; 1980) A continuous function f is in I p BMO q if and only if ¸ 1 { 2 ż | I | ˜ 1 ż | ∆ 2 f p x , h q| 2 dh dx sup ă 8 | I | h I Ď I 0 I 0 Approximation in the Zygmund Class CAFT 2018 9 / 21

Main Result The Main Result Given ε ą 0 and f P Λ ˚ , consider A p f , ε q “ tp x , h q P R ˆ R ` : | ∆ 2 f p x , h q| ą ε u Theorem Let f P Λ ˚ be compactly supported on I 0 . For each ε ą 0 consider ż | I | 1 ż χ A p f ,ε q p x , h q dh dx C p f , ε q “ sup . | I | h I Ď I 0 I 0 Then, dist p f , I p BMO qq » inf t ε ą 0: C p f , ε q ă 8u . (1) Denote by ε 0 the infimum in (1) Approximation in the Zygmund Class CAFT 2018 10 / 21

Main Result Generalisation to Zygmund Measures A measure µ on R d is a Zygmund measure, µ P Λ ˚ p R d q , if | Q | ´ µ p Q ˚ q ˇ ˇ µ p Q q ˇ ˇ � µ � ˚ “ sup ˇ ă 8 ˇ ˇ | Q ˚ | ˇ Q A measure ν on R d is I p BMO q if it is absolutely continuous and d ν “ b p x q dx , for some b P BMO For p x , h q P R d ˆ R ` , let Q p x , h q be a cube with centre x and l p Q q “ h , and denote ∆ 2 µ p x , h q “ µ p Q p x , h qq | Q p x , h q| ´ µ p Q p x , 2 h qq | Q p x , 2 h q| Given ε ą 0 and µ P Λ ˚ , consider A p µ, ε q “ tp x , h q P R d ˆ R ` : | ∆ 2 µ p x , h q| ą ε u Approximation in the Zygmund Class CAFT 2018 11 / 21

Main Result Generalisation to Zygmund Measures Theorem Let µ P R d be compactly supported on Q 0 . For each ε ą 0 consider ż l p Q q 1 ż χ A p µ,ε q p x , h q dh dx C p µ, ε q “ sup . | Q | h Q Ď Q 0 Q 0 Then, dist p µ, I p BMO qq » inf t ε ą 0: C p µ, ε q ă 8u . Approximation in the Zygmund Class CAFT 2018 12 / 21

Main Result Further Results and Open Problem Generalisation for Zygmund measure µ on R d , d ě 1 that is for µ with | Q | ´ µ p Q ˚ q ˇ ˇ µ p Q q ˇ ˇ � µ � ˚ “ sup ˇ ă 8 ˇ ˇ | Q ˚ | ˇ Q Application to functions in the Zygmund class that are also W 1 , p , for 1 ă p ă 8 We can’t generalise the results for functions on R d for d ą 1 Approximation in the Zygmund Class CAFT 2018 13 / 21

Ideas for the Proof The Idea for the Proof A p f , ε q “ tp x , h q P R ˆ R ` : | ∆ 2 f p x , h q| ą ε u Theorem Let f P Λ ˚ . For each ε ą 0 consider ż | I | 1 ż χ A p f ,ε q p x , h q dh dx C p f , ε q “ sup . | I | h I Ď I 0 0 I Then, dist p f , I p BMO qq » ε 0 “ inf t ε ą 0: C p f , ε q ă 8u . The easy part is to show that dist p f , I p BMO qq ě ε 0 Approximation in the Zygmund Class CAFT 2018 14 / 21

Ideas for the Proof Our Tools I is dyadic if it is I “ r k 2 ´ n , p k ` 1 q 2 ´ n q , with n ě 0 and 0 ď k ă 2 n ´ 1 D the set of dyadic intervals and D n the set of dyadic intervals of size 2 ´ n S “ t S n u n ě 0 is a dyadic martingale if S n “ S n p I q constant on any I P D n for n ě 0 and S n p I q “ 1 2 p S n ` 1 p I ` q ` S n ` 1 p I ´ qq ∆ S p I q “ S n p I q ´ S n ´ 1 p I ˚ q , for I P D n and I Ď I ˚ P D n ´ 1 For f continuous, define the average growth martingale S by S n p I q “ f p b q ´ f p a q I P D n , , 2 ´ n for each n ě 1 and where I “ p a , b q Approximation in the Zygmund Class CAFT 2018 15 / 21

Ideas for the Proof if f P Λ ˚ , its average growth martingale S satisfies � S � ˚ “ sup | ∆ S p I q| ă 8 I P D if b P BMO , its average growth martingale B satisfies 1 { 2 ¨ ˛ ˝ 1 ÿ | ∆ B p J q| 2 | J | � B � BMO “ sup ă 8 ‚ | I | I P D J P D p I q These are related to the dyadic versions of our spaces... Approximation in the Zygmund Class CAFT 2018 16 / 21

Ideas for the Proof A function f : I 0 Ñ R belongs to the dyadic Zygmund class Λ ˚ d if it is continuous and � f � ˚ d “ sup | ∆ 2 f p I q| ă 8 , I P D to BMO d if it is locally integrable and ˆ 1 ˙ 1 { 2 ż | f p t q ´ f I | 2 dt � f � BMO d “ sup ă 8 , | I | I P D I to I p BMO q d if it is continuous and f 1 P BMO d (distributional) Approximation in the Zygmund Class CAFT 2018 17 / 21

Ideas for the Proof The Easy Version of the Theorem Theorem Let f P Λ ˚ d . For each ε ą 0 consider 1 ÿ D p f , ε q “ sup | J | . | I | I P D J P D p I q | ∆ 2 f p J q|ą ε Then, dist p f , I p BMO q d q “ inf t ε ą 0: D p f , ε q ă 8u . We can construct a function that approximates f using martingales Approximation in the Zygmund Class CAFT 2018 18 / 21

Ideas for the Proof The Last Pieces Theorem (Garnett-Jones, 1982) Let α ÞÑ b p α q be measurable from R to BMO d , all b p α q supported on a � � b p α q � BMO d ď 1 and compact I 0 , with � ż b p α q p t q dt “ 0 . Then, ż R b R p t q “ 1 b p α q p t ` α q d α 2 R ´ R is in BMO and there is C ą 0 such that � b R � BMO ď C for any R ą 0 . Approximation in the Zygmund Class CAFT 2018 19 / 21

Ideas for the Proof The Last Pieces Theorem Let α ÞÑ h p α q be measurable from R to Λ ˚ d , all h p α q supported on a � h p α q � � compact I 0 , with ˚ d ď 1 . Then, � ż R h R p t q “ 1 h p α q p t ` α q d α 2 R ´ R is in Λ ˚ and there is C ą 0 such that � h R � ˚ ď C for any R ą 0 . Approximation in the Zygmund Class CAFT 2018 20 / 21

Ευχαριστώ πολύ Approximation in the Zygmund Class CAFT 2018 21 / 21