Approximation in the Zygmund Class

Transcription

1 Approximation in the Zygmund Class Odí Soler i Gibert Joint work with Artur Nicolau Universitat Autònoma de Barcelona New Developments in Complex Analysis and Function Theory, 02 July 2018 Approximation in the Zygmund Class CAFT / 21

2 Introduction A Short Motivation Consider the spaces of functions f : I 0 r0, 1s Ñ R L p, for 1 ď p ă 8, with L 8, with and BMO, with ˆż f L p f ptq p dt I 0 f L 8 sup tpi 0 f ptq, 1{p, ˆ ż 1 f BMO sup f ptq f I dt 1{2 2, I ĎI 0 I I where f I ş I f ptq dt Approximation in the Zygmund Class CAFT / 21

3 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 / 21

4 Introduction A Different Setting Consider the spaces of continuous functions f : I 0 Ñ R Lip α, for 0 ă α ď 1, with and the Zygmund class Λ, with f pxq f pyq ď C x y α, x, y P R, f px ` hq 2f pxq ` f px hq f sup ă 8 xpi 0 h hą0 Approximation in the Zygmund Class CAFT / 21

5 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 / 21

6 Introduction Our Concepts A function f : I 0 Ñ R belongs to the Zygmund class Λ if it is continuous and f px ` hq 2f pxq ` f px hq f sup ă 8, xpi 0 h hą0 BMO if it is locally integrable and ˆ ż 1 f BMO sup I ĎI 0 I I f ptq f I 2 dt ă 8, IpBMOq if it is continuous and f 1 P BMO (distributional) Note that IpBMOq Ĺ Λ Approximation in the Zygmund Class CAFT / 21

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

8 Introduction Notation Given x P R, h ą 0, denote 2 f px, hq If I px h, x ` hq, we say Given f, g P Λ, consider and given X Ď Λ, we say f px ` hq 2f pxq ` f px hq h 2 f pi q 2 f px, hq distpf, gq f g, distpf, X q inf f g gpx Approximation in the Zygmund Class CAFT / 21

9 Introduction A Characterisation for IpBMOq Theorem (R. Strichartz; 1980) A continuous function f is in IpBMOq if and only if sup I ĎI 0 1 I ż I ż I 0 1{2 2 dh dx 2 f px, hq ă 8 h Approximation in the Zygmund Class CAFT / 21

10 Main Result The Main Result Given ε ą 0 and f P Λ, consider Apf, εq tpx, hq P R ˆ R` : 2 f px, hq ą εu Theorem Let f P Λ be compactly supported on I 0. For each ε ą 0 consider ż 1 Cpf, εq sup I ĎI 0 I I ż I 0 χ Apf,εq px, hq dh dx h. Then, distpf, IpBMOqq» inftε ą 0: Cpf, εq ă 8u. (1) Denote by ε 0 the infimum in (1) Approximation in the Zygmund Class CAFT / 21

11 Main Result Generalisation to Zygmund Measures A measure µ on R d is a Zygmund measure, µ P Λ pr d q, if µ sup µpqq ˇ µpq q Q Q Q ˇ ă 8 A measure ν on R d is IpBMOq if it is absolutely continuous and dν bpxq dx, for some b P BMO For px, hq P R d ˆ R`, let Qpx, hq be a cube with centre x and lpqq h, and denote 2 µpx, hq Given ε ą 0 and µ P Λ, consider µpqpx, hqq Qpx, hq µpqpx, 2hqq Qpx, 2hq Apµ, εq tpx, hq P R d ˆ R` : 2 µpx, hq ą εu Approximation in the Zygmund Class CAFT / 21

12 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 1 Cpµ, εq sup QĎQ 0 Q Q ż lpqq 0 χ Apµ,εq px, hq dh dx h. Then, distpµ, IpBMOqq» inftε ą 0: Cpµ, εq ă 8u. Approximation in the Zygmund Class CAFT / 21

13 Main Result Further Results and Open Problem Generalisation for Zygmund measure µ on R d, d ě 1 that is for µ with µ sup µpqq ˇ µpq q Q Q Q ˇ ă 8 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 / 21

14 Ideas for the Proof The Idea for the Proof Apf, εq tpx, hq P R ˆ R` : 2 f px, hq ą εu Theorem Let f P Λ. For each ε ą 0 consider ż 1 Cpf, εq sup I ĎI 0 I I ż I 0 χ Apf,εq px, hq dh dx h. Then, distpf, IpBMOqq» ε 0 inftε ą 0: Cpf, εq ă 8u. The easy part is to show that distpf, IpBMOqq ě ε 0 Approximation in the Zygmund Class CAFT / 21

15 Ideas for the Proof Our Tools I is dyadic if it is I rk2 n, pk ` 1q2 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 ts n u ně0 is a dyadic martingale if S n S n pi q constant on any I P D n for n ě 0 and S n pi q 1 2 ps n`1pi`q ` S n`1 pi qq SpI q S n pi q S n 1 pi 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 pi q f pbq f paq 2 n, I P D n, for each n ě 1 and where I pa, bq Approximation in the Zygmund Class CAFT / 21

16 Ideas for the Proof if f P Λ, its average growth martingale S satisfies S sup SpI q ă 8 I PD if b P BMO, its average growth martingale B satisfies B BMO sup I PD 1 I ÿ JPDpI q BpJq 2 J These are related to the dyadic versions of our spaces... 1{2 ă 8 Approximation in the Zygmund Class CAFT / 21

17 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 pi q ă 8, I PD to BMO d if it is locally integrable and ˆ ż 1 f BMO d sup f ptq f I dt 1{2 2 ă 8, I PD I I to IpBMOq d if it is continuous and f 1 P BMO d (distributional) Approximation in the Zygmund Class CAFT / 21

18 Ideas for the Proof The Easy Version of the Theorem Theorem Let f P Λ d. For each ε ą 0 consider Dpf, εq sup I PD 1 I ÿ JPDpI q 2 f pjq ąε J. Then, distpf, IpBMOq d q inftε ą 0: Dpf, εq ă 8u. We can construct a function that approximates f using martingales Approximation in the Zygmund Class CAFT / 21

19 Ideas for the Proof The Last Pieces Theorem (Garnett-Jones, 1982) Let α ÞÑ b pαq be measurable from R to BMO d, all b pαq supported on a compact I 0, with b pαq BMO d ď 1 and Then, ż b pαq ptq dt 0. b R ptq 1 ż R b pαq pt ` αq dα 2R 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 / 21

20 Ideas for the Proof The Last Pieces Theorem Let α ÞÑ h pαq be measurable from R to Λ d, all h pαq supported on a compact I 0, with h pαq d ď 1. Then, h R ptq 1 ż R h pαq pt ` αq dα 2R 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 / 21

21 Ευχαριστώ πολύ Approximation in the Zygmund Class CAFT / 21