Difference between revisions of "VMOA-space"
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''space of analytic functions of vanishing mean oscillation'' | ''space of analytic functions of vanishing mean oscillation'' | ||
| − | The class of analytic functions on the unit disc that are in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130080/v1300802.png" /> (see also [[BMO-space| | + | The class of analytic functions on the unit disc that are in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130080/v1300802.png"/> (see also [[BMO-space|$\operatorname{BMO}$-space]]; [[BMOA-space|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130080/v1300804.png"/>-space]]; [[VMO-space|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130080/v1300805.png"/>-space]]). |
| − | Fefferman's duality theorem (see [[BMO-space| | + | Fefferman's duality theorem (see [[BMO-space|$\operatorname{BMO}$-space]]) gives the characterization that an [[Analytic function|analytic function]] in $\operatorname{BMO}$ is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130080/v1300808.png"/> if and only if its boundary values can be expressed as the sum of a [[Continuous function|continuous function]] and the harmonic conjugate (cf. also [[Harmonic function|Harmonic function]]) of a continuous function. This suggests that functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130080/v1300809.png"/> are close to being continuous, but one has to be careful because their behaviour can be quite wild. For example, it can be show that any [[Conformal mapping|conformal mapping]] onto a region of finite area is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130080/v13008010.png"/>. |
| − | D. Sarason [[#References|[a5]]] used the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130080/v13008011.png" /> is the closure of the disc algebra | + | D. Sarason [[#References|[a5]]] used the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130080/v13008011.png"/> is the closure of the disc algebra $A$ in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130080/v13008013.png"/> to prove that $H ^ { \infty } + C$, with $C$ the class of continuous functions, is a closed subalgebra of $L^{\infty}$ and consequently the simplest example of a Douglas algebra (see [[VMO-space|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130080/v13008017.png"/>-space]]). |
| − | The distance between a function | + | The distance between a function $f$ in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130080/v13008019.png"/> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130080/v13008020.png"/> has attracted some interest, [[#References|[a1]]], [[#References|[a2]]], [[#References|[a4]]]. Let $f$ be an analytic function on the unit disc, $\zeta$ a point on the boundary $T$ and write $K _ { \zeta }$ for the [[Cluster set|cluster set]] $\operatorname{Cl} ( f , \zeta )$. Using an assortment of tools from [[Functional analysis|functional analysis]], S. Axler and J. Shapiro [[#References|[a1]]] proved that |
| − | + | \begin{equation*} \| f + \operatorname {VMOA} \| _ { * } \leq C \operatorname { lim sup } _ { \zeta \in T } \sqrt { \operatorname { area } ( K _ { \zeta } ) }. \end{equation*} | |
This led to a search for the optimal geometric condition for the right-hand side above, see [[#References|[a4]]] for the answer. | This led to a search for the optimal geometric condition for the right-hand side above, see [[#References|[a4]]] for the answer. | ||
====References==== | ====References==== | ||
| − | <table>< | + | <table><tr><td valign="top">[a1]</td> <td valign="top"> S. Axler, J. Shapiro, "Putnam's theorem, Alexander's spectral area estimate and VMO" ''Math. Ann.'' , '''271''' (1985) pp. 161–183</td></tr><tr><td valign="top">[a2]</td> <td valign="top"> J. Carmona, J. Cufi, "On the distance of an analytic function to VMO" ''J. London Math. Soc. (2)'' , '''34''' (1986) pp. 52–66</td></tr><tr><td valign="top">[a3]</td> <td valign="top"> C. Fefferman, "Characterization of bounded mean oscillation" ''Bull. Amer. Math. Soc.'' , '''77''' (1971) pp. 587–588</td></tr><tr><td valign="top">[a4]</td> <td valign="top"> K. Stephenson, D. Stegenga, "Sharp geometric estimates of the distance to VMOA" ''Contemp. Math.'' , '''137''' (1992) pp. 421–432</td></tr><tr><td valign="top">[a5]</td> <td valign="top"> D. Sarason, "Functions of vanishing mean oscillation" ''Trans. Amer. Math. Soc.'' , '''207''' (1975) pp. 391–405</td></tr></table> |
Revision as of 17:00, 1 July 2020
space of analytic functions of vanishing mean oscillation
The class of analytic functions on the unit disc that are in 
(see also $\operatorname{BMO}$-space; 
-space; 
-space).
Fefferman's duality theorem (see $\operatorname{BMO}$-space) gives the characterization that an analytic function in $\operatorname{BMO}$ is in 
if and only if its boundary values can be expressed as the sum of a continuous function and the harmonic conjugate (cf. also Harmonic function) of a continuous function. This suggests that functions in 
are close to being continuous, but one has to be careful because their behaviour can be quite wild. For example, it can be show that any conformal mapping onto a region of finite area is in 
.
D. Sarason [a5] used the fact that 
is the closure of the disc algebra $A$ in 
to prove that $H ^ { \infty } + C$, with $C$ the class of continuous functions, is a closed subalgebra of $L^{\infty}$ and consequently the simplest example of a Douglas algebra (see 
-space).
The distance between a function $f$ in 
and 
has attracted some interest, [a1], [a2], [a4]. Let $f$ be an analytic function on the unit disc, $\zeta$ a point on the boundary $T$ and write $K _ { \zeta }$ for the cluster set $\operatorname{Cl} ( f , \zeta )$. Using an assortment of tools from functional analysis, S. Axler and J. Shapiro [a1] proved that
\begin{equation*} \| f + \operatorname {VMOA} \| _ { * } \leq C \operatorname { lim sup } _ { \zeta \in T } \sqrt { \operatorname { area } ( K _ { \zeta } ) }. \end{equation*}
This led to a search for the optimal geometric condition for the right-hand side above, see [a4] for the answer.
References
| [a1] | S. Axler, J. Shapiro, "Putnam's theorem, Alexander's spectral area estimate and VMO" Math. Ann. , 271 (1985) pp. 161–183 |
| [a2] | J. Carmona, J. Cufi, "On the distance of an analytic function to VMO" J. London Math. Soc. (2) , 34 (1986) pp. 52–66 |
| [a3] | C. Fefferman, "Characterization of bounded mean oscillation" Bull. Amer. Math. Soc. , 77 (1971) pp. 587–588 |
| [a4] | K. Stephenson, D. Stegenga, "Sharp geometric estimates of the distance to VMOA" Contemp. Math. , 137 (1992) pp. 421–432 |
| [a5] | D. Sarason, "Functions of vanishing mean oscillation" Trans. Amer. Math. Soc. , 207 (1975) pp. 391–405 |
How to Cite This Entry:
VMOA-space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=VMOA-space&oldid=16311
VMOA-space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=VMOA-space&oldid=16311
This article was adapted from an original article by D. Stegenga (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article