Chapter 1. Comple x Geometr y an d Application s
1.1. Nonisotropi c Spaces . Nonisotropi c spaces arise naturally i n many dif-
ferent contexts—ofte n i n connectio n wit h a partia l differentia l equation . I n th e
classical setting , th e hea t equatio n give s ris e t o suc h spaces :
Consider th e (parabolic ) hea t equatio n
ut = u xx. (1)
on th e half-plan e K+ . Th e fundamenta l solutio n fo r thi s equatio n i s
# (
M
)
=
* .
e
-*
2
/(4) .
V ' 2V7T t
In fact H i s annihilated b y the heat operato r and , a s t 0
+
, th e function H(-, t)
tends i n th e wea k sta r topolog y o f measure s t o th e Dira c mas s a t th e origi n i n
R. Se e [WID] . I t follow s tha t i f / i s a testing functio n o n R = dR 2^ the n
u{x, t) = H * / = / H(x - w, t)f(w) dw
JR
provides a solution t o (1) whic h assumes , i n a natural sense , the boundar y limi t
/•
We ca n ge t a glimps e o f th e nonisotropi c regularit y o f th e hea t equatio n
by noticin g tha t (1) mandate s tha t on e derivativ e i n t count s th e sam e a s tw o
derivatives in x. I n particular, i f u = i J * / i s a solution of the heat equatio n the n
uxx wil l hav e just th e sam e growt h an d smoothnes s a s u
t
. On e ma y formulat e
these statement s mor e precisel y usin g specia l functio n spaces . W e shal l no t
indulge ourselve s i n th e detail s here , bu t refe r th e reade r t o the pape r [FBI] .
An elementary contex t i n which nonisotropi c analysi s arise s in complex func -
tion theor y i s a s follows . Conside r th e uni t bal l 5 i n C 2. Le t P = (Pi,0 ) b e
a poin t o f distanc e 8 fro m th e boundar y poin t 1 = (1,0). [I n wha t follow s i t
will b e convenien t t o le t 6 SQ(P) denot e th e distanc e o f the poin t P fro m th e
boundary o f fi.] Assum e tha t 8 1. Refe r t o Figur e 1.1 . The n trivia l algebr a
shows that th e point s o f the disc s
dn = {K,0):K-Pi|tf }
l
http://dx.doi.org/10.1090/cbms/081/01
Previous Page Next Page