Abstract
Let Φ be an entire function on
C
n
, and for any
h > 0 and
r > 0 define
F
r = ¦Φ(z)¦
r e
−2π¦z¦
2
h
. Let
dμ
h
denote
h
−
n
times Lebesgue measure on
C
n
.
∝ ¦▽F
r
s
2
¦
2 dμ
h =
nπs
h
∝ F
r
sdμ
h
. From this and a logarithmic Sobolev inequality we easily deduce
q
n
q
∥F
r∥
q ⩽ p
n
p
∥F
r∥
p
for all 0 <
p ⩽
q <
t8 where the
L
p
norms are taken with respect to the measure
dμ
h
above. We apply these results to the study of the spaces
A
p
consisting of all entire functions Φ satisfying
∝ ¦Φ(z)¦
pe
−2π¦z¦
2
h
dμ
h < ∞
obtaining sharp bounds for some associated operators and proving denseness of analytic polynomials in
A
p
for 1 ⩽
p < ∞. We then apply our results to the coherent state transform, extending and simplifying some previously known results.