Hilbert matrix norms on weighted Bergman spaces: even exponents and a counterexample to the beta formula · arXivDesk
2607.23540Jul 26, 202622 pages. We disprove Karapetrović's conjectured beta-function formula for the norm of the Hilbert matrix operator on weighted Bergman spaces
Hilbert matrix norms on weighted Bergman spaces: even exponents and a counterexample to the beta formula
Let Aαp be the weighted Bergman space on the unit disk, where α>−1. For f(z)=∑k=0∞akzk∈Aαp
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, consider the Hilbert matrix operator
Hf(z)=∑n=0∞(∑k=0∞n+k+1ak)zn=∫011−tzf(t)dt
. For even exponents
p=2m
, we prove that
∥H∥Aα2m→Aα2m=B(a,1−a)
, where
a=(α+2)/(2m)
, whenever
0<a≤m/(2m−1)
. For
p=2,4,6,8,10
, the same formula holds throughout the admissible range. We also show that the beta-function norm formula does not hold for all admissible parameters. Set
a0=800001/1000000
and
αp=a0p−2
. Then, for every real
p≥1100000
,
∥H∥Aαpp→Aαpp>B(a0,1−a0)
. The counterexample is based on the fixed function
f0(z)=(1−z2)−4/5=∑k=0∞k!(4/5)kz2k
. A rigorous interval estimate at
p=1100000
, together with monotonicity in
p
, yields the result on the entire half-line. In particular, the formula fails for every even exponent
p=2m
with
m≥550000
.
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