LIST OF CORRECTIONS
SOME NEW ESTIMATES FOR THE HELGASON FOURIER
TRANSFORM ON RANK ONE SYMMETRIC SPACES
by R. DAHER
and
S. EL OUADIH
• For (1) to (9): See the revised version
• Answer of (10): We changed the Laplace Beltrami operator Λ by the operator ∆.
• Answer of (11): The action of the Laplace Beltrami operator ∆ on f ∈ L2 (X)
is understood in the sense of distributions
• Answer of (13): Denote by W2r (∆), r = 0, 1, 2..., the class of functions
f ∈ L2 (X) that have generalized derivatives in the sense of distributions satisfying ∆r f ∈ L2 (X).
• Answer of (14) and (15): Yes, we made a mistake in Lemma 3.1 (2r instead of r)
Lemma 3.1. If f ∈ W2r (∆), then
Z +∞ Z
k∆kt ∆r f k22 =
(λ2 + ρ2 )2r |1 − ϕλ (t)|2k |fb(λ, b)|2 dµ(λ)db.
0
B
2
(f ) instead of the
• Answer of (16): In Theorem 3.2 we used the symbol JN
symbol JN (f )
Theorem 3.2. Given k, r and f ∈ W2r (∆). Then there exist a constant c > 0 such
that, for all N > 0,
2
JN
(f ) = O(N −4r ω(∆r f, cN −1 )22,k ).
• For (17) and (18): See the revised version
• Answer of (19): With the change of 2r instead r there is no need of inequality
(λ2 + ρ2 )r ≤ (λ2 + ρ2 )2r .
we obviously the inequality
Z +∞
Z
2
2 2r
2k
(λ +ρ ) |1−ϕλ (t)| dη(λ) ≤
N
+∞
(λ2 +ρ2 )2r |1−ϕλ (t)|2k dη(λ) = k∆kt ∆r f k22 .
0
• Answer of (20): Yes N = [t−1 ].
1
2
• For (21): See the revised version (we agree with your suggestion)
• Answer of (22): Yes we made a mistake. we do not need the inequality
N = [t−1 ]. (So we suprime Line 37, page 7)
• For (23): See the revised version (we agree with your suggestion).
• For (24): Yes t instead of h.
• For (25) and (26): We rewrite the proof of the theorem 3.4 by replacing
r with 2r (See the revised version)
Acknowledgements
The authors would like to thank the referee for his valuable comments and suggestions.