Twisted spherical means in annular regions in \mathbb{C}^n and support theorems

Abstract

Let Z(Ann(r, R)) be the class of all continuous functions f on the annulus Ann(r, R) in C n with twisted spherical mean f × µ s (z) = 0, whenever z ∈ C n and s > 0 satisfy the condition that the sphere S s (z) ⊆ Ann(r, R) and ball B r (0) ⊆ B s (z). In this paper, we give a characterization for functions in Z(Ann(r, R)) in terms of their spherical harmonic coefficients. We also prove support theorems for the twisted spherical means in C n which improve some of the earlier results.

Sets of Injectivity for Weighted Twisted Spherical Means and Support Theorems

“…In this section, we prove Theorem 1.4, which is an analogue of the author's support theorem ([13], Therorem 1.2) for the TSM to the WTSM on C n . Our previous result ( [13], Theorem 1.2) is a special case of Theorem 1.4, when for p = q = 0. We would like to quote support theorem for the case n = 1.…”

“…P 1 (Ã)ϕ n−1 k (z) = (−2) −p−q P 1 (z)ϕ n+p+q−1 k−q (z), moreover P 1 (Ã) is right invariant, therefore it follows that P 1 (Ã)(ãP s,t × ϕ n−1 k )(z) = 0, ∀ k ≥ q and |z| = R.Using Hecke-Bochner identity (Lemma 2.2), we getã, ϕ n+s+t−1 k−s P 1 (Ã)P s,t ϕ n+s+t−1 k−s (z) = 0, ∀ k ≥ max(q, s) and |z| = R. Ifà * (P s,t ϕ n+s+t−1 k−s )(R) = 0 for some k ≥ max(q, s), then a computation similar as done for Z * j f in[13], p.2516-17, we havẽ |z| = R and γ = n + s + t. Since {P s,t | S 2n−1 : s, t ≥ 0} form an orthonormal basis for L 2 (S 2n−1 ). An inductive process, then gives the coefficient of highest degree polynomial P p+s,q+t as1 1 k−s (R) = 0.…”

“…In a recent work [8], Thangavelu and Narayanan prove a support theorem, for the TSM for certain subspace of Schwartz class functions on C n . In our previous work [13], we have given an exact analogue of Helgason's support theorem for the TSM on C n (n ≥ 2). For n = 1, we have proved a surprisingly stronger result where we do not need any decay condition.…”

“…[13] A necessary and sufficient condition for f ∈ Z B,∞ (C n ) is that for all p, q ∈ Z + , 1 ≤ j ≤ d(p, q), the spherical harmonic coefficientsã p,q j of f satisfy the following conditions:…”

“…[13] Let f be a continuous function on C. Then f is supported in |z| ≤ B if and only if f × µ r = µ r × f = 0 for s > B + |z| and ∀z ∈ C.…”