Characterization of the Poisson integrals for the non-tube bounded symmetric domains

Abstract

We characterize the L p −range, 1 < p < +∞, of the Poisson transform on the Shilov boundary for non-tube bounded symmetric domains. We prove that this range is a Hua-Hardy type space for harmonic functions satisfying a Hua system. 1991 Mathematics Subject Classification. 43A85; 32A25; 32M15.

L^p-Poisson integral representations of the generalized Hua operators on line bundles over SU(n,n)/S(U(n)xU(n))

“…For 0 ≤ r < 1, we write F r (U) = F (rU). Then Following the same method as in the proof of the trivial line bundle case (see [2], [3]) we can show that f j lies in L p (S) with…”

“…We compute the above Hua-type integral in a manner similar to the case ν = 0 [3]. We give an outline of the proof.…”

“…(ii) For ν = 0 the first author and Koufany [2,3] generalized Theorem 1.1 to all Hermitian symmetric spaces.…”

On Poisson transforms for differential forms on real hyperbolic spaces

“…Hence, for each m, F m ∈ E 2 q,λ (G/K; τ p ) and from Theorem 5.2 it follows that there exists f m ∈ L 2 (K/M ; σ q ) such that F m = P p q,λ f m . To prove that f m ∈ L r (K/M ; σ q ) we will follow the same method as in [5]. According to Theorem 5.5 we have, for any ϕ ∈ C ∞ (K/M ; σ q ), K f m (k), ϕ(k) Λ q C n−1 dk = lim t→∞ K g t m (k), ϕ(k) Λ q C n−1 dk, where g t m (k) := c −2 p,q |c q (λ, p)| −2 e 2(ρ− (iλ))t π q p K P λ (ha t , k) * F m (ha t )dh.…”

Characterization of the Lp-range of the Poisson transform on the Octonionic Hyperbolic Plane

“…More precisely, we will discuss a uniform L p -boundedness of a family of Calderon-Zygmund operators (Ψ r (λ)) r∈[0,1[ ( see Section 4) on the boundary ∂B(O 2 ) considered as a space of homogeneous type in the sense of Coifman and Weiss [9]. To prove the sufficiency condition for p = 2, we follow the method we used in [5] and [6], to characterize Poisson integrals of L p -functions on the Shilov boundary of bounded symmetric domains. Although the techniques we use here may seem to be similar to those in [4], however in working in the exceptional case we encounter a prime difficulty, due to the fact that the algebra of Octonions O is not associative.…”