H^p-theory for generalized M-harmonic functions in the unit ball

Ahern, Patrick; Bruna, Joan; Cascante, Carme

doi:10.1512/iumj.1996.45.1961

Cited by 44 publications

(51 citation statements)

References 2 publications

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“…have been considered by many authors; see, e.g., [8,1,5]. Among other important results, it was proved by Graham in [9] that, despite the fact that the Dirichlet problem for ∆ 0,0 is solvable for arbitrary continuous boundary data, in order that the solution is infinitely differentiable up to the boundary it is necessary and sufficient that the C ∞ data be the boundary value of a pluriharmonic function (the real part of a holomorphic function).…”

Section: Congwen Liu and Lizhong Pengmentioning

confidence: 99%

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Boundary regularity in the Dirichlet problem for the invariant Laplacians $\Delta _\gamma $ on the unit real ball

Liu

Peng

2004

Proc. Amer. Math. Soc.

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Abstract. We study the boundary regularity in the Dirichlet problem of the differential operatorsOur main result is: if γ > −1/2 is neither an integer nor a half-integer not less than n/2 − 1, one cannot expect global smooth solutions of ∆γ u = 0; if u ∈ C ∞ (Bn) satisfies ∆γu = 0, then u must be either a polynomial of degree at most 2γ + 2 − n or a polyharmonic function of degree γ + 1.

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Section: Congwen Liu and Lizhong Pengmentioning

confidence: 99%

“…The proof of the theorem is much like that of Theorem 2.1 in [1], and we will only sketch it. For each 0 < r < 1 the L 2 -decomposition in spherical harmonics of u r (ζ) = u(rζ) gives that…”

Section: Preliminariesmentioning

confidence: 99%

Boundary regularity in the Dirichlet problem for the invariant Laplacians $\Delta _\gamma $ on the unit real ball

Liu

Peng

2004

Proc. Amer. Math. Soc.

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“…In [ABC,Theorem 4.13], the first three statements of Theorem 4.1 are proved to be equivalent. So we prove the following to fulfill the proof of Theorem 4.1.…”

Section: Area Function G-function and The Maximal Functionmentioning

confidence: 99%

Hyperbolic mean growth of bounded holomorphic functions in the ball

Kwon¹

2002

Trans. Amer. Math. Soc.

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Abstract. We consider the hyperbolic Hardy class H p (B), 0 < p < ∞. It consists of φ holomorphic in the unit complex ball B for which |φ| < 1 andwhere denotes the hyperbolic distance of the unit disc. The hyperbolic version of the Littlewood-Paley type g-function and the area function are defined in terms of the invariant gradient of B, and membership of H p (B) is expressed by the L p property of the functions. As an application, we can characterize the boundedness and the compactness of the composition operator C φ , defined by C φ f = f • φ, from the Bloch space into the Hardy space H p (B).

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“…In [1] it is shown that every generalized -harmonic function on the complex unit ball in n has a series expansion in homogeneous polynomials. In this section we prove an analogous result for hyperharmonic functions.…”

Section: Hyperharmonic Expansionsmentioning

confidence: 99%

“…A motivation for a series expansion (1.3) is paper [1] of Ahern et al, who considered the case of generalized -harmonic functions on the complex unit ball.…”

Section: Introductionmentioning

confidence: 99%