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A remark on gradients of harmonic functions.

Wen Sheng Wang (1995)

Revista Matemática Iberoamericana

In any C1,s domain, there is nonzero harmonic function C1 continuous up to the boundary such that the function and its gradient on the boundary vanish on a set of positive measure.

Asymptotic behavior of the invariant measure for a diffusion related to an NA group

Ewa Damek, Andrzej Hulanicki (2006)

Colloquium Mathematicae

On a Lie group NA that is a split extension of a nilpotent Lie group N by a one-parameter group of automorphisms A, the heat semigroup μ t generated by a second order subelliptic left-invariant operator j = 0 m Y j + Y is considered. Under natural conditions there is a μ ̌ t -invariant measure m on N, i.e. μ ̌ t * m = m . Precise asymptotics of m at infinity is given for a large class of operators with Y₀,...,Yₘ generating the Lie algebra of S.

Boundary behavior of subharmonic functions in nontangential accessible domains

Shiying Zhao (1994)

Studia Mathematica

The following results concerning boundary behavior of subharmonic functions in the unit ball of n are generalized to nontangential accessible domains in the sense of Jerison and Kenig [7]: (i) The classical theorem of Littlewood on the radial limits. (ii) Ziomek’s theorem on the L p -nontangential limits. (iii) The localized version of the above two results and nontangential limits of Green potentials under a certain nontangential condition.

Boundary behaviour of harmonic functions in a half-space and brownian motion

D. L. Burkholder, Richard F. Gundy (1973)

Annales de l'institut Fourier

Let u be harmonic in the half-space R + n + 1 , n 2 . We show that u can have a fine limit at almost every point of the unit cubs in R n = R + n + 1 but fail to have a nontangential limit at any point of the cube. The method is probabilistic and utilizes the equivalence between conditional Brownian motion limits and fine limits at the boundary.In R + 2 it is known that the Hardy classes H p , 0 < p < , may be described in terms of the integrability of the nontangential maximal function, or, alternatively, in terms of the integrability...

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