Inferior Mean of Measures on Curves and Subspaces

In [1], Wu showed that, for a positive harmonic function u defined on the unit disc, the same result holds, if the boundary measure μ of u is absolutely continuous with respect to the arclength measure of the unit circle; while, for an arbitrary positive harmonic function u in the disc, (2/π)μ(∂Ω) ≤ IM(u) ≤ μ(∂Ω). In [2], these results of Wu were extended to positive functions u in the harmonic Hardy space h(Ω), which is the space of harmonic functions defined onΩ by finite (positive) Borel boundary measures μ:


Introduction
Let Ω be the unit ball,   , or the upper half-space,  +1 + , in  +1 , and let Ω be the boundary of Ω.The notion of the inferior mean, IM(), of a positive function  is due to Maurice Heins.Definition 1.Let  be a positive function defined on Ω and let   = { ∈ Ω : dist(, Ω) < }.The inferior mean of  is where Γ is the volume measure of a continuous, piecewise differentiable, orientable surface Γ that separates boundaries of   .
The study of the inferior mean began with the following unpublished result of Heins, which is obtained in connection with   spaces.
If a positive function  has a subharmonic logarithm on the annulus { < || < 1} and Γ are rectifiable Jordan curves in { < || < 1} separating 0 from ∞, then lim In [1], Wu showed that, for a positive harmonic function  defined on the unit disc, the same result holds, if the boundary measure  of  is absolutely continuous with respect to the arclength measure of the unit circle; while, for an arbitrary positive harmonic function  in the disc, (2/)(Ω) ≤ IM() ≤ (Ω).In [2], these results of Wu were extended to positive functions  in the harmonic Hardy space ℎ 1 (Ω), which is the space of harmonic functions defined on Ω by finite (positive) Borel boundary measures : where   is the area of the unit sphere   in  +1 .Equality on the right holds for functions whose boundary measures are absolutely continuous with respect to the Lebesgue measure of Ω.This paper refines the upper bound in (3) for  ∈ ℎ 1 (Ω) defined on the upper half-space Ω =  +1 + by a finite positive Borel boundary measure  that lies either on a subspace or on a smooth curve  ⊂ Ω.With () = (Ω), we will prove the following theorems (throughout smooth means  differentiable,  ≥ 1).

Journal of Mathematics
In Theorem 7 we show that equality in (4) holds for boundary measures  that are absolutely continuous with respect to the Lebesgue measure of .
In Theorem 9 we show that if  lies on a smooth curve , then (4) is true with  = 1.
In Theorem 10 we show that equality in (4) holds for the boundary measure  that is the arclength measure of .

Tubular Coordinates
In this section, we recall a few facts about spherical transformations, obtain the volume elements of hypersurfaces in tubular coordinates, and evaluate three constants occurring in the integration over the tubes.In all sections, depending on the context, we use the same notation for points and position vectors, vectors and line segments, and so forth.
Define Λ  by Λ 0 = 1, Λ  = ∏  =1 sin   .The spherical transformation from Cartesian to spherical coordinates is given by the identities: We denote by   the unit sphere in  +1 and by   + =   ∩  +1 + its upper-half in  +1 .The volume element of a sphere of constant radius  (see e.g., [3,Section 676]) is where We recall recurrence relations for the area,   , and area element of   : (9)
Lemma 2. If a hypersurface Γ is locally given by a  1differentiable function  = (, z), then its volume element admits this form: ) . ( When the chain rule is applied to each Jacobian in the second sum of (12), the definition (7) Journal of Mathematics 3 Then, Ψ is a subspace of coordinate space (, , z) in  +1 + and can be regarded as an -dim manifold without boundary.
A projection Γ → Ψ given by (, , z) → (1, , z) → (, z) is continuous, onto, and smooth on each smooth component of Γ.Under this projection by Sard's theorem [4], the image of points  ∈ Γ, where (, z) is defined in some neighborhood of , has full measure in Ψ.This will justify integration over Γ in coordinates ((, z), , z) in Theorems 7 and 10.
With the help of ( 9) and (8), we evaluate the constants:

Measures on Subspaces
Here, we show that if  is defined by a boundary measure  lying on a subspace   , then IM() is bounded by the integrals of  over the half-tubes    ∩ Ω.If  is absolutely continuous with respect to the volume measure of   , these integrals approach IM() as  → 0. 4) for  on   Theorem 5. Let Ω be the upper half-space  +1 + .Let a harmonic function  be defined on Ω by a finite positive Borel boundary measure  that lies on a subspace   ⊂ Ω.Then IM () ≤  ,  (  ) .

Conjectures
With methods of Theorems 9 and 10 and triangulation of mdimensional manifolds, we conjecture that these theorems are true for  ≥ 1.

Lemma 6 .
* =  = /z by Step Functions.Let z ∈   .Denote by (z, ℎ) a cube of side 2ℎ and by (z, ℎ) a ball of radius ℎ both centered at z. Let   = |(0, 1)|.When a cube (0, ) is broken into   equal cubes   of side  = 2/, the smaller cubes of side  − 2 at the centers of   are denoted by Δ  .Let  ∈
85) Moreover, equality in (85) holds, if  is the volume measure of .The next conjecture, if true, combines the results of Theorems 7 and 10.With the assumptions of Conjecture 1, equality in (85) holds, if  is absolutely continuous with respect to the volume measure of .