Abstract
We obtain new sharp upper bounds of the inferior mean for positive harmonic functions defined by finite boundary measures that lie on curves or subspaces of the boundary of the half-space.
“Dedicated to L. D. Ivanov and my mother, Faina M. Movshovich”
1. Introduction
Let be the unit ball, , or the upper half-space, , in , and let be the boundary of . The notion of the inferior mean, , of a positive function is due to Maurice Heins.
Definition 1. Let be a positive function defined on and let . The inferior mean of is where is the volume measure of a continuous, piecewise differentiable, orientable surface 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The study of the inferior mean began with the following unpublished result of Heins, which is obtained in connection with spaces.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 and are rectifiable Jordan curves in separating from , then 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, . In [2], these results of Wu were extended to positive functions in the harmonic Hardy space , which is the space of harmonic functions defined on by finite (positive) Borel boundary measures : where is the area of the unit sphere in . Equality on the right holds for functions whose boundary measures are absolutely continuous with respect to the Lebesgue measure of .
This paper refines the upper bound in (3) for defined on the upper half-space by a finite positive Borel boundary measure that lies either on a subspace or on a smooth curve . With , we will prove the following theorems (throughout smooth means -differentiable, ).
In Theorem 5 we show that if lies on a subspace , then
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 .
In Theorem 10 we show that equality in (4) holds for the boundary measure that is the arclength measure of .
2. 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.
2.1. Spherical Coordinates
Let . In Cartesian coordinates , one has and . In local spherical coordinates with the origin at , spherical radius , the first spherical angle is measured from to , and the remaining spherical angles lie in with the range: , , . A -tuple will be denoted by .
Define by . The spherical transformation from Cartesian to spherical coordinates is given by the identities: We denote by the unit sphere in and by its upper-half in . 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
2.2. Tubular (Cylindrical) Coordinates
In tubular coordinates, points have with , and . The volume element of a tubular surface about of constant radius is given by (6) and (7): The next result is a corollary to the surface measure lemma in [2].
Lemma 2. If a hypersurface is locally given by a -differentiable function , then its volume element admits this form:
Proof. By Cramer’s rule (see e.g., [2]), where and indicate that columns are omitted. The surface measure lemma in [2] computes the first sum in (12) as When the chain rule is applied to each Jacobian in the second sum of (12), the definition (7) for allows us to write this sum as
2.3. A Remark on Integration in Tubular Coordinates
For , let , where is the boundary of . Consider a continuous, piecewise smooth, orientable hypersurface separating boundaries of . We assume that each smooth component of has a piecewise smooth boundary. Denote by the following set:
Then, is a subspace of coordinate space in and can be regarded as an -dim manifold without boundary. A projection given by is continuous, onto, and smooth on each smooth component of . Under this projection by Sard’s theorem [4], the image of points , where is defined in some neighborhood of , has full measure in . This will justify integration over in coordinates in Theorems 7 and 10.
2.4. Constants , , and
The first set, , is defined for . These are Eulerian integrals (see e.g., [3, Section 534]): The second set, , is defined for , where and The next lemma is used in Theorems 7 and 10 with small or .
Lemma 3. Let , , and . Then,
Proof. We split the integral into three parts and then use and in the middle integral:
The third set, , is also defined for . These are the integrals of over (see (22) below). In the half-space , it is customary to denote by . If is given in tubular coordinates of Section 2.2, then With the help of (9) and (8), we evaluate the constants:
3. Poisson Kernel, Invariants, and Herglotz Representation of Harmonic Functions
3.1. Poisson Integral Invariants
The Poisson kernel, , is a function of and
Lemma 4. Let . The integrals of over half-tubes do not depend on , denoted by ; they are evaluated as
Proof. Let and . Define . Then, (21) implies that .
The invariants are computed with the help of (10), (18), and (22):
Herglotz Representation (see [5–7]). A positive harmonic function is defined by a finite positive Borel measure on . It can be expressed as the Poisson-Stieltjes integral of
3.2. Integration over Hyperplanes with Holes
Let and , . Then, (see e.g., [2, Section 5]) Now, consider a subset of with holes above the support of : Fubini’s Theorem and (27) yield
4. Measures on Subspaces
Here, we show that if is defined by a boundary measure lying on a subspace , then is bounded by the integrals of over the half-tubes . If is absolutely continuous with respect to the volume measure of , these integrals approach as .
4.1. Inequality (4) for on
Theorem 5. Let be the upper half-space . Let a harmonic function be defined on by a finite positive Borel boundary measure that lies on a subspace . Then
Proof. Fix and let . Consider hypersurfaces separating the two boundaries of : Then, defined by (28) and defined in Section 2.2. Now, (30) follows from (1), Fubini’s Theorem, Lemma 4, and (29):
Next, we prove that when is absolutely continuous with respect to the Lebesgue measure of . But first, we recall necessary facts about the derivative of such .
4.2. Approximation of z by Step Functions
Let . Denote by a cube of side and by a ball of radius both centered at . Let . When a cube is broken into equal cubes of side , the smaller cubes of side at the centers of are denoted by .
Lemma 6. Let , and a positive is given. For sufficiently large and , there exist positive numbers , , and and sets such that
Proof. If the support set, , of is unbounded, there exists such that the integral over the bounded set, , satisfies Let . By Luzin’s theorem, there exists a sequence of closed sets approaching , so that and the function is (uniformly) continuous on . Let for such that Now, find so that, for all , if , then We break into equal cubes with diameters less than taking . Then, (38) implies that, for , We observe next that, since as , there exits small enough to yield Let be the set of points of density for . Lebesgue’s Theorem states that . Therefore, for , the sets approach as . Then, (41) and (37) imply that there exists such that, for all , This allows us to take in (34). Next, we apply (39) to each term in the sum: It can be seen that and , and also . Hence, The last inequality becomes (35) when (42) (with ) and (40) are applied.
4.3. Equality in (4) for Absolutely Continuous
Theorem 7. Let be the upper half-space . Let a harmonic function be defined on by a finite positive Borel boundary measure that lies on a subspace and is absolutely continuous with respect to the Lebesgue measure of . Then,
Proof. By hypothesis, , where . Given , we obtain constants , sets , and cubes of sides from Lemma 6. We may assume that . Let . In view of Theorem 5, it is enough to show that for any continuous, piecewise smooth, orientable surface in that separates boundaries of . Consider
Since the range of on is , the range of on is . When is fixed, the ball because . When with , we have by (33). Inequality (34) implies that ; that is,
Let ; then , so we may write
The change of variables defines a map and a set whose size by (47). Now, Lemma 3 estimates the factor to the right of in (48) as follows:
If is given by , we have by Lemma 2. Then, (35) and (22) estimate the sum:
where if , and otherwise. Finally, (48)–(50) and Lemma 4 yield the desired inequality:
with .
Since were arbitrary and , the inferior mean satisfies
The opposite inequality of Theorem 5 gives us equality in (52).
5. Measures on Curves
In this section, we show that if is defined by a boundary measure lying on a smooth curve, , then is bounded by the integrals of over half-tubes about . If is the arclength measure of , these integrals approach as . We assume that the dimension of is greater than 1. The case , where or , is covered by Theorems 5 and 7.
5.1. On Linear Approximation of -Curves
Let be a simple -curve parameterized by arclength. Let be ’s injectivity radius (radius of the largest embedded normal tube around ).
Lemma 8. For any , there exists so that, for , one has the following: For any , there exists such that If and is sufficiently close to , then
Proof. Since is continuous, there exists such that, for , there exist numbers , , so that This implies (53) because we have . Denote by the unit vector . Using (57), we write vectors below as where . Then, (54) follows from the estimates of : One can check that . Let , and be as in (55) and . If , then . Now, (56) is obtained from
5.2. Inequality (4) for Measures on Curves
Since measures are finite, we may assume that curves are bounded (see (36)).
Theorem 9. Let be the upper half-space and let be a simple -curve. Let a harmonic function be defined on by a finite positive Borel boundary measure that lies on . Then,
Proof. We assume that is parameterized by the arclength parameter . Thus, . Let . Consider a curve with injectivity radius and as in Lemma 8. We will show that, for , there exists a surface such that
Let and let be a polygonal approximation of with line segments connecting with . We take
Note that and . It follows from (53) and (54) that
Let and be segments of length extending and . Denote by lines containing . We build a Quonset hut above with the axis along planted in the hyperplane (See Figure 1). It consists of straight cylindrical huts glued together where meets and two spherical doors and . The part of outside the hut is called and our surface is . By (64), (subset of is defined in (28)). Thus by (29),
We apply Fubini’s theorem to and estimate for a fixed . The main contribution to will come from .
Integration over (and similarly ) is replaced with that over its copy, , on obtained by swiveling to until the two contours are aligned horizontally (see Figure 1). Under this rotation, the distance of to is preserved and the distance to its image is
If is the projection of on , then, by (64), . So
That gives us and now, with the assumption that is closer to , Lemma 4 yields the main estimate:
Next, we integrate over as far as (where (53)–(59) of Lemma 8 are valid). Let be as in Figure 2.
By (59), the angle , so is closer to than to the projection of on . But by (53). Hence, by the triangle inequality, we have a lower bound for
Recall that and . These, combined with (69), estimate the integrals of over the two remaining parts of as follows:
Finally, from (65), (68), (70), and Fubini’s theorem, we can obtain the estimate (62):
5.3. Equality in (4) for Arclength Measures
Theorem 10. Let be the upper half-space and let be a simple -curve. Let a harmonic function be defined on by the boundary measure , that is, the arclength measure of . Then,
Proof. We assume that is parameterized by the arclength parameter . Thus, and . Let . Consider a curve with injectivity radius and as in Lemma 8. We fix an integer , and, for , define
Note that . Consider segments connecting endpoints of ; then, by (54) and (73),
In view of Theorem 9, it suffices to show that, for any continuous piecewise smooth surface in separating its boundaries,
The can be quite small; therefore, we break into equal segments and place their copies, , at points on so that are parallel sides in rectangles (Figure 3). We require that
In local tubular coordinates where the short -axes lie on , let
Fix . Recall that . Then, (76) yields
Next, we claim that (and thus ) do not overlap. Indeed,
prove the claim
Now, the first estimate below is analogous to (49) of Theorem 7. It is obtained from (56), (80), the change of variables , and Lemma 3 with and
Since the range of on is , the range of on is . If is given by , then by Lemma 2. The second estimate is analogous to (50) of Theorem 7. It is obtained from (22) and (74) as follows:
where and we estimated using (74).
Finally, , Lemma 4, and (82)-(83) yield the inequality (75):
6. Conjectures
With methods of Theorems 9 and 10 and triangulation of m-dimensional manifolds, we conjecture that these theorems are true for .
Conjecture 1. Let be the upper half-space . Let a harmonic function be defined on by a finite positive Borel boundary measure that lies on an -dimensional manifold , and then Moreover, equality in (85) holds, if is the volume measure of .
The next conjecture, if true, combines the results of Theorems 7 and 10.
Conjecture 2. With the assumptions of Conjecture 1, equality in (85) holds, if is absolutely continuous with respect to the volume measure of .
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The author is very grateful to P. G. Andrews, D. Berg, C. I. Delman, J. F. Glazebrook, and C. M. Hawker for their assistance.