Compl. anal. oper. theory 1 (2007), 301–315c© 2007 Birkhauser Verlag Basel/Switzerland1661-8254/030301-15, published online April 2, 2007DOI 10.1007/s11785-007-0015-0

Complex Analysisand Operator Theory

Minkowski Dimension and Cauchy Transformin Clifford Analysis

Ricardo Abreu Blaya, Juan Bory Reyes, and Tania Moreno Garcıa

Abstract. In the present paper we provide some conditions of a geometricalcharacter for continuous extendibility of the Clifford–Cauchy transform tothe boundary of a domain in the Euclidean space of higher dimensions if itsdensity satisfies a Holder condition. The criterion obtained in this work is anextension to a very general class of domains of a result, which has already be-come classical, obtained by Viorel Iftimie, who proved in 1965, for the case ofa domain with compact Liapunov boundary, that the Clifford–Cauchy trans-form has Holder–continuous limit values for any Holder–continuous density.

Mathematics Subject Classification (2000). 30E20, 30E25, 30G35.

Keywords. Clifford analysis, geometric measure theory, Cauchy transform.

1. Introduction

Let γ be an oriented rectifiable Jordan curve in the complex plane and let Ω+

and Ω− denote its two mutually complementary regions. For any defined on γcontinuous function u the complex Cauchy transform (called also Cauchy typeintegral) Cγu defined by

(Cγu)(z) :=12π

∫γ

1ζ − z

n(ζ)u(ζ)ds , z /∈ γ ,

where n(ζ) is the outward pointing unit normal at ζ ∈ γ and ds is an arc-lengthdifferential, exists and represents a function holomorphic in C\γ. In the last severalyears there has been traditional interest to the question concerning the existenceand properties of the limit boundary values of Cγu for z approaching to t ∈ γfrom Ω±.

The immediate cause of this interest is that the complex Cauchy transformis closely related to the potentials of simple and double layers which are involveddirectly to the solution of the Dirichlet problem for harmonic functions with bound-ary data on closed curves.

302 R. Abreu Blaya et al. Comp.an.op.th.

On the other hand, it should be noted that the existence of the limit bound-ary values of the complex Cauchy transform leads to numerous and importantconsequences and applications to global theory of boundary value problems andsingular integral equations both for holomorphic functions in the complex plane,see [23, 28].

There are numerous investigations on the evolution of the limit boundaryvalues of the complex Cauchy transform in smoothly bounded domain. A well-known first result is that Cγu possesses continuous limit boundary values on γ ifits density u satisfies a Holder condition with exponent ν ∈ (0, 1].

It seems to be that this result was essentially associated at the beginning tothe relevant investigation due by Sokhotski (1873), Harnack (1885) and Morera(1889). Therefore it was called in [29] by HMS-Theorem.

The next important results were those of Plemelj published in a short note in1908. Further very general discussion of the result was devoted apparently unawareof Plemelj’s work by Privalov in 1918.

To get an overview of this historical background we recomend the wonderfulmonograph by Muskhelishvili [28] and the extensive references therein.

In more recently times the above mentioned result has been essentially en-riched by E, Dyn’kin in [18, 19], which reflects the author’s attempt to develop amethod based on a direct estimate technique of the modulus of continuity of theCauchy type integral via the corresponding modulus of its density and some otherquantities characterizing the metric properties of the rectifiable boundary of thedomain in general non-smooth.

The most interesting consequences of the Dyn’kin’s work are the follows:

(a) If u satisfies a Holder condition with exponent ν > 1/2, then the Cauchytype integral Cγu is continuous on the closure of the domain bounded by anarbitrary rectifiable Jordan curve γ and this condition cannot be improved.In particular, for arbitrary fixed ν ∈ (0, 1/2], in [19] a rectifiable curve γν

and a ν-Holder continuous function uν were constructed such that Cγu withγ = γν and u = uν loses the continuity at one of the points of the integrationboundary.

(b) If γ is a regular (in the sense of Ahlfors–David) closed curve, i.e., iff thereis a constant c > 0 such that, for all balls B in the complex plane, the totallengh of γ ∩ B is less than c times the radius of B, then the limit boundaryvalues of the Cauchy type integral Cγu with a ν−Holder continuous densityexist for arbitrary ν ∈ (0, 1].

When investigating non-rectifiable version of the HMS-Theorem, because the com-plex Cauchy transform loses its meaning, there becomes necessary to use a newmethods which does not use contour of integration and can thus be used on fractalcurves. A complete approach to this situation has been shown by B. A. Kats in [30].See also Harrison and Norton [25, 26] who defined integration along non-smoothboundaries in Rn+1, for n ≥ 1.

An analogous effect does carry over to this more general situation; the criticalvalue of the exponent ν is here α/2, where α is the Minkowski dimension of the

Vol. 1 (2007) Minkowski Dimension and Cauchy Transform 303

curve (the notion of this fractal dimension will be posed explicitly below). Thecondition ν > α/2 cannot be weakened in terms of the Minkowski dimension ofthe curve.

The construction of the example developed in [30] is a ”non-rectifiable” ver-sion of Dyn’kin construction from [19]. The result thus obtained confirmed insome sense a remark made by Kats where he noticed that Dyn’kin believed thatthe number 1/2 is half the Hausdorff dimension of the integration contour. Kats’sresult imply that 1/2 is half of its Minkowski dimension. However, for rectifiablecurves these dimensions coincide.

The interested reader should refer to the work of B.A. Kats and A. Y. Pogo-dina [31], where a brief historical outline of such development could be found.

As Clifford analysis provides a direct generalization of classical analysis inthe complex plane to Euclidean space of higher dimensions, see [14]; the questionarises about the existence of the Dyn’kin approach of the HMS-Theorem in Cliffordanalysis. Clifford analysis is, roughly speaking, centered around the notion of amonogenic function, i.e., a null solution to the Dirac operator, which represents adirect generalization of the complex Cauchy–Riemann operator.

The present paper deals with this situation and workout some fine details ofthe higher dimensional version of the HMS-Theorem and Kats’s attempt as well. Itis a continuation of a series of papers by the Cuban Clifford research group [1–4,11]where we have been expressed our vision of the role of the Cauchy transformin generalizations of the one-dimensional complex analysis to a great extent. Inparticular, properties of the boundary values of the Cliffordian–Cauchy transformunder weaker assumptions imposed on the boundary of the domains than almostall reference until nowadays (to the authors’s knowledge) were examined.

In connection with various traditional theories the subject also has beentreated for some classes of harmonic functions, as for instance, with the Laplacianvector fields theory, HodgedeRham system theory and multivector fields theory.Key references where the interested reader can find material concerning the topicsjust mentioned are [5–10].

In Geometric Measure Theory, see [21] there is an interesting divorce betweensets of R

n+1 with Hausdorff dimension either m = 1 or 1 < m ≤ n. Every compactconnected set E which has finite Hausdorff measure H1(E) is rectifiable (see [15],Th 1.8) and thus it can be parameterized nicely by a Lipschitz function, howeverfor sets with finite m-dimensional Hausdorff measure Hm (m > 1) one cannot ingeneral find such a nice parameterization. This is because for one-dimensional setsthe notion of connectedness plays a very special role.

A quantitative and scale-invariant version of having Hausdorff dimension mis the notion of Ahlfors David regularity with dimension m (ADm-regularity forshort). To be more precise, an m-dimensional closed subset E of Rn+1 is said tobe ADm-regular if there exists a constant C > 0 such that for all x ∈ E and0 < r ≤ diam(E)

C−1rm ≤ Hm(E ∩ B(x, r)

)≤ C rm , (1)

304 R. Abreu Blaya et al. Comp.an.op.th.

where B(x, r) denotes the open ball with center x and radius r.When m = 1 the lower estimate in (1) is automatic, but when m > 1 it is a

restriction.Many curves as well as countable unions of curves and surfaces are ADm-

regular. Of particular concern to us here and further on is the boundary of boundedJordan domains in Rn+1. Recall that Ω ⊂ Rn+1 is called a Jordan domain (see [25,26]) if it is a bounded oriented connected open set whose boundary is a compacttopological surface.

It is worth noting that for m > 1 situation the ADm-regular sets need not beeven remotely like rectifiable sets. However, AD1-regular compact connected setsare automatically rectifiable and even more: are all uniformly rectifiable. Amongmonographs mainly devoted to this topic, we quote [15].

Many standard facts from complex analysis have reasonable counterpartsin Clifford analysis [14]. For instance, there is a natural version of the Cauchytransform in this context. Let Ω be a Jordan domain in Rn+1 with Hn-finiteboundary Γ and let u be a continuous Clifford algebra valued function on Γ, thenthe Cliffordian Cauchy transform is formally defined by

(CΓu)(x) :=∫

Γ

y − x

σn|y − x|n+1n(y)u(y)dHn(y) , x /∈ Γ (2)

where σn is the area of the unit sphere in Rn+1, and n(y) denotes the outward

pointing unit normal vector on Γ in the Federer’s sense (see [21], Chapter three).A great deal of research in the framework of Clifford analysis has been devoted

to the study of the existence of continuous limit values of the Cliffordian Cauchytransform (2) over surfaces in Euclidean spaces of higher dimensions, assumingrougher conditions on the smoothness of the surface.

The question on the existence of the limit boundary values of the Clifford–Cauchy transform with preset density is a question on the resolvability of theboundary value problem on reconstruction of a function monogenic in Rn+1 \ Γby its gap on the surface Γ. In [6, 7] there is a suitable matter of how to treatthis question which go far beyond the classical complex case already indicated byE. Dyn’kin in the late of 1990’s.

As was already indicated, contrary to what happens in dimension 1, for mdimensional sets (m > 1) the notion of rectifiability has nothing to do with thenotion of ADm-regularity. In large part because of this, in generalizing to higherdimension classical results concerning a nice behavior of singular integrals andthe Cauchy transform, has been needed to consider the above operators on setssatisfying the both conditions at once. For instance, in [15] the authors showedthat a large class of m-dimensional Calderon–Zygmund singular integral operatorsare bounded on Lp(E) if and only if E is uniformly rectifiable (something strongerthan rectifiable + ADm-regular) and this generalizes a result by Guy David [16]for curves in the complex plane.

Vol. 1 (2007) Minkowski Dimension and Cauchy Transform 305

In [11] the question on the existence of continuous limit values of CΓu(x) onrectifiable + ADn-regular surfaces is optimally answered and this generalizes aresult by Salaev and Tokov [36] for regular curves in the complex plane.

In [2] the authors discuss a reasonable interesting alternative of the CliffordianCauchy transform in connection with the Plemelj–Sokhotski formulae involving ge-ometry of sets, measures and densities ratios. This time they are interested in thestudy of the boundary values of a kind of weighted Clifford–Cauchy transform onrectifiable surfaces which satisfy a regularity condition with respect to a weightedHausdorff measure, the so-called ω-regular surfaces, see [17]. This attempt gener-alizes the case treated in the aforementioned paper [11].

The main goal of this paper is to prove that if ν > nn+1 with density u

satisfying the Holder condition with exponent ν on the integration surface, thenthe Clifford–Cauchy transform (2) has continuous limit boundary values for anyrectifiable surface as well as for any ADn-regular surface in Rn+1. Indeed, we onlyneed the lower bound in the ADn-regularity condition (1) for our purposes.

As the reader may have noticed, the new issue of this result is that it isvalid either for rectifiable or for ADn-regular surfaces, without assuming the bothconditions simultaneously. A main point for the proof is that the Minkowski andHausdorff dimensions of such sets agree and are equal to n. Many references in [35]deal with the Minkowski dimension and compare it to the Hausdorff dimension.

2. Preliminaries

2.1. Clifford algebras and monogenic functions

The real Clifford algebra associated with Rn endowed with the Euclidean metric

is the minimal enlargement of Rn to a real linear associative algebra R0,n withidentity such that x2 = −|x|2, for any x ∈ Rn.

It thus follows that if {ej}nj=1, is the standard basis of Rn, then we must have

that eiej + ejei = −2δij. Every element a ∈ R0,n is of the form a =∑

A⊆N aAeA,N = {1, . . . , n}, aA ∈ R, where e∅ = e0 = 1, e{j} = ej , and eA = eα1 · · · eαk

forA = {α1, . . . , αk} where αj ∈ {1, . . . , n} and α1 < · · · < αk. The conjugation isdefined by a :=

∑A aAeA, where

eA = (−1)keik· · · ei2ei1 , if eA = ei1ei2 · · · eik

.

Notice that for x ∈ Rn+1, we thus have that

xx = x x = |x|2 .

By means of the conjugation R0,n may be endowed with the natural Euclideannorm |a|2 = [aa]0. An algebra norm is defined by taking |a|20 = 2n|a|2.

We consider functions u defined in some subset Ω of Rn+1 and taking valuesin R0,n:

u(x) =∑A

uA(x)eA ,

306 R. Abreu Blaya et al. Comp.an.op.th.

where uA are R00,n-valued functions. We say that u belongs to some classical class

of function on Ω if each of its components uA belongs to that class. In particular,for a closed subset E ⊂ Rn+1, we call C0,ν(E) the class of R0,n-valued functionssatisfying on E the Holder condition with exponent ν, 0 < ν ≤ 1.

In [14] a theory of monogenic functions with values in Clifford algebras isconsidered which generalizes in a natural way the theory of analytic functionsof one complex variable to the (n + 1)-dimensional Euclidean space. Monogenicfunctions are null solutions of the generalized Cauchy–Riemann operator in Rn+1:

D :=n∑

i=0

ei∂

∂xi.

It is a first order elliptic operator whose fundamental solution is given by

e(x) =1σn

x

|x|n+1,

where σn is the area of the unit sphere in Rn+1. If Ω is open in Rn+1 and u ∈ C1(Ω),then u is said to be left (resp. right) monogenic in Ω if D u = 0 (resp. uD = 0)in Ω.

Notice that the fundamental solution e is both side monogenic in Rn+1 \ {0}.Other basic examples of monogenic functions are obtained by means of the

Cliffordian Cauchy transform (2). Clearly CΓu is monogenic in Rn+1 \ Γ.Actually, any a priori smoothness assumption for Γ is not necessary. For

instance there is a very general notion of the unit normal n(y) introduced by Fed-erer [21] such that the Stokes’s Theorem still holds for boundaries with Hn(Γ) <+∞. It is exactly this version of Stokes’s Theorem we need to establish basic for-mulas in Clifford analysis such as the following Borel–Pompeiu formula: Here werefer the reader to [1,12] for a detailed presentation of the easiest way to make thedesired formula true.

For smooth Clifford algebra valued functions u ∈ C1(Ω) ∩ C(Ω)

u(x) = CΓu(x) + TΩDu(x) for x ∈ Ω ,

where

TΩϕ(x) = −∫

Ω

e(y − x)ϕ(y)dLn+1(y)

is the so-called Teodorescu transform of ϕ. This operator has the following goodproperty, see [33, 37]: if ϕ ∈ Lp(Ω), p > n + 1, has a compact support, then TΩϕsatisfies in R

n+1 a Holder condition with exponent 1 − n+1p .

Using the properties of the Whitney decomposition of Rn+1 \E, the followingWhitney extension type theorem is obtained, see [38, p. 174].

Theorem 2.1 (Whitney extension theorem). If the function u is defined on a com-pact set E ⊂ Rn+1 and satisfies the Holder condition with index ν on E. Thenthere exists a compactly supported function u ∈ C0,ν(Rn+1), called the Whitneyextension operator of u, satisfying

Vol. 1 (2007) Minkowski Dimension and Cauchy Transform 307

(i) u|E = u|E,(ii) u ∈ C∞(Rn+1 \ E),(iii) |Du(x)| ≤ Cdist(x,E)ν−1 for x ∈ Rn+1 \ E.

2.2. Rectifiable and regular sets in Rn+1

In what follows m will be a fixed integer such that 1 ≤ m ≤ n. If E ⊂ Rn+1 isan Hm measurable set and 0 < Hm(E) < ∞ we say that E is an m-set.

The geometric condition 0 < Hm(E) < ∞ is a natural condition withoutany quantitative estimates on the size of the set E. In particular, if H1(E) < ∞,then E is contained in a rectifiable curve of length at most 2H1(E). In that case, Ecan be parameterized nicely by a Lipschitz function. The fact that any curve γ inthe plane has finite length if and only if it has a Lipschitz parameterization playsan essential role in several deep results in complex analysis.

Notice however that for m-dimensional subsets of Rn+1 (m > 1) one cannot,in general, find such a nice parameterization.

In fact, the rectifiable surfaces class, in the sense of Federer [21], describe awide class of surfaces containing classes classically considered in the literature.

Definition 2.1. An m-set E is said to be m-rectifiable if it is the Lipschitz imageof some bounded subset of Rm.

The above definition is a special case of the more general Federer’s definitionof (Hm, m)-rectifiability (see [21, p. 251]). An m-set E is said to be (Hm, m)-rectifiable if it is the union of an Hm-zero set and countable many Lipschitz imagesof subsets of Rm. The notion of (Hm, m)-rectifiability is surprisingly stable inthe sense that many other conditions give an equivalent definition. For instance,instead of Lipschitz images one can chose C1-submanifolds or Lipschitz graphs.Notice that rectifiability expresses a qualitative condition about the set E.

We say that the m-set E is left ADm-regular if it satisfies just the lowerestimate in (1), i.e., if there exists a constant C > 0 such that for all x ∈ E and0 < r ≤ diam(E)

C−1 rm ≤ Hm(E ∩ B(x, r)

). (3)

2.3. Hausdorff and Minkowski dimensions

Let E ⊂ Rn+1 then the Hausdorff measure Hs(E) is defined by

Hs(E) := limδ→0

inf

{ ∞∑k=1

(diam Bk)s : E ⊂ ∪k Bk, diam Bk < δ

},

where the infimum is taken over all countable δ-coverings {Bk} of E with open orclosed balls.

Note that Hn+1 coincides with the Lebesgue measure Ln+1 in Rn+1 up to apositive multiplicative constant.

Let E be a bounded set in Rn+1. The Hausdorff dimension of E, denoted byH(E), is the infimum of the numbers s ≥ 0 such that Hs(E) < ∞. For more detailsconcerning the Hausdorff measure and dimension we refer the reader to [20, 22].

308 R. Abreu Blaya et al. Comp.an.op.th.

Frequently, see [34], the Minkowski dimension (also called box dimension) ismore appropiated dimension than the Hausdorff dimension to measure the rough-ness of E. Let us recall the definition of that dimension (see [20, 22]).

Suppose M0 denotes a grid covering Rn+1 and consisting of n+1-dimensional

cubes with sides of length 1 and vertices with integer coordinates. The grid Mk

is obtained from M0 by division of each of the cubes in M0 into 2(n+1)k differentcubes with side length 2−k. Denote by Nk(E) the number of cubes of the grid Mk

which intersect E.The upper Minkowski dimension, of E ⊂ Rn+1, denoted by M(E), is defined

by

M(E) = lim supk→∞

log Nk(E)k log(2)

.

The upper Minkowski dimension can also easily be seen to be determined withballs (see [35]): let P (E, ε) be the greatest number of disjoint balls of radius ε andcenters in E. Then

M(E) = lim supε→o

log P (E, ε)log(1/ε)

.

For n-dimensional set E ⊂ Rn+1, we have in general the situation: n ≤ H(E) ≤M(E) ≤ n+1. If the set E has H(E) > n, then it is called a fractal set in the senseof Mandelbrot.

The statements in the following lemma are probably well-known, but forthe reader’s convenience and for their crucial role: it makes possible an effectiverealization of the suggestion in the open question as mentioned earlier, we shallinclude their proofs in our paper.

Lemma 2.1. Let E ⊂ Rn+1 be an n-set. The upper Minkowski dimension has thefollowing properties:

(i) if E is n-rectifiable, then M(E) = H(E) = n.(ii) if E is left ADn-regular, then M(E) = H(E) = n.

Proof of (i). Let E = ρ(G), where G is a bounded set of Rn and ρ is a Lipschitzfunction, i.e., there exists c > 0 such that |ρ(x)− ρ(y)| ≤ c|x− y| for all x, y ∈ Rn

(c = c(ρ) is called Lipschitz coefficient).Let Q be an n-dimensional cube such that G ⊂ Q, let d be the diameter

of Q and put δk := 2−k/c, where c is the Lipschitz coefficient of ρ. Then Q can bedivided into (d/δk+1)n n-dimensional cubes Qj of diameter not greater than δk.Here x:= the largest integer ≤ x. Therefore ρ(Qj ∩G) intersects not more than2n+1 cubes of the grid Mk, whence

Nk(Γ) ≤ 2n+1(d/δk + 1

)n ≤ C 2nk ,

where the constant C only depends on E. This proofs (i).

Vol. 1 (2007) Minkowski Dimension and Cauchy Transform 309

Proof of (ii). Choose disjoint balls B(xi, ε), xi ∈ E. Then taking into account (3)we have

P (E, ε)εn ≤∑

i

H(E ∩ B(xi, ε)

)≤ C Hn(E) ,

which implies (ii). �

In concluding this section we should note that the quantities H(E) and M(E)do not generally coincide. In this connection, we remark that an exact descriptionof this phenomenon is given by an example in [4].

3. Cauchy transform of a Holder continuous function

The procedure treated in this section is in analogy to what is done in [3, 4] todefine a higher dimensional version of the Kats’s method. But we will analyze amore precise characterization of the possibility to arrive at a rather satisfactoryClifford–Cauchy-like transform, in the case where a very wider setting concerningthe boundary could be examined, caring over to a more advanced and generalcontext of the peculiarities of the boundary; latter is due to a delicate characterof the generalizations of the rectifiability concept from curves to surfaces.

Through the rest of the paper we suppose Ω+ be a bounded Jordan domainin R

n+1 with a boundary Γ := ∂Ω+, by Ω− we denote the complement domain ofΩ+ := Ω+ ∪ Γ.

Let X be the characteristic function of the closed domain Ω+. For u ∈ C(Γ),put uw(x) := X (x)u(x), where as before u is the Whitney extension operator of u.

If u ∈ C0,ν(Γ) then uw ∈ C0,ν(Ω+) and the function uw is differentiablein Ω+. Also, the degree of summability of Duw depends in the following manneron ν an Γ:

|Duw(x)| ≤ c(dist(x, Γ)

)ν−1. (4)

Note that, if ν = 1 then Duw(x) is bounded in Ω+ for any M(Γ).We shall need a preliminary fact.

Lemma 3.1. Suppose m = M(Γ) < n + 1 and let u ∈ C0,ν(Γ) (0 < ν < 1). ThenDuw ∈ Lp(Ω+) for p < (n + 1 − m)/(1 − ν).

Proof. Before starting the proof, let us recall the notion of Whitney partition.Introduce the layers

Ωk :={x ∈ R

n+1 : 2√

n + 1 2−k ≤ dist(x, Γ) ≤ 4√

n + 1 2−k}

,

and consider the collection of cubes Vk of the grid Mk intersecting the layer Ωk.After removing from the set V ′ :=

⋃k≥0 Vk those cubes which are contained in

larger cubes of V ′ the Whitney partition V is obtained.Denote by vk the number of cubes of the grid Mk appearing in V . Then

vk ≤ Nk(Ωk). Let Q be a cube of Mk intersecting Ωk. Then there exits a cubeQ′ of the same grid intersecting Γ such that Q lies inside the ball with radius

310 R. Abreu Blaya et al. Comp.an.op.th.

11/2√

n + 1 2−k, the center of which coincides with the center of Q′. Hence vk ≤mk(Ωk) ≤ (2(11/4)n + 5 + 3)n+1mk(Γ).

Let m′ ∈ (m, n + 1). Then Nk(Γ) ≤ c 2k m′for some positive constant C. To

prove the lemma it is sufficient to establish the convergence of the series∑Q∈V

∫Q

|Duw(x)|pdLn+1(x) .

From (4) and taking into account that for all Q ∈ V

dist(x, Γ) ≤ 5 diam Q , x ∈ Q ,

we get∫Q

|Duw(x)|pdLn+1(x) ≤ c 2pk(1−ν)

∫Q

dLn+1(x) = c 2k(p(1−ν)−(n+1)) , Q ∈ Vk .

Hence∑Q∈V

∫Q

|Duw(x)|pdLn+1(x) ≤ c

∞∑k=0

vk2k(p(1−ν)−(n+1))

≤ c

∞∑k=0

2k(p(1−ν)−(n+1)+m′) .

For p < (n + 1 − m′)/(1 − ν) this series converges. In view of the arbitrary choiceof m′ the lemma is proved. �Theorem 3.1. Let Γ be an n-set with M(Γ) = n. Then, under the condition

n

n + 1< ν ≤ 1 (5)

the Clifford–Cauchy transform CΓu has continuous limit values for anyu ∈ C0,ν(Γ).

Proof. By applying the Borel–Pompeiu formula to uw we have:

CΓu(x) = uw(x) − TΩ+Duw(x) for x ∈ Ω+ ,

andCΓu(x) = −TΩ+Duw(x) for x ∈ Ω− .

Hence, it is suffices to prove that under condition (5) the function TΩ+Duw(x) iscontinuous through Γ.

In fact, under condition (5) we have 11−ν > n + 1. Then in accordance with

Lemma 3.1 there exists a p > n + 1 such that Duw ∈ Lp(Ω+). Consequently,TΩ+Duw(x) ∈ C0,1−n+1/p(Rn+1) and it completes the proof. �Remark. In [4] the authors have shown that for any pair of numbers ν and m ∈[n, n + 1) connected by the inequality 0 < ν ≤ m

n+1 < 1 (we use notation of thepresent paper) there is constructed a surface Γ∗ of Minkowski dimension M(Γ∗) =m, Hausdorff dimension H(Γ∗) = n as well as a function u∗ ∈ C0,ν(Γ∗) such thatthe representation of the Cliffordian Cauchy transform CΓ∗u∗ (in accordance with

Vol. 1 (2007) Minkowski Dimension and Cauchy Transform 311

the above used Borel–Pompeiu formula) which can serve now as definition for it,has not limit boundary values everywhere on Γ∗.

The precise statement concerning our main assertion reads as follows:

Theorem 3.2. Under condition (5) the Clifford–Cauchy transform of a functionu ∈ C0,ν(Γ) has continuous limit values for either any n-rectifiable or any leftADn-regular surface Γ.

This statement is an an immediate consequence of Theorem 3.1 and Lem-ma 2.1.

3.1. The case ν = 1N. A. Davydov was among the fist mathematicians who investigated the questionof the continuity of the Cauchy type integral along a non-smooth curve. In 1949,he proved that over an arbitrary Jordan rectifiable curve, in general non-smooththe Cauchy type integral can be continuously extended to this curve from bothsides if its density belongs to the Lipschitz class, i.e., Holder class for ν = 1.

Subsequent generalizations of the Davydov result has been the subject ofresearch in a number of papers, see [24, 32], both in complex and hypercomplexcontext. Recalling the above said about the Clifford–Cauchy transform, it becomesalmost compulsory to try to prove an exact analogue of Davydov result in theClifford analysis setting, and this was the main goal of the paper [5]. On theother hand, note that in fact, certain higher dimensional version of Davydov result(to our knowledge new) for Clifford–Cauchy transform fall into the cases whichTheorem 3.2 leads easily. We conclude that:

Theorem 3.3. The Clifford–Cauchy transform CΓu over a compact topological sur-face Γ, satisfying the condition given in the Theorem 3.2, is continuous in theclosures of the domains Ω± into which the surface divides the space R

n+1 if ubelongs to the Lipschitz class.

4. Concluding remarks and observations

First, it is easily to see that our results are significant extension of the Iftimie’sresult [27] to more general classes of domains considered here. Moreover, the resultsprovide to reaffirm that the nature of the assertions in higher dimensions ensuringthe continuity of the Cliffordian Cauchy transform on Holder classes turn in anatural way to be of the same kind as those attained in the complex analysis.

We conclude this paper by remarking that there are many aspects of the sub-ject under consideration, open questions and conjectures which are still unansweredfor the problem we are dealing with. In particular, analogous generalizations of theremaining results of [18, 19] as well.

Seemingly, we supposes that one can deduce the most general attempts andapproaches made by Dyn’kin in [18,19] under the assumption of a density satisfyinga generalized Holder classes, i.e., continuous function for which properties of the

312 R. Abreu Blaya et al. Comp.an.op.th.

corresponding modulus of continuity are regulated by a majorant (i.e., a defined,finite, positive, non-decreasing function ω in (0, +∞) with limδ→0+ ω(δ) = 0). Forexample, ω(δ) = δν , 0 < ν ≤ 1 is a majorant and we have the usual Holdercondition.

A conjecture reads as follows:

For a suitable choice of a majorant, necessary and sufficient condition (insome global sense) for the Clifford–Cauchy transform to be continuously extendableto the clousure of the domain, replacing the Holder condition by that of generalizedcharacter, is possible to verify. Particularly, the authors supposes that the conditionof its continuity (ν > n

n+1 ) can be expressed in term of the majorant ω as

1∫

0

ωn+1

n (τ)dτ

τ2< +∞ . (6)

More precisely, the above guesswork will be fully demonstrated and illustratedin the forthcoming paper [13].

Notice that (6) in the case n = 1 is exactly the Dyn’kin condition consideredfor the main result obtained in [18, 19].

References

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Ricardo Abreu Blaya and Tania Moreno GarcıaDepartamento de MatematicaFacultad de Informatica y MatematicaUniversidad de HolguınHolguın 80100Cuba

e-mail: rabreu@facinf.uho.edu.cu

tania.moreno@facinf.uho.edu.cu

Vol. 1 (2007) Minkowski Dimension and Cauchy Transform 315

Juan Bory ReyesDepartamento de MatematicaFacultad de Matematica y ComputacionUniversidad de OrienteSantiago de Cuba 90500Cubae-mail: jbory@rect.uo.edu.cu

Communicated by Daniel Alpay.

Received: August 15, 2006.

Accepted: November 2, 2006.