Marcinkiewicz exponent and boundary value problems in fractal domains of R n + 1

This paper aims to study the jump problem for monogenic functions in fractal hyper-surfaces of Euclidean spaces. The notion of the Marcinkiewicz exponent has been taken into consideration. A new solvability condition is obtained, basing the work on speciﬁc properties of the Teodorescu transform in Clifford analysis. It is shown that this condition improves those involving the Minkowski dimension


Introduction
The Riemann boundary value problem in Complex Analysis is widely used in many branches of Mathematics and Physics.The classical references here are [10,16,20].In the solution to the Riemann boundary value problem, the Cauchy type integral is used as the main mathematical apparatus.It is well known that for every Hölder continuous function u with exponent ν > 1 2 , (C γ u)(z) has continuous limit values on a rectifiable closed Jordan curve γ; hence the jump problem is solvable.For a thorough description of old and recent results concerning the geometric conditions on a Jordan curve in the plane that imply the boundedness of the Cauchy type integral boundary behavior, the reader is referred to [2].However, in the context of non-rectifiable curves, the Cauchy type integral (1) has no sense.In contrast, the Riemann boundary value problem is still completely valid.In the early eighties, a complete treatment of this topic was given by B. A. Kats in [13].There is shown that under the condition ν > dim M (γ)

Clifford Algebras and Monogenic Functions
This subsection has compiled some basic facts concerning Clifford Algebras and Clifford Analysis.For a discussion of this topic, as was mentioned before, the reader is referred to [8,11,19].Definition 1.The Clifford algebra associated with R n , endowed with the usual Euclidean metric, is the extension of R n to a unitary, associative algebra C (n) over the reals, which is generated, as an algebra, by R n .It is not generated by any proper subspace of R n and satisfies Definition 1 is equivalent to the next construction.Let {e j } n j=1 be the standard basis of R n .Multiplication is defined through the basic rule e i e j + e j e i = −2δ ij , where δ ij is the Kronecker delta.
Every a ∈ C (n) is of the form a = An algebra norm is defined on C (n) through |b| = ( A b 2 A ) 1 2 .With this, C (n) becomes a Euclidean space.Every point in R n+1 can be identified with the Clifford number x = n i=0 x i e i , named paravector.
Functions to be studied will be the ones defined in a domain G ⊂ R n+1 valued in C (n).They have the form where u A (x) are real-valued.From now on, unless the opposite is specified, all functions will be considered Clifford-valued.We say that u In general, we say that a Clifford-valued function belongs to a class if all its components belong to that class.The cliffordian Cauchy-Riemann operator in R n+1 is defined as where ∂ i := ∂ ∂x i is the partial derivative with respect to x i .If needed, we shall specify the variable we are applying the operator to, i.e., D x and ∂ i,x .The fundamental solution of this first-order elliptic operator is where σ n is the area of the unit sphere in R n+1 .Let G be a domain in R n+1 and u ∈ C 1 (G) we say that u is a left-monogenic function (respectively right-monogenic) if Du = 0 (respectively uD = 0) in G.

Hausdorff and Minkowski Dimensions
In order to deal with domains with fractal boundaries, we should refresh some basic notions about fractal dimensions.The books [9,17,18] are recommended as references on this topic.We shall present the notions of Minkowski and Hausdorff dimensions, which are essential tools in this theory.Here we restrict ourselves to the definition of the upper Minkowski dimension.Definition 2. (Upper Minkowski dimension) Let E be a non-empty bounded subset of R n+1 and let N δ (E) be the smallest number of sets of diameter at most δ, covering E. The upper Minkowski dimension of E is defined as The upper Minkowski dimension can also easily be seen to be determined with cubes in a grid, see [9].Suppose M 0 denotes a grid covering R n+1 consisting of (n + 1)-dimensional cubes with sides of length one and vertices with integer coordinates.The grid M k is obtained from M 0 by dividing each of the cubes in M 0 into 2 (n+1)k different cubes with side lengths 2 −k .Denote by N k (E) the number of cubes of the grid M k which intersect E. Then Now the concept of the Hausdorff dimension will be introduced.To do that, we need some previous definitions.Definition 3. Let E be an arbitrary non-empty set in R n+1 .For any δ > 0 and s ≥ 0, H s δ (E) is defined as, where the infimum is taken over all countable δ-coverings U i of E with open or closed balls.We write The Hausdorff dimension of E is defined as When s = n + 1 there is a relation between the (n + 1)-dimensional Lebesgue and Hausdorff measure as we can see in the next theorem.See for instance [9, pp 28].
where ρ n+1 is the volume of a (n + 1)-dimensional ball of diameter one.
In [18, pp 77] is given the next theorem relating the Hausdorff and Minkowski dimensions.
Theorem 2. For the bounded set E ⊂ R n+1 with topological dimension n, we have Definition 4. If an arbitrary set E ⊂ R n+1 with topological dimension n has dim H E > n, then E is called a fractal set in the sense of Mandelbrot.
From Definitions 3 and 4, we know that a fractal set in the sense of Mandelbrot E satisfies that H n (E) = ∞.Besides, we should note that a bounded set E with dim H E = n can have H n (E) = ∞.However, classical methods cannot be applied to this kind of set.The ideas developed in this paper are intended to deal with these sets and fractals from Definition 4.

Teodorescu Transform and Whitney Extension
In starting this subsection, we take up some basic properties of the Teodorescu Transform, which will play an essential role in the method developed below, see [11] for more details.
Definition 5. Let G ⊂ R n+1 be a domain and let u ∈ C 1 (G), the operator defined by where dV (y) is the volume element, is called the Teodorescu transform.
The next theorem gives us sufficient conditions for the Hölder continuity of the Teodorescu Transform.
(ii) For x, y ∈ R n+1 , and x = y, we have the inequality .
The following theorem provides conditions for the derivability of the operator T G u over the domain G.
Theorem 4. Let G be a domain and let u be a continuously differentiable function in G. Then T G u is also a differentiable function for every x ∈ G with Particularly, we have the identity In [4] can be found the Whitney extension theorem for Clifford valued functions.It is based on the result [22, pp 174] for real-valued functions, which was stated originally by H. Whitney.This result has enormous importance in this research.
Theorem 5. (Whitney extension theorem) Let E ⊂ R n+1 be a compact set and let u ∈ C 0,ν (E), with 0 < ν ≤ 1.Then there exists a function u ∈ C 0,ν (R n+1 ), named Whitney extension operator of u, that satisfies (i) The following theorem is a corollary of a more general result called the Dolzhenko theorem.For the proof, we refer the reader to [6].Theorem 6.Let G be a domain in R n+1 and E ⊂ G be a compact set.Let be H n+µ (E) = 0 where 0 < µ ≤ 1.If u ∈ C 0,µ (G), and it is monogenic in G \ E, then u is also monogenic in G.

Marcinkiewicz Exponent
From now on, let S be a topologically compact surface, which is the boundary of a Jordan domain in R n+1 that divides it into two domains, the bounded component G + and the unbounded component G − respectively.Let D ⊂ R n+1 be a bounded set, which does not intersect the surface S, fractal in general.We define the integral .
When p = 0, this integral is the volume of D. However, when p is large enough, the integral could diverge.
We define the domain G * := G − {x : |x| < r}, where r is selected such that S is wholly contained inside the ball of radius r.The inner and outer Marcinkiewicz exponent are defined as follows.The following lemma plays a significant role in proving the relationship between the Minkowski dimension and the Marcinkiewicz exponent.Here we shall use the Whitney extension decomposition, see [22].Lemma 1.Let S be a topologically compact surface which is the boundary of a Jordan domain in R n+1 .Let w k be the number of cubes with edges equal to 2 −k in the Whitney extension decomposition of R n+1 \ S, then Proof.Denote by w k the number of cubes in the grid M k , appearing in the Whitney extension decomposition F (see [22]).We need to remember that where Ω k is defined as follows ) be the number of cubes of the grid M k , which intersect Ω k , and thus w k ≤ m k (Ω k ).Suppose that x ∈ Ω k , then we can find a point x ∈ S separated from x by a distance not greater than C 0 2 −k+1 .If Q is a cube of M k , containing x, and Q is a cube of the same grid containing x .Then Q intersects a sphere of radius C 0 2 −k+1 with the center in Q .Cubes of M k intersecting with such spheres lie inside the cube Q with edges large enough.Indeed, if we take the cube where y is the center of Q , we obtain a cube Q thicker than Q by C 0 2 −k+1 .Hence Q contains all the spheres centered in a point x in Q and the radius equal to 4( √ n + 1)2 −k .Let us notice that when x is in the boundary of Q , the ball with the center in x , and the radius equal to C 0 2 −k+1 , touches the boundary of Q .Therefore, we need to make Q a bit thicker than Q in order to get all the balls completely contained in Q .See Figure 1.
It is convenient to have a value in the form m2 −k , where m is an integer number, in order From (2) we get that there exists a N 0 such that for all k > N 0 we have where d ∈ (dim M (S), n + 1] is fixed.Consequently, In [15, Lemma 1], it is shown, using other tools, a more general result which particularly implies the next theorem when we restrict ourselves to Lebesgue measure over R n+1 .Here it is shown in a direct way using Lemma 1.
Theorem 7. Let S be a topologically compact surface which is the boundary of a Jordan domain in R n+1 , then m(S) ≥ n + 1 − dim M (S).
Proof.Let us consider the Whitney extension decomposition (3) again.We know from [22] that the cubes Q satisfy the inequality These cubes have edges with lengths equal to 2 −k where k ∈ Z in general.For a fixed cube Q with edge 2 −k in this decomposition, we infer, from (4) and since diam( We define the values w k as follows where w k is the number of cubes with edges equal to 2 −k in the Whitney extension decomposition.
Then we have .
However, there is only a finite amount of cubes with edges of length 2 −k such that k ≤ 0. Indeed, if k ≤ 0, then 2 −k ≥ 1, and if there are infinitely many cubes with an edge more than or equal to 1, then the (n+1)-dimensional Lebesgue measure of G + would be infinite.
In contradiction with the fact that G + is a bounded set in R n+1 .Therefore, Let d ∈ (dim M (S), n + 1], and then from Lemma 1, we have that for all k ≥ m d , with m d large enough, and the constant B only depends on n + 1. Hence we have Therefore, if the series on the right hand converges, the series on the left side converges.That occurs when is fulfilled the condition Consequently, (n + 1) − dim M (S) ≤ m + (S).
An analogous analysis can be done with G * and m − (S).

A Class of Surfaces in Three Dimensions
In this section, we construct a class of surfaces in R 3 .For every possible value of the Minkowski dimension in (2, 3), it is shown that there is a non-numerable class of surfaces with that dimension and such that inequality in Theorem 7 is strict.This construction is similar in spirit to a two-dimensional curve developed in [14].That idea on the complex plane goes back at least as far as [13].The construction follows the simple idea of adding infinitely many three-dimensional rectangles with suitable dimensions to a three-dimensional cube.This begins with a cube Let us fix α ≥ 1 and β ≥ 2. First, we look at the segment [0, 1] in the x 1 axis.We divide it into infinitely many segments of the form [2 −n , 2 −n+1 ] for each n ∈ N.Then, for each n ∈ N, we divide the segments [2 −n , 2 −n+1 ] into 2 [nβ] equally spaced segments where [nβ] is the integer part of nβ.We denote by x nj , where j = 1, 2, ..., 2 [nβ] , the points determined at the right side of these segments.See Figure 2. Let a n be the distance between x nj and x n(j+1) , i.e. a n = 2 −n− [nβ] and C n = 1 2 a α n .Then let R nj be the following three-dimensional rectangles: We define the set We take the surface S α,β = ∂T α,β .See Figure 3, which was generated using MATLAB, as an illustration.We should note here that the parameter β only affects the number of rectangles R nj for each n ∈ N, while α only affects the width of the rectangles R nj .

Minkowski Dimension of the Surfaces S α,β
Now, let us compute the Minkowski dimension of S α,β .In order to do that, we shall use the grid M k defined in 2.2.Many straightforward steps are omitted in order to reduce the exposition.We need to find a lower and an upper bound such that they are equal.To calculate the lower bound, we shall construct a set A β such that A β ⊂ S α,β and therefore dim M (A β ) ≤ dim M (S α,β ).Let P nj be the two-dimensional rectangles defined as:  From construction, we know that A β ⊂ S α,β .We are going to find a lower bound for dim M (A β ).To do that, let us focus on the distance between P nj and P nj+1 .It is equal to a n = 2 −n− [nβ] .If k > n, and a n > 2 −k , a cube cannot touch two of these rectangles.The quantity of cubes in M k that cover a single two-dimensional rectangle P nj is 2( 2 −n 2 −k ) 2 because these two-dimensional rectangles have lengths of 2 −n+1 and widths of 2 −n .There are 2 [nβ] rectangles P nj for a fixed n.Therefore 2 [nβ]+1 ( 2 −n 2 −k ) 2 cubes are needed to cover all the P nj for a fixed n.Then we have where N k (A β ) is the minimal number of cubes of the grid M k which cover A β .Denote by B k the integer defined by the condition It is not difficult to show that the condition a n > 2 −k is fulfilled if and only if n = 1, 2, ..., B k .Next, we get where C does not depend on k.Therefore We need to find an accurate upper bound for dim M (S α,β ).In order to do that, we define the sets Λ n := we can observe that S α,β = Q ∪ Λ.We shall focus first on Λ.
We are going to consider three cases.The first one is when n ≥ k, the second one will be if n < k and C n > 2 −k , and the last one is if n < k and C n ≤ 2 −k .From construction, the surfaces Λ n , with n > k, are covered by one cube of the grid M k .While the surface Λ k is covered by two of these cubes.As above, if n < k and cubes are needed to cover the two-dimensional rectangles in Λ n parallel to the coordinate plane x 2 = 0.In addition, 2( 2 −n 2 −k ) 2 cubes are enough to cover the two-dimensional rectangles in Λ n parallels to cubes are needed to cover the two-dimensional rectangles in Λ n parallel to the coordinate plane x 2 = 0. Besides, ( 2 −n 2 −k ) 2 cubes are enough to cover the two-dimensional rectangles in Λ n parallels to x 3 = 0. Finally, with 6( 1 2 −k ) 2 cubes of M k , we can cover Q.As a consequence, we get ) is a positive function, the integral in the left hand converges if and only if the seven integrals in the sum in the right hand converge.It is possible to show through direct computations that if and only if p < 1. Hence we only need to analyze these values of p in the following integrals.Analogous computations can be done to obtain that the integrals over the regions G + i , where i = 2, ..., 6 converge when p < 1.On the other hand, for the integral over the region G + 7 , we have that .
In order to compute the integral over R nj , we divide that region in the same way that in the cube Q.By drawing the planes that bisect the dihedral angles at the edges, we get regions where the function dist p (x, S α,β ) can be represented through elementary functions.
After doing the tedious calculations, we can reduce the convergence of the integral over G + 7 when p < 1 to the convergence of the series This geometric series converges if and only if the condition , is fulfilled.Thus, we have that the inner Marcinkiewicz exponent is .
Also, we obtain that the absolute Marcinkiewicz exponent satisfy .

Remarks about the Surfaces S α,β
Now we are able to prove Theorem 8.
On the other hand, as a trivial conclusion we see that when α = 1 or β = 2 we have that m + (S α,β ) = 3 − dim M (S α,β ).We can also note that when β = 2 then 2 ≤ dim H (S α,2 ) ≤ dim M (S α,2 ) = 2 and consequently dim H (S α,2 ) = 2. On the other hand, the 2-Hausdorff measure is H 2 (S α,2 ) = ∞, because H 2 (S α,2 ) ≥ H 2 (A 2 ) and from Theorem 1 we have that H 2 (A 2 ) = ∞ .Therefore, S α,2 is not a fractal in the sense of Mandelbrot.However, classical methods cannot be applied to it, even those developed for non-smooth surfaces.Even though it is impossible to draw a hypersurface like this example in dimensions higher than three, we are able to describe it analytically.Indeed, let Q 0] be a (n+1)-dimensional cube.Additionally, let R mj be the (n+1)-dimensional rectangles given by a product of (n + 1) segments.Then we analogously define where the hypersurface S n+1 α,β = ∂T n+1 α,β .We should note that for n + 1 = 3 this surface is pretty similar to the one in the Figure 3.

Jump Problem in Fractal Domains
Throughout this section, the following temporary notation will be used.Let S be a topologically compact surface that is the boundary of a Jordan domain G + ⊂ R n+1 , and let be G − := (R n+1 {∞}) \ (G + S).The Jump Problem in Clifford Analysis is stated as follows: Given a continuous function f defined on S, to find two functions Φ + (x), monogenic in the domain G + , and Φ − (x) monogenic in the domain G − , such that they have continuous limit values in the boundary S and there satisfy the relation with Φ(∞) = 0.If S is a fractal surface, it is impossible to use the cliffordian Cauchy type integral to solve the problem (6).In the context of Clifford Analysis, we have the following result, which generalizes [14, Theorem 2] to higher dimensions.
Theorem 9. Let S be a topologically compact surface which is the boundary of a Jordan domain in R n+1 , and let f ∈ C 0,ν (S).If then the jump problem (6) is solvable.
Proof.First, we consider the inner Marcinkiewicz exponent m + (S).We look for sufficient conditions such that the Whitney extension f of f satisfies that D f ∈ L p (G + ) with p > n + 1.Indeed, from Theorem 5, we have From Definition 6, we have that the above right-hand integral converges for p < m + (S) 1−ν .Then we need that n + 1 < m + (S) 1−ν , or equivalently Note that this is a sufficient condition for D f ∈ L p (G + ) with p > n + 1. Next, let us consider the function where χ(x) is the characteristic function of G + .We shall show that, under condition (7), function ( 8) is a solution to the jump problem.Indeed, we have that Φ − (x) = −(T G + D f )(x), x ∈ G − .From Theorem 3, we get that Φ − (x) is a monogenic function over G − , vanish at infinity, and also Φ − From Theorem 7, it follows that Theorem 9 improves the existing conditions for the solvability of the jump problem.Additionally, using Theorem 6, we obtain the following unicity conditions.
Then the solution to the jump problem ( 6) is unique in the classes C 0,µ (G + ) and C 0,µ (G − ).
The unicity in Theorem 10 is assumed when there exists a value of µ such that condition (9) is fulfilled.
Further results 1.The results obtained here are also valid in the context of vectorial Clifford analysis.There we need to use the properties of the Teodorescu transform written in the vectorial sense that can be found in [6,12] and analogous reasoning.We should note that in the theorems with this approach, we need to change the dimension of the space from (n + 1) to n.

Definition 6 .
Let S be a topologically compact surface which is the boundary of a Jordan domain in R n+1 .We define the inner and outer Marcinkiewicz exponent of S, respectively, as m + (S) = sup{p : I p (G + ) < ∞}, m − (S) = sup{p : I p (G * ) < ∞}, and the (absolute) Marcinkiewicz exponent of S as, m(S) = max{m + (S), m − (S)}.
for each k ≥ m d for m d large enough, where d ∈ (dim M (S), n + 1], and C is a constant that only depends on n + 1.

Figure 1 :
Figure 1: Two dimensional representation of the cubes Q , Q , and Q .

Theorem 8 .
Let α ≥ 1 and β ≥ 2. For each value d ∈ (2, 3), there exists a non-numerable class of topologically compact surfaces S α,β , which are the boundary of a Jordan domain in R 3 such that d = dim M (S α,β ) and m(S α,β ) > 3 − d for suitable values of α and β.

Figure 2 :
Figure 2: Distribution of some x nj in the x 1 axes for β = 2.1.