A higher Dimensional Marcinkiewicz Exponent and the Riemann Boundary Value Problems for Polymonogenic Functions on Fractals Domains

In this section, we recall some basic facts about Clifford analysis which will be needed in the sequel. It could be seen in more detail in the literature, for instance [9, 14].

It thus follows that, the standard orthonormal basis of $\mathbb{R}^{n}$, denoted by $\{e_{j}\}_{j=1}^{n}$, is subject to the basic multiplication rules

were $\delta_{ij}$ is the Kronecker delta.

For arbitrary $a\in\mathcal{C}\ell(n)$ we have $a=\sum\limits_{A\subseteq N}a_{A}e_{A}$, $N=\{1,\ldots,n\}$, $a_{A}\in\mathbb{R}$ where $e_{\emptyset}=e_{0}=1$, $e_{\{j\}}=e_{j}$ and $e_{A}=e_{\beta_{1}}\cdots e_{\beta_{k}}$ for $A=\{\beta_{1},\ldots,\beta_{k}\}$ with $\beta_{j}\in\{1,\ldots,n\}$ and $\beta_{1}<\ldots<\beta_{k}$.

An important subspace of $\mathcal{C}\ell(n)$ is the so-called space of paravectors and its elements have the form $x=\sum_{j=0}^{n}x^{j}e_{j},$. Each $x=(x^{0},x^{1},\ldots,x^{n})\in\mathbb{R}^{n+1}$ will be identified with a paravector.

The algebra norm of arbitrary $a$ is defined by $\arrowvert a\arrowvert:=\left(\sum_{A}a_{A}^{2}\right)^{\frac{1}{2}}$ and $\mathcal{C}\ell(n)$ becomes an Euclidean space.

The conjugation in $\mathcal{C}\ell(n)$ is defined as the anti-involution $a\mapsto\overline{a}:=\sum_{A}a_{A}\overline{e}_{A}$ such that

For every paravector $x$ the relation

holds.

An $\mathcal{C}\ell(n)-$valued function $u$ defined over $\Omega\subset\mathbb{R}^{n+1}$ has the representation

where $u_{A}$ are $\mathbb{R}$-valued components.

Unless otherwise states we assume that $u$ to be $\mathcal{C}\ell(n)-$valued. Properties such as continuity or differentiability have to be understood component-wise.

Clifford analysis is mainly centered around the concept of monogenic functions which are null solutions of the generalized Cauchy-Riemann operator in $\mathbb{R}^{n+1}$ defined by

Let $\Omega\subset\mathbb{R}^{n+1}$ be an open set and $u\in C^{1}(\Omega)$ then $u$ will be called left (respectively right) monogenic in $\Omega$ if $\mathcal{D}u=0$ (respectively $u\mathcal{D}=0$) in $\Omega$.

The fundamental solution for the Cauchy-Riemann operator is

where $\sigma_{n+1}$ stands for the hypersurface area of the unit sphere in $\mathbb{R}^{n+1}$.

Polymonogenic functions $f\in C^{k}(\Omega)$ of order $k$, emerge as the solutions of the iterated action of the operator $\mathcal{D}$

in $\Omega\subset\mathbb{R}^{n+1}$. This notion goes back as far as [6, 8, 11].

For arbitrary $k\in N$, let us introduce

Note that for $k=1$

as it is easy to check.

A direct computation shows that

Consequently, by decreasing induction on $k$,

Let us now introduce the Teodorescu transform related to the theory of monogenic functions, see [14] for more details.

Sufficient conditions for the Hölder continuity of $T_{\Omega}u$ are established by our next theorem.

Conditions for the derivability of $T_{\Omega}u$ on $\Omega$ are given as follows

Now, we introduce a kind of polymonogenic Teodorescu transform following [6]. Due to its good properties, this integral operator will play an essential role in the strategies described below. For a recent account of the subject we refer the reader to [22].

By derivability properties of $T^{k}_{\Omega}u(x)$, the following equality is achieved

Therefore, decreasing induction on $k$ combining with Theorems 1 and 2 gives

In order to state the main problems that we address, it is necessary to define the suitable classes where the data functions will be considered. These are the higher order Lipschitz classes, on the basis of which some Whitney-type extension theorems will be presented.

As a first step, we shall recall the class of $p$-integrables functions, see [14].

Let ${\bf E}$ be a closed subset of $\mathbb{R}^{n+1}$. We write $j=(j_{0},j_{1},\cdots,j_{n})$ a n-dimensional multi-index of order $|j|=j_{0}+j_{1}+\cdots+j_{n}$, where $j_{0},j_{1},\dots,j_{n}$ are non-negative integers. In addition, we have, $j!=j_{0}!j_{1}!\cdots j_{n}!$ and $x^{j}=x_{j_{0}}x_{1}^{j_{1}}\cdots x_{n}^{j_{n}}$.

The space of all these functions is named higher order Lipschitz spaces, see for instance [29]. When $k=1$ and ${\bf E}$ is a compact set, these functions reduce to the classical Hölder continuous functions.

A norm in ${\mbox{Lip}}({\bf E},k+\nu)$ is defined as the smallest $M$ satisfying (3). In [31] was proved that ${\mbox{Lip}}({\bf E},k+\nu)$ endowed with this norm is a Banach space. Also, conditions for continuous and compact embeddings of generalized higher-order Lipschitz classes on a compact subset of Euclidean spaces were obtained, showing that these spaces are not only a generalization but also a refinement of the classical Lipschitz classes.

In general, an element of ${\mbox{Lip}}({\bf E},k+\nu)$ should be interpreted as a collection

In order to present a kind of a Whitney type extension theorem for polymonogenic functions, let us recall the following classical theorem by Whitney, see [29, pag. 177].

Here and subsequently the symbol

stands for the higher-order partial derivatives.

Let $f\in{\mbox{Lip}}(\mathcal{S},k-1+\nu)$ be a $\mathcal{C}\ell(n)$-valued function, interpreted as the collection

with $f^{(0)}=f$ satisfying (2) and (3).

In order to present a suitable version of Whitney extension theorem for $\mathcal{C}\ell(n)$-valued function, in [1] are constructed the following functions,

Here $\textbf{1}_{r_{i}}$ is the multi-index $(j_{0},j_{1},\cdots,j_{n})$ with

We should note that the functions $\mathbf{f}^{(i)}$ are an appropriate arranged of every function $f^{(j)}$ with $\left|j\right|=i$. In addition, $\mathbf{f}^{(0)}$ = $f^{(0)}$ = $f$.

Let us mention an important consequence of Theorem 3. This can be found in [1].

To make the presentation self-contained, we review some basic ideas about fractal dimensions of sets that will be required to work with domains with fractal boundaries. For a deeper discussion of this topic, we refer the reader to [12, 24, 25].

Let $\textbf{E}\in\mathbb{R}^{n+1}$ a non-empty set. For any $s\geq 0$ and $\delta>0$, $\mathcal{H}_{\delta}^{s}(\textbf{E})$ is defined as,

where $\mathrm{diam}(U)$ is the diameter of the set $U$. The infimum here is taken over all countable $\delta$-coverings ${U_{i}}$ of E for open or closed balls. Now, we can present the Hausdorff measure.

It can be shown that the $s$-dimensional Hausdorff measure of a set E is almost always 0 or $\infty$. There is only one value of $s$ where the measure change between these two values. Therefore, it looks natural to define the Hausdorff dimension as this value.

The following theorem can be found in [5].

The following definition of fractal set is due to Mandelbrot [24]

The results presented in this paper are intended to deal with sets such that $\mathcal{H}^{n}(\textbf{E})=\infty$ as well as fractals from Definition 8. Throughout this paper, the expression ‘fractal domain’ will always refer to a domain with a fractal boundary.

The Minkowski dimension is widely used when working with fractals. That is due to the fact that computations are easier than with other fractal dimensions. We will only consider the upper Minkowski dimension.

We denote $\mathcal{M}_{0}$ the grid covering $\mathbb{R}^{n+1}$ which consists of $(n+1)$-dimensional cubes with vertices with integer coordinates and edges of length one. From $\mathcal{M}_{0}$ is generated the grid $\mathcal{M}_{k}$ by dividing every cube in $\mathcal{M}_{0}$ into $2^{(n+1)k}$ different cubes with edges lengths $2^{-k}$. Let $N_{k}(\textbf{E})$ be the amount of cubes of the grid $\mathcal{M}_{k}$ which intersect E. Then as can be found in [12] we have

In [25, pp 77] is given the next theorem relating the Hausdorff and Minkowski dimensions.

The following new metric characteristics of a fractal set in $\mathbb{R}^{n+1}$ are mainly included to keep the exposition as self-contained as possible. These can be found in [30].

Let $\mathcal{S}$ be a topologically compact hypersurface in $\mathbb{R}^{n+1}$, which bounds a Jordan domain $\Omega^{+}$. We write $\Omega^{-}$ for the unbounded complement. It is assumed $\mathcal{S}$ to be fractal.

Let $D\subset\mathbb{R}^{n+1}$ be a bounded set, which does not touch the hypersurface $\mathcal{S}$. We will consider the integral

For completeness, we recall:

Here, the domain $\Omega^{*}:=\Omega^{-}\bigcap\{x:\arrowvert x\arrowvert<r\}$, where $r$ is chosen in a way that $\mathcal{S}$ is completely contained inside the ball of radius $r$. We should note that the value of $\mathfrak{m}^{-}(\mathcal{S})$ does not depend on the selection of the radius $r$ when constructing $\Omega^{*}$, due to the fact that only the points closest to $\mathcal{S}$ influence the convergence of the integral $I_{p}(D)$.

The following theorem expresses the relationship between the Minkowski dimension with the Marcinkiewicz exponent, it was proved in [19] and in a different way in [30].