Hypercomplex Analysis Selected Topics

Remark 6. In the case when m > 1, a definition of fine partial derivatives is
not straightforward because
V := B(˜z, r) \
m
{˜z + tel; t ∈ R, t = 0}
l=0
is a fine neighbourhood of a point ˜z ∈ R m+1 for any r > 0. Here B(˜z, r) is
the ball in R m+1 with center ˜z and radius r.
Now we introduce some function spaces. Let us denote by fine-C 1 (U)
the set of all functions f finely differentiable everywhere on U whose fine
differential dff is finely continuous on U. As usual we can define inductively
the spaces fine-C k (U) for all k ∈ N0. In particular, the space fine-C 0 (U) =
fine-C(U) is the set of finely continuous functions on U and the space fine-C 2 (U)
consists of functions f ∈ fine-C 1 (U) whose first fine derivatives belong to
fine-C 1 (U) as well. Finally, put
fine-C ∞ (U) =
∞
fine-C k (U).
k=0
See [62] for details. Moreover, for k ∈ N0 ∪ {∞}, we denote by Ck f-loc (U) the
set of all functions f on U such that, for each z ∈ U, there is V ∈ Fz and
F ∈ Ck (Rm+1 ) with F = f on V. It is easy to see that Ck f-loc (U) ⊂ fine-Ck (U).
A question whether these spaces coincide or not is discussed later on, see
Section 4.3.
Finally, let us recall that the Sobolev space W 1,2 (Rm+1 ) consists of (Lebesgue)
measurable functions F whose second power is integrable on Rm+1 together
(U) the set
with second powers of its first weak derivatives. Denote by W 1,2
f-loc
of functions f on U satisfying that, for each z ∈ U, there exist V ∈ Fz and
F ∈ W 1,2 (Rm+1 ) such that F = f on V. For an account of the Sobolev spaces
on fine domains, we refer to [73].
Finely holomorphic functions are closely related with finely harmonic ones
(see [54]). For our purposes, let us recall one of their characterizations.
A real-valued function f is finely harmonic on a finely open set U ⊂ Rm+1 if
and only if for every z ∈ U there is V ∈ Fz such that f|V , the restriction of f
to V , is a uniform limit of functions fn harmonic on open sets Vn containing
V . Let us remark that f is harmonic on a usual open set Ω ⊂ Rm+1 if and
only if f is finely harmonic and locally bounded (from above or below) on Ω.
In case of R2 we need not assume local boundedness of f. Moreover, finely
harmonic functions are finely continuous but, in general, have the first fine
differential only almost everywhere (a.e.), see e.g. [58]. Next finely harmonic
functions need not possess the unique continuation property. Indeed, by [81],
there is a non-trivial finely harmonic function f in a fine domain U which
vanish in some fine neighbourhood of a point of U.
Now we are ready to state some basic facts about finely holomorphic
functions, see e.g. [56], [57] and [61]. Let U ⊂ C be finely open and let f :
U → C. Then there are several equivalent definitions of finely holomorphic
functions available. Indeed, a function f is finely holomorphic if one of the
following (equivalent) conditions holds:
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