Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 109, pp. 1–10.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

HEAT AND LAPLACE TYPE EQUATIONS WITH COMPLEX

SPATIAL VARIABLES IN WEIGHTED FOCK SPACES

CIPRIAN G. GAL, SORIN G. GAL

Communicated by Jerome A. Goldstein

Abstract. In a recent book co-authored by the authors of this article, we
studied by semigroup theory methods several classical evolution equations, in-

cluding the heat and Laplace equations, with real time variable and complex
spatial variable, under the hypothesis that the boundary function belongs to

the space of analytic functions in the open unit disk and continuous in the

closed unit disk, endowed with the uniform norm. Also, in a subsequent pa-
per, the authors have extended the results for the heat and Laplace equations

in weighted Bergman spaces on the unit disk. The purpose of this article is to

show that the semigroup theory methods work for these two evolution equa-
tions of complex spatial variables, under the hypothesis that the boundary

function belongs to the weighted Fock space on C, F pα(C), with 1 ≤ p < +∞,

endowed with the Lp-norm. Also, the case of several complex variables is con-
sidered. The proofs use the Jensen’s inequality, Fubini’s theorem for integrals

and the Lp-integral modulus of continuity.

1. Introduction

Extending the method of semigroups of operators in solving the evolution equa-
tions of real spatial variable, a way of “complexifying” the spatial variable in the
classical evolution equations is to “complexify” their solution semigroups of op-
erators, as it was summarized in the book [4]. In the cases of heat and Laplace
equations, the results obtained can be summarized as follows.

Let D = {z ∈ C : |z| < 1} be the open unit disk and A(D) = {f : D → C;
f is analytic on D, continuous on D}, endowed with the uniform norm ‖f‖ =
sup{|f(z)|; z ∈ D}. It is well-known that (A(D), ‖ · ‖) is a Banach space. Let
f ∈ A(D) and consider the operator

Wt(f)(z) =
1√
2πt

∫ +∞

−∞
f(ze−iu)e−u

2/(2t)du, z ∈ D. (1.1)

In [3] (see also [4, Chapter 2], for more details) it was proved that (Wt, t ≥ 0)
is a (C0)-contraction semigroup of linear operators on A(D) and that the unique

2010 Mathematics Subject Classification. 47D03, 47D06, 47D60.
Key words and phrases. Complex spatial variable; semigroups of linear operators;

heat equation; Laplace equation; weighted Fock space.
c©2020 Texas State University.

Submitted April 1, 2020. Published October 30, 2020.

1



2 C. G. GAL, S. G. GAL EJDE-2020/109

solution u(t, z) (that belongs to A(D), for each fixed t ≥ 0) of the Cauchy problem

∂u

∂t
(t, z) =

1

2

∂2u

∂ϕ2
(t, z), (t, z) ∈ (0,+∞)×D, z = reiϕ, z 6= 0, (1.2)

u(0, z) = f(z), z ∈ D, f ∈ A(D), (1.3)

is exactly

u(t, z) = Wt(f)(z). (1.4)

In the same contribution [3], setting

Qt(f)(z) :=
t

π

∫ +∞

−∞

f(ze−iu)

u2 + t2
du, z ∈ D, (1.5)

we proved that (Qt, t ≥ 0) is a (C0)-contraction semigroup of linear operators on
A(D). Consequently, the unique solution u(t, z) (that belongs to A(D), for each
fixed t) of the Cauchy problem

∂2u

∂t2
(t, z) +

∂2u

∂ϕ2
(t, z) = 0, (t, z) ∈ D × (0,+∞), z = reiϕ, z 6= 0, (1.6)

u(0, z) = f(z), z ∈ D, f ∈ A(D), (1.7)

is exactly

u(t, z) = Qt(f)(z). (1.8)

In a recent paper [2] we proved that the well-posedness of the above problems in
the space A(D), can be replaced by well-posedness in (larger) weighted Bergman
spaces defined as follows. For 0 < p < +∞, 1 < α < +∞ and ρα(z) = (α+ 1)(1−
|z|2)α, the weighted Bergman space Bpα(D), is the space of all analytic functions in
D, such that [ ∫

D

|f(z)|pdAα(z)
]1/p

< +∞,

where dAα(z) = ρα(z)dA(z), with

dA(z) =
1

π
dx dy =

1

π
r dr dθ, z = x+ iy = reiθ,

the normalized Lebesgue area measure on the unit disk of the complex plane.
The goal of this note is to show that the well-posedness of the above problems

in the space A(D), can be replaced by well-posedness in weighted Fock spaces too,
defined as follows.

Definition 1.1 (see, e.g. [7, p. 36]). Let 0 < p < ∞ and α > 0. The Fock space
F pα(C) is defined as the space of all entire functions in C with the property that

αp

2π

∫
C

∣∣f(z)e−α|z|
2|/2∣∣pdA(z) < +∞,

where dA(z) = 1
π dx dy = 1

π r dr dθ, z = x + iy = reiθ, is the area measure in the
complex plane.

Remark 1.2. Endowed with

‖f‖pp,α =
αp

2π

∫
C

∣∣f(z)e−α|z|
2/2
∣∣pdA(z),



EJDE-2020/109 HEAT AND LAPLACE EQUATIONS IN COMPLEX VARIABLES 3

it is known (see, e.g., [7, p. 36]) that F pα is a Banach space for 1 ≤ p < ∞, and a
complete metric space for ‖ · ‖pp,α with 0 < p < 1. Also, if p = +∞, then endowed
with

‖f‖∞,α = ess sup{|f(z)|e−α|z|
2|/2 : z ∈ C},

F∞α is a Banach space.

It is worth mentioning that the Fock spaces have been introduced in quantum
mechanics via tensor products, to describe the quantum states space of variables
belonging to a Hilbert space. Later one, it was observed that in fact this description
corresponds to the spaces of holomorphic functions of several variables which are
square integrable with respect to a Gaussian measure. These spaces are involved in
harmonic analysis on the Heisenberg group, PDE, infinite dimensional analysis and
free analysis. In more detail, these spaces are related with the white noise space,
with the theory of stochastic distributions, see [6], and the parametrized Berezin
transform on the Fock space provides a solution to the initial value problem on the
complex plane for the heat equation, while weighted translation operators give rise
to a unitary representation of the Heisenberg group on the Fock space.

For other details in the theory of Fock spaces one can consult, for example, the
book [7].

The results obtained in this paper can be considered as a kind of complex ana-
logues of those for the classical heat and Laplace equations in Lp(R) spaces (see,
e.g., [5, p. 23]). In Section 2, we reconsider (1.1), (1.2), (1.3), (1.4) assuming that
the boundary function f ∈ F pα(C) with 1 ≤ p < +∞. Section 3 treats (1.5), (1.6),
(1.7), (1.8) under the same hypothesis for the boundary function f . It is worth
mentioning that since the uniform norm used in the case of the space A(D) is now
replaced with the Lp-type norm in the Fock space F pα(C), the proofs of these results
require new tools, like the Jensen’s inequality, the Fubini’s theorem for integrals
and the Lp-integral modulus of continuity.

2. Heat-type equations with complex spatial variables

The first main result of this section is concerned with the heat equation of com-
plex spatial variable.

Theorem 2.1. Let 1 ≤ p < +∞ and consider Wt(f)(z) given by (1.1), for z ∈ C.
Then, (Wt, t ≥ 0) is a (C0)-contraction semigroup of linear operators on F pα(C)
and the unique solution u(t, z) that belongs to F pα(C) for each fixed t, of the Cauchy
problem (1.2) (with D replaced there with C) with the initial condition

u(0, z) = f(z), z ∈ C, f ∈ F pα(C),

is given by u(t, z) = Wt(f)(z).

Proof. Since f is entire function, reasoning exactly as in [3, Theorem 2.1] (see also
[4, Theorem 2.2.1, p. 27]), (with the unit disk D replaced by C), we obtain that
Wt(f)(z) is entire as function of z and for all t, s ≥ 0, we have

Wt(f)(z) =

∞∑
k=0

ake
−k2t/2zk

and Wt+s(f)(z) = Wt[Ws(f)](z).



4 C. G. GAL, S. G. GAL EJDE-2020/109

Next, we apply the following well-known Jensen type inequality for integrals: if∫ +∞
−∞ G(u)du = 1, G(u) ≥ 0 for all u ∈ R and ϕ(t) is a convex function over the

range of the measurable function of real variable F , then

ϕ
(∫ +∞

−∞
F (u)G(u)du

)
≤
∫ +∞

−∞
ϕ(F (u))G(u)du.

Now, since
∫ +∞
−∞

1√
2πt

e−u
2/(2t)du = 1, by the above Jensen’s inequality with

ϕ(t) = tp, F (u) = |f(ze−iu)|, G(u) =
1√
2πt

e−u
2/(2t)du,

we find

|Wt(f)(z)|p ≤ 1√
2πt

∫ +∞

−∞
|f(ze−iu)|pe−u

2/(2t)du.

Multiplying this inequality by αp
2π [e−α|z|

2/2]p, integrating on C with respect to
dA(z) (the normalized Lebesgue’s area measure) and taking into account the Fu-
bini’s theorem, we obtain

αp

2π

∫
C

[
|Wt(f)(z)|e−α|z|

2/2
]p
dA(z)

≤ 1√
2πt

∫ +∞

−∞

[αp
2π

∫
C

[
|f(ze−iu)|e−α|z|

2/2
]p
dA(z)

]
e−u

2/(2t)du.

But writing z = reiθ in polar coordinates and taking into account that dA(z) =
1
π r dr dθ, some simple calculations lead to the equality∫

C

[
|f(ze−iu)|e−α|z|

2/2
]p
dA(z) =

∫
C

[
|f(z)|e−α|z|

2/2
]p
dA(z), (2.1)

for all u ∈ R. This immediately implies

‖Wt(f)‖p,α ≤ ‖f‖p,α.

So Wt(f) ∈ F pα(D) and Wt is a contraction. Next, for f ∈ F pα(D) let us introduce
the integral modulus of continuity of the form (see, e.g., [1])

ω1(f ; δ)Fpα = sup
0≤|h|≤δ

(∫
C

[
|f(zeih)− f(z)|e−α|z|

2/2
]p
dA(z)

)1/p
.

We obtain

|Wt(f)(z)− f(z)|p =
∣∣∣ 1√

2πt

∫ +∞

−∞
[f(ze−iu)− f(z)]e−u

2/(2t)du
∣∣∣p

and repeating the reasonings as above, for |Wt(f)(z)|p, we obtain∫
C

[
|Wt(f)(z)− f(z)|e−α|z|

2/2
]p
dA(z)

≤ 1√
2πt

∫ +∞

−∞

[ ∫
C

[
|f(ze−iu)− f(z)|e−α|z|

2/2
]p
dA(z)

]
e−u

2/(2t)du

≤ 1√
2πt

∫ +∞

−∞
ω1(f ; |u|)p

Fpα
e−u

2/(2t) du

≤ 1√
2πt

∫ +∞

−∞
ω1(f ;

√
t)p
Fpα

(
|u|√
t

+ 1)pe−u
2/(2t) du



EJDE-2020/109 HEAT AND LAPLACE EQUATIONS IN COMPLEX VARIABLES 5

= 2

√
t√

2πt

∫ ∞
0

ω1(f ;
√
t)p
Fpα

(v + 1)pe−v
2/2dv

= Cpω1(f ;
√
t)p
Fpα
.

Consequently,

‖Wt(f)− f‖p,α ≤ C ′pω1(f ;
√
t)Fpα ,

and so it yields

lim
t↘0
‖Wt(f)− f‖p,α = 0.

Since by [7, Corollary 2.8, p. 38], we obtain

|f(z)−Wt(f)(z)| ≤ eα|z|
2/2‖f −Wt(f)‖p,α, for all z ∈ C,

combining this with the previous limit, it easily follows the uniform convergence on
any compact in C of Wt(f) to f , that is

lim
t↘0

Wt(f)(z) = f(z), for all z ∈ C.

It is worth mentioning here that this last limit property can be proved in a
different way by the same method used in the proof of next Theorem 3.1 for proving
that

lim
t↘0

Qt(f)(z) = f(z), for all z ∈ C.

Now, let s ∈ (0,+∞), Vs be a small neighborhood of s, both fixed, and take an
arbitrary t ∈ Vs, t 6= s. Applying the reasonings in the proof of [3, Theorem 2.1,
(iii)] (see also [4, Theorem 2.2.1, (iii)]), we obtain

|Wt(f)(z)−Ws(f)(z)| ≤
∫ +∞

−∞
|f(ze−iu)|

∣∣∣e−u2/(2t)

√
2πt

− e−u
2/(2s)

√
2πs

∣∣∣du
≤ 1√

2π
|t− s|

∫ +∞

−∞
|f(ze−iu)|e−u

2/c2
∣∣2u2
c4
− 1

c2
∣∣ du,

where c depends on s (and not on t).
Denoting

Ks =

∫ +∞

−∞
e−u

2/c2
∣∣2u2
c4
− 1

c2
∣∣ du,

where 0 < Ks < +∞, by the proof of [4, Theorem 2.2.1, (iii)], it follows that∫
C

[
|Wt(f)(z)−Ws(f)(z)|e−α|z|

2/2
]p
dA(z)

≤
( 1√

2π

)p
|t− s|pKp

s

∫
C

(∫ +∞

−∞
|f(ze−iu)| 1

Ks
e−u

2/c2
∣∣2u2
c4
− 1

c2
∣∣ du)p

× e−pα|z|
2/2dA(z)

and reasoning exactly as at the beginning of the proof, we obtain∫
C

[
|Wt(f)(z)−Ws(f)(z)|e−α|z|

2/2
]p
dA(z) ≤

( 1√
2π

)p
|t− s|pKp

s ‖f‖pp,α .

This implies

‖Wt(f)−Ws(f)‖p,α ≤
1√
2π
|t− s|Ks‖f‖p,α .



6 C. G. GAL, S. G. GAL EJDE-2020/109

Therefore, (Wt, t ≥ 0) is a (C0)-contraction semigroup of linear operators on F pα(C).
Also, since the series representation for Wt(f)(z) is uniformly convergent in any
compact disk included in C, it can be differentiated term by term with respect to
t and ϕ. We then easily obtain that Wt(f)(z) satisfies the Cauchy problem in the
statement of the theorem. We also note that in equation we must take z 6= 0 simply
because z = 0 has no polar representation, namely z = 0 cannot be represented as
function of ϕ. This completes the proof. �

The partial differential equation (1.2) can equivalently be expressed in terms of
t and z as follows.

Corollary 2.2. Let 1 ≤ p < +∞. For each f ∈ F pα(C), the initial value problem

∂u

∂t
+

1

2

(
z
∂u

∂z
+ z2

∂2u

∂z2

)
= 0, (t, z) ∈ R+ × C\{0},

u(0, z) = f(z), z ∈ C,

is well-posed and its unique solution is Wt(f) ∈ C∞(R+;F pα(C)).

Proof. By [3, Theorem 2.1, (i)], we can compute the generator associated with
Wt(f), to obtain ( d

dt
Wt(f)(z)

)∣∣∣
t=0

= −
∞∑
k=0

k2

2
akz

k

= −
∞∑
k=0

(k(k − 1)

2
+
k

2

)
akz

k

= −z
2

2
f ′′(z)− z

2
f ′(z).

Therefore, the statement is an immediate consequence of a classical result of Hill
(see, e.g., [4, Theorem 1.2.1, p. 8]). �

Remark 2.3. Theorem 2.1 can be easily extended to functions of several complex
variables, as follows. For 0 < p < ∞, let f : Cn → C, f(z1, z2, . . . , zn) be entire
function with respect to each variable zi ∈ C, i = 1, . . . , n, such that∫

Cn

[
|f(z1, . . . , zn)|e−α|z1|

2/2 · · · e−α|zn|
2/2
]p
dA(z1) · · · dA(zn) <∞.

We write this by f ∈ F pα(Cn) and

‖f‖Fpα(Cn) =
(∫

Cn

[
|f(z1, . . . , zn)|e−α|z1|

2/2 · · · e−α|zn|
2/2
]p
dA(z1) · · · dA(zn)

)1/p
becomes a norm on F pα(Cn). We call the later the weighted Fock space in several
complex variables. Following now the model for the semigroup attached to the real
multivariate heat equation (see, e.g., [5, pg. 69]), for z1, z2, . . . , zn ∈ C, we may
define the integral of Gauss-Weierstrass type by

Ht(f)(z1, . . . , zn) = (2πt)−n/2
∫
Rn
f(z1e

−iu1 , . . . , zne
−iun)e−|u|

2/(2t)du1 · · · dun,

where u = (u1, . . . , un) and |u| =
√
u21 + · · ·+ u2n. One can reason exactly as in the

proof of Theorem 2.1 to deduce that (Ht, t ≥ 0) is (C0)-contraction semigroup of
linear operators on F pα(Cn) and that

u(t, z1, . . . , zn) = Ht(f)(z1, . . . , zn)



EJDE-2020/109 HEAT AND LAPLACE EQUATIONS IN COMPLEX VARIABLES 7

is the unique solution of the Cauchy problem

∂u

∂t
(t, z1, . . . , zn) =

1

2

[ ∂2u
∂ϕ1

2
(t, z1, . . . , zn) + · · ·+ ∂2u

∂ϕn2
(t, z1, . . . , zn)

]
, t > 0,

z1 = r1e
iϕ1 , . . . , zn = rne

iϕn ∈ C, z1, . . . , zn 6= 0,

u(0, z1, . . . , zn) = f(z1, . . . , zn).

Remark 2.4. Reasoning as in the proof of Corollary 2.2 and calculating( d
dt
Ht(f)(z1, . . . , zn)

)∣∣∣
t=0

,

we easily find that the initial value problem

∂u

∂t
+

1

2

n∑
k=1

(
zk
∂u

∂zk
+ z2k

∂2u

∂z2k

)
= 0, (t, z1, . . . , zn) ∈ R+ × (C\{0})n,

u(0, z1, . . . , zn) = f(z1, . . . , zn), z1, . . . , zn ∈ C,

is well-posed and its unique solution is Ht(f) ∈ C∞(R+;F pα(Cn)).

3. Laplace-type equations with complex spatial variables

The first main result of this section is concerned with the Laplace equation of a
complex spatial variable.

Theorem 3.1. Let 1 ≤ p < +∞ and consider Qt(f)(z) given by (1.5), for z ∈ C.
Then (Qt, t ≥ 0) is a (C0)-contraction semigroup of linear operators on F pα(C) and
the unique solution u(t, z) that belongs to F pα(C) for each fixed t, of the Cauchy
problem (1.6) (with D replaced there by C) with the initial condition

u(0, z) = f(z), z ∈ C, f ∈ F pα(C),

is given by u(t, z) = Qt(f)(z).

Proof. Since f is entire function, reasoning exactly as in [3, Theorem 3.1] (see also
[4, Theorem 2.3.1, p. 27]) (with the unit disk D replaced by C), Qt(f)(z) is entire
function as function of z and for all t, s ≥ 0 we have

Qt(f)(z) =

∞∑
k=0

ake
−ktzk, Qt+s(f)(z) = Qt[Qs(f)](z).

Now, since t
π

∫ +∞
−∞

1
u2+t2 du = 1, by Jensen’s inequality, we obtain

|Qt(f)(z)|p ≤ t

π

∫ +∞

−∞
|f(ze−iu)|p 1

u2 + t2
du,

which multiplied on both sides by [e−α|z|
2/2]p, then integrated on C with respect

to the Lebesgue’s area measure dA(z) and applying the Fubini’s theorem, gives∫
C
[|Qt(f)(z)|e−α|z|

2/2]pdA(z)

≤ t

π

∫ +∞

−∞

[ ∫
C

[
|f(ze−iu)|e−α|z|

2/2
]p
dA(z)

] 1

u2 + t2
du.

As in the proof of Theorem 3.1, writing z = reiθ (in polar coordinates) and taking
into account that dA(z) = 1

π r dr dθ, some simple calculations lead to the same



8 C. G. GAL, S. G. GAL EJDE-2020/109

equality (2.1). Hence, we obtain ‖Qt(f)‖p,α ≤ ‖f‖p,α. This implies that Qt(f) ∈
F pα(C) and that Qt is a contraction.

To prove that
lim
t↘0

Qt(f)(z) = f(z),

for any f ∈ F pα(C) and z ∈ C, let f = U + iV , z = reix be fixed with 0 < r and
denote

F (v) = U [r cos(v), r sin(v)], G(v) = V [r cos(v), r sin(v)].

We can write

Qt(f)(z) =
t

π

∫ +∞

−∞
F (x− u)

1

t2 + u2
du+ i

t

π

∫ +∞

−∞
G(x− u)

1

t2 + u2
du.

From the maximum modulus principle, passing to limit as t→ 0+ and taking into
account the property in the real case (see, e.g., [5, p. 23, Exercise 2.18.8]), we find

lim
t↘0
|Qt(f)(z)− f(z)| ≤ lim

t↘0

∣∣∣ t
π

∫ +∞

−∞
F (x− u)

1

t2 + u2
du− F (x)

∣∣∣
+ lim
t↘0

∣∣∣ t
π

∫ +∞

−∞
G(x− u)

1

t2 + u2
du−G(x)

∣∣∣ = 0,

which holds uniformly with respect to |z| ≤ r. Consequently,

lim
t↘0

Qt(f)(z) = f(z),

uniformly in any compact subset of C.
Now, let s ∈ (0,+∞), Vs be a small neighborhood of s, both fixed, and take an

arbitrary t ∈ Vs, t 6= s. Applying the same reasoning as in the proof of [3, Theorem
3.1, (ii)] (see also [4, Theorem 2.3.1, (ii)]), we obtain

|Qt(f)(z)−Qs(f)(z)| ≤ 1

π

∫ +∞

−∞
|f(ze−iu)|

∣∣ t

t2 + u2
− s

s2 + u2
∣∣ du

=
1

π
|t− s|

∫ +∞

−∞
|f(ze−iu)|

∣∣ u2 − ts
(t2 + u2)(s2 + u2)

∣∣ du.
Setting

Ks =

∫ +∞

−∞

∣∣ u2 − ts
(t2 + u2)(s2 + u2)

∣∣ du,
i.e.,

1 =
1

Ks

∫ +∞

−∞

∣∣ u2 − ts
(t2 + u2)(s2 + u2)

∣∣ du
(where 0 < Ks < +∞), by the proof of [4, Theorem 2.3.1, (ii)], it follows that∫

C
[|Qt(f)(z)−Qs(f)(z)|e−α|z|

2/2]pdA(z)

≤
( 1

π

)p|t− s|pKp
s

∫
C

(∫ +∞

−∞
|f(ze−iu)| 1

Ks

∣∣ u2 − ts
(t2 + u2)(s2 + u2)

∣∣ du)p
× e−pα|z|

2/2dA(z).

Applying Jensen’s inequality and then Fubini’s theorem, we obtain∫
C

[
|Qt(f)(z)−Qs(f)(z)|e−α|z|

2/2
]p
dA(z) ≤

( 1

π

)p|t− s|pKp
s ‖f‖pp,α



EJDE-2020/109 HEAT AND LAPLACE EQUATIONS IN COMPLEX VARIABLES 9

and so

‖Qt(f)−Qs(f)‖p,α ≤
1

π
|t− s|Ks‖f‖p,α .

Then (Qt, t ≥ 0) is a (C0)-contraction semigroup of linear operators on F pα(C).
Also, since the series representation for Qt(f)(z) is uniformly convergent in any
compact disk included in C, it can be differentiated term by term, with respect to
t and ϕ. We then easily obtain that Qt(f)(z) satisfies the Cauchy problem in the
statement of the theorem. We also note that in equation we must take z 6= 0 simply
because z = 0 has no polar representation. This completes the proof. �

Reasoning exactly as in the proof of [4, Theorem 2.3.1, (v), pp. 53-54], we im-
mediately get the following.

Corollary 3.2. Let 1 ≤ p < +∞. For each f ∈ F pα(C), the initial value problem

∂u

∂t
+ z

∂u

∂z
= 0, (t, z) ∈ R+ × C\{0},

u(0, z) = f(z), z ∈ C,

is well-posed and its unique solution Qt(f) belongs to C∞(R+;F pα(C)).

Remark 3.3. The above results can be easily extended to several complex vari-
ables. We may define the complex Poisson-Cauchy integral by

Pt(f)(z1, . . . , zn) =
Γ((n+ 1)/2)

π(n+1)/2
t

∫ +∞

−∞
· · ·
∫ +∞

−∞
f(z1e

−iu1 , . . . , zne
−iun)

× 1

(t2 + u21 + · · ·+ u2n)(n+1)/2
du1 · · · dun.

Using similar arguments (following now the model for the semigroup attached to
the real multivariate Laplace equation, see, e.g., [5, p. 69]), as in the univariate
complex case we can prove that the unique solution of the Cauchy problem

∂2u

∂t2
(t, z1, . . . , zn) +

∂2u

∂ϕ1
2

(t, z1, . . . , zn) + · · ·+ ∂2u

∂ϕn2
(t, z1, . . . , zn) = 0, t > 0,

z1 = r1e
iϕ1 , . . . , zn = rne

iϕn ∈ C, z1, . . . , zn 6= 0,

u(0, z1, . . . , zn) = f(z1, . . . , zn), for z1, . . . , zn ∈ C, f ∈ F pα(Cn)

is given by

u(t, z1, . . . , zn) = Pt(f)(z1, . . . , zn).

Remark 3.4. Reasoning as in the proof of Corollary 3.2 and calculating( d
dt
Pt(f)(z1, . . . , zn)

)∣∣∣
t=0

,

we easily obtain that for each f ∈ F pα(Cn), the initial value problem

∂u

∂t
+

n∑
k=1

(
zk
∂u

∂zk

)
= 0, (t, z1, . . . , zn) ∈ R+ × (C\{0})n,

u(0, z1, . . . , zn) = f(z1, . . . , zn), z1, . . . , zn ∈ C,

is well-posed and its unique solution Qt(f) belongs to C∞(R+;F pα(C)).



10 C. G. GAL, S. G. GAL EJDE-2020/109

References

[1] O. Blasco; Modulus of continuity with respect to semigroups of analytic functions and appli-
cations, J. Math. Anal. Appl., 435 (2016), 616–626.

[2] C. G. Gal, S. G. Gal; Heat and Laplace type equations with complex spatial variables in

weighted Bergman spaces, Electronic J. Diff. Equ., Vol. 2017 (2017), Art. no. 236, 1–8.
[3] C. G. Gal, S. G. Gal, J. A. Goldstein; Evolution equations with real time variable and complex

spatial variables, Complex Variables Elliptic Equations, 53 (2008), 753–774.

[4] C. G. Gal, S. G. Gal, J. A. Goldstein; Evolution Equations with a Complex Spatial Variable,
World Scientific, New Jersey-London-Singapore-Beijing-Shanghai-Hong Kong-Taipei-Chennai,

2014.
[5] J. A. Goldstein; Semigroups of Linear Operators and Applications, Expanded second edition,

Dover Publications, New York, 2017.

[6] D. V. Voiculescu, K. J. Dykema, A. Nica; Free Random Variables, volume 1, CRM Monograph
Series, American Mathematical Society, Providence, RI, 1992.

[7] K. Zhu; Analysis on Fock Spaces, Graduate Texts in Mathematics, vol. 263, Springer, New

York, 2012.

Ciprian G. Gal
Department of Mathematics, Florida International University, Miami, FL 33199, USA

Email address: cgal@fiu.edu

Sorin G. Gal

University of Oradea, Department of Mathematics and Computer Science, Str. Uni-

versitatii Nr. 1, 410087 Oradea, Romania
Email address: galso@uoradea.ro


	1. Introduction
	2. Heat-type equations with complex spatial variables
	3. Laplace-type equations with complex spatial variables
	References