




































©2025 Ada Academica https://adac.eeEur. J. Math. Anal. 5 (2025) 21doi: 10.28924/ada/ma.5.21
New Characterization of Hardy-Fofana Spaces and Temperature Equation

Martial Agbly Dakoury1, Justin Feuto2,∗

1Laboratoire de Mathématiques et Applications, UFR Mathématiques et Informatique, Université
Félix Houphouët-Boigny Abidjan-Cocody, 22 B.P 582 Abidjan 22. Côte d’Ivoire

dakourymartial@gmail.com
2Laboratoire de Mathématiques et Applications, UFR Mathématiques et Informatique, Université

Félix Houphouët-Boigny Abidjan-Cocody, 22 B.P 1194 Abidjan 22. Côte d’Ivoire
justfeuto@yahoo.fr

∗Correspondence: justfeuto@yahoo.fr

Abstract. The aim of this paper is to give a characterization of Hardy-Fofana spaces via Riesztransforms. This characterization allows us to describe the distributions belonging to these spacesas a bounded solutions of Cauchy-Riemann’s general temperature equations.

1. Introduction
Let Rd (d is a positive integer) be the Euclidean space of dimension d equipped with theLebesgue measure dx and the Euclidean norm. The classical Hardy space Hp(Rd) (0 < p < ∞)is defined as the space of tempered distributions f satisfying ‖Mf ‖p < ∞, where the maximalfunction Mf is defined by

Mf (x) = sup
t>0
|(f ∗ ϕt)(x)|, (1.1)

with ϕ in the Schwartz class S(Rd) having non vanish integral, and ϕt(x) = t−dϕ(t−1x).It is well known that not only this space does not depends on ϕ, but one can replaced Schwartzfunction by Poisson kernel in the definition of the maximal function (1.1).In [1], Ablé and the second author studied Hardy-amalgam spaces H(p,q)(Rd) (0 < p, q < ∞)by taking in the above maximal characterization of classical Hardy space the Wiener amalgamquasi-norm ‖·‖p,q instead of Lebesgue’s.
Received: 14 May 2025.
Key words and phrases. Amalgam spaces, Hardy-Amalgam spaces, Generalized Hardy-Morrey spaces, Calderón-Zygmund operators, Molecular decomposition. 1

https://adac.ee
https://doi.org/10.28924/ada/ma.5.21


Eur. J. Math. Anal. 10.28924/ada/ma.5.21 2A locally integrable function u belongs to the amalgam space (Lp, `q)(Rd) if
‖u‖p,q :=

∑
k∈Zd

‖uχQk‖
q
p

 1
q

<∞,

where for k ∈ Zd , Qk = k + [0, 1)d and χQk stands for the characteristic function of Qk .Multiple characterizations of H(p,q)(Rd) spaces including atomic and Poisson kernel character-ization, were given in [1]. We notice that the atoms in this context are exactely the one used inclassical Hardy space.Recently, Assaubay et al in [3] characterized this spaces by using first-order classical Riesztransforms and composition of first-order Riesz transformations. They also describe the distributionsinH(p,q)(Rd) as the boundary values of solutions of harmonic and caloric Cauchy-Riemann systems.Here we intend to prove that similar characterizations are possible in the context of Hardy-Fofanaspaces.It is well known that for 0 < p,α, q <∞ and r > 0, there exists a constant Cr ;α > 0 dependingon r and α such that
C−1
r ;α‖u‖p,q ≤ ‖Stαr u‖q,p ≤ Cr ;α‖u‖p,q, u ∈ (Lp, `q)(Rd), (1.2)

where (Stαr u)(x) = r−
d
α u(r−1x). It follows from the above relation that for u ∈ (Lp, `q)(Rd),we have Stαr u ∈ (Lp, `q)(Rd) for α > 0 and r > 0. Unfortunately, the family {Stαr u}r>0 is notbounded in (Lp, `q)(Rd). Ibrahim Fofana considered in [7], the spaces (Lp, `q)α(Rd) defined for

0 < p, q, α ≤ ∞ by
(Lp, `q)α(Rd) =

{
f ∈ (Lp, `q)(Rd)/ ‖f ‖p,q,α <∞

}
where

‖f ‖p,q,α := sup
r>0
‖Stαr f ‖p,q . (1.3)

These spaces known as Fofana’s spaces are non trivial if and only if p ≤ α ≤ q (see [7]). In therest of the paper we will always assume that this condition is fulfilled. It is proved in [6] that for
u ∈ (Lp, `q)α(Rd), we have ‖Stαr u‖p,q,α = ‖u‖p,q,α and that (Lp, `q)α(Rd) (1 ≤ p ≤ α ≤ q) is thebiggest norm space which is continuously embedded in (Lp, `q)(Rd) and for which the translation
Stαr is an isometry. These spaces can also be viewed as some generalized Morrey spaces since for
p < α, the space (Lp, `∞)α(Rd) is exactly the classical Morrey space Lp,d pα (Rd).For 0 < p ≤ α ≤ q <∞, Hardy-Fofana space H(p,q,α)(Rd), introduced by the authors in [4], isa subspace of Hardy-amalgam spaces consists of tempered distributions f satisfying

‖f ‖H(p,q,α) := ‖Mf ‖p,q,α <∞.

https://doi.org/10.28924/ada/ma.5.21


Eur. J. Math. Anal. 10.28924/ada/ma.5.21 3The purpose of this article is twofold. We first characterize these spaces via Riesz transforms andsecondly, we describe the distributions belonging to these spaces as bounded solutions of certaingeneral temperature equations of CAUCHY-RIEMANN.This paper is organized as follow:The next Section is devoted to the prerequisites on Hardy-Fofana spaces. In Section 3, wegive the characterizations of Hardy-Fofana spaces with Riesz transforms. In the last section,we characterize distributions belonging to our spaces as bounded solutions of certain generaltemperature equations of CAUCHY-RIEMANN.In this work, S := S(Rd) will denote the Schwartz class of rapidly decreasing smooth functionsequipped with its usual topology. The dual space of S is the space of tempered distributionsdenoted by S ′ := S ′(Rd). The pairing between S ′ and S is denoted by 〈·, ·〉.We denote by |E|, the Lebesgue measure of a measurable subset E of Rd . The notation A ≈ Bmeans that there exist two constants 0 < C1 and 0 < C2 such that A ≤ C1B and B ≤ C2A, while
A := B means that B is the definition of A.

2. Prerequisites for Hardy-Fofana Spaces
Fofana’s spaces have among others, the following properties (see for example [6] and [7]):(1) let 0 < p,α, q ≤ ∞. The space ((Lp, `q)α(Rd), ‖·‖p,q,α

) is a Banach space if 1 ≤ p ≤
α ≤ q and a quasi-Banach space if 0 < p < 1;(2) if α ∈ {p, q} then (Lp, `q)α(Rd) = Lα(Rd) with equivalent norms;(3) if p < α < q then Lα(Rd) ( (Lp, `q)α(Rd) ( (Lp, `q)(Rd);(4) let f and g be two measurable functions on Rd . If |f | ≤ |g|, then ‖f ‖p,q,α ≤ ‖g‖p,q,α.For many operators including the maximal Hardy-Littlewood operator, norm inequalities aregiven in these spaces for 1 ≤ p ≤ α ≤ q.Let f be a locally integrable function and M(f ) be the centered Hardy-Littlewood maximalfunction defined by

M(f )(x) := sup
r>0
|B(x, r)|−1

∫
B(x,r)

|f (y)|dy, ∀ x ∈ Rd .

It is proved in [6, Proposition 4.2] that M is bounded on (Lp, `q)α(Rd), whenever 1 < p ≤ α ≤ q ≤
∞. Using [8, Proposition 11.12], it is easy to extablish the following result whose proof is omitted.
Proposition 2.1. Let 1 < p ≤ α ≤ q < +∞ and 1 < u ≤ +∞. For all sequences {fn}n≥0 of
measurable functions, we have∥∥∥∥∥∥∥

∑
n≥0

|M(fn)|u
 1

u

∥∥∥∥∥∥∥
p,q,α

≈

∥∥∥∥∥∥∥
∑
n≥0

|fn|u
 1

u

∥∥∥∥∥∥∥
p,q,α

,

with the equivalence constants not depending on the sequence {fn}n≥0.

https://doi.org/10.28924/ada/ma.5.21


Eur. J. Math. Anal. 10.28924/ada/ma.5.21 4As Hardy-Fofana spaces are concerned, we have among others, the following properties whichcan be found in [4].
Proposition 2.2. Let 1 ≤ p ≤ α ≤ q <∞.(1) If 1 < p then the space H(p,q,α)(Rd) and (Lp, Lq)α(Rd) are equal with equivalence norms.(2) The space H(1,q,α)(Rd) is continuously embedded in (L1, `q)α(Rd).

Notice that for p < 1, we have as in the classical Hardy and Hardy-amalgam spaces, that thespaces (H(p,q,α)(Rd), ‖ · ‖H(p,q,α)

) are quasi-Banach and for f , g ∈ H(p,q,α)(Rd),
‖f + g‖pH(p,q,α) ≤ ‖f ‖

p

H(p,q,α) + ‖g‖pH(p,q,α) .

We can also define (see [5]) these spaces as subspaces of Hardy-amalgam spaces for which thefamilly of dilations {Stαρ
}
ρ>0

is locally bounded.More precisely, for a tempered distribution f , ρ > 0 and α two real numbers we put〈
Stαρ f , ϕ

〉
:=
〈
f ,Stα

′

ρ−1ϕ
〉
,

where 1
α′ + 1

α = 1. We have (see [4]) that for 0 < p ≤ α ≤ q ≤ ∞,
‖f ‖H(p,q,α) = sup

ρ>0
‖Stαρ f ‖H(p,q) . (2.1)

Just as Hardy-amalgam spaces was characterized in [1] with Poisson kernel, so are Hardy-Fofana’sspaces. In fact, a tempered distribution f belonging to Hardy-amalgam spaces is bounded; i.e
f ∗ ψ ∈ L∞(Rd) for all ψ ∈ S(Rd). A convolution of such distribution with integrable functionscan be defined in term of distribution. More precisely, if f ∈ S ′(Rd) is bounded and u ∈ L1(Rd),then the convolution f ∗ u is defined as a tempered distribution acting on S(Rd) by the pairing

〈f ∗ u, ϕ〉 := 〈f ∗ ϕ̃, ũ〉(L∞,L1) ϕ ∈ S(Rd)

where ũ(x) = u(−x) and 〈f ∗ ϕ̃, ũ〉(L∞,L1) is the pairing between L∞(Rd) and L1(Rd). But if wetake as u the Poisson kernel P defined by
P (x) :=

Γ(d+1
2 )

π
d+1

2

1

(1 + |x |2)
d+1

2

x ∈ Rd ,

then f ∗ Pt can be identified for all t > 0, to a well defined bounded function. As we can see forexample in [9], there exist ϕ,ψ ∈ S(Rd) such that
f ∗ Pt = (f ∗ ϕt) ∗ Pt + f ∗ ψt for t > 0.

It is proved in [1] that for an element f ∈ H(p,q)(Rd), we have
‖x 7→ sup

t>0
sup
|x−y |<t

|f ∗ Pt(y)|‖p,q ≈ ‖Mf ‖p,q (2.2)

https://doi.org/10.28924/ada/ma.5.21


Eur. J. Math. Anal. 10.28924/ada/ma.5.21 5where Mf is the maximal function defined in Relation (1.1). It follows that
‖x 7→ sup

t>0
sup
|x−y |<t

|f ∗ Pt(y)|‖p,q,α ≈ ‖Mf ‖p,q,α, (2.3)
thanks to Relations (2.1) and (2.2) and the fact that Stαρ commute with the maximal function M.
Lemma 2.3. Let f ∈ S ′(Rd), ϕ ∈ S(Rd), ρ and α positive real numbers. We have

Stαρ (f ∗ ϕt) =
(

Stαρ f
)
∗ ϕρt , t > 0.

Infact,
ρ
−d
α (f ∗ ϕt) (ρ−1x) = ρ

−d
α

〈
f , ρdϕρt(x − ρ·)

〉
= ρ

d
α′ 〈f , ϕρt(x − ρ·)〉

=
〈
f ,Stα

′

ρ−1 [ϕρt(x − ·)]
〉

=
(

Stαρ f ∗ ϕρt
)

(x).

Lemma 2.4. Let f ∈ S ′(Rd), ϕ ∈ S(Rd), ρ and α positive real numbers. We have

Stαρ [(f ∗ ϕt) ∗ Pt ] =
(

Stαρ f ∗ ϕρt
)
∗ Pρt . (2.4)

Relation (2.4) follows from the fact that
ρ
−d
α

(
f ∗ ϕρ−1t

)
∗ Pρ−1t(ρ

−1x) = ρ
−d
α

∫
Rd

(
f ∗ ϕρ−1t

)
(ρ−1x − y)Pρ−1t(y)dy

=

∫
Rd

〈
f , ρ

d
α′ϕt(x − z − ρ·)

〉
Pt(z)dz

=

∫
Rd

〈
Stαρ f , ϕt(x − z − ·)

〉
Pt(z)dz

=

∫
Rd

(
Stαρ f ∗ ϕt

)
(x − z)Pt(z)dz =

(
Stαρ f ∗ ϕt

)
∗ Pt(x).

It comes from Lemma 2.3 and 2.4 that for a bounded tempered distribution f and u(x, t) = f ∗Pt(x),(
Stαρ u

)
(x, t) =

[(
Stαρ f

)
∗ Pt

]
(x), ρ > 0 and α > 0 (2.5)

for all t > 0.
3. Cauchy-Riemann equations, Riesz transforms and Hardy-Fofana spaces

Let u be a harmonic function on Rd+1
+ ; i.e, u ∈ C2(Rd+1

+ ) and ∆u :=
∑d+1
j=1

∂2u
(∂xj )2 = 0, where

xd+1 = t and Rd+1
+ := Rd×]0,+∞[. We define its non tangential maximal function u∗ by

u∗(x) := sup
t>0

sup
|x−y |<t

|u(y , t)| ∀x ∈ Rd . (3.1)
Let f be a bounded tempered distribution, and u(x, t) = Pt ∗ f (x). As we can see in [4],
u∗ ∈ (Lp, `q)α (Rd) whenever f ∈ H(p,q,α)(Rd). We give in the next result a necessary and

https://doi.org/10.28924/ada/ma.5.21


Eur. J. Math. Anal. 10.28924/ada/ma.5.21 6sufficient conditions for a harmonic function u in Rd+1 to have its non tangential maximal func-tion in (Lp, `q)α(Rd). The proof is based on the dilation characterization of Hardy-Fofana spacesand [3, Proposition 2.1 ].
Proposition 3.1. Let 0 < p ≤ α ≤ q < +∞ and u an harmonic function on Rd+1

+ . The maximal
function u∗ belongs to (Lp, `q)α(Rd) if and only if there exists f ∈ H(p,q,α)(Rd) such that

u(x, t) := f ∗ Pt(x), (x, t) ∈ Rd+1
+ .

Moreover, ‖f ‖H(p,q,α) ≈ ‖u∗‖p,q,α .

Proof. Let u be an harmonic function on Rd+1
+ , and u∗ the associate non tangential maximal functionas defined in Relation (3.1).We suppose that there exists f ∈ H(p,q,α)(Rd) such that u(x, t) := f ∗Pt(x) for all (x, t) ∈ Rd+1

+ .From the Poisson characterization of Hardy-Fofana spaces (see [4, Theorem 2.3.8 ]), we deducethat ‖u∗‖p,q,α ≤ C ‖f ‖H(p,q,α) .For the converse, let us suppose that u∗ ∈ (Lp, `q)α(Rd) ⊂ (Lp, `q)(Rd). It comes from [3,Proposition 2.1 ] that there exists f ∈ H(p,q)(Rd) and a constant C > 0 such that
u(x, t) = (f ∗ Pt)(x), (x, t) ∈ Rd+1

+ (3.2)
and

1

C
‖f ‖H(p,q) ≤ ‖u∗‖p,q ≤ C‖f ‖H(p,q) .Since Stαρ f ∈ H(p,q)(Rd) for all ρ > 0, Stαρ u harmonic on Rd+1

+ and (Stαρ u
)

(x, t) = (Stαρ f )∗Pt(x),it comes that (Stαρ u)∗ ∈ (Lp, Lq)(Rd) and
1

C
‖Stαρ f ‖H(p,q) ≤ ‖(Stαρ u)∗‖p,q ≤ C‖Stαρ f ‖H(p,q) .

This relation being thrue for all ρ > 0, we have
1

C
‖f ‖H(p,q,α) ≤ ‖u∗‖p,q,α ≤ C‖f ‖H(p,q,α) ,

where we use the trivial identity (Stαρ u)∗ = Stαρ u
∗, ρ > 0 and 0 < α <∞. �

We say that a vector values function F := (u1, u2, ..., ud+1), with uj : Rd+1
+ → R, j ∈ {1, 2, ..., d+

1}, satisfies the generalized Cauchy-Riemann equations (in short F ∈ CR(Rd+1
+ )) if

∂uj
∂xk

=
∂uk
∂xj

, 1 ≤ j, k ≤ d + 1 and d+1∑
j=1

∂uj
∂xj

= 0, (3.3)
where we set xd+1 = t. Also recall that for j ∈ {1, 2, ...d}, the j-th Riesz transform Rj(g) of ameasurable function g is formally defined by

Rj(g)(x) := lim
ε→0+

∫
|x−y |>ε

Kj(x − y)g(y)dy a.e x ∈ Rd .

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Eur. J. Math. Anal. 10.28924/ada/ma.5.21 7

where Kj(x) :=
Γ( d+1

2
)

π
d+1

2

xj
|x |d+1 , x ∈ Rd\{0}.In [2, Corollary 4.19], Ablé and Feuto demonstrated that Riesz transformations are extendableinto bounded linear operators on Hardy-amalgam spaces H(p,q)(Rd) for 0 < p ≤ 1. We will keepthe notations Rj , j = 1, · · · , d for these extentions. Assaubay et al proved the following result.

Proposition 3.2 ( [3], Proposition 2.3). Let d−1
d < min {p, q} < +∞. Suppose that u is harmonic

function in Rd+1
+ . Then u∗ ∈ (Lp, `q)(Rd) if and only if there exists an harmonic vector F :=

(u1, ..., ud+1) ∈ CR(Rd+1
+ ) such that ud+1 := u and supt>0 ‖|F (., t)|‖p,q < +∞.

Furthermore, supt>0 ‖|F (., t)|‖p,q ≈ ‖u∗‖p,q .

Since u∗ ∈ (Lp, `q)(Rd) if and only if u = f ∗ Pt for some f ∈ H(p,q)(Rd), they proved that onecan take uj(x, t) = Rj(f ) ∗ Pt(x), j = 1, · · · , d .In the case of Hardy-Fofana’s spaces, we have the following.
Proposition 3.3. Assume that d−1

d < p ≤ α ≤ q < +∞ and u is an harmonic function in Rd+1
+ .

Then u∗ ∈ (Lp, `q)α(Rd) if and only if there exists an harmonic vector F := (u1, ..., ud+1) ∈
CR(Rd+1

+ ) such that ud+1 := u and supt>0 ‖|F (., t)|‖p,q,α < +∞. Furthermore

sup
t>0
‖|F (., t)|‖p,q,α ≈ ‖u∗‖p,q,α (3.4)

Proof. Let d−1
d < p ≤ α ≤ q < +∞ and u an harmonic function on Rd+1

+ .We suppose that u∗ ∈ (Lp, `q)α(Rd). Since (Lp, `q)α(Rd) ⊂ (Lp, `q)(Rd), Proposition 3.2assert that there exists f ∈ (Lp, `q)(Rd) so that:
• u(x, t) = f ∗ Pt(x),
• the harmonic vector F = (u1, · · · , ud+1) with uj(x, t) = Rj(f ) ∗ Pt(x) for j ∈ {1, · · · , d}and ud+1 = u belongs to CR(Rd+1

+ ),
• supt>0 ‖|F (·, t)|‖p,q ≈ ‖u∗‖p,q.Since u∗ ∈ (Lp, `q)α (Rd) we have that the tempered distribution f belongs to H(p,q,α)(Rd), thanksto Proposition 3.1. All we have to prove now is that x 7→ F (x, t) belongs to (Lp, `q)α(Rd) for

t > 0 and that Relation (3.4) is satisfies.Fix t > 0 and ρ > 0. Since u∗ ∈ (Lp, `q)α (Rd) and (Stαρ u
)∗

= Stαρ (u∗), we have that for ρ > 0,
‖
(

Stαρ u
)∗ ‖p,q ≤ ‖u∗‖p,q,α. Hence (Stαρ u

)∗ ∈ (Lp, `q) (Rd) so that there exists fρ ∈ H(p,q)(Rd)satisfying
(Stαρ u)(x, t) = (fρ ∗ Pt)(x),with

Fρ(x, t) := (R1(fρ) ∗ Pt(x), · · · ,Rd(f ρ) ∗ Pt(x), (fρ) ∗ Pt)(x)) (3.5)belonging to CR+(Rd+1
+ ) and

sup
t>0
‖|Fρ(·, t)|‖p,q ≈ ‖Stαρ (u∗)‖p,q. (3.6)

https://doi.org/10.28924/ada/ma.5.21


Eur. J. Math. Anal. 10.28924/ada/ma.5.21 8Moreover (Stαρ u)(x, t) = (Stαρ f )∗Pt(x), thanks to Relation (2.5). It follows that fρ∗Pt = (Stαρ f )∗Ptfor all t > 0 so that fρ = Stαρ f . We recall that the last equality comes from the fact that for
f ∈ H(p,q)(Rd), f ∗ Pt tends to f in S ′(Rd) as t goes to 0. Replacing fρ by Stαρ f in Relation (3.5)yields Fρ(x, t) =

(
R1(Stαρ f ) ∗ Pt(x), · · · ,Rd(Stαρ f ) ∗ Pt(x), (Stαρ f ) ∗ Pt)(x)

). Since the operator
Stαρ commute with Rj we have that

Fρ(·, t) =
(

Stαρ (R1f ) ∗ Pt(·), · · · ,Stαρ (Rd f ) ∗ Pt(·), (Stαρ f ) ∗ Pt)(·)
)

=
(

Stαρ
(
u1(·, ρ−1t)

)
, · · · ,Stαρ

(
ud(·, ρ−1t)

)
,Stαρ

(
ud+1(·, ρ−1t)

))
= Stαρ

(
F (·, ρ−1t)

)
.

If we take this expression of Fρ in Relation (3.6) we obtain that
sup
t>0
‖|Stαρ (F (·, ρ−1t)|‖p,q ≈ ‖Stαρ (u∗)‖p,q.

But supt>0 ‖|Stαρ (F (·, ρ−1t))|‖p,q = supt>0 ‖|Stαρ (F (·, t))|‖p,q and the result follow from the defi-nition of Hardy-Fofana space. �

The next result gives a characterization of H(p,q,α)(Rd) via Riesz transforms Rj(f ∗ φ). Sincewe need to use the characterization of H(p,q)(Rd) given in [3], we give the following definition.
Definition 3.4. Let 0 < p ≤ α ≤ q <∞. A tempered distribution f is said to be :

• (p, q)-restricted at infinity if there exists µ0 ≥ 1 such that for µ ≥ µ0, we have

f ∗ φ ∈ (Lpµ, `qµ)(Rd), φ ∈ S(Rd).

• (p, q, α)-restricted at infinity if there exists µ0 ≥ 1 such that for µ ≥ µ0, we have

f ∗ φ ∈ (Lpµ, `qµ)αµ(Rd), φ ∈ S(Rd).

It is easy to see that tempered distributions which are (p, q, α)-restricted for p ≤ α ≤ q, are also
(p, q)-restricted. Theorem 1.1 in [3] assert that a tempered distribution f belongs to H(p,q)(Rd)for d−1

d < min (p, q) <∞, if and only if it is (p, q)-restricted at infty and, for φ ∈ S(Rd) with nonvanish integral,
sup
t>0

‖f ∗ φt‖p,q +

d∑
j=1

‖(Rj f ) ∗ φt‖p,q

 <∞.

When this is the case,
‖f ‖H(p,q) ≈ sup

t>0

‖f ∗ φt‖p,q +

d∑
j=1

‖(Rj f ) ∗ φt‖p,q

 .
In the case of Hardy-Fofana space, we have the following result.

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Eur. J. Math. Anal. 10.28924/ada/ma.5.21 9

Theorem 3.5. Let d−1
d < p ≤ α ≤ q < ∞, f ∈ S ′(Rd).Then f ∈ H(p,q,α)(Rd) if and only if f is

(p, q, α)-restricted at infinity and, for φ ∈ S(Rd) with non vanish integral,

sup
t>0

‖f ∗ φt‖p,q,α +

d∑
j=1

∥∥(Rj f ) ∗ φt
∥∥
p,q,α

 < +∞. (3.7)
Moreover,

‖f ‖H(p,q,α) ≈ sup
t>0

‖f ∗ φt‖p,q,α +

d∑
j=1

∥∥(Rj f ) ∗ φt
∥∥
p,q,α

 . (3.8)
Proof. Let d−1

d < p ≤ α ≤ q <∞ and f ∈ S ′(Rd).We suppose that f is (p, q, α)-restricted at infinity and satisfies (3.7) for non vanishing Schwartzfunction φ. There exists µ0 > 1 (large enought) such that for µ > µ0, we have
f ∗ φ ∈ (Lpµ; `qµ)αµ (Rd), φ ∈ S(Rd). (3.9)

It comes from the definition of Fofana spaces that
Stαµρ (f ∗ φ) ∈ (Lpµ, `µq) (Rd) φ ∈ S(Rd), ρ > 0.

Taking ρ = 1, we obtain that f is (p, q)-restricted at infinity. Since for all φ ∈ S(Rd) with nonvanishing integral we also have that
A = sup

t>0

sup
ρ

∥∥Stαρ (f ∗ φt)
∥∥
p,q

+

d∑
j=1

sup
ρ>0

∥∥Stαρ
(

(Rj f ) ∗ φt
)∥∥
p,q

 <∞,

it follows that
sup
t>0

‖f ∗ φt‖p,q +

d∑
j=1

∥∥(Rj f ) ∗ φt
∥∥
p,q

 ≤ A.
Thus f ∈ H(p,q)(Rd) thanks to [3, Theorem 1.1]. It remains to prove that the familly {Stαρ f

}
ρ>0

isuniformly bounded in H(p,q)(Rd).Fix ρ > 0. We have Stαρ f ∈ H(p,q)(Rd) so that
‖Stαρ f ‖H(p,q) ≈ sup

t>0

∥∥Stαρ (f ) ∗ φt
∥∥
p,q

+

d∑
j=1

∥∥Rj(Stαρ f ) ∗ φt
∥∥
p,q


thanks once more to [3, Theorem 1.1]. But we have in one hand that
Rj(f ) ∗ φt = Rj (f ∗ φt), so that

Stαρ
[(
Rj f

)
∗ φt

]
= Stαρ

[
Rj (f ∗ φt)

]
= Rj

[
Stαρ (f ∗ φt)

]
, (3.10)

where the last equality comes from the fact that dilation comute with Riesz transforms. In the otherhand we have that
sup
t>0
‖Stαρ (f ∗ φt) ‖p,q = sup

t>0
‖Stαρ (f ) ∗ φt‖p,q, (3.11)

https://doi.org/10.28924/ada/ma.5.21


Eur. J. Math. Anal. 10.28924/ada/ma.5.21 10thanks to Lemma 2.3. Therefore, we have
sup
ρ>0

sup
t>0
‖Stαρ (f ) ∗ φt‖p,q = sup

ρ>0
sup
t>0
‖Stαρ (f ∗ φt) ‖p,q ≤ A

and
sup
ρ>0

sup
t>0

d∑
j=1

‖Rj
[
Stαρ (f ∗ φt)

]
‖p,q = sup

ρ>0
sup
t>0

d∑
j=1

‖Stαρ
[(
Rj f

)
∗ φt

]
‖p,q ≤ A.

We deduce that supρ>0 ‖Stαρ f ‖H(p,q) <∞, which prove that f ∈ H(p,q,α)(Rd).For the converse, we suppose that f ∈ H(p,q,α)(Rd). It follows that Stαρ f ∈ H(p,q)(Rd) with
‖Stαρ f ‖H(p,q) ≤ ‖f ‖H(p,q,α) < ∞ for all ρ > 0. It comes from [3, Theorem 1.1] that Stαρ f is (p, q)-resticted at infinity and

‖Stαρ f ‖H(p,q) ≈ sup
t>0

∥∥Stαρ (f ) ∗ φt
∥∥
p,q

+

d∑
j=1

∥∥Stαρ (Rj f ) ∗ φt
∥∥
p,q


for all φ ∈ S(Rd) with non vanish integral.From Relations (3.11) and (3.10), and the definitions of ‖ · ‖p,q,α and of ‖ · ‖H(p,q,α) , we have that

‖f ‖H(p,q,α) ≈ sup
t>0

(
‖f ∗ φt‖p,q,α +

d∑
j=1

∥∥(Rj f ) ∗ φt
∥∥
p,q,α

)
<∞.

Let φ ∈ S(Rd). We have ‖f ∗ φ‖p,q,α ≤ C ‖f ‖H(p,q,α) . For µ ≥ 1 we have f ∗ φ ∈ (Lpµ, `qµ)αµ.In fact assuming that ‖f ∗ ϕ‖∞ 6= 0 we have
f ∗ φ ∈ (Lp, `q)α and ‖f ∗ φ‖pµ,qµ,αµ ≤ C ‖f ∗ φ‖1− 1

µ
∞ ‖f ∗ φ‖

1
µ
p,q,α

and then f is (q, p, α)-restricted at infinity. �

4. Temperature Cauchy-Riemann Equations and Hardy-Fofana Spaces
A vector F = (u1, u2, · · · , ud+1) of functions in Rd+1

+ satisfy the generalized temperature Cauchy-Riemann equations, if it satisfies the following conditions :(1) ∑d
j=1

∂uj
∂xj

= i∂
1/2
t ud+1(2) ∂uj

∂xk
= ∂uk

∂xj
for j, k = 1, 2, · · · , d(3) ∂ud+1

∂xj
= −i∂1/2

t uj , j = 1, 2, · · · , d , with
(∂

1/2
t g)(t) :=

e iπ/2

√
π

∫ ∞
t

g′(s)√
s − t

ds, t > 0

when g is a smooth enough function on (0,∞)In [3], the authors defined the space Hp,q(Rd+1
+ ) (0 < p, q < ∞) as the vector space of vectorfunctions F = (u1, u2, · · · , ud+1) satisfying generalized temperature Cauchy-Riemann equationsand such that

‖F‖Hp,q(Rd+1
+ ) := sup

t>0
‖|F (·, t)|‖p,q <∞.

https://doi.org/10.28924/ada/ma.5.21


Eur. J. Math. Anal. 10.28924/ada/ma.5.21 11

They also proved that under appropriate conditions on the exponents p and q, the space Hp,q(Rd+1
+ )is topologically isomorphic to Hp,q(Rd). To carry out the proof of this result, they use a subspaceof what they call the temperature space T (Rd+1

+ ); that is the space of functions u ∈ C2(Rd+1
+ ),satisfying

∂u

∂t
=

d∑
j=1

∂2u

∂x2
j

in Rd+1
+ .

More precisely, for 0 < p, q <∞, they put
T p,q(Rd+1

+ ) :=
{
u ∈ T (Rd+1

+ ) : ||u||T (p,q) <∞
}

where
||u||T (p,q) := sup

t>0
||u(., t)||q,p.

They proved [3, Proposition 3.2 (i)] that for d−1
d < p, q <∞, F = (u1, u2, · · · , ud+1) ∈ Hp,q(Rd+1

+ )implies that u := ud+1 ∈ T p,q(Rd+1
+ ) and uj(·, t) = Rj(u(·, t)), t > 0, j = 1, · · · , d .We claim that for 0 < p ≤ α ≤ q < ∞ and r > 0, the space T p,q(Rd+1

+ ) is stable under thedilation Stαr . This is due to the fact that for f ∈ (Lp, `q) (Rd), there exists a constant C(α, r, p, q) >

0 such that
C(α, r, p, q)−1‖f ‖p,q ≤ ‖Stαr f ‖p,q ≤ C(α, r, p, q)‖f ‖p,q,and this dilation commute with Riesz transforms. It follows that if F = (u1, · · · , ud+1) ∈ Hp,q(Rd+1

+ )then Stαr F ∈ Hp,q(Rd+1
+ ).We put

‖F‖H(p,q,α) := sup
r>0
‖Stαr F‖Hp,q(Rd+1

+ )

and defined the space H(p,q,α)(Rd+1
+ ) as the subspace of Hp,q(Rd+1

+ ) consits of F satisfying
‖F‖H(p,q,α) <∞. We have the following result in Hardy-Fofana spaces.
Theorem 4.1. Let d−1

d < p ≤ α ≤ q <∞, and Wt the heat kernel defined by

Wt(x) =
e−|x |

2/4t

(4πt)d/2
.

The map L define on H(p,q,α)(Rd) by

L(f )(x, t) :=
(

((R1f ) ∗Wt)(x), · · · , ((Rd f ) ∗Wt)(x), (f ∗Wt)(x)
)

for all x ∈ Rd and t > 0, is a topological isomorphism from H(p,q,α)(Rd) onto H(p,q,α)(Rd+1
+ ).

Proof. Let f ∈ H(p,q,α)(Rd). For r > 0 we have Stαr f ∈ Hp,q(Rd), thanks to the definition of
H(p,q,α)(Rd). It comes from [3, Theorem 1.3] that

L(Stαr f ) ∈ Hp,q(Rd+1
+ ), with ‖L(Stαr f )‖Hp,q(Rd+1

+ ) ≤ C‖Stαr f ‖Hp,q(Rd ) (4.1)

https://doi.org/10.28924/ada/ma.5.21


Eur. J. Math. Anal. 10.28924/ada/ma.5.21 12for all r > 0. Since Rj(f ∗Wt) = Rj(f ) ∗Wt and Stαr commute with Riesz transforms, we havethat L(Stαr f ) = Stαr (L(f )), r > 0. Thaking this remark in (4.1) we obtain that
‖L(f )‖H(p,q,α)(Rd+1

+ ) ≤ C‖f ‖H(p,q,α)(Rd ).Let now F = (u1, u2, · · · , ud , ud+1) belonging to H(p,q,α)(Rd+1
+ ). This implies that Stαr F =

(Stαr u1,Stαr u2, · · · ,Stαr ud ,Stαr ud+1) ∈ Hp,q(Rd+1
+ ) for all r > 0. As we can see in the proofof [3, Theorem 3.1] this implies that for all r > 0, there exists fr ∈ Hp,q(Rd) so that Stαr ud+1 ∈

T p,q(Rd+1
+ ) with Stαr ud+1(x, t) = fr ∗Wt(x) and

‖fr‖Hp,q ≤ C sup
t>0
‖Stαr F (·, t)‖p,q ≤ C‖F‖H(p,q,α) , (4.2)

and Stαr uj(·, t) = Rj(Stαr ud+1(·, t)), t > 0, j = 1, · · · , d.We put f := f 1. We have Stαr f = fr for all r > 0. Taking this in estimate (4.2) yields
‖Stαr f ‖Hp,q ≤ C‖F‖H(p,q,α)

wich prove that f ∈ H(p,q,α)(Rd).The vector
G(x, t) = ((R1(f ) ∗ Pt)(x), · · · ,Rd(f ) ∗ Pt)(x), (f ∗ Pt)(x)), x ∈ Rd , t > 0

is harmonic, satisfies the generalized Cauchy-Riemann equation, and
sup
t>0
‖|G(·, t)|‖p,q,α ≤ C‖F‖H(p,q,α) .

�

References
[1] Z.V.d.P. Ablé and J. Feuto, Atomic decomposition of Hardy-amalgam spaces, J. Math. Anal. Appl. 455 (2017), 1899–1936.[2] Z.V.d.P. Ablé and J. Feuto, Duals of Hardy amalgam spaces and norm inequalities, Anal. Math. 45 (2019), 647–686.[3] A.-T. Assaubay, J.J. Betancor, A.J. Castro and J.C. Farina, Riesz transforms, Cauchy-Riemann systems, and Hardy-amalgam spaces, Banach J. Math. Anal. 3 (2019), 697–725.[4] M.A. Dakoury and J. Feuto, Norm inequality for intrinsic square functions in generalized Hardy-Morrey spaces,Open Access Libr. J. 9 (2022), e8463.[5] M.A. Dakoury and J. Feuto, Norm inequalities for Calderón–Zygmund operators in some generalized Hardy–Morreyspaces, Vietnam J. Math. (2024). https://doi.org/10.1007/s10013-024-00703-0.[6] J. Feuto, Norm inequalities in some subspaces of Morrey space, Ann. Math. Blaise Pascal 21 (2014), 21–37.[7] I. Fofana, Étude d’une classe d’espaces de fonctions contenant les espaces de Lorentz, Afrika Mat. 2 (1988), 29–50.[8] Y. Liang, Y. Sawano, T. Ullrich, D. Yang and W. Yuan, A new framework for generalized Besov-type and Triebel-Lizorkin-type spaces, Diss. Math. (Rozprawy Mat.) 489 (2013), 1–114.[9] L. Grafakos, Modern Fourier Analysis, 2nd ed., Grad. Texts Math., 250, Springer, New York, 2009.

https://doi.org/10.28924/ada/ma.5.21

	1. Introduction
	2. Prerequisites for Hardy-Fofana Spaces
	3. Cauchy-Riemann equations, Riesz transforms and Hardy-Fofana spaces
	4. Temperature Cauchy-Riemann Equations and Hardy-Fofana Spaces
	References

