Let d ≥ 1 be a fixed integer, ℝ d the classic Euclidean space with the structure of a vector space. Let Δ be the set of cubes of ℝ d and S the set of countable families { Q i : i ∈ I } of the elements of Δ such as Q i ∩ Q j = 0 if i ≠ j . We defined the variation measures

and

Let p and α be two real numbers such that 1 ≤ α ≤ p ≤ + ∞ . We introduced

a new classes of functions B V p , α ( ℝ d ) = { f ∈ L loc 1 ( ℝ d ) / ‖ | D f | ‖ T p , α < + ∞ } in our paper entitled “A Poincaré Inequality for Functions with Locally Bounded Variation in ℝ d ” [1].

In this paper, we establish some important properties of B V p , α ( ℝ d ) .

It is worth noting that these amalgam-type spaces were introduced in 2017 with the aim of studying in depth the relationships between functions and their partial derivatives. The purpose is to relax some conditions when solving partial differential equations (PDEs). Furthermore, an extension of classical functional spaces should allow a greater number of PDEs to have solutions. Unlike spaces of Radon measures T p , α ( ℝ d ) and function spaces F ( 1 , p , α ) ( ℝ d ) , which are part of a general framework, B V p , α ( ℝ d ) spaces focus exclusively on Radon measures derived from the variation measure of locally integrable functions on ℝ d .

Studies conducted on B V p , α ( ℝ d ) spaces, defined between L loc 1 ( ℝ d ) , space of locally integrable functions on ℝ d , and M ( ℝ d ) , space of Radon measures on ℝ d , have enabled us to:

Provide a characterization of the B V p , α ( ℝ d ) spaces through the rate of growth;Establish an approximation of an element in B V p , α ( ℝ d ) ;Investigate the continuity of the translation operator.

In the first part of the paper, we give some properties of B V p , α ( ℝ d ) and provide some necessary notations and definitions. The second part we give a characterization of the B V p , α ( ℝ d ) spaces. This characterization is a very practical tool in analysis, as it provides valuable information about the nature of often quite complex spaces. In the last part, we study the continuity of the translation operator as well as the density of the class of smooth functions within these spaces.

In the next, d ≥ 1 is a fixed integer, ℝ d is the classic Euclidean space with the structure of a vector space, p and α are reals numbers such that: 1 ≤ α ≤ p ≤ + ∞ , Δ is the set of cubes of ℝ d and S the set of countable families { Q i : i ∈ I } of the elements of Δ two by two disjoints, M ( ℝ d ) is the set of Radon measures on ℝ d , M 1 ( ℝ d ) is the set of Radon measures bounded on ℝ d , L 0 ( ℝ d ) is the vector space of functions modulo Lebesgue equality almost everywhere, L loc α ( ℝ d ) are locally Lebesgue spaces, W 1 , α ( ℝ d ) are classical Sobolev spaces, W 1 , α ( ℝ d ) are homogeneous Sobolev spaces, C c k ( ℝ d ) k ≥ 1 integer is the set of functions of class C k with compact support on ℝ d , B V ( ℝ d ) is the space of functions with bounded variation.

For any μ ∈ M ( ℝ d ) ,

where | μ | is the total variation of μ and

For any f ∈ L 0 ( ℝ d ) ,

For more details on these spaces, see [2]-[4].

The following proposition establishes a close link between the measure spaces T p , α ( ℝ d ) and the function spaces F ( 1 , p , α ) ( ℝ d ) , a connection that is a crucial step in our research.

Proposition 2.1([2], Proposition 2.4.1)

If f is an element of L loc 1 ( ℝ d ) and μ f the Radon measure on ℝ d such that d μ f ( x ) = f ( x ) d x , then

Definition 2.2The variation measure of f ∈ L l o c α ( ℝ d ) is defined by

where d i v φ = ∑ j = 1 d ∂ φ j ∂ x j is the divergence of φ and | φ | = ( ∑ j = 1 d   φ j 2 ) 1 2 .

| D f | of f ∈ L loc 1 ( ℝ d ) is a Radon measure that does not necessarily belong to T p , α ( ℝ d ) .

Thus, L loc 1 ( ℝ d ) decomposes into two disjoint subsets:

the elements whose variation measure is unbounded for the ‖   ⋅   ‖ T p , α norm;the elements whose measure variation is bounded for the ‖   ⋅   ‖ T p , α norm.

By focusing on elements of L loc 1 ( ℝ d ) whose measure variation is bounded for the ‖   ⋅   ‖ T p , α norm, we introduce B V p , α ( ℝ d ) spaces defined as the set functions of L loc 1 ( ℝ d ) such that the measure variation belongs to T p , α ( ℝ d ) .

We have shown through the following proposition that B V p , α ( ℝ d ) is a Banach space.

Proposition 2.3([1], Proposition 6.2).

B V p , α ( ℝ d ) with application f ↦ ‖ f ‖ B V p , α = ‖ | D f | ‖ T p , α + | f ∞ | are Banach spaces.

B V p , α ( ℝ d ) spaces include Sobolev spaces, homogeneous Sobolev spaces, and the space of functions with bounded variation. Additionally, B V p , α ( ℝ d ) is equivalent to Morrey space when p = + ∞ .

When α = 1 , B V p , α ( ℝ d ) spaces include B V ( ℝ d ) space.

Proposition 2.4The space of functions with bounded variation B V ( ℝ d ) isansubspace of B V p , 1 ( ℝ d ) spaces.

Proof. Let f ∈ B V ( ℝ d ) = { f ∈ L 1 ( ℝ d ) / | D f | ∈ M 1 ( ℝ d ) } .

It is known that L 1 ( ℝ d ) ⊂ L loc 1 ( ℝ d ) and M 1 ( ℝ d ) = T 1 , 1 ( ℝ d ) and furthermore, from ([2] Proposition 2. 4.5), we have T 1 , 1 ( ℝ d ) ⊆ T p , 1 ( ℝ d ) . Thus f belongs to B V p , 1 ( ℝ d ) .

W 1 , α ( ℝ d ) are vector subspaces of W 1 , α ( ℝ d ) which are themselves vector subspaces of the B V p , α ( ℝ d ) spaces.

Proposition 2.5 W 1 , α ( ℝ d ) spaces are vector subspaces of W 1 , α ( ℝ d ) spaces.

Proof. Let f ∈ W 1 , α ( ℝ d ) = { f ∈ L α ( ℝ d ) / ∇ f ∈ ( L α ( ℝ d ) ) d } then f and ∂ f ∂ x j ( 1 ≤ j ≤ d ) belongs of L α ( ℝ d ) . Since, L α ( ℝ d ) is included to L loc 1 ( ℝ d ) , then f belongs to L loc 1 ( ℝ d ) .

Proposition 2.6 W 1 , α ( ℝ d ) spaces are vector subspaces of B V p , α ( ℝ d ) spaces.

Proof. Let f ∈ W 1 , α ( ℝ d ) . We have f ∈ L loc 1 ( ℝ d ) and | ∇ f | ∈ L α ( ℝ d ) ⊂ L loc 1 ( ℝ d ) , then f ∈ W loc 1 , 1 ( ℝ d ) and therefore | D f | is the Radon measure of density | ∇ f | with respect to the Lebesgue measure

Moreover, according to ([2], proposition 2.2.12) L α ( ℝ d ) is included in F ( 1 , p , α ) ( ℝ d ) , thus | ∇ f | ∈ F ( 1 , p , α ) ( ℝ d ) and from([2], Proposition 2.4.1) | D f | belongs to T p , α ( ℝ d ) . Therefore f ∈ B V p , α ( ℝ d ) .

Remark 2.7

For more details on the B V ( ℝ d ) space, W 1 , α ( ℝ d ) and W 1 , α ( ℝ d ) spaces see [5]-[10].

B V p , α ( ℝ d ) spaces grow as the p coefficient increases.

Proposition 2.8Suppose that 1 ≤ α ≤ p 1 ≤ p 2 ≤ + ∞ . Then B V p 1 , α ( ℝ d ) is included in B V p 2 , α ( ℝ d ) .

Proof. Consider an element f of B V p 1 , α ( ℝ d ) . Then f belongs to L loc 1 ( ℝ d ) and | D f | to T p 1 , α ( ℝ d ) . According to ([2], proposition 2.4.5), T p 1 , α ( ℝ d ) is included in T p 2 , α ( ℝ d ) . So | D f | belongs to T p 2 , α ( ℝ d ) . So f belongs to B V p 2 , α ( ℝ d ) .

In this section, we are inspired by the work of Haïm Brezis [8] and Luigi Ambrosio, Nicola Fusco and Diego Pallara [7] respectively on the characterization of Sobolev spaces W 1 , α ( ℝ d ) and the space of functions with bounded variation B V ( ℝ d ) to establish a characterization of the spaces B V p , α ( ℝ d ) . Given Proposition 2.1.3 and Remark 2.1.6, it is clear that the result of this section is a generalization of Proposition 1.3.3 and Proposition 1.3. 23 which characterize by means of the rate of increase, respectively, the spaces W 1 , α ( ℝ d ) ( 1 < α ≤ + ∞ ) and B V ( ℝ d ) when α = 1 .

We begin by establishing some lemmas necessary for the result of this section and the results of the next section.

Let φ be a positive element of C c ∞ ( ℝ d ) with support included in the unit ball such that its integral over ℝ d is equal to one. For any strictly positive real ε and any f ∈ L loc 1 ( ℝ d ) , we posit:

where

The following lemma is a classic result known in the literature.

Lemma 3.1([11], Theorem 2.5.3) Let f be an element of L loc 1 ( ℝ d ) . We have

1. ∀ ε ∈ ℝ + * , f ε ∈ C ∞ ( ℝ d ) .

2. For any bounded measurable subset A of ℝ d and any element u of ℝ d ,

In the following, we will establish the continuity of the translation operator on C ∞ ( ℝ d ) .

Lemma 3.2Let f be an element of L loc 1 ( ℝ d ) and ε a strictly positive real.

For any bounded open subset Ω of ℝ d ,

where

Proof. Let Ω be an open bounded subset of ℝ d .

1. Inequality (3.3). Inspired by the proof ([8], proposition IX.3).

Consider an element y of Ω, an element u of ℝ d and set:

Then

and therefore

As f ε ( y − u ) = τ u f ε ( y ) , we have

As a result, we have

and integrating each member of the previous inequality over Ω, we obtain

Applying the Fubini-Tonelli theorem to the second member of the previous inequality, we obtain

By changing the variable z = y − t u , we obtain

2. Inequality (3.4).

(a) ϕ = ( ϕ i ) 1 ≤ i ≤ d be an element of [ C c 1 ( Ω ) ] d such that | ϕ | ≤ 1 . We now have

By changing the variable z = x − y , we get

Note that, for any element y of ℝ d , z ↦ − ϕ ( y + z ) belongs to [ C c 1 ( Ω − y ) ] d and satisfies | ϕ ( y + z ) | ≤ 1 , z ∈ ℝ d . Then, note that, for any element y of ℝ d , z ↦ − ϕ ( y + z ) belongs to [ C c 1 ( Ω − y ) ] d and satisfies | ϕ ( y + z ) | ≤ 1 , z ∈ ℝ d .

Then,

(b) Note also that, for any element y of ℝ d ,

Thus

So

In what follows, we’ll establish the continuity of the translation operator on L loc 1 ( ℝ d ) .

Lemma 3.3Let f be an element of L loc 1 ( ℝ d ) and u an element of ℝ d . Then

1. for any cube Q of ℝ d ,

2. for any countable family { Q i : i ∈ I } of cubes of ℝ d ,

Proof.

1. Let ( x , r ) be an element of ℝ d × ] 0 ; + ∞ [ .

Suppose 0 < δ < r and 0 < ε < r − δ 2 .

From the inequalities (3.3) and (3.4) of the lemma 3.2, we have

Since δ + 2 ε < r , we have

On the other hand,

and according to the triangular inequality

Integrating each member of the inequality (3.8) over Q ( x , δ ) , we have

By going to the limit, when ε tends to zero, in this last inequality and using (3.2) of the lemma 3.1, we deduce that

From inequality (3.7),

Using the fact that f belongs to L loc 1 ( ℝ d ) , χ Q ( x , r ) ∘ = χ Q ( x , r ) = χ Q ¯ ( x , r ) almost everywhere and lim δ → r χ Q ( x , δ ) = χ Q ( x , r ) ∘ , we obtain the inequality (3.5) from the inequality (3.10).

2. Let { Q i : i ∈ I } be a countable family of ℝ d cubes. From inequality (3.5), we have

So

Applying Minkowski’s inequality for integrals to the second member of the previous inequality, we have

So,

In all that follows, we will note ( e 1 , e 2 , ⋯ , e d ) the canonical basis of ℝ d . The following lemma allows us to define the derivative by transposition in L loc 1 ( ℝ d ) .

Lemma 3.4Let f be an element of L loc 1 ( ℝ d ) , Ωan open subset of ℝ d and ϕ = ( ϕ j ) 1 ≤ j ≤ d an element of [ C c 1 ( ℝ d ) ] d with support K included inΩ. Then

Proof. Let Ω be an open ℝ d and ϕ = ( ϕ j ) 1 ≤ j ≤ d an element of [ C c 1 ( ℝ d ) ] d with support K included in Ω. Suppose that 0 < δ < d i s t ( K , ∂ Ω ) and j an element of { 1 , 2 , ⋯ , d } , t ∈ ℝ .

For 0 < | t | < δ , we have

Therefore,

where K δ = { x ∈ ℝ d / d i s t ( K , x ) ≤ δ } . Note that ‖ D j ϕ j ‖ ∞ < + ∞ and K δ is a compact subset of ℝ d . Therefore, ‖ D j ϕ j ‖ ∞ χ K δ f belongs to L 1 ( ℝ d ) .

We also have for almost all x ∈ ℝ d

Therefore, according to the dominated convergence theorem, we have

Summing the previous equality over j = 1 , 2 , ⋯ , d , we obtain the equality (3.11).

Lemma 3.5Suppose that f belongs to L loc 1 ( ℝ d ) and t ∈ ℝ .

1. Let Q be an element of Δ and ϕ = ( ϕ j ) 1 ≤ j ≤ d an element of [ C c 1 ( ℝ d ) ] d with support K included in Q and such that | ϕ | ≤ 1 . Then, for 0 < | t | < d i s t ( K , ∂ Q )