Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces

1 Introduction and main results

This work studies the Schrödinger differential operator of second order in R n , where n ≥ 3 , as described by

For some exponent q ≥ n ⁄ 2 , the function V is non-negative, V ≠ 0 , and it belongs to a reverse Hölder class R H q . Assume that V is a nonnegative locally L q ( R n ) integrable function on R n , then we say that V belongs to R H q ( 1 < p ≤ ∞ ) if there exists a positive constant C such that the reverse Hölder inequality

holds for every x ∈ R n , where B ( x , r ) denotes the ball centered at x with radius r . The nonnegative polynomial V ∈ R H ∞ , and specifically ∣ x ∣ ∈ R H ∞ , should be noted (see, e.g., [1,2]).

Let the potential V ∈ R H q with q ≥ n ⁄ 2 , and the critical radius function ρ ( x ) be defined as

Clearly, 0 < m V ( x ) < ß when V ≠ 0 , and m V ( x ) = 1 when V = 1 and m V ( x ) ≈ 1 + ∣ x ∣ with V ( x ) = ∣ x ∣ 2 .

For sufficient good function f , the heat diffusion semigroup e − t L allows the negative powers L − β ⁄ 2 ( β > 0 ) associated with the Schrödinger operators L to be written as

Applying Lemma 3.3 in [3] for enough good function f holds that

The commutator of ℐ β L is defined by

Note that if L = − △ is the Laplacian on R n , then ℐ β L and [ b , ℐ β L ] are the Riesz potential I β and the commutator of the Riesz potential [ b , I β ] , respectively, i.e.,

Let θ > 0 and 0 < ν < 1 ; in view of [4], the Campanato class, associated with Schrödinger operator, Λ ν θ ( ρ ) consists of the locally integrable functions b such that

for all x ∈ R n and r > 0 . A seminorm of b ∈ Λ ν θ ( ρ ) , denoted by [ b ] ν θ , is given by the infimum of the constants in the aforementioned inequality.

Note that if θ = 0 , Λ ν θ ( ρ ) is the classical Campanato space; if ν = 0 , Λ ν θ ( ρ ) is exactly the space BMO θ ( ρ ) introduced in [1].

Throughout this article, the letter p → denotes n-tuples of the numbers in ( 0 , ∞ ] , n ≥ 1 , p → = ( p 1 , p 2 , … , p n ) , 1 ≤ p → < ∞ means 1 ≤ p i < ∞ for each i . For 1 ≤ p → ≤ ∞ , we denote p → ′ = ( p 1 ′ , … , p n ′ ) , where p i ′ satisfies 1 ⁄ p i + 1 ⁄ p i ′ = 1 .

In 2019, Nogayama [5] considered a new Morrey space, with the L p norm replaced by the mixed Lebesgue norm L p → ( R n ) , which is called mixed Morrey spaces.

We first recall the definition of mixed Lebesgue space defined in [6].

Let p → = ( p 1 , … , p n ) ∈ ( 0 , ∞ ] n . Then, the mixed Lebesgue norm ‖ ⋅ ‖ L p → or ‖ ⋅ ‖ L ( p 1 , … , p n ) is defined by

where f : R n → R is a measurable function. If p j = ∞ for some j = 1 , … , n , then we have to make appropriate modifications. We define the mixed Lebesgue space L p → ( R n ) = L ( p 1 , … , p n ) ( R n ) to be the set of all f ∈ L 0 ( R n ) with ‖ f ‖ L p → < ∞ , where L 0 ( R n ) denotes the set of measureable functions on R n .

The following analog of Hölder’s inequality for L p → is well known (see, e.g., [7]).

By elementary calculations, we have the following property.

By Theorem 1.1 and Lemma 1.1, we obtain the following estimate.

The following lemma shows the Lebesgue differential theorem in mixed-norm Lebesgue spaces as follows.

Morrey [8] proposed the traditional Morrey spaces L p , λ to examine the local behavior of solutions to elliptic partial differential equations (PDEs). Actually, a higher degree of regularity in the solutions to certain elliptic and parabolic boundary problems may be obtained thanks to the improved inclusion between the Morrey and the Hölder spaces. Generalized Morrey spaces M p , φ were separately introduced by Guliyev et al. [9–11] (see also [12–14]). Generally speaking, local Morrey spaces were also introduced separately by Guliyev [9] and Garcia-Cuerva and Herrero [15] (see also [16]). It should be noted that Guliyev introduced and analyzed integral operators in local Morrey-type spaces, including generalized local Morrey spaces, in [9].

We now present the definition of local generalized mixed Morrey space L M p → , φ α , V , { x 0 } and generalized mixed Morrey spaces M p → , φ α , V ( R n ) associated with Schrödinger operator, which in the case of p → = ( p , … , p ) introduced by Guliyev [17].

For brevity, in the sequel, we use the notations

The vanishing space V M p → , φ α , V ( R n ) is Banach space with respect to the norm

In the case of α = 0 , p → = ( p , … , p ) , and φ ( x , r ) = r ( λ − n ) ⁄ p , V M p → , φ α , V ( R n ) is the vanishing Morrey space V M p , λ introduced in [23], where applications to PDE were considered.

We refer to [24–27] for some properties of vanishing generalized Morrey spaces.

When b ∈ BMO , Chanillo proved in [28] that [ b , I β ] is bounded from L p ( R n ) to L q ( R n ) with 1 ⁄ q = 1 ⁄ p − β ⁄ n , 1 < p < n ⁄ β . When b belongs to the Campanato space Λ ν , 0 < ν < 1 , Paluszyński [29] showed that [ b , I β ] is bounded from L p ( R n ) to L q ( R n ) with 1 ⁄ q = 1 ⁄ p − ( β + ν ) ⁄ n , 1 < p < n ⁄ ( β + ν ) . When b ∈ BMO θ ( ρ ) , Bui [30] obtained the boundedness of [ b , ℐ β L ] from L p ( R n ) to L q ( R n ) with 1 ⁄ q = 1 ⁄ p − β ⁄ n , 1 < p < n ⁄ β .

Inspired by the aforementioned results, we are interested in the boundedness of [ b , ℐ β L ] generalized mixed Morrey spaces M p → , φ α , V ( R n ) and the vanishing generalized mixed Morrey spaces V M p → , φ α , V ( R n ) , when b belongs to the new Campanato class Λ ν θ ( ρ ) .

In this article, we consider the boundedness of the commutator of ℐ β L on the local generalized mixed Morrey spaces L M p → , φ α , V , { x 0 } , the generalized mixed Morrey spaces M p → , φ α , V ( R n ) , and the vanishing generalized mixed Morrey spaces V M p → , φ α , V ( R n ) . When b belongs to the new Campanato space Λ ν θ ( ρ ) , 0 < ν < 1 , we show that [ b , ℐ β L ] are bounded from L M p → , φ 1 α , V , { x 0 } to L M q → , φ 2 α , V , { x 0 } , from M p → , φ α , V ( R n ) to M q → , φ α , V ( R n ) , and from V M p → , φ α , V ( R n ) to V M q → , φ α , V ( R n ) with ∑ i = 1 n 1 ⁄ p i − ∑ i = 1 n 1 ⁄ q i = β + ν , 1 < p → < n ⁄ ( β + ν ) .

Our main results are the following.

In this article, we shall use the symbol A ≲ B to indicate that there exists a universal positive constant C , independent of all important parameters, such that A ≤ C B . A ≈ B means that A ≲ B and B ≲ A .

2 Some technical lemmas and propositions

We would like to recall the important properties concerning the critical function.

According to [1], the new BMO space BMO θ ( ρ ) with θ ≥ 0 is defined as a set of all locally integrable functions b such that

for all x ∈ R n and r > 0 , where b B = 1 ∣ B ∣ ∫ B b ( y ) d y and BMO ≡ BMO _θ (ρ). A norm for b ∈ BMO θ ( ρ ) , denoted by [ b ] θ , is given by the infimum of the constants in the aforementioned inequalities. Clearly, B M O ⊂ BMO θ ( ρ ) .

Let θ > 0 and 0 < ν < 1 ; a seminorm on Campanato class Λ ν θ ( ρ ) is denoted by [ b ] ν θ

The Lipschitz space, associated with Schrödinger operator (see [4]), consists of the functions f satisfying

It is easy to see that this space is exactly the Lipschitz space when θ = 0 .

Note that if θ = 0 in (2.2), Λ ν θ ( ρ ) is exactly the classical Campanato space; if ν = 0 , Λ ν θ ( ρ ) is exactly the space BMO θ ( ρ ) ; if θ = 0 and ν = 0 , it is exactly the John-Nirenberg space BMO .

The following relation between Lip ν θ ( ρ ) and Λ ν θ ( ρ ) was proved in [4, Theorem 5].

We list some properties involving Campanato space, associated with Schrödinger operator Λ ν θ ( ρ ) .

Let K β be the kernel of ℐ β L . The following result gives the estimate on the kernel K β ( x , y ) .

Finally, we recall a relationship between essential supremum and essential infimum.

It is natural, first of all, to find conditions ensuring that the spaces L M p → , φ α , V , { x 0 } and M p → , φ α , V are nontrivial, which consist not only of functions equivalent to 0 on R n .