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Stein-Weiss inequality for local mixed radial-angular Morrey spaces

  • Mingquan Wei EMAIL logo , Fangming Su and Lanyin Sun
Published/Copyright: October 20, 2022
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Open Mathematics
From the journal Open Mathematics Volume 20 Issue 1

Abstract

In this article, a generalization of the well-known Stein-Weiss inequality for the fractional integral operator on functions with different integrability properties in the radial and the angular direction in local Morrey spaces is established. We find that some conditions can be relaxed for the Stein-Weiss inequality for local mixed radial-angular Morrey spaces.

Keywords: Stein-Weiss inequality; fractional integral operator; mixed radial-angular space; local Morrey space
MSC 2010: 42B20; 42B25; 42B35

1 Introduction

In this article, we devote to extending the celebrated Stein-Weiss inequality to local mixed radial-angular Morrey spaces.

The Stein-Weiss inequality [1] gives the two power-weighted norm inequalities for fractional integral operator. After that, the two-weighted inequalities for fractional integral operator were extended to general weight functions. For instance, the weighted norm inequalities with Muckenhoupt weights for fractional integral operator were considered in [2,3, 4,5]. Note that, if we consider the power weights, De Nápoli et al. [6] and Hidano and Kurokawa [7] proved that some conditions in [1] can be relaxed. And more generally, if the functions under consideration have different integrability properties in the radial and the angular direction, i.e., mixed radial-angular spaces, D’Ancona and Luca’ [8] pointed out that the conditions in [1,6,7] can be extended to a more general setting. Obviously, the functions with different integrability properties in the radial and the angular direction may not be radial functions, and the result in [8, Theorem 1.3] essentially improves the results of [6,7]. Note that recently the mixed radial-angular spaces have been successfully used to study Strichartz estimates and partial differential equations to improve the corresponding results (see [9,10,11, 12,13], etc.).

As we know, Morrey spaces, initially introduced by Morrey in [14], are natural generalizations of Lebesgue spaces. Many important results in harmonic analysis, such as the mapping properties of some important integral operators, have been extended to Morrey spaces. Particularly, the boundedness of the fractional integral operator on Morrey-type spaces was established in [15,16,17, 18,19]. Nowadays, the Stein-Weiss inequality has been successfully generalized to weighted Morrey spaces by Ho [20]. Recently, based on the results of [6, Theorem 1.2] and [7, Theorem 2.1], Ho [21] considered the Stein-Weiss inequality for radial functions in local Morrey spaces and showed that some conditions can be relaxed. Inspired by [8,21], we will consider the Stein-Weiss inequality for functions with different integrability properties in the radial and the angular direction in local Morrey spaces.

In view of [8, Theorem 1.3], it is expectable that when we consider functions with different integrability properties in local Morrey spaces, the conditions of [21, Theorem 3.1] can also be relaxed. As the central versions of the classical Morrey spaces, the local Morrey spaces are also important function spaces to study the mapping properties of integral operators. We refer the readers to [22,23, 24,25] for more studies of local Morrey spaces. See also [26] for the extrapolation theory on local Morrey spaces with variable exponents.

The main result of this article can be seen as a complement of the mapping properties of fractional integral operator acting on local Morrey spaces. As applications of the main result, we obtain the Poincaré and Sobolev inequalities for local mixed radial-angular Morrey spaces.

The organization of the remainder of this article is as follows. Section 2 contains the definitions of local Morrey spaces and local mixed radial-angular Morrey spaces used in this article. The celebrated Stein-Weiss inequality and some extensions of it to radial functions will also be presented in this section. The main result of this article, Stein-Weiss inequality for local mixed radial-angular Morrey spaces, is proved in Section 3. As applications, we give the Poincaré and Sobolev inequalities for functions with different integrability properties in the radial and the angular direction in local Morrey spaces in Section 4.

2 Definitions and preliminaries

Throughout the article, we use the following notations.

For any r > 0 and x ∈ R n , let B ( x , r ) = { y : ∣ y − x ∣ < r } be the ball centered at x with radius r . Let B = { B ( x , r ) : x ∈ R n , r > 0 } be the set of all such balls. We use χ E and ∣ E ∣ to denote the characteristic function and the Lebesgue measure of a measurable set E . Let ℳ ( E ) be the class of Lebesgue measurable functions on E . For a nonnegative function w ∈ L loc 1 ( R n ) and 1 ≤ p < ∞ , the weighted Lebesgue space L w p ( R n ) consists of all f ∈ ℳ ( R n ) such that ‖ f ‖ L w p ≔ ∫ R n ∣ f ( x ) ∣ p w ( x ) d x 1 ∕ p < ∞ . In particular, if w ( x ) = ∣ x ∣ α p , then we write L w p ( R n ) = L α p ( R n ) . By A ≲ B , we mean that A ≤ C B for some constant C > 0 .

Let 0 < γ < n , the fractional integral operator I γ is defined by

I γ f ( x ) = ∫ R n f ( y ) ∣ x − y ∣ γ d y , x ∈ R n ,

where f ∈ L loc 1 ( R n ) .

We first recall the famous Stein-Weiss inequality for the fractional integral operator.

Theorem 2.1

Let n ≥ 1 , 0 < γ < n , 1 < p < ∞ , α < n p ′ , β < n q , α + β ≥ 0 , and 1 q = 1 p + γ + α + β n − 1 . If p ≤ q < ∞ , then for all f ∈ L α p ( R n ) , we have

‖ I γ f ‖ L − β q ≲ ‖ f ‖ L α p .

The reader is referred to [1, Theorem B*], for the proof of the above theorem.

Define by

ℛ ( R n ) = { f ∈ ℳ ( R n ) : f ( x ) = f ( y ) , if ∣ x ∣ = ∣ y ∣ } .

For any 1 ≤ p < ∞ and α ∈ R , define L α , rad p ( R n ) = L α p ( R n ) ∩ ℛ ( R n ) . Then the Stein-Weiss inequality for the radial functions can be stated as follows.

Theorem 2.2

Let n ≥ 1 , 0 < γ < n , 1 < p < ∞ , α < n p ′ , β < n q , α + β ≥ ( n − 1 ) 1 q − 1 p , and 1 q = 1 p + γ + α + β n − 1 . If p ≤ q < ∞ , then for all f ∈ L α , rad p ( R n ) , we have

‖ I γ f ‖ L − β q ≲ ‖ f ‖ L α p .

The proof of Theorem 2.2 can be found in [6, Theorem 1.2] and [7, Theorem 2.1].

Next, we give the definition of local Morrey spaces.

Definition 2.1

Let 1 ≤ p < ∞ , w : R n → [ 0 , ∞ ) , and u : ( 0 , ∞ ) ⟶ ( 0 , ∞ ) be Lebesgue measurable functions. The local Morrey spaces L M w p , u ( R n ) consist of all f ∈ ℳ ( R n ) such that

‖ f ‖ L M w p , u = sup r > 0 1 u ( r ) ‖ f χ B ( 0 , r ) ‖ L w p < ∞ .

In particular, if w ( x ) = ∣ x ∣ α p , then we write L M w p , u ( R n ) = L M α p , u ( R n ) .

We refer the readers to [22,23, 24,25] for the mapping properties for various integral operators on local Morrey spaces. For 1 ≤ p < ∞ , α ∈ R , and u : ( 0 , ∞ ) ⟶ ( 0 , ∞ ) , the radial local Morrey spaces L M α , rad p , u ( R n ) consist of all radial functions f ∈ L M α p , u ( R n ) , i.e.,

L M α , rad p , u ( R n ) = L M α p , u ( R n ) ∩ ℛ ( R n ) .

The Stein-Weiss inequality for radial local Morrey spaces is as follows, see [21], Theorem 3.1] for the proof.

Theorem 2.3

Let n ≥ 1 , 0 < γ < n , 1 < p < ∞ , α < n p ′ , β < n q , α + β ≥ ( n − 1 ) 1 q − 1 p , and 1 q = 1 p + γ + α + β n − 1 and u : ( 0 , ∞ ) ⟶ ( 0 , ∞ ) . If p ≤ q < ∞ and there exists a constant C > 0 such that for any r > 0 , u satisfies

(2.1) u ( 2 r ) ≤ C u ( r ) ,

(2.2) ∑ j = 0 ∞ 2 − ( j + 1 ) ( n q − β ) u ( 2 j + 1 r ) ≤ C u ( r ) ,

then for any f ∈ L M α , rad p , u ( R n ) , we have

‖ I γ f ‖ L M − β q , u ≲ ‖ f ‖ L M α p , u .

Note that when n = 1 , Theorem 2.2 coincides with Theorem 2.1. And in this case, the conditions imposed on p , q , α , β of Theorems 2.1, 2.2, and 2.3 are the same. However, when n ≥ 2 , the condition α + β ≥ ( n − 1 ) 1 q − 1 p in Theorems 2.2 and 2.3 implies that α + β may be negative, while Theorem 2.1 requires α + β ≥ 0 . This observation yields that the Stein-Weiss inequality imposed on radial functions has better performance.

The mixed radial-angular spaces, including the functions with different integrability properties in the radial and the angular direction may not be radial functions, are extensions of Lebesgue spaces. Now we recall their definitions.

Definition 2.2

For n ≥ 1 , 1 ≤ p , p ¯ ≤ ∞ , the mixed radial-angular space L ∣ x ∣ p L θ p ¯ ( R n ) consist of all functions f ∈ ℳ ( R n ) for which

‖ f ‖ L ∣ x ∣ p L θ p ¯ ≔ ∫ 0 ∞ ∫ S n − 1 ∣ f ( r , θ ) ∣ p ¯ d θ p ∕ p ¯ r n − 1 d r 1 ∕ p < ∞ ,

where S n − 1 denotes the unit sphere in R n . If p = ∞ or p ¯ = ∞ , then we have to make appropriate modifications.

Similar to the power-weighted Lebesgue spaces, we can define the power-weighted mixed radial-angular spaces.

Definition 2.3

For n ≥ 1 , α ∈ R , 1 ≤ p , p ¯ ≤ ∞ , the power-weighted mixed radial-angular spaces L ∣ x ∣ p , α L θ p ¯ ( R n ) consists of all f ∈ ℳ ( R n ) such that

‖ f ‖ L ∣ x ∣ p , α L θ p ¯ ≔ ‖ f v ‖ L ∣ x ∣ p L θ p ¯ < ∞ ,

where v ( x ) = ∣ x ∣ α .

The mixed radial-angular spaces were initially introduced to improve some classical results in PDE, see [9,10, 11,12]. Later, the mixed radial-angular spaces were frequently used in harmonic analysis. For instance, Liu et al. [27,28,29] considered the mapping properties of various operators with rough kernels on mixed radial-angular spaces. See also [30] for the extrapolation theorems on mixed radial-angular spaces. One can see that the mixed radial-angular spaces are particular cases of mixed-norm Lebesgue spaces studied by Benedek and Panzone [31]. The readers are referred to [32,33, 34,35,36, 37,38] for more studies on mixed-norm Lebesgue spaces.

When we consider function with different integrability properties in the radial and the angular direction, D’Ancona and Luca’ [8, Theorem 1.3] obtained the following extensions of Stein-Weiss inequality for mixed radial-angular spaces.

Theorem 2.4

Let n ≥ 1 , 0 < γ < n , 1 < p < ∞ , α < n p ′ , β < n q , α + β ≥ ( n − 1 ) 1 q − 1 p + 1 p ¯ − 1 q ¯ , and 1 q = 1 p + γ + α + β n − 1 . If p ≤ q < ∞ and 1 ≤ p ¯ ≤ q ¯ ≤ ∞ , then for all f ∈ L ∣ x ∣ p , α L θ p ¯ ( R n ) , we have

‖ I γ f ‖ L ∣ x ∣ q , − β L θ q ¯ ≲ ‖ f ‖ L ∣ x ∣ p , α L θ p ¯ .

By combining the definitions of local Morrey spaces and mixed radial-angular spaces, we can define local mixed radial-angular Morrey spaces as follows:

Definition 2.4

Let 1 ≤ p , p ¯ ≤ ∞ , and u : ( 0 , ∞ ) ⟶ ( 0 , ∞ ) be a Lebesgue measurable function. The local mixed radial-angular Morrey spaces L M u p , p ¯ ( R n ) consist of all f ∈ ℳ ( R n ) such that

‖ f ‖ L M u p , p ¯ = sup r > 0 1 u ( r ) ‖ f χ B ( 0 , r ) ‖ L ∣ x ∣ p L θ p ¯ < ∞ .

Roughly speaking, the local mixed radial-angular Morrey spaces are just the functions in local Morrey spaces which have different integrability properties in the radial and the angular direction. Therefore, local mixed radial-angular Morrey spaces are extensions of radial local Morrey spaces, since the integrability in the angular direction holds automatically for radial functions. Consequently, it is meaningful to consider Stein-Weiss inequality for local mixed radial-angular Morrey spaces.

3 Main result

This section establishes the Stein-Weiss inequality on local mixed radial-angular Morrey spaces. To do this, we first give the definition of power-weighted local mixed radial-angular Morrey spaces, which are combinations of power-weighted mixed radial-angular spaces and local Morrey spaces.

Definition 3.1

Let α ∈ R , 1 ≤ p , p ¯ ≤ ∞ , and u : ( 0 , ∞ ) ⟶ ( 0 , ∞ ) be a Lebesgue measurable function. The power-weighted local mixed radial-angular Morrey space L M α , u p , p ¯ ( R n ) consists of all f ∈ ℳ ( R n ) such that

‖ f ‖ L M α , u p , p ¯ = sup r > 0 1 u ( r ) ‖ f χ B ( 0 , r ) ‖ L ∣ x ∣ p , α L θ p ¯ < ∞ .

Now we are in a position to state our main result in this article.

Theorem 3.1

Let n ≥ 1 , 0 < γ < n , 1 < p < ∞ , α < n p ′ , β < n q , α + β ≥ ( n − 1 ) 1 q − 1 p + 1 p ¯ − 1 q ¯ , 1 q = 1 p + γ + α + β n − 1 , and u : ( 0 , ∞ ) ⟶ ( 0 , ∞ ) be a Lebesgue measurable function. If p ≤ q < ∞ and 1 ≤ p ¯ ≤ q ¯ ≤ ∞ and there is a constant C > 0 such that for any r > 0 , u satisfies (2.1) and (2.2) in Theorem 2.3, then for any f ∈ L M α , u p , p ¯ ( R n ) , we have

‖ I γ f ‖ L M − β , u q , q ¯ ≲ ‖ f ‖ L M α , u p , p ¯ .

Proof

Let f ∈ L M α , u p , p ¯ ( R n ) . For any r > 0 , define

D k = B ( 0 , 2 k + 1 ) \ B ( 0 , 2 k ) , k ∈ N \ { 0 } .

Denote by f = ∑ k = 0 ∞ f k , where f k = f χ D k , k ≥ 1 , and f 0 = f χ B ( 0 , 2 r ) .

Noting that I γ is a sublinear operator, there holds

(3.1) ∣ I γ f ∣ ≤ ∑ k = 0 ∞ I γ ∣ f k ∣ .

Since f ∈ L M α , u p , p ¯ ( R n ) , we have ∣ f ∣ ∈ L M α , u p , p ¯ ( R n ) . As a consequence, Theorem 2.4 guarantees

1 u ( r ) ‖ χ B ( 0 , r ) I γ ∣ f 0 ∣ ‖ L ∣ x ∣ q , − β L θ q ¯ ≤ 1 u ( r ) ‖ I γ ∣ f 0 ∣ ‖ L ∣ x ∣ q , − β L θ q ¯ ≲ 1 u ( r ) ‖ f 0 ‖ L ∣ x ∣ p , α L θ p ¯ .

Consequently, (2.1) implies

(3.2) 1 u ( r ) ‖ χ B ( 0 , r ) I γ ∣ f 0 ∣ ‖ L ∣ x ∣ q , − β L θ q ¯ ≲ 1 u ( 2 r ) ‖ f χ B ( 0 , 2 r ) ‖ L ∣ x ∣ p , α L θ p ¯ ≲ ‖ f ‖ L M α , u p , p ¯ .

Next we consider the terms I γ f k , where k ≥ 1 . From the definitions of I γ , for any x ∈ B ( 0 , r ) , we have

I γ ∣ f k ∣ ( x ) ≲ 1 2 k γ r γ ∫ D k ∣ f k ( y ) ∣ d y .

By using Hölder’s inequality on mixed Lebesgue spaces (see [31]), we obtain

(3.3) I γ ∣ f k ∣ ( x ) ≲ 1 2 k γ r γ ‖ f k ‖ L ∣ x ∣ p , α L θ p ¯ ‖ χ B ( 0 , 2 k + 1 r ) ‖ L ∣ x ∣ p ′ , − α L θ p ¯ ′ .

A direct calculation yields

(3.4) ‖ χ B ( 0 , 2 k + 1 r ) ‖ L ∣ x ∣ p ′ , − α L θ p ¯ ′ = w n 1 ∕ p ¯ ′ ∫ 0 2 k + 1 r r − p ′ α + n − 1 d r 1 ∕ p ′ = w n 1 ∕ p ¯ ′ ( 2 k + 1 r ) − α + n ∕ p ′ − p ′ α + n = C 2 ( k + 1 ) ( − α + n ∕ p ′ ) r − α + n ∕ p ′

for some constant C > 0 , where w n is the induced Lebesgue measure of S n − 1 .

From the assumption 1 q = 1 p + γ + α + β n − 1 , we obtain

(3.5) − α + n p ′ − γ = − α + n 1 − 1 p − γ = β − n q .

Inequality (3.3), together with (3.4) and (3.5), shows that

(3.6) I γ ∣ f k ∣ ( x ) ≲ 2 ( k + 1 ) ( − α + n ∕ p ′ − γ ) r − α + n ∕ p ′ − γ ‖ f k ‖ L ∣ x ∣ p , α L θ p ¯ = 2 ( k + 1 ) ( β − n ∕ q ) r β − n ∕ q ‖ f k ‖ L ∣ x ∣ p , α L θ p ¯ .

By multiplying χ B ( 0 , r ) on both sides of inequality (3.3), and then using (3.4) and (3.6), we arrive at

‖ χ B ( 0 , r ) I γ ∣ f k ∣ ‖ L ∣ x ∣ q , − β L θ q ¯ ≲ 2 ( k + 1 ) ( β − n ∕ q ) r β − n ∕ q ‖ f k ‖ L ∣ x ∣ p , α L θ p ¯ ‖ χ B ( 0 , r ) ‖ L ∣ x ∣ q , − β L θ q ¯ ≲ 2 ( k + 1 ) ( β − n ∕ q ) r β − n ∕ q ‖ f k ‖ L ∣ x ∣ p , α L θ p ¯ × r − β + n ∕ q ≲ 2 ( k + 1 ) ( β − n ∕ q ) ‖ f k ‖ L ∣ x ∣ p , α L θ p ¯ .

As a consequence,

(3.7) 1 u ( r ) ‖ χ B ( 0 , r ) I γ ∣ f k ∣ ‖ L ∣ x ∣ q , − β L θ q ¯ ≲ u ( 2 k + 1 r ) u ( r ) 2 ( k + 1 ) ( β − n ∕ q ) 1 u ( 2 k + 1 r ) ‖ f χ B ( 0 , 2 k + 1 r ) ‖ L ∣ x ∣ p , α L θ p ¯ ≲ u ( 2 k + 1 r ) u ( r ) 2 ( k + 1 ) ( β − n ∕ q ) ‖ f ‖ L M α , u p , p ¯ .

As a result of (2.2), (3.1), (3.2), and (3.7), there holds

∑ k = 0 ∞ 1 u ( r ) ‖ χ B ( 0 , r ) I γ ∣ f k ∣ ‖ L ∣ x ∣ q , − β L θ q ¯ ≲ 1 + ∑ k = 0 ∞ u ( 2 k + 1 r ) u ( r ) 2 ( k + 1 ) ( β − n ∕ q ) ‖ f ‖ L M α , u p , p ¯ ≲ ‖ f ‖ L M α , u p , p ¯ ,

since β < n ∕ q .

By taking the supremum over r > 0 on the above inequalities, one obtains

‖ I γ f ‖ L M − β , u q , q ¯ = sup r > 0 1 u ( r ) ‖ χ B ( 0 , r ) I γ ∣ f ∣ ‖ L ∣ x ∣ q , − β L θ q ¯ ≲ sup r > 0 ∑ k = 0 ∞ 1 u ( r ) ‖ χ B ( 0 , r ) I γ ∣ f k ∣ ‖ L ∣ x ∣ q , − β L θ q ¯ ≲ ‖ f ‖ L M α , u p , p ¯ ,

which finishes the proof.□

Remark 3.1

  1. When the functions under consideration in Theorem 3.1 are radial, we can choose p ¯ = q ¯ = s for some 1 < s < ∞ . In this sense, we obtain Theorem 2.3 as a consequence of Theorem 3.1.

  2. Conditions (2.1) and (2.2) are satisfied for many functions u . For instance, if we take u ( r ) = r σ for some 0 < σ < n q − β , then u fulfills (2.1) and (2.2).

4 Applications

By applying Theorem 3.1, we will establish the Poincaré and Sobolev inequalities for local mixed radial-angular Morrey spaces.

The Poincaré inequality is closely related to the Harnack’s inequalities, see [39]. Now we give the first result of this section.

Theorem 4.1

Let n > 1 , 1 < p < ∞ , α < n p ′ , β < n q , α + β ≥ ( n − 1 ) 1 q − 1 p + 1 p ¯ − 1 q ¯ , 1 q = 1 p + α + β − 1 n , and u : ( 0 , ∞ ) ⟶ ( 0 , ∞ ) be a Lebesgue measurable function. If p ≤ q < ∞ and 1 ≤ p ¯ ≤ q ¯ ≤ ∞ and there is a constant C > 0 such that for any r > 0 , u satisfies (2.1) and (2.2) in Theorem 2.3, then for any D ∈ B and any continuous differentiable function f satisfying either ∫ D f ( x ) d x = 0 or supp f ⊆ D , there holds

sup r > 0 1 u ( r ) ‖ f χ D ∩ B ( 0 , r ) ‖ L ∣ x ∣ q , − β L θ q ¯ ≲ sup r > 0 1 u ( r ) ‖ ∇ f χ B ( 0 , r ) ‖ L ∣ x ∣ p , α L θ p ¯ .

where ∇ is the gradient operator.

Proof

By virtue of [40, (4.34) and (4.35)], we know that for any x ∈ D ,

∣ f ( x ) ∣ ≲ I n − 1 ( ∣ ∇ f ( x ) ∣ χ D ( x ) ) .

By using Theorem 3.1 with γ = n − 1 , we obtain the desired result.□

The second application is connected with the Sobolev inequality. As we know, the Sobolev inequality gives a two-weighted norm inequality for the Laplacian operator Δ . Next we present the Sobolev inequality for local mixed radial-angular Morrey spaces.

Theorem 4.2

Let n > 2 , 1 < p < ∞ , α < n p ′ , β < n q , α + β ≥ ( n − 1 ) 1 q − 1 p + 1 p ¯ − 1 q ¯ , 1 q = 1 p + α + β − 2 n , and u : ( 0 , ∞ ) ⟶ ( 0 , ∞ ) be a Lebesgue measurable function. If p ≤ q < ∞ and 1 ≤ p ¯ ≤ q ¯ ≤ ∞ and there is a constant C > 0 such that for any r > 0 , u satisfies (2.1) and (2.2) in Theorem 2.3, then for any D ∈ B and any twice continuous differentiable function f satisfying supp f ⊆ D , there holds

(4.1) sup r > 0 1 u ( r ) ‖ f χ D ∩ B ( 0 , r ) ‖ L ∣ x ∣ q , − β L θ q ¯ ≲ sup r > 0 1 u ( r ) ‖ Δ f χ B ( 0 , r ) ‖ L ∣ x ∣ p , α L θ p ¯ .

Proof

Noting that f = I n − 2 ( Δ f ) , (4.1) is a consequence of Theorem 3.1 with γ = n − 2 .□

Acknowledgements

The authors would like to express their deep gratitude to the anonymous referees for their careful reading of the manuscript and their comments and suggestions.

  1. Funding information: This work was supported by the Science and Technology Project of Henan Province (212102210394), the Natural Science Foundation of Henan Province (202300410338), the Program for Science Technology Innovation Talents in Universities of Henan Province (22HASTIT021), and the Nanhu Scholar Program for Young Scholars of Xinyang Normal University.

  2. Conflict of interest: The authors state no conflict of interest.

  3. Data availability statement: No data, models, or code are generated or used during the study.

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Received: 2021-11-18
Revised: 2022-04-29
Accepted: 2022-09-09
Published Online: 2022-10-20

© 2022 Mingquan Wei et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Regular Articles
  2. A random von Neumann theorem for uniformly distributed sequences of partitions
  3. Note on structural properties of graphs
  4. Mean-field formulation for mean-variance asset-liability management with cash flow under an uncertain exit time
  5. The family of random attractors for nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients
  6. The intersection graph of graded submodules of a graded module
  7. Isoperimetric and Brunn-Minkowski inequalities for the (p, q)-mixed geominimal surface areas
  8. On second-order fuzzy discrete population model
  9. On certain functional equation in prime rings
  10. General complex Lp projection bodies and complex Lp mixed projection bodies
  11. Some results on the total proper k-connection number
  12. The stability with general decay rate of hybrid stochastic fractional differential equations driven by Lévy noise with impulsive effects
  13. Well posedness of magnetohydrodynamic equations in 3D mixed-norm Lebesgue space
  14. Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems
  15. Generic uniqueness of saddle point for two-person zero-sum differential games
  16. Relational representations of algebraic lattices and their applications
  17. Explicit construction of mock modular forms from weakly holomorphic Hecke eigenforms
  18. The equivalent condition of G-asymptotic tracking property and G-Lipschitz tracking property
  19. Arithmetic convolution sums derived from eta quotients related to divisors of 6
  20. Dynamical behaviors of a k-order fuzzy difference equation
  21. The transfer ideal under the action of orthogonal group in modular case
  22. The multinomial convolution sum of a generalized divisor function
  23. Extensions of Gronwall-Bellman type integral inequalities with two independent variables
  24. Unicity of meromorphic functions concerning differences and small functions
  25. Solutions to problems about potentially Ks,t-bigraphic pair
  26. Monotonicity of solutions for fractional p-equations with a gradient term
  27. Data smoothing with applications to edge detection
  28. An ℋ-tensor-based criteria for testing the positive definiteness of multivariate homogeneous forms
  29. Characterizations of *-antiderivable mappings on operator algebras
  30. Initial-boundary value problem of fifth-order Korteweg-de Vries equation posed on half line with nonlinear boundary values
  31. On a more accurate half-discrete Hilbert-type inequality involving hyperbolic functions
  32. On split twisted inner derivation triple systems with no restrictions on their 0-root spaces
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  34. Bifurcation and chaos in a discrete predator-prey system of Leslie type with Michaelis-Menten prey harvesting
  35. A posteriori error estimates of characteristic mixed finite elements for convection-diffusion control problems
  36. Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect
  37. An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem
  38. Connectivity with respect to α-discrete closure operators
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  41. New properties for the Ramanujan R-function
  42. Shooting method in the application of boundary value problems for differential equations with sign-changing weight function
  43. Ground state solution for some new Kirchhoff-type equations with Hartree-type nonlinearities and critical or supercritical growth
  44. Existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delays
  45. Ambrosetti-Prodi-type results for a class of difference equations with nonlinearities indefinite in sign
  46. Research of cooperation strategy of government-enterprise digital transformation based on differential game
  47. Malmquist-type theorems on some complex differential-difference equations
  48. Disjoint diskcyclicity of weighted shifts
  49. Construction of special soliton solutions to the stochastic Riccati equation
  50. Remarks on the generalized interpolative contractions and some fixed-point theorems with application
  51. Analysis of a deteriorating system with delayed repair and unreliable repair equipment
  52. On the critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields
  53. The exact solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities using the dynamical system and the first integral methods
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  55. Multiplicity solutions for a class of p-Laplacian fractional differential equations via variational methods
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  61. Orbital stability and Zhukovskiǐ quasi-stability in impulsive dynamical systems
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  65. Double domination in maximal outerplanar graphs
  66. Existence and uniqueness of solutions to the norm minimum problem on digraphs
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  68. Robust estimation for varying coefficient partially functional linear regression models based on exponential squared loss function
  69. Hessian equations of Krylov type on compact Hermitian manifolds
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  75. Dependence of eigenvalues of Sturm-Liouville problems on time scales with eigenparameter-dependent boundary conditions
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  77. Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces
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  79. Spectra and reticulation of semihoops
  80. Stein-Weiss inequality for local mixed radial-angular Morrey spaces
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  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
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  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
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  122. Admissible congruences on type B semigroups
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  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
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  143. Jensen-type inequalities for m-convex functions
  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
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https://doi.org/10.1515/math-2022-0500
Keywords for this article
Stein-Weiss inequality; fractional integral operator; mixed radial-angular space; local Morrey space
Creative Commons
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Articles in the same Issue

  1. Regular Articles
  2. A random von Neumann theorem for uniformly distributed sequences of partitions
  3. Note on structural properties of graphs
  4. Mean-field formulation for mean-variance asset-liability management with cash flow under an uncertain exit time
  5. The family of random attractors for nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients
  6. The intersection graph of graded submodules of a graded module
  7. Isoperimetric and Brunn-Minkowski inequalities for the (p, q)-mixed geominimal surface areas
  8. On second-order fuzzy discrete population model
  9. On certain functional equation in prime rings
  10. General complex Lp projection bodies and complex Lp mixed projection bodies
  11. Some results on the total proper k-connection number
  12. The stability with general decay rate of hybrid stochastic fractional differential equations driven by Lévy noise with impulsive effects
  13. Well posedness of magnetohydrodynamic equations in 3D mixed-norm Lebesgue space
  14. Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems
  15. Generic uniqueness of saddle point for two-person zero-sum differential games
  16. Relational representations of algebraic lattices and their applications
  17. Explicit construction of mock modular forms from weakly holomorphic Hecke eigenforms
  18. The equivalent condition of G-asymptotic tracking property and G-Lipschitz tracking property
  19. Arithmetic convolution sums derived from eta quotients related to divisors of 6
  20. Dynamical behaviors of a k-order fuzzy difference equation
  21. The transfer ideal under the action of orthogonal group in modular case
  22. The multinomial convolution sum of a generalized divisor function
  23. Extensions of Gronwall-Bellman type integral inequalities with two independent variables
  24. Unicity of meromorphic functions concerning differences and small functions
  25. Solutions to problems about potentially Ks,t-bigraphic pair
  26. Monotonicity of solutions for fractional p-equations with a gradient term
  27. Data smoothing with applications to edge detection
  28. An ℋ-tensor-based criteria for testing the positive definiteness of multivariate homogeneous forms
  29. Characterizations of *-antiderivable mappings on operator algebras
  30. Initial-boundary value problem of fifth-order Korteweg-de Vries equation posed on half line with nonlinear boundary values
  31. On a more accurate half-discrete Hilbert-type inequality involving hyperbolic functions
  32. On split twisted inner derivation triple systems with no restrictions on their 0-root spaces
  33. Geometry of conformal η-Ricci solitons and conformal η-Ricci almost solitons on paracontact geometry
  34. Bifurcation and chaos in a discrete predator-prey system of Leslie type with Michaelis-Menten prey harvesting
  35. A posteriori error estimates of characteristic mixed finite elements for convection-diffusion control problems
  36. Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect
  37. An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem
  38. Connectivity with respect to α-discrete closure operators
  39. Khasminskii-type theorem for a class of stochastic functional differential equations
  40. On some new Hermite-Hadamard and Ostrowski type inequalities for s-convex functions in (p, q)-calculus with applications
  41. New properties for the Ramanujan R-function
  42. Shooting method in the application of boundary value problems for differential equations with sign-changing weight function
  43. Ground state solution for some new Kirchhoff-type equations with Hartree-type nonlinearities and critical or supercritical growth
  44. Existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delays
  45. Ambrosetti-Prodi-type results for a class of difference equations with nonlinearities indefinite in sign
  46. Research of cooperation strategy of government-enterprise digital transformation based on differential game
  47. Malmquist-type theorems on some complex differential-difference equations
  48. Disjoint diskcyclicity of weighted shifts
  49. Construction of special soliton solutions to the stochastic Riccati equation
  50. Remarks on the generalized interpolative contractions and some fixed-point theorems with application
  51. Analysis of a deteriorating system with delayed repair and unreliable repair equipment
  52. On the critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields
  53. The exact solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities using the dynamical system and the first integral methods
  54. Regularity of models associated with Markov jump processes
  55. Multiplicity solutions for a class of p-Laplacian fractional differential equations via variational methods
  56. Minimal period problem for second-order Hamiltonian systems with asymptotically linear nonlinearities
  57. Convergence rate of the modified Levenberg-Marquardt method under Hölderian local error bound
  58. Non-binary quantum codes from constacyclic codes over 𝔽q[u1, u2,…,uk]/⟨ui3 = ui, uiuj = ujui⟩
  59. On the general position number of two classes of graphs
  60. A posteriori regularization method for the two-dimensional inverse heat conduction problem
  61. Orbital stability and Zhukovskiǐ quasi-stability in impulsive dynamical systems
  62. Approximations related to the complete p-elliptic integrals
  63. A note on commutators of strongly singular Calderón-Zygmund operators
  64. Generalized Munn rings
  65. Double domination in maximal outerplanar graphs
  66. Existence and uniqueness of solutions to the norm minimum problem on digraphs
  67. On the p-integrable trajectories of the nonlinear control system described by the Urysohn-type integral equation
  68. Robust estimation for varying coefficient partially functional linear regression models based on exponential squared loss function
  69. Hessian equations of Krylov type on compact Hermitian manifolds
  70. Class fields generated by coordinates of elliptic curves
  71. The lattice of (2, 1)-congruences on a left restriction semigroup
  72. A numerical solution of problem for essentially loaded differential equations with an integro-multipoint condition
  73. On stochastic accelerated gradient with convergence rate
  74. Displacement structure of the DMP inverse
  75. Dependence of eigenvalues of Sturm-Liouville problems on time scales with eigenparameter-dependent boundary conditions
  76. Existence of positive solutions of discrete third-order three-point BVP with sign-changing Green's function
  77. Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces
  78. Generalized 4-connectivity of hierarchical star networks
  79. Spectra and reticulation of semihoops
  80. Stein-Weiss inequality for local mixed radial-angular Morrey spaces
  81. Eigenvalues of transition weight matrix for a family of weighted networks
  82. A modified Tikhonov regularization for unknown source in space fractional diffusion equation
  83. Modular forms of half-integral weight on Γ0(4) with few nonvanishing coefficients modulo ℓ
  84. Some estimates for commutators of bilinear pseudo-differential operators
  85. Extension of isometries in real Hilbert spaces
  86. Existence of positive periodic solutions for first-order nonlinear differential equations with multiple time-varying delays
  87. B-Fredholm elements in primitive C*-algebras
  88. Unique solvability for an inverse problem of a nonlinear parabolic PDE with nonlocal integral overdetermination condition
  89. An algebraic semigroup method for discovering maximal frequent itemsets
  90. Class-preserving Coleman automorphisms of some classes of finite groups
  91. Exponential stability of traveling waves for a nonlocal dispersal SIR model with delay
  92. Existence and multiplicity of solutions for second-order Dirichlet problems with nonlinear impulses
  93. The transitivity of primary conjugacy in regular ω-semigroups
  94. Stability estimation of some Markov controlled processes
  95. On nonnil-coherent modules and nonnil-Noetherian modules
  96. N-Tuples of weighted noncommutative Orlicz space and some geometrical properties
  97. The dimension-free estimate for the truncated maximal operator
  98. A human error risk priority number calculation methodology using fuzzy and TOPSIS grey
  99. Compact mappings and s-mappings at subsets
  100. The structural properties of the Gompertz-two-parameter-Lindley distribution and associated inference
  101. A monotone iteration for a nonlinear Euler-Bernoulli beam equation with indefinite weight and Neumann boundary conditions
  102. Delta waves of the isentropic relativistic Euler system coupled with an advection equation for Chaplygin gas
  103. Multiplicity and minimality of periodic solutions to fourth-order super-quadratic difference systems
  104. On the reciprocal sum of the fourth power of Fibonacci numbers
  105. Averaging principle for two-time-scale stochastic differential equations with correlated noise
  106. Phragmén-Lindelöf alternative results and structural stability for Brinkman fluid in porous media in a semi-infinite cylinder
  107. Study on r-truncated degenerate Stirling numbers of the second kind
  108. On 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq
  109. Some new characterizations of finite p-nilpotent groups
  110. A Billingsley type theorem for Bowen topological entropy of nonautonomous dynamical systems
  111. F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface
  112. On modules related to McCoy modules
  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
  114. Solvability for a nonlocal dispersal model governed by time and space integrals
  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
  117. On the connection between Sp-almost periodic functions defined on time scales and ℝ
  118. On a class of Harada rings
  119. On regular subgroup functors of finite groups
  120. Fast iterative solutions of Riccati and Lyapunov equations
  121. Weak measure expansivity of C2 dynamics
  122. Admissible congruences on type B semigroups
  123. Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
  124. Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator
  125. Data transmission mechanism of vehicle networking based on fuzzy comprehensive evaluation
  126. Dual uniformities in function spaces over uniform continuity
  127. Review Article
  128. On Hahn-Banach theorem and some of its applications
  129. Rapid Communication
  130. Discussion of foundation of mathematics and quantum theory
  131. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part II)
  132. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function
  133. Representations by degenerate Daehee polynomials
  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
  142. A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN
  143. Jensen-type inequalities for m-convex functions
  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
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