A Note about Young’s Inequality with Different Measures

e key purpose of this paper is to work on the boundedness of generalized Bessel–Riesz operators dened with doubling measures in Lebesgue spaces with dierent measures. Relating Bessel decaying the kernel of the operators is satisfying some elementary properties. Doubling measure, Young’s inequality, and Minköwski’s inequality will be used in proofs of boundedness of integral operators. In addition, we also explore the relation between the parameters of the kernel and generalized integral operators and see the norm of these generalized operators which will also be bounded by the norm of their kernel with dierent measures.

is operator is de ned by for every f ∈ L p loc (R n ), with 1 ≤ p < ∞. Here, K α is called fractional integral kernel or Riesz kernel [1].
Studies about Riesz potentials T α were started since 1920's. Hardy-Littlewood [2,3] proved the boundedness of Riesz potentials on Lebesgue spaces for n 1. After 50's, Hardy-Littlewood and Sobolev [4] proved the boundedness of T α for n ∈ N. [5] (see also D. Edmunds [[6], Chapter 6]) Kokilashvili had a complete description of nondoubling measure μ guaranteeing the boundedness of fractional integral operator T α from L p (μ, X) to L q (μ, X), 1 < p < q < 1. We notice that this result was derived in [7] for potentials on Euclidean spaces. In [4], theorems of Sobolev and Adams type for fractional integrals de ned on quasimetric measure spaces were established.
Some two-weight norm inequalities for fractional operators on R n with nondoubling measure were studied in [8]. e boundedness of the Riesz potential in Lebesgue and Morrey spaces de ned on Euclidean spaces was studied in Peter and Adams's paper [9,10]. e same problem for fractional integrals on R n with nondoubling measure was investigated by Sawano in [11]. Eridani ([12], eorem 4, eorem 3.1, eorem 3.3) established the boundedness of fractional integral operators and mention the necessary and su cient conditions for the boundedness of maximal operators.
Since 1930s, some researchers [2,3] have studied the boundedness of T α on some function spaces.
Theorem 1 (Hardy-Littlewood-Sobolev) (see [4]). If f ∈ L p (R n ), with 1 < p < n/α, then there exists C p,q > 0 such that From now on, C, C p,q , C p,q,s > 0 will be serve as a positive constant, not necessarily the same one.
The purpose of this paper is to work on the boundedness of generalized Bessel-Riesz operators de ned in Lebesgue spaces with di erent measures. Role of Young's inequality and Minköwski's inequality will be used in proofs of boundedness of integral operators.
Here, we define and f ∈ L p � L p (R n ), as a collection of f such that ‖f: L p ‖ < ∞. Next, for a given K α,c : We define Bessel-Riesz operator as The origin of Bessel-Riesz operators is Schröndinger equation. e Schrödinger equation is a linear partial differential equation that describes the wave function or state function of a quantum-mechanical system. In quantum mechanics, the analogue of Newton's law is Schrödinger's equation. In 1999, Kazuhiro Kurata, Seiichi Nishigaki, and Satoko Sugano [13] studied boundedness of integral operators on Lebesgue and generalized Morrey spaces and its application to estimate in Morrey spaces for the Schrödinger operator L 2 � −Δ + V(x) + W(x) with nonnegative V ∈ (RH) ∞ (reverse Hölders class) and small perturbed potentials W.
Eridani et al. [12] presented the boundedness of fractional integral operators defined on quasimetric measure spaces. Moreover, Idris et al. [14] have investigated the boundedness of generalized Morrey spaces with weight and presented the boundedness of these operators on Lebesgue spaces and Morrey spaces for Euclidean spaces.
Since Euclidean spaces are the simplest example of measure metric spaces. Kurata et al. [16] have investigated the boundedness of Bessel-Riesz operators on generalized Morrey spaces with weight. e boundedness of these operators on Lebesgue spaces in Euclidean settings will be proved using Young's inequality and Minköwski's inequality.
Moreover, we will also find the norm of the generalized Bessel-Riesz operators bounded by the norm of the kernels. Saba et al. [17] used Young's inequality, to prove the boundedness of Bessel-Riesz operators on Lebesgue spaces in measure metric spaces, which are easy consequences of Young's inequality. e second consequence after the fact Bessel-Riesz operator is bounded on Lebesgue spaces; we entered to next phenomena of Morrey spaces. In Young's inequality, we have the best constant known as 1. But at this point, we still have no information about the best constant in Morrey spaces. So in this paper, we move towards generalized Bessel-Riesz operators in Lebesgue spaces.
We will also mention the case when the measure satisfies the doubling condition. The derived conditions are simultaneously necessary and sufficient for appropriate inequalities that were derived in [18,19]. We also have the following result [14] about the boundedness of such an operator as follows: For some functions f: R n ⟶ R and g: R n ⟶ R, we define We know that f⋆g is a generalization of T α and T α,c . Moreover, the above result is a particular case of the following [20].
Kurata et al. [16] have shown that W · T α,c is bounded on generalized Morrey spaces where W is any real functions. For applications of the above operators in Euclidean spaces setting, see [16].

The Kernel
For 1 ≤ s < ∞, 0 < c < ∞, and R + : � (0, ∞), we define functions ρ: R + ⟶ R + , with the following conditions: From (ρI) we will have at is, there exists 0 < c 1 ≤ c 2 such that Since 2 International Journal of Mathematics and Mathematical Sciences r sc then (ρII) and (ρIII) are equivalent, with the following explanations.
It is easy to see that (ρII)⇒(ρIII). Suppose (ρIII) is true. en, for 0 < r < 1, we have On the other hand, for 1 < r < ∞, then and we already prove that both of the conditions are equivalent.
With the same technique as used to estimate the kernel defined by equation (15) or every y ∈ R n , we also have As a classical example, we can consider the following: If we consider then as a consequences of Young's inequality, we also have the following.
From the above lemma, there exists C > 0 such that Suppose μ is an arbitrary measure on R n . We define μ ∈ (GC) (growth condition), if and only if there exists C 2 > 0 such that ror every open balls B(a, R): For more information about this kind of measure, see [21].
Based on the above definitions, we try to estimate For 1 ≤ s < ∞, and R > 0, we consider the following: and also where letter C > 0 shall always denote a constant, not necessarily the same one. At this point, for 1 ≤ s < ∞, and μ ∈ (GC), we define

Main Results
For any measure ] on R n , any measurable functions f: R n ⟶ R, and x ∈ R n , we define Before we state our main results, about the boundedness of T ρ,c , we consider the following simple result [13].
Lemma 3 (Minköwski's inequality). Suppose 1 ≤ p < ∞, and we are given F: R n × R n ⟶ R. For any measure ] and μ on R n , then Now, we state the following: Theorem 4. Suppose μ ∈ (GC) and ] is any measure on R n .
If f ∈ L 1 (]) and K ρ,c ∈ L s (μ), then there exists C s > 0 such that Proof. By Minköwski's inequality, with 1 ≤ s < ∞, then As a corollary of the above result, we also have □ Corollary 1. Suppose μ ∈ (GC) and ] is any measure on R n .
If f ∈ L 1 (]) and K ρ,c ∈ L 1 (μ), then there exists C > 0 such that Suppose we have the It is noted that After the above simple calculations, we have for some C * > 0 and for every open balls B on R n .
If f ∈ L p (]) and K ρ,c ∈ L q (μ), then there exists C > 0 such that Proof. With Hölder's inequality, we start with the following: If we define then for every x, we come to Now, we want to estimate the right hand side, especially, for x ∈ R n and R > 0, we will have (41) We start with and also Up to now, for every x ∈ R n , we already have By the previous fact, we know that T ρ,c f: L 1 (μ) � � � � � � � � � � ≤ C 1 K ρ,c : L 1 (μ) � � � � � � � � � � · f: L 1 (]) � � � � � � � �, and after this, we come to And finally,