Weighted Morrey Spaces Related to Schrödinger Operators with Nonnegative Potentials and Fractional Integrals

LetL � − Δ + V be a Schrödinger operator onR, d≥ 3, whereΔ is the Laplacian operator onR, and the nonnegative potentialV belongs to the reverse Hölder class RHs with s≥ d/2. For given 0< α< d, the fractional integrals associated with the Schrödinger operator L is defined by Iα � L − α/2. Suppose that b is a locally integrable function on R and the commutator generated by b andIα is defined by [b.Iα]f(x) � b(x) · Iαf(x) − Iα(bf)(x). In this paper, we first introduce some kinds of weightedMorrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class RHs with s≥d/2. )en, we will establish the boundedness properties of the fractional integrals Iα on these new spaces. Furthermore, weighted strong-type estimate for the corresponding commutator [b,Iα] in the framework of Morrey space is also obtained. )e classes of weights, the classes of symbol functions, as well as weighted Morrey spaces discussed in this paper are larger than Ap,q, BMO(R), and Lp,κ(μ, ]) corresponding to the classical case (that is V ≡ 0).


Critical Radius Function ρ(x).
Let d ≥ 3 be a positive integer and R d be the d-dimensional Euclidean space, and let V: R d ⟶ R, d ≥ 3, be a nonnegative locally integrable function that belongs to the reverse Hölder class RH s for some exponents 1 < s < ∞; i.e., there exists a positive constant C > 0 such that the following reverse Hölder inequality holds for every ball B in R d . For given V ∈ RH s with s ≥ d/2, we introduce the critical radius function ρ(x) � ρ(x; V) which is given by where B(x, r) denotes the open ball centered at x and with radius r. It is well known that this auxiliary function satisfies 0 < ρ(x) < ∞ for any x ∈ R d under the above condition on V (see [1]). We need the following known result concerning the critical radius function (2).
Lemma 1 (see [1]). If V ∈ RH s with s ≥ d/2, then there exist two constants C 0 ≥ 1 and N 0 > 0 such that, for all x and y in R d , As a straightforward consequence of (3), we can see that, for each integer k ≥ 1, the following estimate 1 + 2 k r ρ(y) is valid for any y ∈ B(x, r) with x ∈ R d and r > 0, and C 0 is defined in (3).

Fractional Integrals Associated with Schrödinger
Operators. Let V ∈ RH s with s ≥ d/2. For such a potential V, we consider the Schrödinger differential operator on R d , d ≥ 3, and its associated semigroup (6) where p t (x, y) denotes the kernel of the operator e − tL , t > 0. From the Feynman-Kac formula, it is well known that where h t is the classical heat kernel; i.e., Moreover, this estimate (7) can be improved when V belongs to the reverse Hölder class RH s for some s ≥ d/2 (see [2,3], for instance). e auxiliary function ρ(x) arises naturally in this context.

Proposition 1.
Let V ∈ RH s with s ≥ d/2. For every positive integer N ≥ 1, there exists a positive constant C N > 0 such that, for all x and y in R d , For given 0 < α < d, the L-Riesz potential or L-fractional integral operator is defined by e − tL f(x)t α/2− 1 dt. (10) In this work, we shall be interested in the behavior of the L-fractional integral operator I α � L − α/2 .

A ρ,∞ p
and A ρ,∞ p,q Weights. A weight will always mean a nonnegative function which is locally integrable on R d . Given a Lebesgue measurable set E and a weight w, |E| will denote the Lebesgue measure of E and Given B � B(x 0 , r) and t > 0, we will write tB for the tdilate ball, which is the ball with the same center x 0 and radius tr. As in [2] (see also [4,5]), we say that a weight w belongs to the class A ρ,θ p for 1 < p < ∞ and 0 < θ < ∞, if there is a positive constant C > 0 such that, for all balls B � B(x 0 , r) ⊂ R d with x 0 ∈ R d and r > 0, 1 |B| B w(x)dx where p ′ is the dual exponent of p such that (1/p) + (1/p ′ ) � 1. For p � 1 and 0 < θ < ∞, we also say that a weight w belongs to the class A ρ,θ 1 , if there is a positive constant C > 0 such that, for all balls B � B(x 0 , r) ⊂ R d with x 0 ∈ R d and r > 0, For given 1 ≤ p < ∞, we define For any given θ > 0, let us introduce the maximal operator which is given in terms of the critical radius function (2): |B(x, r)| B(x,r) |f(y)|dy, x ∈ R d . (15) Observe that a weight w belongs to the class A ρ, ∞ 1 if and only if there exists a positive number θ > 0 such that for 0 < θ 1 < θ 2 < ∞, then for given p with 1 ≤ p < ∞, one has where A p denotes the classical Muckenhoupt's class (see [6], Chapter 7) and hence A p ⊂ A ρ,∞ p . In addition, for some fixed θ > 0 (see [7]), Obviously, for any fixed θ > 0, To establish weighted norm inequalities for fractional integrals, we need to introduce another weight class A ρ,∞ p,q . As in [7], we say that a weight w satisfies the condition A ρ,θ p,q for 1 < p < q < ∞ and 0 < θ < ∞, if there exists a positive constant C > 0 such that, for any ball where p ′ � p/(p − 1). We also say that a weight w satisfies the condition A ρ,θ 1,q for 1 < q < ∞ and 0 < θ < ∞, if there exists a positive constant C > 0 such that, for any ball

(21)
Similarly, for given p, q with 1 ≤ p < q < ∞, by (16), one has whenever 0 < θ 1 < θ 2 < ∞. Here, A p,q denotes the classical Muckenhoupt-Wheeden class (see [8]). We also define p,q . e following results (Lemmas 2-5) are extensions of well-known properties of A p and A p,q weights. We first present an important property of the classes of weights in A ρ,θ p with 1 ≤ p < ∞, which was given by Bongioanni, Harboure, and Salinas in Lemma 5 of [2].

Lemma 3. If w ∈ A
ρ,θ p with 0 < θ < ∞ and 1 ≤ p < ∞, then there exist two positive numbers δ > 0 and η > 0 such that for any measurable subset E of a ball B � B(x 0 , r), where C > 0 is a constant which does not depend on E and B.
Proof. For any given ball B � B(x 0 , r) with x 0 ∈ R d and r > 0, suppose that E ⊂ B, then by Hölder's inequality with exponent 1 + ϵ and (24), we can deduce that is gives (25) with δ � ϵ/(1 + ϵ). Here and in the sequel, the characteristic function of E is denoted by χ E .
In view of Lemma 2, we now define the reverse Höldertype class RH ρ,θ q that is given in terms of the critical radius function (2). We say that w ∈ RH ρ,θ q for some 1 < q < ∞ and 0 < θ < ∞, if there exists a positive constant C > 0 such that the following reverse Hölder-type inequality Clearly, one has RH q ⊂ RH ρ,∞ q . Let 1 < q < ∞ and A q 1 � w: w q ∈ A 1 . It is known that, for the classical case (see [9]), Proof. e conclusion w ∈ A ρ,θ/q 1 follows easily by Hölder's inequality and the definition of A ρ,θ 1 . Indeed, for any given ball B � B(x 0 , r) with x 0 ∈ R d and r > 0, On the contrary, fix a ball B � B(x 0 , r) ⊂ R d and take y ∈ B. Let E be a ball centered at y 0 and with radius h which contains y. By picking h small enough so that E ⊂ B, then by the condition w q ∈ A ρ,θ 1 , we can deduce that which is equivalent to Since this holds for all E ⊂ B and y ∈ E, then by taking h ⟶ 0 + and using Lebesgue differentiation theorem,

Journal of Function Spaces
(33) us, by raising both sides to the power 1/q and integrating over B, we get is amounts to w ∈ RH ρ,θ/q q . A subtle interplay between these two classes of weights, A ρ,∞ p and A ρ,∞ p,q , is expressed by the following lemma: □ Lemma 5. Suppose that 1 ≤ p < q < ∞. en, the following statements are true: Proof. In fact, when t � 1 + q/p ′ , then a simple computation shows that If w ∈ A ρ,θ p,q with 1 < p < q < ∞ and 0 < θ < ∞, then we have which means that w q ∈ A ρ,θ·(1/((1/q)+(1/p′))) t with t � 1 + q/p ′ . Here and in the sequel, for any positive number c > 0, we denote w c (x) � w(x) c by convention. Analogously, it can be easily shown that w − p′ ∈ A ρ,θ·(1/((1/q)+(1/p′))) t′ with t ′ � 1 + p ′ /q. On the contrary, if w ∈ A ρ,θ 1,q with 1 < q < ∞ and 0 < θ < ∞, then we have which means that w q ∈ A ρ,θ·q 1 . Given a weight w on R d , as usual, the weighted Lebesgue space L p (w) for 1 ≤ p < ∞ is defined to be the set of all functions f such that We also denote by WL p (w) the weighted weak Lebesgue space consisting of all measurable functions f for which When p � 1 and w ∈ A 1,q , I α is bounded from L 1 (w) into WL q (w q ) (see [8]). Recently, Bongioanni et al. ( [2], eorem 4) established the weighted boundedness for fractional integral operators I α associated with Schrödinger operators defined in (10). ey showed that the same estimates also hold for weights in the class A ρ,∞ p,q , which is larger than Muckenhoupt-Wheeden's class (another proof was later given by Tang in eorem 3.8 of [7]). eir results can be summarized as follows: □ Theorem 1 (see [2]). Let 0 < α < d, 1 < p < d/α, Theorem 2 (see [2]).
. For a locally integrable function b on R d (usually called the symbol), we will also consider the commutator operator Recently, Bongioanni et al. [5] introduced a kind of new spaces BMO ρ,∞ (R d ) defined by where for 0 < θ < ∞, the space BMO ρ,θ (R d ) is defined to be the set of all locally integrable functions b satisfying A norm for b ∈ BMO ρ,θ (R d ), denoted by ‖b‖ BMO ρ,θ , is given by the infimum of the constants satisfying (42), or equivalently where the supremum is taken over all balls B(x 0 , r) with x 0 ∈ R d and r > 0. With the above definition in mind, one has [4,5] for more examples). We need the following key result for the space BMO ρ,θ (R d ), which was proved by Tang in Proposition 4.2 of [7].
en, there exist two positive constants C 1 and C 2 such that, for any given ball B(x 0 , r) in R d and for any λ > 0, we have where θ * � (N 0 + 1)θ and N 0 is the constant appearing in Lemma 1.
As a consequence of Proposition 2 and Lemma 3, we have the following result: en, there exist positive constants C 1 , C 2 and η > 0 such that, for any given ball B(x 0 , r) in R d and for any λ > 0, we have where θ * � (N 0 + 1)θ and N 0 is the constant appearing in Lemma 1. Notice is corresponds to the norm inequalities satisfied by I α . In eorem 4.4 of [7], Tang obtained weighted strong-type estimate for the commutator [b, I α ] of fractional integrals associated with Schrödinger operators, when b in a larger space than BMO(R d ), that is the space BMO ρ,∞ (R d ). More precisely, he gave the following weighted result (see also [11], eorem 3.5).
In this paper, firstly, we will define several classes of weighted Morrey spaces related to certain nonnegative potentials satisfying appropriate reverse Hölder inequality. Secondly, we establish weighted estimates of fractional integrals I α associated with L on these new spaces. Finally, we also study the boundedness property for the commutators [b, I α ] of fractional integrals with the new BMO functions defined above.
Throughout this paper, C denotes a positive constant not necessarily the same at each occurrence, and a subscript is added when we wish to make clear its dependence on the parameter in the subscript. We also use a ≈ b to denote the equivalence of a and b; that is, there exist two positive constants C 1 and C 2 independent of a and b such that

Main Results
In this section, we introduce some types of weighted Morrey spaces related to the nonnegative potential V and then give our main results.
, is given by the infimum of the constants in (48), or equivalently,

Journal of Function Spaces 5
where the supremum is taken over all balls B in R d and x 0 and r denote the center and radius of B, respectively. Define For any given f ∈ L p,κ ρ,∞ (μ, ]), let Now, define It is easy to see that ‖ · ‖ ⋆ satisfies the axioms of a norm; i.e., for f, g ∈ L p,κ ρ,∞ (μ, ]) and λ ∈ R, we have Note that this definition does not coincide with the one given in [13] (see also [14] for the unweighted case), but in view of the space BMO ρ,∞ (R d ) and the class A ρ,∞ p,q defined above, it is more natural in our setting. Obviously, if we take θ � 0 or V ≡ 0, then this new space is just the familiar weighted Morrey space L p,κ (μ, ]) (or L p,κ (w)), which was first introduced by Komori and Shirai in [15] (see also [16]). Definition 2. Let 1 ≤ p < ∞, 0 ≤ κ < 1, and w be a weight on R d . For given 0 < θ < ∞, the weighted weak Morrey space WL p,κ ρ,θ (w) is defined to be the set of all measurable functions f on R d such that Correspondingly, we define For any given f ∈ WL p,κ ρ,∞ (w), let Similarly, we define By the definition, we can easily show that ‖ · ‖ ⋆⋆ satisfies the axioms of a (quasi) norm, and WL p,κ ρ,∞ (w) is a (quasi) normed linear space. Clearly, if we take θ � 0 or V ≡ 0, then this space is just the weighted weak Morrey space WL p,κ (w) (see [17]). According to the above definitions, one has ρ,θ (w)) could be viewed as an extension of weighted (or weak) Lebesgue space (when κ � θ � 0). Naturally, one may ask the question whether the above conclusions (i.e., eorems 1 and 2 as well as eorem 3) still hold if replacing the weighted Lebesgue spaces by the weighted Morrey spaces. In this work, we will give a positive answer to this question. We now state our main results as follows.
For weighted strong-type and weak-type estimates of fractional integrals associated with L on L p,κ ρ,∞ (w p , w q ), where w belongs to the new classes of weights, we have the following.
If V ∈ RH s with s ≥ d/2 and 0 < κ < 1/q, then the L-fractional integral operator I α is bounded from Concerning the boundedness property of the commutators [b, I α ] in the setting of weighted Morrey space, where b is in the new BMO-type space, we shall prove the following.
Moreover, for the extreme case κ � p/q of eorem 4, we will show that the fractional integrals associated with L maps L p,κ ρ,∞ (w p , w q ) continuously into the new space BMO ρ,∞ (R d ).
is result may be regarded as a supplement of eorem 4.
If V ∈ RH s with s ≥ d/2 and κ � p/q, then the L-fractional integral operator I α is bounded from L p,κ ρ,∞ (w p , w q ) into BMO ρ,∞ (R d ).

Remark 1.
It is worth pointing out that, in the classical case when V ≡ 0, eorems 4, 5, and 6 have been proved by Komori and Shirai in [15], while eorem 7 has been shown by the author in [18].

Proofs of Theorems 4 and 5
In this section, we will prove the conclusions of eorems 4 and 5. Let us remind that the L-fractional integral operator of order α ∈ (0, d) can be written as e kernel of the fractional integral operator I α will be denoted by K α (x, y). en, (see [19,20]) e following lemma plays a key role in the proof of our main theorems, which can be found in Proposition 8 of [20] (see also [7], Lemma 3.7).
Lemma 6 (see [20]). Let V ∈ RH s with s ≥ d/2 and 0 < α < d. For every positive integer N ≥ 1, there exists a positive constant C N > 0 such that, for all x and y in R d , Proof of eorem 4. By definition, we only have to show that for any given ball B � B(x 0 , r) of R d , there is some ϑ > 0 such that holds for any f ∈ L p,κ ρ,∞ (w p , w q ) with 1 < p < q < ∞ and 0 < κ < p/q. Suppose that f ∈ L p,κ ρ,θ (w p , w q ) for some θ > 0 and w ∈ A ρ,θ′ p,q for some θ ′ > 0. We decompose the function f as where 2B is the ball centered at x 0 of radius 2r > 0 and χ 2B is the characteristic function of 2B. en, by the linearity of I α , we write (64) We now analyze each term separately. By eorem 1, we get Since w ∈ A ρ,θ′ p,q with 1 < p < q < ∞ and 0 < θ ′ < ∞, then we know that w q ∈ A ρ,θ′·(1/((1/q)+(1/p′))) t with t � 1 + q/p ′ according to Lemma 5. Now we claim that, for every weight v ∈ A ρ,τ t and every ball B in R d , there exists a dimensional constant C > 0 independent of v and B such that In fact, for 1 < t < ∞, by Hölder's inequality and the definition of A ρ,τ If we take Z(x) � χ B (x), then the above expression becomes which in turn implies (66). Also observe that Journal of Function Spaces erefore, in view of (66) and (19), where ϑ ′ ≔ (qθ ′ κ)/p + θ. For the other term I 2 , notice that, for any x ∈ B and y ∈ (2B) c , one has |x − y| ≈ |x 0 − y|. It then follows from Lemma 6 that, for any x ∈ B(x 0 , r) and any positive integer N, In view of (4) in Lemma 1 and (19), we further obtain We consider each term in the sum of (72) separately. By using Hölder's inequality and A ρ,θ′ p,q condition on w, we obtain that, for each k ≥ 1, Hence, Recall that w q ∈ A ρ,θ′·(1/(1/q)+(1/p′)) t with t � 1 + q/p ′ and 0 < θ ′ < ∞, then there exist two positive numbers δ, η > 0 such that (25) holds. is allows us to obtain us, by choosing N large enough so that N > θ + θ ′ + η((1/q) − (κ/p)), and the last series is convergent, we then have 8 Journal of Function Spaces where the last inequality follows from the fact that (1/q) − (κ/p) > 0. Summing up the above estimates for I 1 and I 2 and letting ϑ � max ϑ ′ , N · (N 0 /(N 0 + 1)) , we obtain our desired inequality (62). is completes the Proof of eorem 4.

□
Proof of eorem 5. To prove eorem 5, by definition, it suffices to prove that, for each given ball holds for any f ∈ L 1,κ ρ,∞ (w, w q ) with 1 < q < ∞ and 0 < κ < 1/q. Now suppose that f ∈ L 1,κ ρ,θ (w, w q ) for some θ > 0 and w ∈ A ρ,θ′ 1,q for some θ ′ > 0. We decompose the function f as en, for any given λ > 0, by the linearity of I α , we can write We first give the estimate for the term I 1 ′ . By eorem 2, we get Since w ∈ A ρ,θ′ 1,q with 1 < q < ∞ and 0 < θ ′ < ∞, then we know that w q ∈ A ρ,θ′·q 1 according to Lemma 5. We now claim that there exists a dimensional constant C > 0 independent of v and B such that, for every weight v ∈ A ρ,τ Similar to the proof of (66), by the definition of A ρ,τ 1 , we can deduce that If we choose Z(x) � χ B (x), then the above expression becomes which in turn implies (81). erefore, in view of (81) and (19), where ϑ ′ ≔ θ ′ qκ + θ. As for the second term I 2 ′ , by using the pointwise inequality (72) and Chebyshev's inequality, we deduce that

Proof of Theorem 6
For the result involving commutators of fractional integrals associated with Schrödinger operators, we need the following properties of BMO ρ,∞ (R d ) functions, which are extensions of well-known properties of BMO(R d ) functions.

10
Journal of Function Spaces Proof. We may assume that b ∈ BMO ρ,θ (R d ) with 0 < θ < ∞. According to Proposition 3, we can deduce that By a change of variables, we thus have which yields the desired inequality (92), if we choose C � [C 1 pΓ(p)] 1/p C − 1 2 ‖b‖ BMO ρ,θ and μ � θ * + η/p. □ Lemma 8. If b ∈ BMO ρ,θ (R d ) with 0 < θ < ∞, then for any positive integer k, there exists a positive constant C > 0 such that, for every ball Proof. For any positive integer k, we have Since for any 1 ≤ j ≤ k + 1, the following estimate holds trivially and hence is is just our desired conclusion. Now, we are ready to prove the main result of this section.

□
Proof of eorem 6. By definition, we only need to prove that, for an arbitrary ball B � B(x 0 , r) of R d and 0 < α < d, there is some ϑ > 0 such that holds for any f ∈ L p,κ ρ,∞ (w p , w q ) with 1 < p < q < ∞ and 0 < κ < p/q, whenever b belongs to BMO ρ,∞ (R d ). Suppose that f ∈ L p,κ ρ,θ (w p , w q ) for some θ > 0, w ∈ A ρ,θ′ p,q for some θ ′ > 0 as well as b ∈ BMO ρ,θ″ (R d ) for some θ ″ > 0. As before, we decompose the function f as en by the linearity of [b, I α ], we write Now, we give the estimates for J 1 and J 2 , respectively. According to eorem 3, we have Moreover, notice that w q ∈ A ρ,θ′·(1/((1/q)+(1/p′))) t with t � 1 + q/p ′ by Lemma 5, and that us, in view of the inequalities (66) and (19), we get where ϑ ′ ≔ (qθ ′ κ)/p + θ. On the other hand, by definition (40), we can see that, for any x ∈ B(x 0 , r) Adding and subtracting b B inside the integral, we write So, we can divide J 2 into two parts: To deal with the term J 3 , since t � 1 + q/p ′ < q, one has From the pointwise estimate (72) and (92) in Lemma 7, it then follows that Following along the same lines as that of eorem 4, we are able to show that e last inequality is obtained by using (25). Next, we estimate ζ(x) for any x ∈ B(x 0 , r). For any positive integer N, similar to the proof of (71) and (72), we can also deduce that 12 Journal of Function Spaces where in the last inequality we have used (4) in Lemma 1.
Hence, by the above pointwise estimate for ζ(x), Let us consider each term in the sum of (111) separately. For each integer k ≥ 1, 1 By using Hölder's inequality, the first term of the expression (112) is bounded by Since w ∈ A ρ,θ′ p,q with 0 < θ ′ < ∞ and 1 < p < q < ∞, then by Lemma 5, we know that w − p′ ∈ A ρ,θ′·(1/((1/q)+(1/p′))) t′ erefore, the first term of the expression (112) can be bounded by a constant times: Since b ∈ BMO ρ,θ″ (R d ) with 0 < θ ″ < ∞, then by Lemma 8, Hölder's inequality, and the A ρ,θ′ p,q condition on w, the latter term of the expression (112) can be estimated by |f(y)| p w p (y)dy Journal of Function Spaces 13 Consequently, Substituting the above inequality (117) into (111), we thus obtain Note that w q ∈ A ρ,θ′·(1/((1/q)+(1/p′))) t with t � 1 + q/p ′ . A further application of (25) yields Hence, combining the above estimates for J 3 and J 4 , we conclude that By choosing N large enough so that N > θ + θ ′ + θ ″ + μ + η((1/q) − (κ/p)), we thus have where the last series is convergent since 0 < κ < (p/q). Finally, collecting the above estimates for J 1 and J 2 , and letting ϑ � max ϑ ′ , μ + N · (N 0 /(N 0 + 1)) , we obtain the desired inequality (99). e proof of eorem 6 is finished. e higher-order commutators generated by BMO ρ,∞ (R d ) functions b and the fractional integrals I α are usually defined by which is just the linear commutator (40) and By induction on m, we are able to show that the conclusion of eorem 6 also holds for the higher-order commutators [b, I α ] m with m ≥ 2. e details are omitted here.

Proof of Theorem 7
e following lemma plays a key role in the proof of our main theorem, which can be found in Proposition 8 of [20] (see also [7], Lemma 3.7).
Lemma 9 (see [20]). Let V ∈ RH s with s ≥ d/2 and 0 < α < d. For every positive integer N ≥ 1, there exists a positive constant C N > 0 such that, for all x and y in R d and for some fixed 0 < ε ≤ 1, whenever |x − y| ≤ |x − z|/2.
Proof of eorem 7. For an arbitrary ball B � B(x 0 , r) in R d and 0 < α < d, it suffices to prove that the following inequality holds for any f ∈ L p,κ ρ,∞ (w p , w q ) with 1 < p < q < ∞ and κ � p/q, where (I α f) B denotes the average of I α f over B. 4r). By the linearity of the L-fractional integral operator I α , the left-hand side of (125) can be divided into two parts. at is, First, let us consider the term K 1 . Applying the weighted (L p , L q )-boundedness of I α (see eorem 1) and Hölder's inequality, we obtain Also observe that w q ∈ A ρ,θ′·(1/((1/q)+(1/p′))) t with t � 1 + q/p ′ . Using the inequalities (66) and (130) and noting the fact that κ � p/q, we have where ϑ ′ ≔ 3θ ′ + θ. Now we turn to estimate K 2 . For any x ∈ B(x 0 , r), Next note that, by a purely geometric argument, one has whenever x, y ∈ B and z ∈ (4B) c . is fact along with Lemma 9 yields Journal of Function Spaces Furthermore, by using Hölder's inequality, (4) and A ρ,θ′ p,q condition on w, we get that, for any x ∈ B(x 0 , r), where the last equality is due to the assumption κ � p/q. From the pointwise estimate (135), it readily follows that Now N can be chosen sufficiently large so that N > θ + θ ′ , and hence, the above series is convergent. erefore, Fix this N and set ϑ � max ϑ ′ , N · (N 0 /N 0 + 1) . Finally, combining the above estimates for K 1 and K 2 , the inequality (125) is proved, and then the proof of eorem 7 is finished.

Data Availability
No data were used to support this study. 16 Journal of Function Spaces

Conflicts of Interest
e author declares that there are no conflicts of interest regarding the publication of this paper.