On the convergence of Kantorovich operators in Morrey spaces

In this paper, we investigate Kantorovich operators on Morrey spaces. We first recall the main tool in the proof of the uniform boundedness of Kantorovich operators in Morrey spaces, namely a pointwise estimate of Kantorovich operators by the Hardy–Littlewood maximal operator. We also investigate the convergence of Kantorovich operators in Morrey spaces under weaker assumptions that is also true for the endpoint case. In addition, we obtain the rate of convergence of Kantorovich operators in Morrey spaces under the condition of Hölder continuity. Our result can be applicable to the function spaces in a recent result by Zeren, Ismailov and Karacam.


Introduction
It is useful to use Kantrovitch operators to approximate functions with polynomials since they can be used when we approximate integrable functions.In this paper, we establish the same applies to Morrey spaces to a large extent.
Let C([0, 1]) be the set of all continuous functions on [0, 1].For n ∈ N and f ∈ C([0, 1]), the Bernstein polynomial of order n, denoted by B n f , is defined by These polynomials were initially introduced by Bernstein [1] in his constructive proof of the Weierstrass Approximation Theorem.More precisely, he proved that the sequence {B n f } converges uniformly back to f if f ∈ C([0, 1]).However, this convergence fails once we remove the continuity condition.In particular, Herzog and Hill [3] established that for every bounded function f on [0, 1] with both lim x→c − f (x) and lim x→c + f (x) finite for some c ∈ (0, 1), the sequence {B n f (c)} converges to the average of lim x→c − f (x) and lim x→c + f (x).This result shows that Bernstein polynomials are not suitable for approximating discontinuous functions.An alternative to Bernstein polynomials for integrable functions on [0, 1] was proposed by Kantorovich in [4].He defined the Kantorovich operator K n by ).Note that K n and B n are related by the identity where ) obviously contains some discontinuous functions.The convergence result for {K n f } in L p ([0, 1]) was extended to the studies on its rate of convergence in [6].
Recently, Burenkov, Ghorbanalizadeh, and the fourth author [2] obtained the uniform boundedness of Kantorovich operators in Morrey spaces M p q ([0, 1]) (see Definition 2.2).They also established the convergence of {K n f } in Morrey spaces under some smoothness condition on the function f .The most important tool in proving these results is the pointwise estimate of the n-th Kantorovich operator K n by the Hardy-Littlewood maximal operator M (see Definition 2.1 for the definition of this operator).Theorem 1.1 [2] Let K n be the Kantorovich operator of order n ∈ N and M be the Hardy-Littlewood maximal operator.Then, for every f ∈ L 1 ([0, 1]), we have In general, the pointwise estimates in Theorem 1.1 is very useful because it can be applied to the uniform boundedness of K n in some function spaces once we have the boundedness of the Hardy-Littlewood maximal operator on these function spaces.For instance, for 1 < q ≤ p < ∞, the uniform boundedness of K n in M p q ([0, 1]) follows directly from Theorem 1.1 and the boundedness of the Hardy-Littlewood maximal operator in M p q ([0, 1]).However, one must modify the proof of the uniform boundedness of {K n f } in the space where M is unbounded.For example, K n is uniformly bounded in L 1 ([0, 1]) although M is not bounded on L 1 ([0, 1]).A similar situation is also found in the Morrey space M p 1 ([0, 1]).We refer the reader to [10] for the use of Theorem 1.1 when we establish the uniform boundedness and the convergence of Kantorovich operators on Morrey spaces with variable exponent.
One of the main results in this paper is an alternative and shorter proof of Theorem 1.1.Our proof uses the fact that K n f (x) can be written as for some function S n on [0, 1] × [0, 1].We combine this identity with integration by parts for the Riemann-Stieltjes integral and some careful estimations of S n to obtain our result.The second result in this paper is the extension of the convergence result of {K n f } in M p q ([0, 1]) to the limiting case q = 1 under a weaker assumption on f .The formulation of this result is given as follows: Observe that Theorem 1.2 can be viewed as a complement to [2,Theorem 1.4] where the assumptions in [2, Theorem 1.4] are q > 1 and f belongs to We also refer the reader to [11] for the extension of the case q > 1 of Theorem 1.2 to more general Banach function spaces, where the authors used a different proof to that of Theorem 1.2.Although the result in [11] covers many function spaces, it does not cover the case q = 1 of Theorem 1.2.
Related to the convergence of {K n f }, Theorem 1.2 also can be seen as a generalization of the convergence result in [5] according to the following remark, which [8,9] for the references about Morrey spaces).Since C([0, 1]) is dense in L p ([0, 1]), we may recover the convergence of Although the convergence of Kantorovich operators in the Lebesgue space [5], the information of its rate of convergence is not given.To the best of our knowledge, the oldest result on the rate of convergence of Kantorovich operators in L p ([0, 1]) (1 < p < ∞) is given in [6].However, the rate of convergence for the case p = 1 is not known.This phenomena is unsurprising because The result on the rate of convergence of Kantorovich operators in Morrey spaces is given in [2,Theorem 1.4] where the information on the rate of convergence is known for the Morrey space M p q ([0, 1]) for q > 1.However, the information on the rate of convergence for the case q = 1 remains unknown.To obtain this information, we first assume the Hölder continuity condition on the function f and investigate the rate of convergence of {K n f } in Morrey spaces.This result is given in the next theorem.
Theorem 1.4 Let 1 ≤ q ≤ p < ∞ and f be a Hölder continuous function with exponent α ∈ (0, 1] on [0, 1].Then, for every n ∈ N, In particular, if f is Lipschitz continuous, we have Remark 1.5 By taking p = q, Theorem 1.4 gives an estimate for the rate of convergence of {K n f } in the Lebesgue space L p ([0, 1]) when f is a Hölder continuous function on [0, 1].Moreover, this result can be seen as a counterpart to the estimate obtained by Mathé [7] for the Bernstein polynomials of Hölder continuous functions.
In [2, Theorem 1.4], Burenkov, Ghorbanalizadeh, and the fourth author obtained an estimate for the rate of convergence of {K n f } when f belongs to the non-homogenous Sobolev-Morrey space W 1 M p q ([0, 1]) that is true for q > 1.Under the same conditions for f as in [2, Theorem 1.4], we give another estimate for in the following corollary.
This corollary follows from Theorem 1.4 and the fact that every function in W 1 M p q ([0, 1]), p = 1 is Hölder continuous with exponent 1 − 1/ p.Although [2, Theorem 1.4] generally gives a better estimate, Corollary 1.6 also applies for the endpoint case q = 1.
The remaining parts of this manuscript are organized as follows: We recall some definitions, notation, and basic facts in Sect. 2. The proof of our main results are given in Sect.3. In particular, an alternative proof of Theorem 1.1 is given in Sect.3.1 and the proofs of Theorems 1.2 and 1.4 and Corollary 1.6 are discussed in Sect.3.2.

Preliminaries
We recall some definitions, notation, and some basic facts.We begin with the definition of the Hardy-Littlewood maximal operator M. Definition For a locally integrable function f : [0, 1] → R, the Hardy-Littlewood maximal operator M of f is defined by Recall that the operator M is bounded on L p ([0, 1]) when 1 < p ≤ ∞.In view of Theorem 1.1, the uniform boundedness of K n on L p ([0, 1]) follows from this fact.
We now recall the definition of Morrey spaces below.
) is finite, where Note that if p = q, then we recover the Lebesgue space L p ([0, 1]).
Next, we recall the definition of the Hölder continuity.For 0 < α ≤ 1, the function f is said to be Hölder continuous with exponent α on [0, 1] if there exists a positive real constant K such that for x, y ∈ [0, 1].For such f , we define the seminorm When α = 1, we say that f is Lipschitz continuous on [0, 1] and we write Finally, we define the nonhomogenous Sobolev-Morrey space ) is defined as the space of all functions f such that f is absolutely continuous on [0, 1] and f ∈ M p q ([0, 1]).This space forms a normed space with the norm 3 Proof of the main results

An alternative proof of Theorem 1.1
We recall an integration by parts formula for functions having bounded variation.We can prove Lemma 3.1 in a standard manner by decomposing equally [a, b] into N small intervals and then letting N → ∞.
Applying Lemma 3.1, we establish a general result that promises some applications to other integral operators.
By a similar argument, we have We will show that Kantrovitch operator falls within the scope of Lemma 3.2.For For x, t ∈ [0, 1] and n ∈ N, define where We are now ready to give an alternative proof of Theorem 1.1.

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for z ∈ C with 0 ≤ Re(z) ≤ 1, by the three line theorem, we can obtain the desired result.In fact, for t ∈ R, Consequently, for every α ∈ (0, 1), we have We now prove an embedding from Sobolev-Morrey spaces into Hölder continuous classes.This result can be viewed as a bridge connecting Theorem 1.4 and Corollary 1.6.
Although we can use the extension operator to apply what is known for the function spaces over R if we disregard the multiplicative constant, we supply a direct proof to make the paper self-contained.

Lemma 3 . 1
If f , g are functions having bounded variation on [a, b], then [a,b)