In this paper we prove the weak and strong type boundedness of the Bn-maximal operator M in Bn-Orlicz spaces L?,? (R+ n). © 2018, Institute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan. All rights reserved

Ekincioglu S.E.

Kırşehir Ahi Evran University Institutional Repository

Transactions of NAS of Azerbaijan, Issue Mathematics, 38 (1), 43–51 (2018)Series of Physical-Technical and Mathematical SciencesOn the boundedness of the Bn-maximal operator on Bn-Orlicz spacesS. Elifnur EkinciogluReceived: 05.05.2017 / Revised: 10.01.2018 / Accepted: 16.03.2018Abstract. In this paper we prove the weak and strong type boundedness of the Bn-maximal operator Mγin Bn-Orlicz spaces LΦ,γ(Rn+).Keywords. Bn-maximal operator · Bn-Orlicz space.Mathematics Subject Classification (2010): 42B20, 42B25, 42B35.1 IntroductionThe fractional integral operators play an important role in the theory of harmonic analy-sis, differentiation theory and PDE’s. The fractional integrals and related topics associatedwith the Laplace-Bessel differential operator have been research areas for many mathemati-cians such as I.A. Kipriyanov [14], L.N. Lyakhov [13], A.D. Gadjiev and I.A. Aliev [1], A.Serbetci and I. Ekincioglu [11], V.S. Guliyev [6,8] and others.The first author [8] have introduced the maximal function, generated by the Laplace-Bessel differential operator (Bn-maximal function) and investigated the boundedness ofBn-maximal operator in Lp,γ-spaces. I. Kipriyanov and M. Klyuchantsev [16], [17] haveintroduced the singular integrals, generated by the Laplace-Bessel differential operator (Bn-singular integrals) and investigated the boundedness ofBn-singular operators inLp,γ-spaces.I. Aliev and A. Gadjiev [3], A. Gadjiev and E.V. Guliyev [5] and I. Ekincioglu [2] havestudied the boundedness of Bn-singular integrals in weighted Lp,γ-spaces.Let Rn+ be the part of the Euclidean space Rn of points x = (x1, . . . , xn) defined by theinequalities xn > 0, and γ is a fixed positive number. Denote by Lp,γ ≡ Lp,γ(Rn+) the setof all classes of measurable functions f with finite norm‖f‖Lp,γ =(∫Rn+|f(x)|pxγndx) 1p, 1 ≤ p <∞.If p =∞, we assumeL∞,γ(Rn+) = L∞(Rn+) = {f : ‖f‖L∞,γ = ess supx∈Rn+|f(x)| <∞}.S. Elifnur EkinciogluDepartment of Mathematics, Ahi Evran University, Kirsehir, TurkeyE-mail: ekinciogluelifnur@gmail.com44 On the boundedness of the Bn-maximal operator on Bn-Orlicz spacesFor measurable setE ⊂ Rn+ let |E|γ =∫E xγndx, then |E(0, r)|γ = ω(n, γ)rn+γ , whereω(n, γ) = |E(0, 1)|γ .The operator of generalized shift (Bn shift operator) is defined by the following way(see [16], [20]):T yf(x) = Cγ∫ π0f(x′ − y′,√x2n − 2xnyn cosαn + y2n)sinγ−1 αdα,where Cγ = π−12Γ(γ + 12)Γ−1(γ).Note that the generalized shift operator T y is closely related to the ∆Bn Laplace-Besseldifferential operator ([14])∆Bn =n−1∑i=1∂2∂x2i+Bn,where Bn = ∂2∂x2n+ γxn∂∂xn, γ > 0 (see [16], [20]).Furthermore, T y generates the corresponding Bn-convolution(f ⊗ g)(x) =∫Rn+f(y)T yg(x) yγndy.The fractional integrals and related topics associated with the Laplace-Bessel differentialoperator have been research areas for many mathematicians such as I.A. Kipriyanov [14],L.N. Lyakhov [13], A.D. Gadjiev and I.A. Aliev [1], A.D. Gadjiev and V.S. Guliyev [4], A.Serbetci and I. Ekincioglu [11], V.S. Guliyev [6]-[10] and others.Let f ∈ Lloc1,γ(Rn+). The Bn-maximal function Mγf is defined by (see[6])Mγf(x) = supr>0|B(0, r)|−1∫B(0,r)T y|f(x)| yγn dy.The following theorems is valid (see [6]).Theorem 1.1 1) Let f ∈ L1,γ(Rn+). Then for any τ > 0∣∣{x ∈ Rn+ :Mγf(x) > τ}∣∣γ ≤ C3τ∫Rn+|f(x)| xγndx,where constant C3 does not depend from f .2) Let f ∈ Lp,γ(Rn+), 1 < p ≤ ∞. Then Mγf(x) ∈ Lp,γ(Rn+) and‖Mγf‖Lp,γ ≤ C4‖f‖Lp,γ ,where constant C4 does not depend from f .Corollary 1.1 If f ∈ Lp,γ(Rn+), 1 ≤ p ≤ ∞, thenlimε→0|B(0, ε)|−1γ∫B(0,ε)T yf(x) yγndy = f(x)for almost all x ∈ Rn+.Remark 1.1 Let’s note, that Theorem 1.1 in a one-dimensional case, that is at n = 1 isprove in [25], and in a multivariate case n ≥ 2 in [6] (see [7]).S.E. Ekincioglu 45It is well known that maximal operator play an important role in harmonic analysis (see[24]). Harmonic analysis associated to the Fourier-Bessel transform and the Laplace-Besseldifferential operator gives rise to convolutions with a relevant generalized translation. In theframework of this analysis we study Hardy-Littlewood maximal functions (Bn-maximalfunctions) in the relevant Orlicz-Bessel space (Bn-Orlicz space). We prove the weak andstrong type boundedness of the Bn-maximal operator MB in Bn-Orlicz spaces LΦ,γ(Rn+).By A . B we mean that A ≤ CB with some positive constant C independent ofappropriate quantities. If A . B and B . A, we write A ≈ B and say that A and B areequivalent.2 On Young Functions and Orlicz SpacesOrlicz space was first introduced by Orlicz in [21,22] as a generalizations of Lebesguespaces Lp. Since then this space has been one of important functional frames in the math-ematical analysis, and especially in real and harmonic analysis. Orlicz space is also anappropriate substitute for L1 space when L1 space does not work.First, we recall the definition of Young functions.Definition 2.1 A function Φ : [0,∞) → [0,∞] is called a Young function if Φ is convex,left-continuous, limr→+0Φ(r) = Φ(0) = 0 and limr→∞Φ(r) =∞.From the convexity and Φ(0) = 0 it follows that any Young function is increasing. If thereexists s ∈ (0,∞) such that Φ(s) = ∞, then Φ(r) = ∞ for r ≥ s. The set of Youngfunctions such that0 < Φ(r) <∞ for 0 < r <∞will be denoted by Y. If Φ ∈ Y , then Φ is absolutely continuous on every closed interval in[0,∞) and bijective from [0,∞) to itself.For a Young function Φ and 0 ≤ s ≤ ∞, letΦ−1(s) = inf{r ≥ 0 : Φ(r) > s}.If Φ ∈ Y , then Φ−1 is the usual inverse function of Φ. It is well known thatr ≤ Φ−1(r)Φ̃−1(r) ≤ 2r for r ≥ 0, (2.1)where Φ̃(r) is defined byΦ̃(r) ={sup{rs− Φ(s) : s ∈ [0,∞)} , r ∈ [0,∞)∞ , r =∞.A Young function Φ is said to satisfy the ∆2-condition, denoted also as Φ ∈ ∆2, ifΦ(2r) ≤ CΦ(r), r > 0for some C > 1. If Φ ∈ ∆2, then Φ ∈ Y . A Young function Φ is said to satisfy the∇2-condition, denoted also by Φ ∈ ∇2, ifΦ(r) ≤ 12CΦ(Cr), r ≥ 0for some C > 1.46 On the boundedness of the Bn-maximal operator on Bn-Orlicz spacesLemma 2.1 [19] Let Φ be a Young function with canonical representationΦ(t) =∫ t0ϕ(s)ds, t > 0.(1) Assume that Φ ∈ ∆2. More precisely Φ(2t) ≤ AΦ(t) for some A ≥ 2. Set β =log2A. If p > β + 1, then the following inequality is valid:∫ ∞tϕ(s)sp.Φ(t)tp, t > 0.(2) Assume that Φ ∈ ∇2. Then the following inequality is valid:∫ t0ϕ(s)s.Φ(t)t, t > 0.Definition 2.2 (Orlicz Space). For a Young function Φ, the setLΦ,γ(Rn+) ={f ∈ Lloc1,γ(Rn+) :∫Rn+Φ(k|f(x)|) xγn dx <∞ for some k > 0}is called Orlicz space. If Φ(r) = rp, 1 ≤ p < ∞, then LΦ,γ(Rn+) = Lp,γ(Rn+). If Φ(r) =0, (0 ≤ r ≤ 1) and Φ(r) =∞, (r > 1), then LΦ,γ(Rn+) = L∞(Rn+). The space LlocΦ,γ(Rn+)is defined as the set of all functions f such that fχB ∈ LΦ,γ(Rn+) for all balls B ⊂ Rn+.LΦ,γ(Rn+) is a Banach space with respect to the norm‖f‖LΦ,γ = inf{λ > 0 :∫Rn+Φ( |f(x)|λ)xγn dx ≤ 1}.For a measurable function f on Rn+ and t > 0, letm(f, t)γ =∣∣{x ∈ Rn+ : |f(x)| > t}∣∣γ .Definition 2.3 The weak Orlicz spaceWLΦ,γ(Rn+) = {f ∈ Lloc1,γ(Rn+) : ‖f‖WLΦ,γ <∞}is defined by the norm‖f‖WLΦ,γ = inf{λ > 0 : supt>0Φ(t)m(fλ, t)γ≤ 1}.We note that ‖f‖WLΦ,γ ≤ ‖f‖LΦ,γ ,supt>0Φ(t)m(f, t)γ = supt>0tm(f, Φ−1(t))γ = supt>0tm(Φ(|f |), t)γand ∫Rn+Φ( |f(x)|‖f‖LΦ,γ)xγn dx ≤ 1, supt>0Φ(t)m( f‖f‖WLΦ,γ, t)γ≤ 1. (2.2)The following analogue of the Hölder’s inequality is well known (see, for example,[23]).S.E. Ekincioglu 47Theorem 2.1 Let the functions f and g measurable on Rn+. For a Young function Φ and itscomplementary function Φ̃, the following inequality is valid∫Rn+|f(x)g(x)| xγn dx ≤ 2‖f‖LΦ,γ‖g‖LΦ̃,γ .By elementary calculations we have the following property.Lemma 2.2 Let Φ be a Young function and B be a balls in Rn+. Then‖χB‖LΦ,γ = ‖χB‖WLΦ,γ =1Φ−1(|B|−1γ) .Proof. Note that, by elementary calculus the following equalities are valid‖χB‖LΦ,γ = inf{λ > 0 :∫BΦ(1λ)yγn dy ≤ 1}= inf{λ > 0 : Φ(1λ)∫Byγn dy ≤ 1}= inf{λ > 0 :1λ≤ Φ−1(|B|−1γ)}= inf{λ > 0 : λ ≥ 1Φ−1(|B|−1γ)}=1Φ−1(|B|−1γ) ,and‖χB‖WLΦ,γ = inf{λ > 0 : supt>0Φ(tλ) ∣∣{x ∈ Rn+ : ∣∣χB (x)∣∣ > t}|γ ≤ 1}= inf{λ > 0 : sup0<t<1Φ(tλ) ∣∣∣{x ∈ Rn+ : ∣∣χB (x)∣∣ > t}∣∣∣γ≤ 1}= inf{λ > 0 : Φ(1λ)≤ |B|−1γ}= inf{λ > 0 : λ ≥ 1Φ−1(|B|−1γ)}=1Φ−1(|B|−1γ) .By Theorem 2.1, Lemma 2.2 and (2.1) we get the following estimate.Lemma 2.3 For a Young function Φ and for the balls B = B(x, r) the following inequalityis valid: ∫B|f(y)| yγn dy ≤ 2|B|γΦ−1(|B|−1γ)‖f‖LΦ,γ(B).48 On the boundedness of the Bn-maximal operator on Bn-Orlicz spaces3 Boundedness of the Bn-maximal operator in Bn-Orlicz spaces LΦ,γ(Rn+)In this section the boundedness of the maximal operator Mγ in Bn-Orlicz spacesLΦ,γ(Rn+) have been obtained.Theorem 3.1 Let Φ any Young function. Then the Bn-maximal operator Mγ is boundedfrom LΦ,γ(Rn+) to WLΦ,γ(Rn+) and for Φ ∈ ∇2 bounded in LΦ,γ(Rn+).Proof. At first proved that the Bn-maximal operator Mγ is bounded from LΦ,γ(Rn+) toWLΦ,γ(Rn+).We take f ∈ LΦ,γ(Rn+) satisfying ‖f‖LΦ,γ = 1 so that the modularρΦ,γ(f) :=∫Rn+Φ(|f(x)|) xγn dx ≤ 1.We know that by Jensen inequalityΦ(1|B|γ∫B|f(y)| yγn dy)≤ 1|B|γ∫BΦ(|f(y)|) yγn dy (3.1)for all balls B. Using (3.1) and definition of Bn-maximal operator we haveΦ(Mγf(x)) ≤Mγ [(Φ ◦ f)(x)]. (3.2)Using (3.2) and weak (1, 1)γ boundedness of the Bn-maximal operator we get∣∣{x ∈ Rn+ :Mγf(x) > t}∣∣γ = ∣∣{x ∈ Rn+ : Φ(Mγf(x)) > Φ(t)}∣∣γ≤∣∣{x ∈ Rn+ :Mγ(Φ ◦ f)(x) > Φ(t)}∣∣γ≤ CΦ(t)∫Rn+Φ(|f(x)|) xγn dx≤ CΦ(t)≤ 1Φ( tC‖f‖LΦ),since ‖f‖LΦ = 1 and 1CΦ (t) ≥ Φ(tC), if C ≥ 1.Since ‖ · ‖LΦ,γ norm is homogeneous the inequality∣∣{x ∈ Rn+ :Mγf(x) > t}∣∣γ ≤ 1Φ( tC‖f‖LΦ,γ )is true for every f ∈ LΦ,γ(Rn+).Now proved that for Φ ∈ ∇2 the Bn-maximal operator Mγ is bounded in LΦ,γ(Rn+).S.E. Ekincioglu 49Let Λ > 0 and f ∈ LΦ,γ(Rn+)\{0}. Then we have∫Rn+Φ(Mγf(x)Λ)xγn dx =∫Rn+∫ Mγf(x)Λ0ϕ(s)ds xγn dx=∫Rn+∫ ∞0χ{s∈[0,∞):Mγf(x)Λ>s}ϕ(s)ds xγn dx=∫ ∞0ϕ(s)∫Rn+χ{x∈Rn+:Mγf(x)>Λs} xγn dxds=1Λ∫ ∞0ϕ(λΛ) ∣∣{x ∈ Rn+ :Mγf(x) > λ}∣∣γ dλ=2Λ∫ ∞0ϕ(2λΛ) ∣∣{x ∈ Rn+ :Mγf(x) > λ}∣∣γ dλ.From the maximal inequality (see [7, Lemma X])∣∣{x ∈ Rn+ :Mγf(x) > 2λ}∣∣γ . 1λ∫{x∈Rn+:|f(x)|>λ}|f(x)| xγn dxand change the order of integration∫Rn+Φ(Mγf(x)Λ)xγn dx .1Λ∫ ∞0ϕ(2λΛ)(∫{x∈Rn+:|f(x)|>λ}|f(x)| xγn dx)dλλ.1Λ∫Rn+|f(x)|(∫ |f(x)|0ϕ(2λΛ)dλλ)xγn dx.1Λ∫Rn+|f(x)|(∫ 2Λ−1|f(x)|0ϕ(λ)dλλ)xγn dx.Now we use Lemma 2.1 which yields(∫ 2Λ−1|f(x)|0ϕ(λ)dλλ). |f(x)|−1ΛΦ(2|f(x)|Λ),if f(x) 6= 0. Recall that kΦ(t) ≤ Φ(kt) for k ≥ 1 and t > 0, assuming Φ convex. Therefore,it follows that ∫Rn+Φ(Mγf(x)Λ)xγn dx ≤ c0∫Rn+Φ(2|f(x)|Λ)xγn dx≤∫Rn+Φ(c0|f(x)|Λ)xγn dx.Here c0 is a constant we would like to shed light on. 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On the boundedness of the Bn-maximal operator on Bn-Orlicz spaces

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