Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

IMPACT FACTOR 2017: 0.831
5-year IMPACT FACTOR: 0.836

CiteScore 2018: 0.90

SCImago Journal Rank (SJR) 2018: 0.323
Source Normalized Impact per Paper (SNIP) 2018: 0.821

Mathematical Citation Quotient (MCQ) 2017: 0.32

ICV 2017: 161.82

Open Access
See all formats and pricing
More options …
Volume 16, Issue 1


Volume 13 (2015)

On the boundedness of square function generated by the Bessel differential operator in weighted Lebesque Lp,α spaces

Simten Bayrakci
Published Online: 2018-07-04 | DOI: https://doi.org/10.1515/math-2018-0067


In this paper, we consider the square function


associated with the Bessel differential operator Bt=d2dt2+(2α+1)tddt, α > −1/2, t > 0 on the half-line ℝ+ = [0, ∞). The aim of this paper is to obtain the boundedness of this function in Lp,α, p > 1. Firstly, we proved L2,α-boundedness by means of the Bessel-Plancherel theorem. Then, its weak-type (1, 1) and Lp,α, p > 1 boundedness are proved by taking into account vector-valued functions.

Keywords: Generalized translation; Generalized convolution; Bessel translation operator; Bessel transform; Bessel Plancherel formula; Bessel differential operator; Square function

MSC 2010: 42B35; 42A85

1 Introduction

The classical square function is defined by


where S(ℝn) is the Schwartz space consisting of infinitely differentiable and rapidly decreasing functions, Rn Φ (x)dx = 0 and Φt(x) = tnΦ(xt), t > 0.

This function plays an important role in Fourier harmonic analysis, theory of functions and their applications. It has direct connection with L2-estimates and Littlewood-Paley theory. Moreover, there are a lot of diverse variants of square functions and their various applications (see, Daly and Phillips [7], Jones, Ostrovskii and Rosenblatt [18], Kim [20], Aliev and Bayrakci [5], Keles and Bayrakci [19], etc.)

The Bessel differential operator Bt,


and the Laplace-Bessel differential operator ΔB,


are known as important technical tools in analysis and its applications.

The relevant Fourier-Bessel harmonic analysis, associated with the Bessel differential operator Bt (or the Laplace-Bessel differential operator ΔB) has been a research area for many mathematicians such as Levitan [24, 25], Kipriyanov and Klyuchantsev [21], Trimeche [32], Lyakhov [26], Stempak [30], Gadjiev and Aliev [10, 11], Aliev and Bayrakci [3, 4], Aliev and Saglik [6], Ekincioglu and Serbetci [9], Hasanov [17], Guliyev [14, 15, 16], and others.

The Bessel translation operator is one of the most important generalized translation operators on the half-line ℝ+ = [0, ∞), [24, 32]. It is used while studying various problems connected with Bessel operators (see, [22], [27] and bibliography therein).

In this paper, the square function associated with the Bessel differential operator Bt is introduced on the half-line ℝ+ = [0, ∞) and its L2,α− boundedness by means of the Bessel-Plancherel theorem is proved. Then, (1, 1) weak-type and Lp,α, 1 < p < ∞ boundedness of this function are obtained by taking into account vector-valued functions. For this, some necessary definitions and auxiliary facts are given in Section 2. The main results of the paper are formulated and proved in Section 3.

2 Preliminaries

Let ℝ+ = [0, ∞), C(ℝ+) be the set of continuous functions on ℝ+, C(k)(ℝ+), the set of even k-times differentiable functions on ℝ+ and S(ℝ) be the Schwartz space consisting of infinitely differentiable and rapidly decreasing functions on ℝ and S+(ℝ+) be the subspace of even functions on S(ℝ).

For a fixed parameter α > −1/2, let Lp,α = Lp,α(ℝ+) be the space of measurable functions f defined on ℝ+ and the norm


is finite. In the case p = ∞, we identify L with C0, the corresponding space of continuous functions vanishing at infinity.

Denoted by Ts, s ∈ ℝ+ the Bessel translation operator acts according to the law




and the following relations are known [25]:


It is not difficult to see the following inequality


that is, Ts is a continuous operator in C0. Moreover, for 1 ≤ p < ∞ and fS+(ℝ+) it is shown that


For this, we define a measure on the [0, π] by d μ (φ) = cα(sin φ)2α dφ, where cα is defined by (3). By using (2) and the Hölder inequality, we have


Further, by using (5) and (4) we obtain


As S+(ℝ+) is dense Lp,α for p < ∞, (6) stays valid for every function in fLp,α.

Note that Ts, s ∈ ℝ+ is closely connected with the Bessel differential operator


It is known that the function u(t, s) = Tsf(t), fC2(ℝ+) is the solution the following Cauchy problem, (see [8, 25]):


The Bessel transform of order α > −1/2 of a function fL1,α is defined by


and the inverse Bessel transform is given by the formula




is the normalized Bessel function and Jα(z) is the Bessel function of the first kind. From the following integral presentation for jα(t) (see[13], Eq. 8.411(8))


we have


and the equality takes place only at t = 0. We also note that, by using (8) and the Riemann-Lebesgue Lemma, we have


Moreover, from (9) we have


and thus ∥𝓑f ≤ ∥f1,α is obtained.

The asymptotic formula for Jα(r) is as follows ([28]):


Then, the following asymptotic formula for jα(r) is obtained easily:


The following Lemmas will be needed in proving the main results containing important properties of Bessel transform.

Lemma 2.1

([25]). Let fL1,α then


Lemma 2.2

([25], Bessel-Plancherel formula). Let fL1,αL2,α then


The generalized convolution generated by the Bessel translation operator for f, gL1,α is defined by


The convolution operation makes sense if the integral on the right-hand side of (13) is defined; in particular, if f, gS+(ℝ+), then the convolution fg also belongs to S+(ℝ+).

Now, we list some properties of generalized convolution as follows: (see details in [25])


Further, by using (6) and the Hölder inequality it is not difficult to prove the corresponding Young inequality


3 Main results and proofs

In this part, the L2,α boundedness of the square function generated by the Bessel differential operator is proved by Bessel-Plancherel formula, then its (1, 1) weak-type and Lp,α, 1 < p < ∞ boundedness is obtained by using vector-valued functions.

Definition 3.1

Let ΦS+(ℝ+) and 0Φ(x)x2α+1dx=0. The square function associated with the Bessel differential operator is defined by


where Φt(x)=t2α2Φ(xt),t>0,α>12.

An important trend in mathematical analysis and applications is to investigate convolution-type operators. Convolution type square functions have a very direct connection with L2-estimates by the Plancherel theorem.

For this reason, we have proved L2,α-boundedness of the square function (15), associated with the Bessel differential operator by using Bessel-Plancherel formula (12) in the following.

Theorem 3.2

Let the square function 𝓢f be defined as (15). If fL2,α then there is c > 0 such that



Firstly, let fS+(ℝ+). By making use of the Fubini theorem and Bessel-Plancherel formula, we have


Taking into account (14) and then using Fubini theorem, we get


Since Φt(x) = t2α2Φ(xt), then using Lemma 1, we have



0|(BΦt)(x)|2dtt=0|(BΦ)(tx)|2dtt(set τ=tx)=0|(BΦ)(τ)|2dττ.

By taking this into account in the formula (16) and using (12) we have


where c=0|(BΦ)(τ)|2dττ. Let us show that c < ∞.


Firstly, let us estimate I1. Since 0Φ(x)x2α+1dx=0, we have


and taking into account (8) for the normalized Bessel function jα(t) we get






Now we estimate I2. For this, we need the following asymptotic formula for jα(r), (cf.(11)):




and we have


For arbitrary fL2,α, we will take into account that the Schwartz space S+(ℝ+) is dense in L2,α. Namely, let (fn) be a sequence of functions in S+(ℝ+), which converges to f in L2,α-norm.

From the “triangle inequality” (∥u2,α − ∥v2,α)2 ||uv||2,α2, we have




Hence, by (3.17) we get


This shows that the sequence (Sfn) converges to (Sf) in L2,α −norm. Thus


and the proof is complete.  □

Now, taking into account vector-valued functions spaces, we will obtain Lp,α(ℝ+), 1 < p < ∞ boundedness of the square function associated with the Bessel differential operator.

For this, necessary definitions and theorems are given below. The first theorem is well known as the Marcikiewicz interpolation theorem for the vector-valued functions. The other theorem is the extension of Benedek-Calderon-Panzone principle.

Let H be a seperable Hilbert space. We say that a function f defined on ℝ+ = [0, ∞) and with values in H is measurable if the scalar valued function (f(x), h) is measurable for every h in H, where (, ) denotes the inner product of H and h denotes an arbitrary vector of H. Throughout the text, the absolute value |.|H denotes the norm in H. Moreover, let H1 and H2 be two seperable Hilbert spaces, and B(H1,H2) denote the Banach spaces of bounded linear operators A from H1 to H2 endowed with the norm


Let Lp,α(ℝ+, H) be the space of measurable functions f(x) from ℝ+ to H with the norm


is finite. If p = ∞, then the norm


is finite, (see for details, [28]; p.27-30, [29]; p.45-46 [31]; p.307-309).

Theorem 3.3

([31], Theorem 2.1, p.307). Let be A a sublinear operator defined on L0(ℝ+,H1), i.e., compactly supported, bounded H1-valued functions, with values in M(ℝ+,H2), i,e., the space of measurable, H2-valued function. Suppose in addition that for fL0(ℝ+,H1)




where c1 and cr are independent of λ and f. Then for each 1 < p < r, we have that AfLp,α(ℝ+,H2) whenever fLp,α(ℝ+, H1) and there is a constant c = c1,r,p independent of f such thatAfp,αcfp,α.

Theorem 3.4

([31], Theorem 2.2, p.307). Suppose a linear operator A defined in L0(ℝ+,H1) and with values in M(ℝ+, H2) verifies


and if f has support in B(x0, R) and integral 0, then there are constants c2,c3 > 1 independent of f so that




Now let H1 = ℝ+ and H2 = L2,α(R+,dtt), α > −1/2 be the Hilbert space of square integrable functions on the half-line with respect to the measure dtt and the norm


Since ΦS+(ℝ+) and 0Φ(x)x2α+1dx=0 then we define K(x) to be the H2-valued function given by


So, the square function associated with the Bessel differential operator (𝓢f)(x) is the linear operator (Af)(x) = (fK)(x) and Af takes its values in H2.

Thus, the condition (18) is equivalent to the following inequality


Now let us calculate (19). For this, since ΦS+(ℝ+), we take


and for 0 < ϵ < min {θ, q} by using Hölder inequality we have




then we get


Finally, by using Theorem 3.4, we see that the square function associated with the Bessel differential operator 𝓢f is of weak-type (1, 1) and since we have already verified the L2,α(ℝ+) -boundedness then by the Marcinkiewicz interpolation theorem for the vector-valued functions, (Theorem 3.3) Sf is also of type (p, p), 1 < p < 2 and consequently, by a simple duality argument 𝓢f is of type (p, p), 1 < p < ∞.


  • [1]

    Aliev I.A., Riesz transforms generated by a generalized shift operator, Izvestiya Akad. Nauk Azerbaijan Rep. (Ser. Fiz. Techn. Mat. Nauk),(In Russian), 1987, 7-13Google Scholar

  • [2]

    Aliev I.A., Gadjiev A.D., Weighted estimates of multidimensional singular integrals generated by the generalized shift operator, Russian Acad. Sci. Sb. Math., 1994, 77(1), 37-55Google Scholar

  • [3]

    Aliev I.A., Bayrakci S., On inversion of B-elliptic potentials associated with the Laplace-Bessel differential operator, Fract. Calc. and Appl. Anal., 1998, 1(4), 365-384 Google Scholar

  • [4]

    Aliev I.A., Bayrakci S., On inversion of Bessel potentials associated with the Laplace-Bessel differential operator, Acta Math. Hungar, 2002, 95(1-2), 125-145 CrossrefGoogle Scholar

  • [5]

    Aliev I.A., Bayrakci S., Square-like functions generated by a composite wavelet transform, Mediterr. J. Math., 2011, 8, 553-561 CrossrefGoogle Scholar

  • [6]

    Aliev I.A., Saglik E., Generalized Riesz potential spaces and their characterization via wavelet-type transform, FILOMAT, 2016, 30(10), 2809-2823 CrossrefGoogle Scholar

  • [7]

    Dayl J.E., Phillips K.L., Walsh multipliers and square functions for the Hardy space H1, Acta Math. Hungar, 1998, 79(4), 311-327 CrossrefGoogle Scholar

  • [8]

    Delsarte J., Sur une extension de la formule de Taylor, J. Math. Pure Appl, 1938, 17, 213-231 Google Scholar

  • [9]

    Ekincioglu I., Serbetci A., On weighted estimates of high order Riesz-Bessel transformations generated by the generalized shift operator, Acta Mathematica Sinica, 2005, 21(1), 53-64 CrossrefGoogle Scholar

  • [10]

    Gadjiev A.D., Aliev I.A., On a class of potential type operator generated by a generalized shift operator, Reports Seminar of I.N. Vekua Inst. of Appl. Math. Tbilisi., (In Russian), 1988, 3(2), 21-24 Google Scholar

  • [11]

    Gadjiev A.D., Aliev I.A., The Riesz and Bessel type potentials generated by a generalized shift operator and their inverce, Proc. IV All-Union Winter Conf. Theory of Functions and Approximation, Saratov, 1998; In the book “Theory of Functions and Approximation”, (In Russian), Saratov, 1990, 47-53 Google Scholar

  • [12]

    Gadjiev A.D., Guliyev V.S., The Stein-Weiss type inequalities for fractional integrals, associated with the Laplace-Bessel differential operator, Frac. Calc. and Appl. Anal., 2008, 11(1), 77-90Google Scholar

  • [13]

    Gradshtein I.S., Ryzhik I.M., Tables of Integrals, Sums, Series and Products, (in Russian), Nauka, Moscow, 1971 Google Scholar

  • [14]

    Guliyev V.S., Sobolev theorems for B-Riesz potentials, Dokl. RAN, (In Russian), 1988, 358(4), 450-451 Google Scholar

  • [15]

    Guliyev V.S., Some properties of the anisotropic Riesz-Bessel potentials, Analysis Mathematica, 2000, 26(2), 99-118 CrossrefGoogle Scholar

  • [16]

    Guliyev V.S., On maximal function and fractional integral, associated with the Bessel differential operator, Mathematical Inequalities and Applications, 2003, 6(2), 317-330 Google Scholar

  • [17]

    Hasanov J.J., Some property for anisotropic Riesz potential, associated with the Laplace-Bessel differential operator, Khazar Journal of Mathematics, 2005, 1, 27-34 Google Scholar

  • [18]

    Jones R.L., Ostrovskii K.L., Rosenblatt J.M., Square functions in ergodic theory, Ergod. Theory Dyn. Syst., 1996, 16, 267-305 CrossrefGoogle Scholar

  • [19]

    Keles S., Bayrakci S., Square-like functions generated by the Laplace-Bessel differential operator, Advances in Difference Equations, 2014, 281, 2-9 Google Scholar

  • [20]

    Kim Y.C., Weak type estimates of square functions associated with quasiradial Bochner-Riesz means on certain Hardy spaces, J.Math. Anal. Appl., 2008, 339(1), 266-280 CrossrefGoogle Scholar

  • [21]

    Kipriyanov I.A., Klyuchantsev M.I., On singular integrals generated by the generalized shift operator, II, Sibirsk. Mat. Zh., 1970, 11, 1060-1083; translation in Siberian Math. J., 1970, 11, 787-804 Google Scholar

  • [22]

    Kipriyanov I.A., Singular elliptic boundary value problems, (In Russian), Nauka, Moscow, 1997 Google Scholar

  • [23]

    Klyuchantsev M.I., On singular integrals generated by the generalized shift operator, I, Sibirsk. Mat. Zh., 1970, 11, 810-821; translation in Siberian Math. J., 1970, 11, 612-620Google Scholar

  • [24]

    Levitan B.M., The theory of generalized translation operators, (In Russian), Nauka, Moscow, 1973 Google Scholar

  • [25]

    Levitan B.M., Expansion in Fourier series and integrals with Bessel functions, Uspekhi Mat. Nauk, (In Russian), 1951, 6, 2(42), 102-143 Google Scholar

  • [26]

    Lyakhov L.N., On a class of spherical functions and singular pseudodifferential operators, Dokl. Akad. Nauk., 1983, 272(4), 781-784; translation in Soviet Math. Dokl., 1983, 28(2), 431-434 Google Scholar

  • [27]

    Platonov S.S., Bessel generalized translations and some problems of approximation theory for functions on the half-line, Siberian Mathematical Journal, 2009, 50(1), 123-140 CrossrefGoogle Scholar

  • [28]

    Stein E.M., Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton New Jersey, 1993 Google Scholar

  • [29]

    Stein E.M., Singular Integrals and Differentiability Properties of functions, Princeton Univ. Press, Princeton New Jersey, 1970 Google Scholar

  • [30]

    Stempak K., The Littlewood-Paley theory for the Fourier-Bessel transform, Mathematical Institute of Wroslaw, Poland, 1985 Google Scholar

  • [31]

    Torchinsky A., Real-variable methods in Harmonic Analysis, Academic Press, London, 1986 Google Scholar

  • [32]

    Trimeche K., Generalized wavelets and hypergroups, Gordon and Breach Sci, 1997 Google Scholar

About the article

Received: 2017-12-01

Accepted: 2018-04-25

Published Online: 2018-07-04

Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 730–739, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0067.

Export Citation

© 2018 Bayrakci, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

Comments (0)

Please log in or register to comment.
Log in