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Open Mathematics

formerly Central European Journal of Mathematics

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


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Volume 16, Issue 1

Issues

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

Abstract

In this paper, we consider the square function

(Sf)(x)=(0|(fΦt)(x)|2dtt)1/2

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

(Sf)(x)=(0|(fΦt)(x)|2dtt)1/2,ΦS(Rn)

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,

Bt=d2dt2+(2α+1)tddt,α>1/2,t>0

and the Laplace-Bessel differential operator ΔB,

ΔB=k=1n12xk2+(2xn2+(2α+1)xnxn),α>1/2,xn>0

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

fp,α=(0|f(x)|px2α+1dx)1/p,1p<(1)

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

Tsf(t)=cα0πf(s22stcosξ+t2)(sinξ)2αdξ,(2)

where

cα=(0π(sinξ)2αdξ)1=Γ(α+1)πΓ(α+12)(3)

and the following relations are known [25]:

Tsf(t)=Ttf(s);TsTτf(t)=TτTsf(t);Tsf(t)=Tsf(t);T0f(t)=f(t);0(Tsf(t))g(t)t2α+1dt=0f(t)(Tsg(t))t2α+1dt.(4)

It is not difficult to see the following inequality

Tsff

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

|Tsf(t)|pTs(|f(t)|p).(5)

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

|Tsf(t)|=|0πf(s22stcosξ+t2)dμ(φ)|(0π|f(s22stcosξ+t2)|pdμ(φ))1/p(0πdμ(φ))1/q=(Ts(|f(t)|p))1/p,1p+1q=1.

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

||Tsf||p,αp=0|Tsf(t)|pt2α+1dt0Ts(|f(t)|p)t2α+1dt=0|f(t)|p(Ts1)t2α+1dt=0|f(t)|pt2α+1dt=||f||p,αp.(6)

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

Bt=d2dt2+(2α+1)tddt,α>1/2,t>0.

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

{Btu(t,s)=Bsu(t,s)u(t,0)=f(t),us(t,0)=0.

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

(Bf)(λ)=0f(t)jα(λt)t2α+1dtλ0,(7)

and the inverse Bessel transform is given by the formula

B1=2αΓ(α+1))2B

where

jα(z)=2αΓ(α+1)zαJα(z),(α>1/2,0<z<)

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))

jα(t)=Γ(α+1)πΓ(α+1/2)11(1u2)α1/2cos(tu)du(8)

we have

|jα(t)|1,tR(9)

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

limλ(Bf)(λ)=0.

Moreover, from (9) we have

|(Bf)(λ)|0|f(t)||jα(λt)|t2α+1dt||f||1,α

and thus ∥𝓑f ≤ ∥f1,α is obtained.

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

Jα(r)=O(r1/2),r.(10)

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

jα(r)=O(rα1/2),r.(11)

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

(Bf(at))(x)=a2α2(Bf)(xa),a>0.

Lemma 2.2

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

||Bf||2,α=||f||2,α.(12)

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

(fg)(s)=0Tsf(t)g(t)t2α+1dt.(13)

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])

fg=gf,(fg)h=f(gh),B(fg)(λ)=(Bf)(λ)(Bg)(λ).(14)

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

||fg||p,α||f||1,α||g||p,α,fL1,α,gLp,α,1p.

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

(Sf)(x)=(0|(fΦt)(x)|2dtt)1/2(15)

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

||Sf||2,αc||f||2,α.

Proof

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

||Sf||2,α2=0(0|(fΦt)(x)|2dtt)x2α+1dx=00|(fΦt)(x)|2x2α+1dxdtt=0||fΦt||2,α2dtt=0||B(fΦt)||2,α2dtt=00|B(fΦt)(x)|2x2α+1dxdtt.

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

||Sf||2,α2=00|(Bf)(x)|2|(BΦt)(x)|2x2α+1dxdtt=0|(Bf)(x)|2(0|(BΦt)(x)|2dtt)x2α+1dx.(16)

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

(BΦt)(x)=t2α2(BΦ(xt))=t2α2t2α+2(BΦ)(tx)=(BΦ)(tx).

Thus

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

||Sf||2,α2=c0|(Bf)(x)|2x2α+1dx=c||f||2,α2(17)

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

0|(BΦ)(τ)|2dττ=01|(BΦ)(τ)|2dττ+1|(BΦ)(τ)|2dττ=I1+I2.

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

|(BΦ)(τ)|0|Φ(t)||jα(τt)1|t2α+1dt

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

|jα(τt)1|Γ(α+1)πΓ(α+1/2)11(1u2)α1/2|cos(τtu)1|du=2Γ(α+1)πΓ(α+1/2)11(1u2)α1/2sin2(τtu2)duc1t2τ2.

Therefore,

|(BΦ)(τ)|c1τ20|Φ(t)|t2α+3dt=c2τ2

and

I1=01|(BΦ)(τ)|2dττ=c2201τ4dττ=c3<.

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

|jα(u)|{c4,0<u1c5uα+12,u>1}c6uα+12,c6=max{c4,c5}.

Hence

|(BΦ)(τ)|0|Φ(t)||jα(τt)|t2α+1dt0|Φ(t)|c6τα+12tα+12t2α+1dt=c7τα+12,α>1/2

and we have

I2=1|(BΦ)(τ)|2dττc711τ2α+1dττ=c8<.

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

((Sfn)(x)(Sfm)(x))2=((0|(fnΦt)(x)|2dtt)1/2(0|(fmΦt)(x)|2dtt)1/2)20|((fnΦt)(fmΦt))(x)|2dtt=0|((fnfm)Φt)|2dtt.

and

|(Sfn(x)(Sfm)(x)|(0|((fnfm)Φt)|2dtt)1/2=S(fnfm)(x).

Hence, by (3.17) we get

||SfnSfm||2,α||S(fnfm)||2,αc||fnfm||2,α.

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

||Sf||2,αc||f||2,α,fL2,α

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

|A|B(H1,H2)=|A|=suphH1(|Ah|H2|h|H1).

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

fLp,α(R+,H)=||f||p,α=(0|f(x)|Hpx2α+1dx)1/p,1p<

is finite. If p = ∞, then the norm

fL(R+,H)=esssupxRn|f(x)|H

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)

λ|{|Af|H2>λ}|c1||f||1,α

and

λr|{|Af|H2>λ}|crr||f||r,αr,

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

λr|{|Af|H2>λ}|c1r||f||r,αr,somer>1

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

Rn/B(x0,c2R)|Af(x)|H2dxc3||f||1,α.(18)

Then

λ|{|Af|H2>λ}|c||f||1,α.

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

|φ|H2=(0|φ(t)|2dtt)1/2.

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

K(x)=t2α2Φ(xt)=Φt(x).

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

x2y|TyK(x)K(x)|H2x2α+1dxc,yR+.(19)

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

|Φ(x)|c(1+x)(q+θ),q=2α+2,θ>0and|Φt(x)|ctθ(1+x)(q+θ)

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

x2y|TyK(x)K(x)|H2x2α+1dx=x2yxϵ+q2|TyΦt(x)Φt(x)|H2xϵ+q2x2α+1dx(x2yx(ϵ+q)x2α+1dx)1/2(x2y|TyΦt(x)Φt(x)|H22xϵ+qx2α+1dx)1/2cyϵ2(x2yxϵ+q(0|TyΦt(x)Φt(x)|2dtt)x2α+1dx)1/2cyϵ2(0t2q(x2yxϵ+q|TytΦ(xt)Φ(xt)|2x2α+1dx)dtt)1/2.

Since

|TytΦ(xt)Φ(xt)|c(tx)q+ϵ

then we get

x2y|TyK(x)K(x)|H2x2α+1dxcyϵ20tq+ϵ0|TytΦ(xt)Φ(xt)|x2α+1dxdtt1/2cyϵ20tq+ϵ(tq2||Φt||1,α)dtt1/2cyϵ20ytϵ1dt+ydttϵ+11/2cyϵ2yϵ2=c.

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 < ∞.

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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.

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© 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

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