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

Simten Bayrakci

doi:10.1515/math-2018-0067

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

Simten Bayrakci

Published/Copyright: July 4, 2018

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$Open Mathematics$

From the journal Open Mathematics Volume 16 Issue 1

Abstract

In this paper, we consider the square function

(Sf)(x)=(∫0∞|(f⊗Φt)(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 L_p,α, p > 1. Firstly, we proved L_2,α-boundedness by means of the Bessel-Plancherel theorem. Then, its weak-type (1, 1) and L_p,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∞|(f∗Φt)(x)|2dtt)1/2,Φ∈S(Rn)

where S(ℝⁿ) is the Schwartz space consisting of infinitely differentiable and rapidly decreasing functions, ∫RnΦ (x)dx = 0 and Φ_t(x) = t−nΦ(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 B_t,

Bt=d2dt2+(2α+1)tddt,α>−1/2,t>0

and the Laplace-Bessel differential operator Δ_B,

ΔB=∑k=1n−1∂2∂xk2+(∂2∂xn2+(2α+1)xn∂∂xn),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 B_t (or the Laplace-Bessel differential operator Δ_B) 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 B_t is introduced on the half-line ℝ₊ = [0, ∞) and its L_2,α− boundedness by means of the Bessel-Plancherel theorem is proved. Then, (1, 1) weak-type and L_p,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 L_p,α = L_p,α(ℝ₊) be the space of measurable functions f defined on ℝ₊ and the norm

‖f‖p,α=(∫0∞|f(x)|px2α+1dx)1/p,1≤p<∞(1)

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

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

Tsf(t)=cα∫0πf(s2−2stcos⁡ξ+t2)(sin⁡ξ)2αdξ,(2)

where

cα=(∫0π(sin⁡ξ)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)=T−sf(−t);T0f(t)=f(t);∫0∞(Tsf(t))g(t)t2α+1dt=∫0∞f(t)(Tsg(t))t2α+1dt.(4)

It is not difficult to see the following inequality

‖Tsf‖∞≤‖f‖∞

that is, T^s is a continuous operator in C₀. Moreover, for 1 ≤ p < ∞ and f ∈ S⁺(ℝ₊) it is shown that

|Tsf(t)|p≤Ts(|f(t)|p).(5)

For this, we define a measure on the [0, π] by dμ (φ) = c_α(sin φ)^2αdφ, where c_α is defined by (3). By using (2) and the Hölder inequality, we have

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

||Tsf||p,αp=∫0∞|Tsf(t)|pt2α+1dt≤∫0∞Ts(|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 L_p,α for p < ∞, (6) stays valid for every function in f ∈ L_p,α.

Note that T^s, 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) = T^sf(t), f ∈ C²(ℝ₊) is the solution the following Cauchy problem, (see [8, 25]):

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

The Bessel transform of order α > −1/2 of a function f ∈ L_1,α is defined by

(Bf)(λ)=∫0∞f(t)jα(λt)t2α+1dtλ≥0,(7)

and the inverse Bessel transform is given by the formula

B−1=2αΓ(α+1))−2B

where

jα(z)=2αΓ(α+1)z−αJα(z),(α>−1/2,0<z<∞)

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

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

we have

|jα(t)|≤1,t∈R(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∥_∞ ≤ ∥f∥_1,α is obtained.

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

Jα(r)=O(r−1/2),r→∞.(10)

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

jα(r)=O(r−α−1/2),r→∞.(11)

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

Lemma 2.1

([25]). Letf ∈ L_1,αthen

(Bf(at))(x)=a−2α−2(Bf)(xa),a>0.

Lemma 2.2

([25], Bessel-Plancherel formula). Letf ∈ L_1,α ∩ L_2,αthen

||Bf||2,α=||f||2,α.(12)

The generalized convolution generated by the Bessel translation operator for f, g ∈ L_1,α is defined by

(f⊗g)(s)=∫0∞Tsf(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, g ∈ S⁺(ℝ₊), then the convolution f ⊗ g also belongs to S⁺(ℝ₊).

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

f⊗g=g⊗f,(f⊗g)⊗h=f⊗(g⊗h),B(f⊗g)(λ)=(Bf)(λ)(Bg)(λ).(14)

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

||f⊗g||p,α≤||f||1,α||g||p,α,f∈L1,α,g∈Lp,α,1≤p≤∞.

3 Main results and proofs

In this part, the L_2,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 L_p,α, 1 < p < ∞ boundedness is obtained by using vector-valued functions.

Definition 3.1

LetΦ ∈ S⁺(ℝ⁺) and∫0∞Φ(x)x2α+1dx=0.The square function associated with the Bessel differential operator is defined by

(Sf)(x)=(∫0∞|(f⊗Φt)(x)|2dtt)1/2(15)

whereΦt(x)=t−2α−2Φ(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 L_2,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 𝓢fbe defined as(15). Iff ∈ L_2,αthen there isc > 0 such that

||Sf||2,α≤c||f||2,α.

Proof

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

||Sf||2,α2=∫0∞(∫0∞|(f⊗Φt)(x)|2dtt)x2α+1dx=∫0∞∫0∞|(f⊗Φt)(x)|2x2α+1dxdtt=∫0∞||f⊗Φt||2,α2dtt=∫0∞||B(f⊗Φt)||2,α2dtt=∫0∞∫0∞|B(f⊗Φt)(x)|2x2α+1dxdtt.

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

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

Since Φ_t(x) = t−2α−2Φ(xt), then using Lemma 1, we have

(BΦt)(x)=t−2α−2(BΦ(xt))=t−2α−2t2α+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=c∫0∞|(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 ∫0∞Φ(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 j_α(t) we get

|jα(τt)−1|≤Γ(α+1)πΓ(α+1/2)∫−11(1−u2)α−1/2|cos⁡(τtu)−1|du=2Γ(α+1)πΓ(α+1/2)∫−11(1−u2)α−1/2sin2⁡(τtu2)du≤c1t2τ2.

Therefore,

|(BΦ)(τ)|≤c1τ2∫0∞|Φ(t)|t2α+3dt=c2τ2

and

I1=∫01|(BΦ)(τ)|2dττ=c22∫01τ4dττ=c3<∞.

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

|jα(u)|≤{c4,0<u≤1c5uα+12,u>1}≤c6uα+12,c6=max{c4,c5}.

Hence

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

and we have

I2=∫1∞|(BΦ)(τ)|2dττ≤c7∫1∞1τ2α+1dττ=c8<∞.

For arbitrary f ∈ L_2,α, we will take into account that the Schwartz space S⁺(ℝ₊) is dense in L_2,α. Namely, let (f_n) be a sequence of functions in S⁺(ℝ₊), which converges to f in L_2,α-norm.

From the “triangle inequality” (∥u∥_2,α − ∥v∥_2,α)²≤||u−v||2,α2, we have

((Sfn)(x)−(Sfm)(x))2=((∫0∞|(fn⊗Φt)(x)|2dtt)1/2−(∫0∞|(fm⊗Φt)(x)|2dtt)1/2)2≤∫0∞|((fn⊗Φt)−(fm⊗Φt))(x)|2dtt=∫0∞|((fn−fm)⊗Φt)|2dtt.

and

|(Sfn(x)−(Sfm)(x)|≤(∫0∞|((fn−fm)⊗Φt)|2dtt)1/2=S(fn−fm)(x).

Hence, by (3.17) we get

||Sfn−Sfm||2,α≤||S(fn−fm)||2,α≤c||fn−fm||2,α.

This shows that the sequence (Sf_n) converges to (Sf) in L_2,α −norm. Thus

||Sf||2,α≤c||f||2,α,∀f∈L2,α

and the proof is complete. □

Now, taking into account vector-valued functions spaces, we will obtain L_p,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|=suph∈H1(|Ah|H2|h|H1).

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

‖f‖Lp,α(R+,H)=||f||p,α=(∫0∞|f(x)|Hpx2α+1dx)1/p,1≤p<∞

is finite. If p = ∞, then the norm

‖f‖L∞(R+,H)=esssupx∈Rn|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 beAa sublinear operator defined onL∞0(ℝ₊,H₁), i.e., compactly supported, boundedH₁-valued functions, with values inM(ℝ₊,H₂), i,e., the space of measurable, H₂-valued function. Suppose in addition that forf ∈ L∞0(ℝ₊,H₁)

λ|{|Af|H2>λ}|≤c1||f||1,α

and

λr|{|Af|H2>λ}|≤crr||f||r,αr,

wherec₁andc_rare independent of λ and f. Then for each 1 < p < r, we have that Af ∈ L_p,α(ℝ₊,H₂) wheneverf ∈ L_p,α(ℝ₊, H₁) and there is a constantc = c_1,r,pindependent offsuch that ∥Af∥_p,α ≤ c∥f∥_p,α.

Theorem 3.4

([31], Theorem 2.2, p.307). Suppose a linear operatorAdefined inL∞0(ℝ₊,H₁) and with values inM(ℝ₊, H₂) verifies

λr|{|Af|H2>λ}|≤c1r||f||r,αr,somer>1

and iffhas support inB(x₀, R) and integral 0, then there are constantsc₂,c₃ > 1 independent offso that

∫Rn/B(x0,c2R)|Af(x)|H2dx≤c3||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 Φ ∈ S⁺(ℝ⁺) and ∫0∞Φ(x)x2α+1dx=0 then we define K(x) to be the H₂-valued function given by

K(x)=t−2α−2Φ(xt)=Φt(x).

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

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

∫x≥2y|TyK(x)−K(x)|H2x2α+1dx≤c,y∈R+.(19)

Now let us calculate (19). For this, since Φ ∈ S⁺(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

Since

|TytΦ(xt)−Φ(xt)|≤c(tx)q+ϵ

then we get

∫x≥2y|TyK(x)−K(x)|H2x2α+1dx≤cy−ϵ2∫0∞t−q+ϵ∫0∞|TytΦ(xt)−Φ(xt)|x2α+1dxdtt1/2≤cy−ϵ2∫0∞t−q+ϵ(tq2||Φt||1,α)dtt1/2≤cy−ϵ2∫0ytϵ−1dt+∫y∞dttϵ+11/2≤cy−ϵ2yϵ2=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 L_2,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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Received: 2017-12-01

Accepted: 2018-04-25

Published Online: 2018-07-04

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https://doi.org/10.1515/math-2018-0067

Keywords for this article

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

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