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Square-like functions generated by the Laplace-Bessel differential operator

Abstract

We introduce a wavelet-type transform associated with the Laplace-Bessel differential operator Δ B = k = 1 n 2 x k + 2 ν k x k x k and the relevant square-like functions. An analogue of the Calderón reproducing formula and the L 2 , ν boundedness of the square-like functions are obtained.

MSC:47G10, 42C40, 44A35.

1 Introduction

The classical square functions f(x) S φ (x)= ( 0 | ( f φ t ) ( x ) | 2 d t t ) 1 2 , where φS, SS( R n ) is the Schwartz test function space and R n φ(x)dx=0, φ t (x)= t n φ( t 1 x), t>0, play important role in harmonic analysis and its applications; see Stein [1]. There are a lot of diverse variants of square functions and their applications; see Daly and Phillips [2], Jones et al. [3], Pipher [4], Kim [5]. Square-like functions generated by a composite wavelet transform and its L 2 estimates are proved by Aliev and Bayrakci [6].

Note that the Laplace-Bessel differential operator Δ B is known as an important operator in analysis and its applications. The relevant harmonic analysis, known as Fourier-Bessel harmonic analysis associated with the Bessel differential operator B t = d 2 d t 2 + 2 ν t d d t , has been the research area for many mathematicians such as Levitan, Muckenhoupt, Stein, Kipriyanov, Klyuchantsev, Löfström, Peetre, Gadjiev, Aliev, Guliev, Triméche, Rubin and others (see [714]). Moreover, a lot of mathematicians studied a Calderón reproducing formula. For example, Amri and Rachdi [15], Guliyev and Ibrahimov [16], Kamoun and Mohamed [17], Pathak and Pandey [18], Mourou and Trimèche [19, 20] and others.

In this paper, firstly we introduce a wavelet-like transform associated with the Laplace-Bessel differential operator,

Δ B = k = 1 n 2 x k 2 + 2 ν k x k x k ,ν=( ν 1 , ν 2 ,, ν n ),ν>0,

and then the relevant square-like function. The plan of the paper is as follows. Some necessary definitions and auxiliary facts are given in Section 2. In Section 3 we prove a Calderón-type reproducing formula and the L 2 , ν boundedness of the square-like functions.

2 Preliminaries

R + n ={x=( x 1 ,, x n ) R n : x 1 >0, x 2 >0,, x n >0} and let S( R + n ) be the Schwartz space of infinitely differentiable and rapidly decreasing functions.

L p , ν = L p , ν ( R + n ) (1p<, ν=( ν 1 ,, ν n ); ν 1 >0,, ν n >0) space is defined as the class of measurable functions f on R + n for which

f p , ν = ( R + n | f ( x ) | p x 2 ν d x ) 1 p <, x 2 ν dx= x 1 2 ν 1 x 2 2 ν 2 x n 2 ν n d x 1 d x 2 d x n .

In the case p=, we identify L L , ν with C 0 the space of continuous functions vanishing at infinity, and set f = sup x R + n |f(x)|.

The Fourier-Bessel transform and its inverse are defined by

f (x)= F ν (f)(x)= R + n f(y) ( k = 1 n j ν k 1 2 ( x k y k ) ) y 2 ν dy,
(2.1)
F ν 1 (f)(x)= c ν (n)( F ν f)(x), c ν (n)= [ 2 2 n k = 1 n Γ 2 ( ν k + 1 2 ) ] 1 ,
(2.2)

where j ν 1 2 is the normalized Bessel function, which is also the eigenfunction of the Bessel operator B t = d 2 d t 2 + 2 ν t d d t ; j v 1 2 (0)=1 and j ν 1 2 (0)=0 (see [10]).

Denote by T y (y R + n ) the generalized translation operator acting according to the law:

T y f ( x ) = π n / 2 k = 1 n Γ ( ν k + 1 2 ) Γ 1 ( ν k ) 0 π 0 π f ( x 1 2 2 x 1 y 1 cos α 1 + y 1 2 , , x n 2 2 x n y n cos α n + y n 2 ) k = 1 n sin 2 ν k 1 α k d α 1 d α n .

T y is closely connected with the Bessel operator B t (see [10]). It is known that (see [11])

T y f p , ν f p , ν y R + n ,1p,
(2.3)
T y f f p , ν 0,|y|0,1p.
(2.4)

The generalized convolution ‘B-convolution’ associated with the generalized translation operator is (fg)(x)= R n + f(y)( T y g(x)) y 2 ν dy for which

( f g ) = f g .
(2.5)

We consider the B-maximal operator (see [8, 21])

M B f(x)= sup r > 0 | E + (0,r) | 2 ν 1 E + ( 0 , r ) T y |f(x)| y 2 ν dy,

where E + (0,r)={y R + n :|y|<r} and | E + (0,r) | 2 ν = E + ( 0 , r ) x 2 ν dx=C r n + 2 ν . Moreover, the following inequalities are satisfied (see for details [22]).

  1. (a)

    If f L 1 , ν ( R + n ), then for every α>0,

    | { x : M B f ( x ) > α } | 2 ν c α R + n |f(x)| x 2 ν dx,

where c>0 is independent of f.

  1. (b)

    If f L p , ν ( R + n ), 1<p, then M B f L p , ν ( R + n ) and

    M B f p , ν C p f p , ν ,

where c p is independent of f.

Furthermore, if f L p , ν ( R + n ), 1p, then

lim r 0 | E + (0,r) | 2 ν 1 E + ( 0 , r ) T y f(x) y 2 ν dy=f(x).

Now, we will need the generalized Gauss-Weierstrass kernel defined as

g ν (x,t)= F ν 1 ( e t | | 2 ) (x)= c ν ( n ) t ( n + 2 | ν | ) 2 e x 2 4 t ,x R + n ,t>0
(2.6)

c ν (n) being defined by (2.2) and |ν|= ν 1 + ν 2 ++ ν n .

The kernel g ν (x,t) possesses the following properties:

(a) F ν ( g ν ( , t ) ) (x)= e t | x | 2 (t>0);
(2.7)
(b) R + n g ν (y,t)dy=1(t>0).
(2.8)

Given a function f: R n + C, the generalized Gauss-Weierstrass semigroup, G t f(x) is defined as

G t f(x)= R + n g ν (y,t) ( T y f ( x ) ) y 2 ν dy,t>0.
(2.9)

This semigroup is well known and arises in the context of stable random processes in probability, in pseudo-differential parabolic equations and in integral geometry; see Koldobsky, Landkof, Fedorjuk, Aliev, Rubin, Sezer and Uyhan (see [2326]).

The following lemma contains some properties of the semigroup { G t f } t 0 . (Compare with the analogous properties of the classical Gauss-Weierstrass integral [1, 27, 28].)

Lemma 2.1 If f L p , ν , 1p ( L C 0 ), then

(a) G t f p , ν c f p , ν ,
(2.10)
(b) lim t 0 G t f(x)=f(x).
(2.11)

The limit is understood in L p , ν norm and pointwise almost all x R + n . If f C 0 , then the limit is uniform on R + n .

(c) sup t > 0 | G t f(x)|c M B f(x),
(2.12)

where M B f is the well-known Hardy-Littlewood maximal function.

Moreover, let h(z) be an absolutely continuous function on [0,) and

α= 0 h ( z ) z dz<.
(2.13)

If we denote w(z)= h (z), we have from (2.13)

h(0)=0andh()=0
(2.14)

(see for details [29]).

Now, we define the following wavelet-like transform:

V t f(x)= 1 α 0 G t z f(x)w(z)dz,
(2.15)

where w(z) is known as ‘wavelet function’, 0 w(z)dz=0, and the function G t z f(x) is the generalized Gauss-Weierstrass semigroup.

Using wavelet-like transform (2.15), we define the following square-like functions:

(Sf)(x)= ( 0 | V t f ( x ) | 2 d t t ) 1 2 .
(2.16)

3 Main theorems and proofs

Theorem 3.1

  1. (a)

    Let f L p , ν , 1p ( L C 0 ), ν>0. We have

    V t f p , ν c 1 c 2 f p , ν (t>0),
    (3.1)

where c 1 = 2 2 | ν | n , |ν|= ν 1 + ν 2 ++ ν n , c 2 = 1 α 0 |w(z)|dz<.

  1. (b)

    Let f L p , ν , 1<p ( L C 0 ). We have

    0 V t f(x) d t t lim ϵ 0 ρ ϵ ρ V t f(x) d t t =f(x),
    (3.2)

where limit can be interpreted in the L p , ν norm and pointwise for almost all x R + n . If f C 0 , the convergence is uniform on R + n .

Theorem 3.2 If f L 2 , ν , then

S f 2 , ν 1 2 f 2 , ν .
(3.3)

Proof of Theorem 3.1 (a) By using the Minkowski inequality, we have

V t f p , ν = 1 α ( R + n | 0 G t z f ( x ) w ( z ) d z | p x 2 ν d x ) 1 p 1 α 0 | w ( z ) | G t z f p , ν d z , G t z f p , ν = ( R + n | R + n g ν ( y , t z ) T y f ( x ) y 2 ν d y | p x 2 ν d x ) 1 p R + n | g ν ( y , t z ) | ( R + n | T y f ( x ) | p x 2 ν d x ) 1 p y 2 ν d y f p , ν R + n | g ν ( y , t z ) | y 2 ν d y = c 1 f p , ν .

Taking into account the following equality for Reμ>0, Reν>0, p>0 (see [[30], p.370])

0 x ν 1 e μ x p dx= 1 p μ ν p Γ ( ν p ) ,

we have

0 x 2 ν e x 2 dx= 1 2 Γ ( ν + 1 2 ) ,ν>0

in one dimension. By using this equality, we get

c 1 = R + n | g ν ( y , t ) | y 2 ν d y = 2 n k = 1 n Γ 1 ( ν k + 1 2 ) t 2 ( n + 2 | ν | ) R + n e | y | 2 4 t y 2 ν d y ( y = 2 t y , d y = 2 n t n 2 d y ) = 2 n k = 1 n Γ 1 ( ν k + 1 2 ) t 2 ( n + 2 | ν | ) R + n e | y | 2 2 2 | ν | t | ν | 2 n t n 2 y 2 ν d y = 2 2 | ν | k = 1 n Γ 1 ( ν k + 1 2 ) R + n e | y | 2 y 2 ν d y = 2 2 | ν | k = 1 n Γ 1 ( ν k + 1 2 ) k = 1 n Γ ( ν k + 1 2 ) 2 n = 2 2 | ν | n .

So we have G t z f p , ν 2 2 | ν | n f p , ν , and then inequality (3.1).

  1. (b)

    Let ( A ϵ , ρ f)(x)= ϵ ρ V t f(x) d t t , 0<ϵ<ρ<. Applying Fubini’s theorem, we get

    ( A ϵ , ρ f ) ( x ) = 1 α ϵ ρ ( 0 G t z f ( x ) w ( z ) d z ) d t t = 1 α 0 w ( z ) ( ϵ ρ G t z f ( x ) d t t ) d z = 1 α 0 w ( z ) ( ϵ z ρ z G t f ( x ) d t t ) d z = 1 α 0 ( t ρ t ϵ w ( z ) d z ) G t f ( x ) d t t = 1 α 0 1 t [ h ( t ϵ ) h ( t ρ ) ] G t f ( x ) d t = 1 α 0 h ( t ) t G ϵ t f ( x ) d t 1 α 0 h ( t ) t G ρ t f ( x ) d t = ( A ϵ f ) ( x ) ( A ρ f ) ( x ) .

By Theorem 1.15 in [[28], p.3], if 1<p ( L C 0 ), then

lim ρ G ρ t f p , ν =0.

Therefore, by the Minkowski inequality and the Lebesgue dominated convergence theorem, taking into account Lemma 2.1, we have

A ρ f p , ν = 1 α ( R n + ( 0 h ( t ) t G ρ t f ( x ) d t ) p x 2 ν d x ) 1 p 1 α 0 h ( t ) t G ρ t f p , ν d t = 1 α 0 h ( t ρ ) t ρ G ρ t f p , ν 1 ρ d t 0 , ρ

and

A ϵ f f p , ν = ( R n + ( 1 α 0 h ( t ) t G ϵ t f ( x ) d t f ( x ) ) p x 2 ν d x ) 1 p = ( 2.13 ) ( R n + ( 1 α 0 h ( t ) t G ϵ t f ( x ) d t 1 α 0 h ( t ) t f ( x ) d t ) p x 2 ν d x ) 1 p 1 α 0 h ( t ) t G ϵ t f f p , ν d t 0 , ϵ 0 .

Finally, for 1<p ( L C 0 ), we get

A ϵ , ρ f f p , ν = A ϵ f f p , ν + A ρ f p , ν 0,ϵ0,ρ.

The a.e. convergence is based on the standard maximal function technique (see [[31], p.60], [29] and [32]). □

Proof of Theorem 3.2 Firstly, let fS( R + n ). By making use of the Fubini and Plancherel (for Fourier-Bessel transform) theorems, we get

S f 2 , ν 2 = R + n ( 0 | V t f ( x ) | 2 d t t ) x 2 ν d x = 0 ( R n + | V t f ( x ) | 2 x 2 ν d x ) d t t = 0 ( R n + | ( V t f ) ( x ) | 2 x 2 ν d x ) d t t

and

( V t f ) ( x ) = F ν ( V t f ) ( x ) = 1 α R n + ( 0 G t z f ( y ) w ( z ) d z ) k = 1 n j ν k 1 2 ( x k y k ) y 2 ν d y = 1 α 0 w ( z ) ( R n + G t z f ( y ) k = 1 n j ν k 1 2 ( x k y k ) y 2 ν d y ) d z = 1 α 0 w ( z ) ( G t z f ) ( x ) d z = ( 2.5 ) 1 α 0 w ( z ) f ( x ) e t z | x | 2 d z .

Now, by using Fubini’s theorem, we have

S f 2 , ν 2 = 1 α 2 0 [ R n + ( f ( x ) ) 2 ( 0 w ( z ) e t z | x | 2 d z ) 2 x 2 ν d x ] d t t = 1 α 2 R n + ( f ( x ) ) 2 0 d t t ( 0 w ( z ) e t z | x | 2 d z ) 2 x 2 ν d x ( t = τ | x | 2 , d t = | x | 2 d τ ) = 1 α 2 R n + ( f ( x ) ) 2 0 d τ τ ( 0 w ( z ) e τ z d z ) 2 x 2 ν d x = C 2 1 α 2 f 2 , ν 2 ,

where

C= ( 0 d τ τ ( 0 e τ z w ( z ) d z ) 2 ) 1 / 2 .

Since w(z)= h (z), h(z)0, h()=h(0)=0, it follows that

C = ( 0 d τ τ ( 0 e τ z w ( z ) d z ) 2 ) 1 / 2 = ( 0 ( 0 τ e τ z h ( z ) d z ) 2 d τ ) 1 / 2 0 h ( z ) ( 0 τ e 2 τ z d τ ) 1 / 2 d z ( 2 z τ = t , 2 z d τ = d t ) = 0 h ( z ) ( 0 t 2 z e t 1 2 z d t ) 1 / 2 d z = 0 h ( z ) 2 z ( 0 t e t d t ) 1 / 2 d z = 1 2 α .

Finally, we get

S f 2 , ν 1 2 f 2 , ν .

For arbitrary f L 2 , ν ( R + n ), the result follows by density of the class S( R + n ) in L 2 , ν ( R + n ). Namely, let ( f n ) be a sequence of functions in S( R + n ), which converge to f in L 2 , ν ( R + n )-norm. That is, lim n f n ( x ) f ( x ) 2 , ν =0, x R + n .

From the ‘triangle inequality’ ( ( u 2 , ν v 2 , ν ) 2 u v 2 , ν 2 ), we have

| ( S f n ) ( x ) ( S f m ) ( x ) | 2 = [ ( 0 | V t f n ( x ) | 2 d t t ) 1 2 ( ( 0 | V t f m ( x ) | 2 d t t ) 1 2 ) ] 2 0 | V t f n ( x ) V t f m ( x ) | 2 d t t = 0 | V t ( f n f m ) | 2 d t t = ( S ( f n f m ) ( x ) ) 2 .

Hence

S f n S f m 2 , ν S ( f n f m ) 2 , ν 1 2 f n f m 2 , ν .

This shows that the sequence (S f n ) converges to Sf in L 2 , ν ( R + n )-norm. Thus

S f 2 , ν 1 2 f 2 , ν ,f L 2 , ν ( R + n )

and the proof is complete. □

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Acknowledgements

The authors would like to thank the referees for their valuable comments. This work was supported by the Scientific Research Project Administration Unit of the Akdeniz University (Turkey).

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Correspondence to Şeyda Keleş.

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Keleş, Ş., Bayrakçı, S. Square-like functions generated by the Laplace-Bessel differential operator. Adv Differ Equ 2014, 281 (2014). https://doi.org/10.1186/1687-1847-2014-281

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Keywords

  • square functions
  • generalized translation
  • wavelet transform
  • Calderón reproducing formula