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

Advances in Difference Equations20142014:281

https://doi.org/10.1186/1687-1847-2014-281

Received: 24 June 2014

Accepted: 19 October 2014

Published: 31 October 2014

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.

Keywords

square functionsgeneralized translationwavelet transformCalderón reproducing formula

1 Introduction

The classical square functions f ( x ) S φ ( x ) = ( 0 | ( f φ t ) ( x ) | 2 d t t ) 1 2 , where φ S , S S ( R n ) is the Schwartz test function space and R n φ ( x ) d x = 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 ) ( 1 p < , 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 d x = 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 d y ,
(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 , 1 p ,
(2.3)
T y f f p , ν 0 , | y | 0 , 1 p .
(2.4)
The generalized convolution ‘B-convolution’ associated with the generalized translation operator is ( f g ) ( x ) = R n + f ( y ) ( T y g ( x ) ) y 2 ν d y 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 d y ,
where E + ( 0 , r ) = { y R + n : | y | < r } and | E + ( 0 , r ) | 2 ν = E + ( 0 , r ) x 2 ν d x = 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 d x ,
     
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 ) , 1 p , then
lim r 0 | E + ( 0 , r ) | 2 ν 1 E + ( 0 , r ) T y f ( x ) y 2 ν d y = 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 ) d y = 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 d y , 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 , 1 p ( 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 d z < .
(2.13)
If we denote w ( z ) = h ( z ) , we have from (2.13)
h ( 0 ) = 0 and h ( ) = 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 ) d z ,
(2.15)

where w ( z ) is known as ‘wavelet function’, 0 w ( z ) d z = 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:
( S f ) ( 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 , 1 p ( 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 ) | d z < .
  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 Re μ > 0 , Re ν > 0 , p > 0 (see [[30], p.370])
0 x ν 1 e μ x p d x = 1 p μ ν p Γ ( ν p ) ,
we have
0 x 2 ν e x 2 d x = 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 , ϵ 0 , 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 f S ( 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. □

Declarations

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).

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, Akdeniz University, Antalya, Turkey

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© Kele¿ and Bayrakç¿; licensee Springer. 2014

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