On estimates for the Fourier transform in the space L p ( R n )

In this paper, we prove two estimates useful in applications for the Fourier transform in the space L p ( R n ) , 1 < p ≤ 2, as applied to some classes of functions characterized by a generalized modulus of continuity.


Introduction and preliminaries
In [2], Abilov et al. proved new estimates for the Fourier transform in the space L 2 (R) on certain classes of functions characterized by the generalized continuity modulus for these estimates, using a Steklov function. In this paper, we prove the generalization of Abilov's results [2] in the Fourier transform for multivariable functions on R n . For this purpose, we use spherical mean operator in the place of the Steklov function.
Assume that L 2 (R n ) the space of integrable functions f with the norm The Fourier transform for the function f ∈ L 1 (R n ) is defined by The inverse Fourier transform is defined by the formula The Plancherel theorem provides an extension of the Fourier transform to L 2 (R n ), i.e, Let j p (z) be a normalized Bessel function of the first kind, i.e., Hamma, R. Daher, N. Djellab and Ch. Khalil Consider in L 2 (R n ) the spherical mean operator (see [3]) where S n−1 is the unit sphere in R n , w n−1 its total surface measure with respect to the usual induced measure dw.
The finite differences of the first and higher orders are defined by ..., k and k = 1, 2, ....., I is the identity operator in L 2 (R n ).
The k th order generalized modulus of continuity of function f ∈ L 2 (R n ) is defined as Denote by L 2 r the class of functions f ∈ L 2 (R n ) such that D r f ∈ L 2 (R n ) r = 1, 2, ... (In the sense of Levi (see [5])).
where the operator D = According to [3], we have and By Parseval's identity, we obtain Proof. We have . By Plancherel identity, we have the result.

Main Result
Befor presenting the theorems and their proofs, for convenience, we intoduce the notation where c > 0 is a fixed constant and N → ∞.
Proof. Let f ∈ L 2 (R n ). Then

we have
New Estimates for the Fourier Transform in the Space L 2 (R n ) The previous inequality implies that This theorem is proved.
converge, then f ∈ L 2 r and where k = 1, 2... and N → +∞ Proof. Let f ∈ L 2 (R n ), we have using an Abel transformation we obtain where C > 0 is a positive constant. Hence We have By formula (2.3), we have New Estimates for the Fourier Transform in the Space L 2 (R n ) 7 i.e Now we estimate I 2 , we have It follows that Applying the relations 2 l−1 N 2 l−2 N y r−1 dy = 1 r N r 2 r(l−2) (2 r − 1) Using the fact the sequence m N (f ), N = 1, 2, ..., is monotonically decreasing, we can show (see [6]) that i 2r−1 m i (f )   then this theorem is proved.