Fundamental iterated convolution inequalities in weighted Lp spaces and their applications

Abstract

In this paper, we obtain the inequalities for the iterated convolution and their applications to physical problems. We also get the inequalityand its applications in L p (R n , |ρ|) space. (2000): 44A35, 35A22, 26D20. Mathematics subject classification

Lp− Boundedness for Integral Transforms Associated with Singular Partial Differential Operators

“…Where I is the identity operator. Using the relation (17) and the inversion formula for F µ− 1 2 that is for every f ∈ L 1 (dν µ ) such that F µ− 1 2 (f ) belongs to L 1 (dν µ ) , we have…”

“…We recall in this context, that studing the L p − boundedness of integral transforms connected with differential systems is an interesting subject because knowing the range of parameters µ, p for which an operator is bounded on Lebesgue space gives quantitative information about the rate of growth of the transformed functions ( [15,16,17]) .…”

A formula for the norm of an averaging operator on weighted Lebesgue space

“…The Lebesgue spaces L p with weights of the form |x| α are a natural collection to consider when boundedness of integral operators is concerned [7,8,10]. This is particularly true when studying integral operators connected with differential systems.…”

Convolutions, integral transforms and integral equations by means of the theory of reproducing kernels

“…Proposition 2.5 was expanded in various directions with applications to inverse problems and partial differential equations through L p (p > 1) versions and converse inequalities. See, for example, [6][7][8][19][20][21][22][23][24][25][26][27][28][29].…”

“…In particular, for the product of two Hilbert spaces, the idea gives generalizations of convolutions and the related natural convolution norm inequalities. These norm inequalities gave various generalizations and applications to forward and inverse problems for linear partial differential equations, see for example, [4,[6][7][8][19][20][21][22][23][24][25][26][27][28][29]. Furthermore, surprisingly enough, for some very general nonlinear systems, we can consider similar problems (see [26] for details).…”