INEQUALITIES IN THE MOST SIMPLE SOBOLEV SPACE AND CONVOLUTIONS OF L(2) FUNCTIONS WITH WEIGHTS

For the most simple Sobolev space on R composed of real-valued and abs olutely continuous functions f(x) on R with finite norms ntegral-infin ity/-infinity(a2f'(x)2+b2f(x)2)dx}1/2 (a,b > 0), we shall apply the th eory of reproducing kernels, and derive natural norm inequalities in t he space and the related inequalities for convolutions of L2 functions with weights.