Hardy Sharp Weighted Norm Inequalities and Norm Estimates for Certain Kernels

Abstract

We study the - estimates for certain kernels on Hilbert space , Schatten – von Neumann classes and the homogenous Besov- spaces . We introduce the sharp maximal Function on spaces of generalized homogenous type . We verified expressing functions such as squares or sums of squares is always possible . Certain arithmetic – geometric means and related inequalities for operators are established . Also a comparison of various means for operators are studied and new technique for proving norm inequalities in operator ideals are discussed . We give a norm estimate of almost Mathieu operators with a splitting planar isoperimetric inequality through produal norms of a conformally invariant space . We also find a sharp weighted transplantation result for Laguerre function expansions in Lebesgue spaces and , off diagonal weighed norm inequalities , estimates and elliptic operators . We study the generalization of the norm inequalities for self – adjoint derivatives and the Cauchy – Schwarz norm inequalities for weak* - integrals of operator valued functions . An ultimate estimate of the upper norm bound for the summation of operators , on Hilbert space , is considered . We give the mean values of regular functions , and power series , in the unit disc . We obtained a well – known Hardy’s inequality and a fractal measures on self – similar sets .