Fourier Fractal Measures and Weak*- Integrals of Operators and an Inequality for Unbounded Domains

Abstract

We study the fractal theory and self- similar fractals in noncommutative geometry .We construct the Fredholm modules on post critically finite self –similar fractals and their conformal geometry . We investigate the norm inequalities far self-adjoint derivations for all unitarily invariant norms . comparison of various natural means for operators are considered . We study the relations of Cauchy – Schwarz and means inequalities for elementary operators into norm ideals . we verified and discussed many bounds and estimates of sharp Moser- Trudging inequalities of extremal functions and unbounded domain in the Euclidean spaces with compact Riemannian manifolds Also the simplification of the existence of the extremal functions .We study the connections between the almost periodic functions and the measure theory of fractal sets that introduced by Besicovitch and Wiener -Fourier expansions. we construct the relations between Fourier asymptotics and fractal measures . Results about reverses of Cauchy-Schwarz inequality in inner product spaces and Cauchy- Bunyakovsky -Schwarz type inequalities for sequences of operators in Hillbert spaces with Cauchy- Schwarz norm inequalities for weak*- integrals of operator valued functions are considered.