Some spectral formulas for functions generated by differential and integral operators in Orlicz spaces
In this paper, we investigate the behavior of the sequence of L^Φ-norm of functions, which are generated by differential and integral operators through their spectra (the support of the Fourier transform of a function f is called its spectrum and denoted by sp(f)). With Q being a polynomial, we introduce the notion of Q-primitives, which will return to the notion of primitives if Q(x)= x, and study the behavior of the sequence of norm of Q-primitives of functions in Orlicz space L^Φ(R^n). We have the following main result: let Φ be an arbitrary Young function, Q(x ) be a polynomial and (Q^mf)_m=0^∞ L^Φ(R^n) satisfies Q^0f=f, Q(D)Q^m+1f=Q^mf for mZ_+. Assume that sp(f) is compact and sp(Q^mf)= sp(f) for all mZ_+. Then _m→ ∞ \|Q^m f\|_Φ^1/m= _x sp(f)