Vector-valued Grand Hardy Classes

Summary: It is considered the vector-valued grand Lebesgue space Lp) (X) ≡ Lp) (J;X), 1 < p < ∞, and the concept of a t-basis, generated by some bilinear map (where J = [−π, π)), is introduced. It is proved that the exponential system E ≡ eint n∈Z forms a t-basis for Np) (X), when X is a UMD space, where Np) (X) is the closure of Xvalued infinitely differentiable functions in Lp) (X). The concept of the t-Riesz property of the system E in Np) (X) is defined. It is established that this system has the t-Riesz property, when X is a UMD space. Using these facts, the X-valued grand Hardy classes mH± p) (X) of X-valued analytic functions are introduced, and some of their properties are proved. The obtained results are applied to establish the t-basicity of the perturbed exponential system in Np) (X).