Real Interpolation Spaces Research Articles

Real Interpolation with Constraints

Given Banach spaces X, a subspace Y, and a finite set G of bounded linear functionals on Y, let YG denote all elements of Y which are annihilated by the functionals in G. We investigate the relation between the real interpolation spaces (X, Y)θ, ρ and (X, YG)θ, ρ. Applications are given to Sobolev spaces, best approximation with constraints, and weighted Lebesgue spaces.

Real Interpolation for Families of Banach Spaces and Convexity

AbstractWe show how the geometrical properties of uniform convexity and uniformly non‐𝓁 are inherited by real interpolation spaces for infinite families.

Results on a mathematical model of thermomechanical phase transitions in shape memory materials

In this paper we prove the local existence of solutions for a mathematical model of structural phase transitions in solids with nonconvex Landau-Ginzburg free energy potentials. The partial differential equations that arise from the conservation laws of linear momentum and energy are sometimes referred to as the equations of thermo-visco-elasto-plasticity. The theory of real interpolation spaces for maximal accretive operators is used to show the well posedness of the system when the initial data are in the domain of the ( 3/4 + epsilon )-power of the associated linear differential operator. This not only improves previous existence results by weakening the assumptions on the initial data, but also allows us to obtain some regularity properties of the solutions. The fact that local existence is obtained for initial data in the domain of a fractional delta -power of the differential operator, with delta strictly less than unity, has several important implications for continuity with respect to the admissible parameters.

Some remarks on the reflexivity of the real interpolation spaces

Beauzamy in [1] proved that the Lions-Peetre spaces A ̄ Θ, p (0 < Θ < 1, 1 < p < ∞) are reflexive if and only if the imbedding i: Δ( A ̄ ) → ∑( A ̄ ) is weakly compact. Later on, the Davis-Figiel-Johnson-Pelczynski factorization technique was again used in [10] to prove that the operator T: A ̄ Θ, p → B ̄ Θ, p is weakly compact if and only if T:Δ A ̄ → ∑( B ̄ ) is weakly compact, thereby generalizing the aforesaid result of Beauzamy. In this paper, a modification of this technique is used to give a characterization of the reflexivity for the more general spaces A ̄ Θ, E .

Real interpolation and compact linear operators

We prove that if T: A 0 → B 0 is compact and T: A 1 → B 1 is compact (or T: A 1 → B 1 is bounded and ▪, then given any θ and q with 0 < θ < 1 and 0 < q⩽ ∞, it follows that T: ( A 0, A 1) 0, q → ( B 0, B 1) 0, q , is also compact. Here ( A 0, A 1) 0, q and ( B 0, B 1) 0, q are the usual real interpolation spaces.

Reiteration theorems for real interpolation and approximation spaces

In the present paper we study a general form of Peetre's J- and K-methods of interpolation. Special emphasis is given to the equivalence theorem for J- and K-spaces and to reiteration theorems.

Quelques propriétés topologiques des espaces d'interpolation dans le cadre général

We study, in the general case, the reflexivity of the real Lions-Peetre interpolation spaces ( A 0, A 1) θ, p (0 < θ < 1, 1 < p < ∞). We prove that these spaces are reflexive if and only if the canonical injection i, from the intersection A 0 ∩ A 1 into the sum A 0 + A 1, is weakly compact.