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Nigel Kalton, Marius Mitrea
2018 Transactions of the American Mathematical Society  
We investigate the stability of Fredholm properties on interpolation scales of quasi-Banach spaces. This analysis is motivated by problems arising in PDE's and several applications are presented. Interpolation scales of quasi-Banach spaces. The abstract setup Throughout the paper, for a quasi-normed space (X, · X ), we shall denote by ρ = ρ(X) its modulus of concavity, i.e. the smallest positive constant for which x,y∈X (note that always ρ(X) ≥ 1). We recall the Aoki-Rolewicz theorem [26], [31]
more » ... theorem [26], [31] , which asserts that X can be given an equivalent r-norm (where 2 1/r−1 = ρ) i.e. a quasinorm which also satisfies the inequality: x + y ≤ ( x r + y r ) 1/r . In general a quasi-norm need not be continuous but an r-norm is continuous. We shall assume however, throughout the paper that all quasi-norms considered are continuous: in fact, of course it would suffice to consider an r-norm for suitable r. Let U be a fixed, (Hausdorff) locally compact, locally connected topological space, referred to in the sequel as the space of parameters, and let δ : U × U → C be a (fixed) continuous function such that δ(z, w) = 0 if and only if z = w. Also, suppose that Z is a complex (Hausdorff) topological vector space, called the ambient space. A family F of functions which map U into Z is called admissible (relative to U , δ and Z) provided the following axioms are satisfied:
doi:10.1090/s0002-9947-98-02008-x fatcat:v6pd2jzi4bggdp366xkiwxfspa