Interpolation of bilinear operators between Banach function spaces

We study bilinear operators between couples of Banach function spaces with the second coordinate $L_\infty$-space. We show an estimate in terms of the $K$-functional. This is used to prove a result on interpolation of bilinear operators between considered couples

Extension and splitting theorems for Fréchet spaces of type 2

Sin resume

Generalized Lions-Peetre methods of constants and means and operator ideals

We establish results on interpolation of Rosenthal operators, Banach-Saks operators, Asplund operators and weakly compact operators by means of generalized Lions-Peetre methods of constants and means. Applications are presented for the $K$-method space generated by the Calderón-Lozanovskii space parameters

The Lions's problem for Gustavsson-Peetre functor

The problem of coincidence of the interpolation spaces obtained by use of the interpolation method of Gustavsson-Peetre generated by (parameters) quasi-concave functions is investigated. It is shown that a restriction of this method to the class of all non-trivial Banach couples gives different interpolation spaces whenever two different parameters satisfying some conditions are used

Factorization theorems for some new classes of multilinear operators

[EN] Two new classes of summing multilinear operators, factorable (q,p)-summing operators and (r;p,q)-summing operators are studied. These classes are described in terms of factorization. It is shown that operators in the first (resp., the second) class admit the factorization through the injective tensor product of Banach spaces (resp., through some Banach lattices). Applications in different contexts related to Grothendieck Theorem and Fourier integral bilinear operators are shown. Motivated by Pisier¿s Theorem on factorization of (q,1)-summing operators from C(K)-spaces through Lorentz spaces Lq,1 on some probability Borel measure spaces, we prove two variants of Pisier¿s Theorem for bilinear operators on the product of C(K)-spaces. We also prove bilinear versions of Mityagin¿Pe¿czy¿ski and Kislyakov Theorems.The research was supported by National Science Centre of Poland, project 2015/17/B/ST1/00064. The research was supported by the Ministerio de Ciencia, Innovacion y Universidades, Agencia Estatal de Investigacion (Spain) and FEDER under project MTM2016-77054-C2-1-P.Mastylo, M.; Sánchez Pérez, EA. (2020). Factorization theorems for some new classes of multilinear operators. Asian Journal of Mathematics. 24(1):1-30. https://doi.org/10.4310/AJM.2020.v24.n1.a1S13024

Towards a unified theory of Sobolev inequalities

We discuss our work on pointwise inequalities for the gradient which are connected with the isoperimetric profile associated to a given geometry. We show how they can be used to unify certain aspects of the theory of Sobolev inequalities. In particular, we discuss our recent papers on fractional order inequalities, Coulhon type inequalities, transference and dimensionless inequalities and our forthcoming work on sharp higher order Sobolev inequalities that can be obtained by iteration.Comment: 39 pages, made some changes to section 1

Notes on non-interpolation spaces

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