Interpolation theorem for the p-harmonic transform

Abstract

Abstract.We establish an interpolation theorem for a class of nonlinear operators in the Lebesgue spaces L s (R n ) arising naturally in the study of elliptic PDEs. The prototype of those PDEs is the second order p-harmonic equation div |∇u| p−2 ∇u = div f. In this example the p-harmonic transform is essentially inverse to div(The core of the matter is that we go beyond the natural domain of definition of this operator. Because of nonlinearity our arguments require substantial innovations as compared with the classical interpolation theory of Riesz, Thorin and Marcinkiewicz. The subject is largely motivated by recent developments in geometric function theory.

An essay on the Interpolation Theorem of Józef Marcinkiewicz – Polish patriot

“…The interested reader may wish to take a note of the interpolation lemmas in [1]. In the present paper I will try to elucidate some new advances of Marcinkiewicz interpolation theorem which arise from a study of the nonlinear p-harmonic type PDEs, [12,13,14,16,18,19,20]. The principal result in this paper can be described as follows:…”