On generalized $K$-functionals in $L_p$ for $0<p<1$

TL;DR

This paper proves that the Peetre $K$-functional between certain $L_p$ spaces with $0<p<1$ and smooth function spaces generated by Weyl-type operators is identically zero, revealing a fundamental property of these function spaces.

Contribution

It establishes that the Peetre $K$-functional is identically zero in this setting, providing new insights into the structure of $L_p$ spaces for $0<p<1$ and their smooth subspaces.

Findings

The $K$-functional between $L_p$ and $W_p^ ext{ extpsi}$ is zero for $0<p<1$.

The proof utilizes properties of de la Vallée Poussin kernels and quadrature formulas.

The result links the geometry of $L_p$ spaces with approximation theory techniques.

Abstract

We show that the Peetre $K$-functional between the space $L_p$ with $0<p<1$ and the corresponding smooth function space $W_p^\psi$ generated by the Weyl-type differential operator $\psi(D)$, where $\psi$ is a homogeneous function of any positive order, is identically zero. The proof of the main results is based on the properties of the de la Vall\'ee Poussin kernels and the quadrature formulas for trigonometric polynomials and entire functions of exponential type.