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

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.