Kolmogorov-type inequalities for hypersingular integrals with homogeneous characteristics

Abstract

In this article we obtain sharp Kolmogorov-type inequalities that estimate the uniform norm of a hypersingular integral operator
$$
D^{w,\Omega}_K f(x): = \int_{C} w(|t|_K) (f(x+t) - f(x))\Omega(t)dt, x\in C,
$$
using the uniform norm of the function $$$f$$$ and either the norm $$$\|f\|_{H^\omega_K(C)}$$$ determined by a modulus of continuity $$$\omega$$$, or the weighted integral norm $$$\| \Omega^{\frac 1p} \cdot |\nabla f|_{K^\circ}\|_{L_p(C)}$$$ of the gradient $$$\nabla f$$$. Here $$$C$$$ is a convex cone in $$${\mathbb R}^d$$$, $$$d\geq 2$$$, $$$\Omega\colon C\to\mathbb R$$$ is a non-negative homogeneous of degree 0 locally integrable function, $$$w\colon (0,\infty)\to [0,\infty)$$$ is some weight function, $$$|\cdot|_K$$$ is an arbitrary norm in $$${\mathbb R}^d$$$, $$$|\cdot|_{K^\circ}$$$ is its polar norm, and $$$p\in (d,\infty]$$$.