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 $p\in (d,\infty]$.

Keywords

• kolmogorov-type inequality
• hypersingular integral operator
• modulus of continuity