Uncertainty Type Principles for Radial Derivatives

Abstract

In this paper, we provide uncertainty type principles for radial derivatives on an open bounded set $\varOmega \subset \mathbb {R}^{n}$. We obtain several inequalities of uncertainty type principles of the form

$$ \int _\varOmega |f(x)|^p \, \phi (|x|)\, dx \le p \, \left( \int _\varOmega \frac{\left| \mathscr {R}_{|x|}f(x)\right| ^p \, |\tilde{\phi }(|x|)|^p}{|x|^{n-p}}\, dx\right) ^\frac{1}{p} \left( \int _\varOmega \frac{|f(x)|^p}{|x|^n}\,dx\right) ^\frac{p-1}{p}, $$

for $1<p<+\infty$, acting on functions with the radial derivative $\mathscr {R}_{|x|}$. As byproduct, we present versions of the Hardy and higher order Steklov inequalities for the radial derivatives.