On fractional inequalities on metric measure spaces with polar decomposition

In this paper, we prove the fractional Hardy inequality on polarisable metric measure spaces. The integral Hardy inequality for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>1</m:mn> <m:mo><</m:mo> <m:mi>p</m:mi> <m:mo>≤</m:mo> <m:mi>q</m:mi> <m:mo><</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:math> 1<p≤ q<∞ is playing a key role in the proof. Moreover, we also prove the fractional Hardy–Sobolev type inequality on metric measure spaces. In addition, logarithmic Hardy–Sobolev and fractional Nash type inequalities on metric measure spaces are presented. In addition, we present applications on homogeneous groups and on the Heisenberg group.