The logarithmic Sobolev inequality on the Heisenberg group and applications to the uncertainty inequality and heat equation

Abstract

Abstract We consider the logarithmic Sobolev inequality on the Heisenberg group. One can derive the logarithmic Sobolev inequality from the Sobolev inequality, and we consider an application to the uncertainty inequality on the Heisenberg group. Moreover, one can also obtain a dissipative estimate of a solution of the heat equation on the Heisenberg group from the logarithmic Sobolev inequality.

Keywords

• Logarithmic Sobolev inequality
• Heisenberg group
• Heat equation
• Shannon inequality
• Uncertainty inequality
• Sobolev inequality