Stratospheric temperatures are greatly influenced by the presence of Ozone.

~The start of the stratosphere is where ozone concentration begins rising significantly
~This creates a massive world wide upper inversion called the tropopause
~A cold ground air is not necessarily under a  high stratosphere deep freeze
~The best temperature level to judge the troposphere is 600 mb

   The tiny bit of atmospheric ozone in our world  has a huge influence ,  not only in keeping us safe from harmful UV light, but seems largely forgotten as the main greenhouse gas warming the stratosphere.  A lack of Ozone basically cools the stratosphere,  the famous CFC driven ozone holes
exist at about -80 C.   Too much ozone warms the stratosphere.  Not to confuse with sudden Stratospheric warming,  which does occur not too often ,  but some place too much emphasis on it.   Stratospheric ozone is key in stratospheric temperatures.   Some people strongly believe that stratospheric temperatures influence the troposphere.  Let's look at this:

April 25 2019,  NOAA daily composites, 600 mb temperature is very close to the temperature of the entire troposphere.  We see here a spring time  Tropospheric Polar Vortex ,  which its coldest vortices in dark purple.  So as a few experts like to say the stratosphere influences this.   Not so, the warmest 50 mb temperatures are right above the coldest Cold Temperature North Poles of the troposphere:

Is warmest, -44 C near the center of the locations having more ozone,  while over Europe,  at 50 mb level, it s -61 C  where there was far less ozone:

       The areas with the least ozone are in this case the warmer surface places.    So the stratosphere in  does not  seem to cool the troposphere.  Therefore emphasis on any explosive stratospheric warming event,  as rare as they may be,  must be taken that what happens in the Stratosphere,  often stays there. WD April 28 ,  2019