Thermal decomposition of K2CO3
This question serves as a prime example of how much fake and scientifically incorrect information, or more famously, fake news is there on the web. This is why ordinary web should not be anyone's first priority to search about scientific topics. There is a famous saying that &quot;paper never refused an ink&quot; which applies to the web as well in the sense that anyone can write anything. Use Google Scholar, which only searches research level articles (relatively more authentic).
With this type of search one can arrive at Thermal stability of potassium carbonate near its melting point. This paper rules out the possibilities of (1) and (2). The authors note that there is significant vapor pressure of carbon dioxide near the melting point, and suggest $\ce{K2O}$ formation. They also advise the readers to consult the following as the most comprehensive study on alkali carbonates. I will let you try Google Scholar for this one.
References

A. Reisman, Reactions of the Group VB pentoxides with alkali oxides and carbonates. IX. A DTA study of alkali metal carbonates, J. Am. Chem. Soc. 80 (1958) 3558-3561.