I rent an apartment in Italy and the

Is it written in your rental contract?   It is not a legal requirement, but could be a requirement underthe condominio 'law' .   Public areas are public areas.  If the stairs from your apartment down to the next level are private then the people below dont have to clean them.  If they are public and anybody can use them, then they should be cleaned as should everywhere else. 

Usually every "condominio" or apartment building has it's own rules, drawn up for each individual the apartment building usually written by the "Amministratore" (if there is one) - these usually explain, for example where you can or cannot leave you bicycle or pram, where the rubbish should go, where kids can or cannot play - if they can or cannot play ball games...etc.However, usually these rules are not on display anywhere and you need to ask  the Amministratore, or whoever administers the running of the apartment block, to send you them - or at least send you or explain you the one that is relevant to you. Ciao

Just do it! Great exercise!! :) 

Life is not always fair!