Peace protesters called on senior judges today to let them stay in a makeshift tent village outside Parliament.
They were ordered to leave Parliament Square Gardens by a High Court judge last week after a nine-day hearing but allowed to remain until today's appeal.
This is being heard by a panel of three judges headed by the Master of the Rolls, Lord Neuberger.
Jan Luba QC, representing one of the peace camp factions, said London Mayor Boris Johnson had no right to evict the protesters because he does not own the land, it belongs to the Queen.
He said the mayor failed to prove any legal title to the land and High Court judge Mr Justice Griffith Williams was wrong to find that he could reclaim it.
"The claimant cannot at will exclude the world from entering or remaining on Parliament Square Gardens and cannot bring possession proceedings."
Mr Luba also argued that even if the mayor could bring the proceedings, a court could not order possession because it would be incompatible with laws relating to rights to free speech and assembly.
Ashley Underwood QC, representing the mayor, said Parliament Square Gardens is an open space which the public have a right to use subject to the control of the mayor and to the Trafalgar Square and Parliament Square by-laws.
Mr Justice Griffith Williams had ruled that the mayor had the power to exclude the demonstrators and had reached a reasoned decision.
He said there was a pressing social need not to permit an indefinite camped protest on the site for the protection of the rights and freedoms of others to access.
The High Court judge said there was also a need to protect health - the camp has no running water, toilets and is attracting homeless people because of the lack of control.
At the conclusion of the hearing, the judges said that they hoped to give their decision next week and extended the stay - which allows the protesters to remain on the site - pending that judgment.