* French, Belgian authorities seek suspects, accomplices
* Belgian hire car linked to Paris attack
* One of three arrested in Brussels was in Paris on Friday
* Brussels' Molenbeek district linked to many past probes (Adds detail, expert comment)
By Robert-Jan Bartunek and Barbara Lewis
BRUSSELS, Nov 15 (Reuters) - Belgian police arrested three people on Saturday in raids in a poor, immigrant quarter of Brussels as they pursued emerging links between the Paris attacks and an Islamist bastion in France's northern neighbour.
Prime Minister Charles Michel said at least one of those held from the inner Brussels neighbourhood of Molenbeek was believed to have spent the previous evening in Paris, where two cars registered in Belgium were impounded close to scenes of some of the violence, including the Bataclan music hall.
"Police operations will go on," Michel told RTL television after late-night police raids in Molenbeek, west of the city centre, which is home to many Muslims, notably families originally from Morocco and Turkey. His interior minister spoke impatiently of going in to "clean up" Molenbeek.
"We're talking about a network," the borough's mayor, Francoise Schepmans, said on Sunday, referring to a total of five arrests in Molenbeek.
A French prosecutor said a car hired in Belgium was linked to the attacks and that a Frenchman living in Brussels rented it and was later stopped early on Saturday at the Belgian border.
A parking ticket issued in Molenbeek was found in the hire car in Paris. Officials declined comment on reports that the attacks may have been largely plotted in Brussels, that one of three attack teams came from there and that at least three of the attackers were based in the European Union capital.
Proportional to its 11 million population, of whom half a million are Muslim, Belgium has been the European country which has contributed the most foreign fighters to the civil war in Syria -- over 300 by official estimates a year ago -- and it has figured in many Islamist attacks and plots across the continent.
Many international security experts have long seen Belgium, with its Muslim population, fragmented state structures resulting from bitter divisions between French- and Dutch-speakers and a history as a market for firearms, both legal and illegal, as an "Achilles heel" of violence Islam across Europe.
A prominent, Moroccan-born member of the group behind the 2004 Madrid train bombings that killed 191 was from Molenbeek.
The area has been connected with two attacks in France this year. Security officials have said the Islamist who killed people at a Paris kosher grocery in January at the time of the attack on the magazine Charlie Hebdo acquired weapons in the district. So too did the man overpowered in August on a Thalys high-speed train from Brussels to Paris before he killed anyone.
An alleged plot to attack Belgian police in January, which was broken up by raids in which two men were killed in the eastern town of Verviers, had connections to Molenbeek. And a Frenchman accused of shooting dead four people last year at the Jewish Museum in Brussels also spent time in the area.
One international security expert said: "Belgium is a bit the Achilles heel of Europe." Another, Rafaello Pantucci of London's Royal United Services Institute, said: "The networks between France and Belgium have been very tight for some time."
Michel said: "Belgium has a central position at the heart of Europe, a small country whose local scale favours the movement of people with hostile intent," he said. But he insisted his centre-right coalition, in power for a year, was tackling that.
The problem of young men returning radicalised, and with skills in handling automatic weapons remained, however, Michel, 39, said: "It's always possible to slip through the net."
His interior minister, Jan Jambon, told Belgian television he believed Brussels and Molenbeek in particular was a problem and that he would personally take charge of sorting out issues in a neighbourhood which conservative critics view as an example of failed left-wing experiments in mass immigration.
Schepmans, Molenbeek's centre-right mayor said many radicals passed through, taking advantage of the anonymity afforded by parts of the borough that are almost entirely Muslim.
Criticism of authorities in Brussels and Molenbeek reflects in part Belgium's deep political and linguistic divide between Dutch- and French-speaking communities.
Interior Minister Jambon is a Flemish nationalist who highlighted the success of efforts to break Islamist radical groups in Antwerp and other cities in Flanders. Only in largely in French-speaking Brussels, and particularly Molenbeek, until recently a fiefdom of the French-speaking Socialist party, was there still a problem, he said.
The extreme local devolution that has been Belgium's answer to centrifugal communal forces pulling it apart for decades has been a factor in weakening its response to threats, said Edwin Bakker, professor at the Centre for Terrorism and Counterterrorism at the University of Leiden in the Netherlands.
"Belgium is a federal state and that's always an advantage for terrorists ... Having several layers of government hampers the flow of information among investigators," he said, adding that Belgium, long a big arms manufacturer, had also become a hub for the traffic of guns from the Balkans and elsewhere.
"In parts of Brussels there are areas where the police have little grip, very segregated areas that don't feel they're part of the Belgian state," he added. "While the neighbours may have seen something going on, they're not passing it to the police."
The Belgian prosecutors would not say whether any of those arrested on Saturday were previously known to authorities.
At least three of those killed in the Paris attacks were Belgian and the country launched its own anti-terrorist investigation into the events as a result. Michel said it would work in close cooperation with the French inquiries.
After a meeting of the national security cabinet, the government raised the level of alert across the country for large events, giving officials the ability to call in troops.
Michel urged Belgians not to travel to Paris unless absolutely necessary. Security checks were stepped up at the French border and at airports and rail stations.
Though there has been violence in Belgium, and Brussels is home to the institutions of the European Union, France, and Paris in particular, has appeared a higher-profile target, more clearly associated with the action in the war in Syria. And, across borders normally unchecked, Paris lies barely three hours by road and under 90 minutes by high-speed train from Brussels. (Additional reporting by Alastair Macdonald and Phil Blenkinsop in Brussels and Mark Hosenball in Washington; Writing by Alastair Macdonald; Editing by Diane Craft and Anna Willard)
Our Standards: The Thomson Reuters Trust Principles.