By Robert-Jan Bartunek and Alastair Macdonald

BRUSSELS (Reuters) - "A breeding ground for violence" the mayor of Molenbeek called her borough on Sunday, speaking of unemployment and overcrowding among Arab immigrant families, of youthful despair finding refuge in radical Islam.

But as the Brussels district on the wrong side of the city's post-industrial canal becomes a focus for police pursuing those behind Friday's mass attacks in Paris, Belgian authorities are asking what makes the narrow, terraced streets of Molenbeek different from a thousand similar neighborhoods across Europe.

Three themes emerge as Molenbeek is again in a spotlight of Islamist violence, home not just to militants among Belgium's own half a million Muslims but, it seems, for French radicals seeking a convenient, discreet base to lie low, plan and arm before striking their homeland across the border:

Security services face difficulties due to Belgium's local devolution and tensions between the country's French- and Dutch-speaking halves; the country has long been open to fundamentalist preachers from the Gulf; and it has a thriving black market in automatic rifles of the kind used in Paris.

"With 500-1,000 euros you can get a military weapon in half an hour," said Bilal Benyaich, senior fellow at Brussels think-tank the Itinera Institute, who has studied the spread of radical Islam in Belgium. "That makes Brussels more like a big U.S. city" in mostly gun-free Europe, he said.

Two of the attackers who killed over 130 people, 170 miles away in Paris on Friday night were Frenchmen resident in Belgium. Belgian police raided Molenbeek addresses and seven people have been arrested in Belgium over the Paris attacks.

"Almost every time, there is a link to Molenbeek," said 39-year-old centrist prime minister Charles Michel, whose year-old coalition is battling radical recruiters who have tempted more than 350 Belgians to fight in Syria - relative to Belgium's 11 million population, easily the biggest contingent from Europe.

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But "preventive measures" of the past few months were not enough, Michel said, describing Molenbeek as a "gigantic problem" and saying: "There has to be more of a crackdown."

His interior minister, Jan Jambon, vowed to "cleanse" the district personally. Conservatives blamed lax oversight on left-wing predecessors, nationally and in Molenbeek town hall, and dueled over whether Dutch-speaking Flanders or mainly French-speaking Brussels and the south did more to curb the radicals.

Such differences, which have translated into a profusion of layers of government and policing in an effort to appease centrifugal forces that long threatened to break Belgium apart, have created problems for intelligence and security services.

Jambon has complained himself of a profusion of police forces across state and language lines, including six in Brussels alone, a city of just 1.8 million.

POLICE LACK "GRIP"

"Belgium is a federal state and that's always an advantage for terrorists," said Edwin Bakker, professor at the Centre for Terrorism and Counterterrorism at the University of Leiden in the Netherlands. "Having several layers of government hampers the flow of information between investigators."

Contrasting Belgium with its centralized Dutch neighbor, he added: "It's much more difficult for groups to disappear from the radar just by moving 10 kilometers."

Given the difficulty of gathering intelligence in places like Molenbeek, a borough of 90,000 where some neighborhoods were up to 80 percent Muslim, any gaps in the information chain were problematic, Bakker said: "In parts of Brussels there are areas on which the police have little grip, very segregated areas that don't feel they're a part of the Belgian state.

"In such a case it's very difficult to get feedback from the community. That means while the neighbors may have seen something going on, they're not passing it to the police. Then it becomes very tough for intelligence agencies as only relying on them and not local police is not sufficient."

Political complication is also blamed for slowing the passing of new laws, for example to rein in the preaching of hate in mosques or recruitment for and travel to the Syrian war.

While some of Molenbeek's old factories - it once enjoyed the industrious nickname "Petit Manchester" - have made it a smart address for bohemian loft living, areas tumbling out from the ship canal, offering halal butchers, street stalls and backstreet mosques are some of the poorest in northwest Europe.

The 25 percent jobless rate, rising to 37 percent among the young, is significantly higher than other parts of Brussels, also home to a thriving, cosmopolitan middle class drawn by the European Union institutions on the other side of the city.

Belgian officials are also increasingly concerned about the influence of radical versions of Islam. They remain a minority taste; the Muslim Executive of Belgium, an umbrella group, spoke of its support for democratic values and condemned "barbarism".

Molenbeek, which notably in 2012 saw street protests against enforcement of Belgian law on Muslim face veils, has, however, been among areas where fundamentalist preachers have flourished.

FUNDAMENTALIST PEDIGREE

George Dallemagne, a center-right opposition member of the federal parliament, traces some problems back to the 1970s when resource-poor, heavily industrial Belgium sought favor with Saudi Arabia by providing mosques for Gulf-trained preachers.

These brought with them fundamentalist teachings then alien to most of Belgium's Moroccan immigrants.

Pointing at Molenbeek, Dallemagne said: "The very strong influence of Salafists ... is one of the particularities that puts Belgium at the center of terrorism in Europe today."

Molenbeek is not unique in Belgium. The highest profile radical group taken on by the state has been sharia4belgium, a social media savvy organization whose leader and dozens of members were convicted early this year in the Flemish city of Antwerp of recruiting dozens to fight in Syria.

But, as Prime Minister Michel said, a Molenbeek connection keeps coming up in cases of Islamist attacks in Europe going back at least to the 2004 train bombings in Madrid, where one of those jailed for planning them was a Moroccan from the borough.

Over little more than a year, it has figured repeatedly. In August 2014, a Frenchman of Algerian origin was living there when he gunned down four people at Brussels' Jewish Museum. In January, when Belgian police killed two men in the eastern town of Verviers, foiling what they said was a plot to kidnap and behead a policeman on camera, many leads led back to Molenbeek.

French police investigating after the shootings in January at Paris magazine Charlie Hebdo and a kosher grocery suspect one of the killers acquired guns via Molenbeek. So too, prosecutors say, did the Spanish-based Moroccan overpowered on a Brussels to Paris train in August. He had an AK-47 and nearly 300 bullets.

"AIRBASE FOR JIHADISTS"

"Molenbeek is a pitstop for radicals and criminals of all sorts," said Benyaich, of the Itinera Institute. "It's a place where you can disappear."

Dallemagne added: "Terrorists are radicalised in France, go to Syria to fight and when they come back they find in Molenbeek the logistical support and the networks they need to carry out terrorist attacks, be it here in Belgium or abroad.

"It's like an airbase for jihadists."

One of the main attractions, investigators say, is weaponry.

Some of that, said Nils Duquet, a researcher at the Flemish Peace Institute, dates back to before 2006 when Belgium, whose state-owned FN Herstal sidearm manufacturer supplied many of the world's armies, also had a relaxed approach to gun ownership.

"With the right connections, it's quite easy to find illegal weapons in Belgium," Duquet said. "Criminals used to come to buy weapons legally. And they kept coming because they found the right networks and people here to get weapons, even after 2006."

Kalashnikov assault rifles of the kind used in the attacks in Paris in January and on Friday, were mostly from stocks left after the war in the former Yugoslavia and mostly reached western Europe in the back of a car, he said. Investigators are looking into links between the Paris attacks and a man from Montenegro arrested with guns in his car in Germany this month.

European Union interior ministers will hold an emergency meeting in Brussels on Friday at France's request and will deal yet again with longstanding concerns about traffic in firearms.

However, just as a lack of coordination among the EU's 28 states is blamed by many for a flourishing trade across their open borders, Belgium's extreme form of decentralized government makes it hard to crack down on dealers even in one small state:

"In Belgium, there's a problem with data management. Nobody knows how many illegal weapons there are in Belgium," said Duquet. "The reality is we have no idea."

(Additional reporting by Emmanuel Jarry in Paris; Writing by Alastair Macdonald; Editing by Dean Yates)