* Three Saudi soldiers dead, four missing

* Yemen reports crash of fighter plane in rebel stronghold

(Adds rebel denial)

By Raissa Kasolowsky

DUBAI, Nov 8 (Reuters) - Saudi Arabia said it had regained control of territory seized by Yemeni rebels in an incursion last week, but the rebels denied the claim and said Yemeni villages were being bombed heavily.

Saudi Arabia launched air strikes on rebels in northern Yemen last week after Shi'ite Muslim insurgents crossed the border and said they had seized an area called Jabal Dukhan.

The rebels on Sunday denied they had lost control of Jabal Dukhan and said the kingdom's offensive was continuing, with Yemeni villages the target of heavy bombing.

The world's top oil exporter has become increasingly anxious about instability in Yemen, which is facing a Shi'ite insurgency in the north, separatist sentiment in the south and a growing threat from resurgent al Qaeda fighters.

"The situation is calm ... especially in Jabal Dukhan, of which full control has been regained," Prince Khaled bin Sultan bin Abdul-Aziz, assistant minister for defence and aviation, said on Saturday, the Saudi Press Agency reported.

Prince Khaled said three members of the Saudi security forces were killed and 15 wounded in fighting on the Saudi-Yemeni border.

Four Saudi soldiers were missing, Prince Khaled said, but he denied they had been taken prisoner. He said Saudi security forces had arrested several rebels.

Instructions from Saudi Arabia's King Abdullah specified that any rebel caught on the Saudi side of the border would be arrested, Prince Khaled said, and the kingdom "has not, and will not interfere inside Yemeni borders".

The rebels, referred to as Houthis after the clan of their leader Abdel-Malik al-Houthi, said on Friday they had captured some Saudi soldiers.

"What is being said about Saudi Arabia seizing Jabal Dukhan is entirely false," a rebel spokesman was quoted as saying in a statement posted on the Houthis' website, adding that the Saudis had not taken control of the area.

Saudi-owned al-Arabiya television on Sunday reported that Saudi military operations near the Saudi-Yemeni border were still in progress.

According to David Bender of the Eurasia Group consultancy, disorder in Yemen was the most serious regional threat to the stability of Saudi Arabia, but the kingdom was wary of becoming too deeply involved in the conflict.

"A large Saudi ground invasion is unlikely; it would be extremely messy and Riyadh's objective is to support the Yemeni government, not crush the Houthis," Bender wrote in a research note published on Thursday.

YEMENI PLANE DOWN

In Sanaa, the Yemeni government said one of its fighter planes crashed in a rebel stronghold in the north of the country on Sunday because of a technical fault.

However, the rebels said they shot it down in Saada, a mountainous province where most recent fighting has taken place.

In the past few weeks the Houthis have accused Saudi Arabia of allowing Yemeni forces to use its territory as a base to launch attacks against them and threatened to respond.

Houthi rebels first took up arms against President Ali Abdullah Saleh's government in 2004, citing political, economic and religious marginalisation by the Saudi- and Western-backed administration.

The conflict intensified in August when Yemen's army launched Operation Scorched Earth.

Aid groups, which have been given limited access to the northern provinces, say up to 150,000 people have fled their homes since 2004. (Additional reporting by Mohamed Sudam in Sanaa; editing by Andrew Roche)

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