* Coalition steps up air strikes as Saleh's son vows revenge
* UAE de facto leader offers condolences
* Wary residents stay indoors fearing anti-Saleh crackdown
* U.S. warns of worsening war, Iran hails Houthi victories (Updates with UAE leader offering condolence to Saleh's son)
ADEN/DUBAI, Dec 6 (Reuters) - A Saudi-led coalition stepped up air strikes on Yemen's Houthis on Wednesday as the Iran-allied armed movement tightened its grip on Sanaa a day after the son of slain former president Ali Abdullah Saleh vowed revenge for his father's death.
Former president Saleh plunged the country deeper into turmoil last week by switching allegiances after years helping the Houthis win control of much of the country's north including the capital. He was killed in an attack on his convoy on Monday.
The pro-Houthi Al Masirah television station said on Wednesday Saudi Arabia and its allies had bombed Saleh's residence and other houses of his family members now controlled by the Houthis. Air strikes also hit northern provinces including Taiz, Hajjah, Midi and Saada, it said.
There was no immediate word on casualties.
The intervention by Saleh's son Ahmed Ali, a former commander of the elite Republican Guard who lives in exile in the United Arab Emirates and was once seen as a successor to his father, has provided the anti-Houthi movement with a potential figurehead.
Abu Dhabi Crown Prince Mohammed bin Zayed al-Nahayan, the de facto leader of the UAE, visited Ahmed Ali at his residence to offer his condolences, according to Sheikh Mohammed's Twitter account. He posted a picture of himself sitting near Ahmed Ali.
Ahmed Ali had been widely expected to leave the UAE, a key member of the Saudi-led coalition fighting the Houthis, for Yemen to help in the war amid media reports that some Saleh loyalists have been switching sides.
Many Sanaa residents were staying indoors on Wednesday out of fear of a Houthi crackdown. On Tuesday, Saleh supporters said his nephew Tareq, another top commander, and the head of his party, Aref Zouka, had both been killed.
"There's a scary calm in the city," said Ali, a 47-year-old businessman who declined to use his full name.
"People are reporting that there are many arrests and they are trying to shoot military men and (Saleh party) members."
Yemen's conflict, pitting the Houthis against the Saudi-led military alliance which backs a government based in the south, has unleashed what the United Nations calls the world's worst humanitarian crisis.
The proxy war between regional arch-rivals Iran and Saudi Arabia - armed and given intelligence by the West -- has killed more than 10,000 people, with more than two million displaced.
Saleh's decision to abandon the Houthis was the most dramatic development in three years of stalemate. Top Houthi officials called it high treason backed by their Saudi enemies.
Tens of thousands of Houthi supporters staged a rally in Sanaa on Tuesday to celebrate what the Houthis had said was the defeat of a major conspiracy by Saleh, chanting slogans against Saudi Arabia and its allies.
Political sources said the Houthis had arrested dozens of Saleh's allies and army officers affiliated with his party in and around the city. Several had been killed in the raids.
On Wednesday, several dozen women gathered in a main Sanaa square holding Saleh's portrait and demanding his body be handed over for burial, but they were forcibly dispersed by Houthi security forces, eyewitnesses said.
The Houthi-controlled interior ministry distributed a video of dozens of seated barefoot men it said were pro-Saleh fighters detained in one of its party headquarters.
Media rights group Reporters Without Borders appealed for the release of 41 journalists it said have been held "hostage" by the group since it overran the headquarters of the Saleh-owned al-Yemen al-Youm TV station on Saturday.
THREAT OF FAMINE
Nearly a million people in Yemen have been hit by a cholera outbreak, and famine caused by warring parties blocking food supplies threatens much of the country.
The UN secretary-general's special envoy for Yemen, Ismail Ould Cheikh Ahmed, called on all parties to show restraint.
"Increased hostilities will further threaten civilian lives and exacerbate their suffering," he said in a briefing to the Security Council on Tuesday.
U.S. Defense Secretary Jim Mattis said on Tuesday that the killing of Saleh would likely worsen an already dire humanitarian situation in the country in the short term.
Speaking with reporters on a military aircraft en route to Washington, Mattis said his death could either push the conflict towards U.N. peace negotiations or make it an "even more vicious war."
The commander of Iran's Islamic Revolution Guards Corps, Major General Mohammad Ali Jafari, praised what he called the Houthis' swift quashing of the "coup against the holy warriors", the semi-official Fars news agency reported.
Much is likely to depend on the future allegiances of Saleh loyalists who previously helped the Houthi group, which hails from the Zaidi branch of Shi'ite Islam that ruled a thousand-year kingdom in northern Yemen until 1962. (Writing by Noah Browning; Editing by Michael Georgy, Richard Balmforth and Sonya Hepinstall)
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