“Late Show With David Letterman” and “The Tonight Show With Jay Leno” will each coast-hop during the May sweeps, with Letterman expected to bring his show to Los Angeles while Leno takes “Tonight” back to New York.NBC, apparently eager to beat CBS to the punch, announced Friday that “Tonight” will originate from New York for a week,marking the first taping of the show there since Johnny Carson moved west in 1972. Production will take place at 30 Rockefeller Plaza in the studio used for “Saturday Night Live.” CBS, meanwhile, figures to cre-ate an enormous stir with five local tapings of “Late Show,” which will emanate from a tailor-made studio adjacent to Television City. Tickets should be a hot item and will doubtless test the show’s policy against wholesale allocation of seats to VIPs. Network officials stated when they landed Letterman, who considered moving the show to Los Angeles and owns a home in the area, that they hoped the host would occasionally take “Late Show” on the road. While at NBC, Letterman did successful out-of-town stints in Burbank and Chicago. NBC took “The Tonight Show” east once before under Leno, shooting what turned out to be a high-rated episode in Boston to capitalize on the final installment of “Cheers” last May. The L.A.-New York crossover happens to coincide with release of “The Late Shift: Letterman, Leno, & the Network Battle for the Night,” New York Times reporter Bill Carter’s account of the often-messy jockeying that followed Carson’s retirement. According to the book (and as first reported in Variety), NBC’s last-minute proposal to keep Letterman included giving him “The Tonight Show” when Leno’s contract expired — an offer Letterman nixed after receiving advice from, among others, Carson. “I’d probably walk,” Carson, who has mostly remained mum on the situation, is quoted as saying. Leno recently signed a two-year extension with NBC. The book also maintains that Jack Welch, chairman of NBC parent GE, told a group of network officials that those who pressed to keep Leno over Letterman — including himself — were wrong in hindsight. “Late Show” has topped “Tonight” every week since its Aug. 30 premiere, including repeats.