How apply a function to a list while maintaining list name?

I use lapply to apply a function to the list 'L'. But of course, the
list names in 'X' is not maintained. I'm wondering if there is a
function that can maintain the list names as well as apply the
function.

$ Rscript lapply.R

L=list(x=c('a','b'), y=c('a','b'))
L

$x
[1] "a" "b"

$y
[1] "a" "b"

X=lapply(1:length(L),function(x){paste(L[[x]],collapse='')})
X

[[1]]
[1] "ab"

[[2]]
[1] "ab"

names(X)=names(L)
X

$x
[1] "ab"

$y
[1] "ab"