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A user wants to search a person by name and enter therefore something like:
"Pe Lu"
"Peter L"
"Lustig P"
All of these should match "Peter Lustig".
The data is indexed with default settings having firstname and lastname split.
I tried to use default prefix search using a bool "should" query having splitted the search term by whitespace.
bool
should
prefix
firstName Pe
prefix
name Pe
prefix
firstName Lu
prefix
name Lu
But I do not get any results for this. What is the best way to achieve this?

Can you try following query:;
{
"query": {
"bool": {
"should": [
{
"match_phrase_prefix": {
"firstName": "Peter"
}
},
{
"match_phrase_prefix": {
"firstName": "Lu"
}
},
{
"match_phrase_prefix": {
"lastName": "Peter"
}
},
{
"match_phrase_prefix": {
"lastName": "Lu"
}
}
]
}
}
}