In the English Grammar Profile, B2 point 163 MODALITY is defined as:
‘may’ in phrases such as ‘as you may know’, or ‘as you may have’ + ‘-ed’ to focus the reader on shared knowledge.
This partly overlaps point 168 MODALITY and PAST AFFIRMATIVE of
‘may have’ + ‘-ed’ to talk about possibility in the past.
A search for collocates of ‘may have’ in COCA:
Past possibility example:
He said he believes drug use or mental illness may have contributed to Carter‘s behavior, but that hasn’t been confirmed.
A search on iWeb for as you may * * * *
:
1 AS YOU MAY OR MAY NOT KNOW 950
As you may or may not know, big things are happening here in Williamsburg, Brooklyn.
2 AS YOU MAY HAVE NOTICED , THE 288
As you may have noticed, the chat function has disappeared from the top of the Forum.
3 AS YOU MAY HAVE GUESSED , THE 227
4 AS YOU MAY HAVE NOTICED , I 201
5 AS YOU MAY HAVE NOTICED , WE 194
6 AS YOU MAY BE ABLE TO TELL 189
7 AS YOU MAY NOT BE ABLE TO 189
8 AS YOU MAY HAVE GUESSED FROM THE 184
9 AS YOU MAY ALREADY KNOW , THE 181
10 AS YOU MAY HAVE ALREADY GUESSED , 171
11 AS YOU MAY BE AWARE , THE 159
12 AS YOU MAY KNOW BY NOW , 143
13 AS YOU MAY HAVE HEARD , THE 136
14 AS YOU MAY OR MAY NOT BE 130
15 AS YOU MAY HAVE GUESSED , I 127
16 AS YOU MAY HAVE GUESSED , THIS 125
17 AS YOU MAY OR MAY NOT HAVE 113
18 AS YOU MAY HAVE GUESSED BY NOW 112
19 AS YOU MAY HAVE ALREADY NOTICED , 106
20 AS YOU MAY ALREADY BE AWARE , 102