Install the YouTube app on your phone, film the video on the phone and then upload direct to YouTube or transfer the file over to your phone and then upload.
You can make the file size of the native video smaller by setting the phones camera to a lower resolution but don't make it to low.
Even at full HD the file size only becomes a problem once the video goes over around 5mins.
Once it's uploaded to YouTube whoever watches the video can select the resolution that they require as when you upload an HD video to YouTube they transcodes the video to all the smaller sizes.
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