Lawn and Lime

I live in Patterson, La and I have some bare spots in my St. Augustine grass under some pecan trees and white oaks that do not get too much direct sunlight. I tested the PH level of the soil with a direct contact probe and I'm averaging around 6 to 6.2 for a ph level. Ironically, I checked the soil under a different area where the lawn is very dense with plenty of sun and the results were around 5 to 5.5 for the Ph level.
Why is the Ph level where the grass is growing nicely so low where I understand the ph level should be close to 7.0 ??
I plan to core aerate my lawn and lime afterwards to bring the reading closer to 7.0 in the bare areas under the tree.
Do you think it's not too late in the summer to core aerate and lime. I plan to use "dolomitic limestone".
Should I use pelletized or ground limestone ?
Your advice is much appreciated. Ray
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I used ground limestone because I thought would seep into the soil easier but when I applied it on a non-windy day - there were clouds of it everywhere - I'll use pelletized next time.
Brandi

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The palletized is ground limestone that has been , guess what, palletized! I don't use any other kind but dolomite (ground marble). Do you really expect St. Augustine grass to grow well in the shade of those trees?
I also suggest you have a soil test done so you know for sure what you need to put on that grass, because it's very easy to over fertilize St. Augustine.
Tom J
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