It's not very clear from your post what you are trying to do :-) Let me start by clarifying a few things and we can take it from there.

All lime is does is neutralize or raise the pH an acidic soil. Depending on the type of lime used it can also supply a couple of nutrients that may or may not be lacking, but it does nothing in the way of supplying nitrogen. If you still live in the Portland area, your soil is most likely slightly to moderately acidic as are most of the soils in the PNW. Except for lawns and a few specific plants, the vast majority of plants will grow best in a slightly to moderately acidic soil so you may not need to lime at all.

Any large tree will draw nutrients out of the soil as they have wide spreading root systems that out-compete smaller plants for any available nutrients and any soil moisture. This is one of the reasons it is often hard to get plants to grow within the root zone of these big trees. But black walnuts also pose some other issues - they emit a chemical through their roots and leaves (juglone) that makes it difficult for certain other plants to thrive or even survive. One needs to pay careful attention to any plants selected to grow in this area and makke sure they are compatible and not subject to the juglone.

Tomatoes are noted for being very hard to grow close to black walnuts. And the rot you describe sounds like blossom end rot, which is usually associated with uneven watering practices while the fruit is forming. It can also be affected by a lack of calcium in the soil. You can purchase products that address this but even watering is really the key. And you need to make sure that you are growing your tomatoes well away from the possible influence of the black walnut.

Your soil probably is clay-based - much of the NW is. Adding good quantities of compost or other organic matter will help to loosen the soil, improve structure, improve drainage and add fertility. NW soils also tend to be nitrogen deficient and compost will help with that as well.

Otherwise, to determine what other nutrients may or may not be needed, you might want to consider a soil test. And this will clarify exactly what pH you are working with and whether or not you need to lime. OSU no longer supplies soil tests but they can direct you to a good lab.

Here is a link that might be useful: OSU extension service

gardengal48, that is great advice, no matter where you live.
I would say get a soil test before you add anything.
Then one year later, get another soil test to see how far you have come & what you need to add.
You can not have to much humus in your soil if it is clay or sand.

The Black Walnut trees roots put a growth suppressor, Juglone, into the soil to prevent competition for nutrients from growing near the tree. Whle there are some plants that can grow near a Black Walnut tomatoes are not one of them. That rotting on the bottom of the fruit was most likely Blossom End Rot and the cause of that is a bit more complicated then uneven watering. The cause is lack of calcium when the fruit is set and there are myriad reasons, including lack of sufficient soil moisture, that cause that.
The only way to know whether lime, and which kind, is needed is from a good, reliable soil test, and if lime is needed it should be spread several months befor you are going to plant there. Calcitic lime provides calcium as well as adjusts the soils pH while Dolomitic lime provides Magnesium as well as Calcium. The Magnesium is needed by plants to use Calcium and Calcium is needed to help plants use Magnesium, so you need to know about both.
Sufficient levels of organic matter in the soil can help buffer the effects of the Juglone, some, but oganic matter in soils does much more so you need to look at how much you have and add more if needed. Nitrogen availability in soil is temperature dependant. Soil bacteria provide it to plants and if the soil is not warm enough and they are not active it won't be there for your plants.
Start with a good, reliable soil test to see where you are and what you might need to add to do what you want and add organic matter, 6 to 8 percent is a good level to work toward.

That rotting on the bottom of the fruit was most likely Blossom End Rot and the cause of that is a bit more complicated then uneven watering.

From Cornell University: "Control of blossom end rot is dependent upon maintaining adequate supplies of moisture and calcium to the developing fruits."

And I believe I stated that quite clearly in my post. NW soils tend not to be calcium deficient (UI plant nutrient fact sheet: "Soils in the Pacific Northwest contain plenty of calcium"), therefore the watering issue becomes paramount.

Nitrogen availability in soil is temperature dependant. Soil bacteria provide it to plants and if the soil is not warm enough and they are not active it won't be there for your plants.

It is also the most unstable and mobile of the primary plant nutrients and the one that is most often lacking. Nitrogen supplementation in NW soils, often through the application of organic matter, is pretty much SOP and not really based on temperature concerns, especially during the months when one is normally gardening.

The only blossom end rot tomatoes I normally get are the very first fruits on a plant....usually only one fruit.

gardengal48, I appreciate your posts which get straight to the chase with knowledgeable information and free of pc trappings.

thanks for all the feedback.

prior to seeing all these responses, i went out back harrowed up the ground a bit and sprinkled the lime around as best as was instructed by the folks at the nursery where i got it. there has been a lot of rain/foul weather & i had a couple hours in the morning to work on jump-starting our compost heap & tilling in the lime. as was mentioned- lime "should be spread several months before you are going to plant there" (kimmsr) so i figured sooner the better.

i thought that the soil in this region was not low in calcium, per se however, that's where i figured the walnut's juglone came in & had inhibited the surrounding plants- particularly tomatoes from getting the nutrients (calcium) necessary. of course my housemate may have done a poor job watering the plants evenly, this is very possible as well.

this year we will do tomatoes in the front yard with peppers, who are also very susceptible to the juglone's affects. as soon as the compost is ready to go i will put lots of it along the fence where the walnut tree grows.

is compost humus enough organic matter to add to the soil or should i also add some steer manure?
i also do not know how much to incorporate working with amounts- but i suppose i will just have to split up whatever we have to work with.

kimmsr- "6 to 8 percent" organic matter added... sorry to be naive, but how deep down does that account for?

Pam - feel like Rodney Dangerfield yet? ;o) Great advice - as usual.

Al

Most usually you will look at about the top 6 inches of your soil. That is about what a good soil test lab will tell you to sample and is where the most bacterial activity will be, so that is where you work. This link may help you understand tomatoe problems better.

Here is a link that might be useful: When good tomatoes go bad