One can indeed write down that solution, but the point is that if is in then defined by the formula does not have to be bounded. The big machinery gives us the existence of a bounded solution. ]]>

A couple of comments:

1. We can directly write down u_1 = \int_{-\infty}^{x} f(t,y) dt and u_2 = 0 as a solution. So there should be some explanation as to why we are using all this big machinary. One reason is this doesn’t give us a continuously differentiable u for a continuous F. (u doesn’t “improve” in the y variable if we just do this).

2. This is related to above. Do you have a similarly elementary proof for the regularity of u obtained this way? i.e. Can it be proved (without using the big machinery) that u is C^1, given that F is continuous and L^2.

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