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However, it is still far from enough for applications in such places as the vorticity equation mentioned above. We refer the reader to the papers [1,11,16,18] and reference therein for results in this direction. The main results there is the continuity of weak solutions with potentials in the Kato class. 1) with V in Morrey or Besov classes are also studied. However, these classes are essentially logarithmic improvements over the standard Lp,q class. In the paper [17], K. Sturm proved Gaussian upper and lower bound for the fundamental solution when the potential belongs to a class of time independent, singular oscillating functions.

1) possesses a positive solution. 1). The third theorem is an application of the second one in the more traditional setting of Lp,q conditions on the potential. It will show that our conditions are genuinely much broader than the traditional ones. It should be made clear that there is no claim on uniqueness in any of the theorems. In the absence of uniqueness how does one define the fundamental solution? 1) when the potential V is truncated from above. 1 below. 2. The fundamental solution GV is defined as the pointwise limit of the (increasing) sequence of the fundamental solution GVi where Vi = min{V, i} with i = 1, 2, ....

Write Vj = min{V, j} with j = 1, 2, .... Then (Vj − b)φ2 dx ≤ |∇φ|2 dx. Notice that Vj − b is a bounded function. 1 in [16] to conclude that there exists uj > 0 such that ∆uj + (Vj − b)uj = 0. Writing fj = − ln uj , we have Vj = ∆fj − |∇fj |2 + b. By definition, this means V = ∆f − |∇f |2 + b. 3. Heat Bounded Functions and the Heat Equation Here we introduce another class of singular functions that has its origin in the Kato type class. As mentioned in the introduction, a function is in a Kato type class if the convolution of the absolute value of the function and the fundamental solution of Laplace or the heat equation is bounded.

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