Suppose $K(x,t)$ is known and $$ \int f(x)K(x,t)dx=0 $$ Are there some known sufficient and \ or necessary conditions on $K(x,t)$ such that the only solution is $f(x)=0$ a.s.? ($f$ can be in a space of your choice but to keep things concrete say $f \in \mathcal{L}^p$)

I've been searching around for some answers to this question but the questions are usually unanswered and / or uniqueness is rare (e.g. Uniqueness of the Solution to Fredholm's Integral Equations of the First Kind and https://math.stackexchange.com/questions/873278/existence-of-solution-of-volterra-integral-equation-of-the-first-kind). Are there some known results or is this type of problem very much a case-by-case type of issue?



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