Given a Linear pde $L$, a distribution $u$ is said to be a fundamental solution if $Lu=\delta$ where $\delta$ is the Dirac delta distribution. A common example is the Newtonian potential which is the fundamental solution of the Laplacian. Are there any examples of differential operators of positive order with non-locally integrable fundamental solutions?
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$\begingroup$ If it's not locally integrable, how would you differentiate it? $\endgroup$– Cameron WilliamsJul 15, 2013 at 23:12
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2$\begingroup$ Any distribution can be differentiated,it doesn't have to be necessarily locally integrable $\endgroup$– user8621Jul 15, 2013 at 23:16
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$\begingroup$ This is true but even in the distributional sense you run into issues since your natural space of test functions is going to be the bump functions. $\frac{d}{dx}$ maps the bump functions to themselves and so the inner product $\langle Du,\phi\rangle = -\langle u,D\phi\rangle$ isn't well-posed since the right side diverges since $u$ is not locally integrable. (I think.) $\endgroup$– Cameron WilliamsJul 15, 2013 at 23:20
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1$\begingroup$ Cameron, $\left\langle \cdot,\cdot\right\rangle$ is not an inner product, $\left\langle \varphi,f\right\rangle$ for $\varphi\in D'$, $f\in D$ stands for $\varphi(f)$. And apart from that a fundamental solution is a distribution, so its maybe better to ask for "non regular fundamental solutions". Interesting question though. But I think yes, as if $E$ is a fundamental solution $\endgroup$– user85461Jul 15, 2013 at 23:36
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1$\begingroup$ $<u,D\phi>$ does make sense since u is a distribution and $D\phi$ a test function $\endgroup$– user8621Jul 15, 2013 at 23:36
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2 Answers
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Short answer is yes. In 1999 Hörmander gave an example of a hypoelliptic operator where the fundamental solution is not locally integrable. He also proves, among other things, that no such counterexample exists in two dimensions.
This is the article: On local integrability of fundamental solutions by Hörmander Arkiv för Matematik Volume 37, Issue 1 , pp 121-140
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The differential operator $1$ has $\delta$ as a fundamental solution, which is known to be non regular (not induced by a locally integrable function).
Also, the differential operator $x\partial/\partial x-1$ has $\delta$ as a fundamental solution, since $((x\partial/\partial x-1)\delta)(f)=\delta(-x\partial f/\partial x+f)=\delta(f)$. Are you looking for the constant coefficient case?
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$\begingroup$ I should have mentioned that I was looking for an example where the order of the differential operator is greater than or equal to 1.I'll make the appropriate edit $\endgroup$– user8621Jul 16, 2013 at 0:01
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$\begingroup$ alrite, I hope this is ok now $\endgroup$– user85461Jul 16, 2013 at 0:37
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$\begingroup$ Now that you mentioned I'm curious if there is a constant coefficient example $\endgroup$– user8621Jul 16, 2013 at 1:05