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A bootstrap argument is given in the proof of Lemma 10.7 of a paper of Mei-Montanari-Nguyen: https://web.stanford.edu/~montanar/RESEARCH/FILEPAP/mean_field.pdf

The authors only show $C^{1,2}$ derivatives exist classically under their set of assumptions, but it should be clear how to proceed with smoothness conditions to get infinite regularity. (Also $L^\infty_\text{loc}$ is nicer than $L^p$ since it's invariant under multiplication.)

I don't quite follow their step 5 and don't understand why equations 10.52 and 10.53 are correct. I think we should have, for example, for 10.52, $$ D_t [\varphi_1 \ast_2 G] = \Delta \varphi_1 \ast_2 G - \varphi_1 $$ by $(D_t - \Delta) G = \delta_{0,0}$. So in order to estimate $D_t^2 \rho$ one needs to estimate $D_t\varphi_1$, which again depends on the time-regularity of the drift. This is no-problem in the JKO case, as the drift is time-independent. Also there may be some sign errors in their proof, but I haven't checked carefully.

This is an answer to a question older than my math career since undergraduate. Hope it helps someone at least!

A bootstrap argument is given in the proof of Lemma 10.7 of a paper of Mei-Montanari-Nguyen: https://web.stanford.edu/~montanar/RESEARCH/FILEPAP/mean_field.pdf

The authors only show $C^{1,2}$ derivatives exist classically under their set of assumptions, but it should be clear how to proceed with smoothness conditions to get infinite regularity. (Also $L^\infty_\text{loc}$ is nicer than $L^p$ since it's invariant under multiplication.)

I don't quite follow their step 5 and don't understand why equations 10.52 and 10.53 are correct. I think we should have, for example, for 10.52, $$ D_t [\varphi_1 \ast_2 G] = \Delta \varphi_1 \ast_2 G - \varphi_1 $$ by $(D_t - \Delta) G = \delta_{0,0}$. So in order to estimate $D_t^2 \rho$ one needs to estimate $D_t\varphi_1$, which again depends on the time-regularity of the drift. This is no-problem in the JKO case, as the drift is time-independent. Also there may be some sign errors in their proof, but I haven't checked carefully.

This is an answer to a question older than my math career since undergraduate. Hope it helps someone at least!

Edit: I don't think the heat kernel estimate 10.51 is correct for $p = \infty$. See the equivalent condition of an $L^1$ or $L^\infty$ multiplier on Wikipedia: https://en.wikipedia.org/wiki/Multiplier_(Fourier_analysis)

A bootstrap argument is given in the proof of Lemma 10.7 of a paper of Mei-Montanari-Nguyen: https://web.stanford.edu/~montanar/RESEARCH/FILEPAP/mean_field.pdf

The authors only show $C^{1,2}$ derivatives exist classically under their set of assumptions, but it should be clear how to proceed with smoothness conditions to get infinite regularity. (Also $L^\infty_\text{loc}$ is nicer than $L^p$ since it's invariant under multiplication.)

I don't quite follow their step 5 and don't understand why equations 10.52 and 10.53 are correct. I think we should have, for example, for 10.52, $$ D_t [\varphi_1 \ast_2 G] = \Delta \varphi_1 \ast_2 G - \varphi_1 $$ by $(D_t - \Delta) G = \delta_{0,0}$. So in order to estimate $D_t^2 \rho$ one needs to estimate $D_t\varphi_1$, which again depends on the time-regularity of the drift. This is no-problem in the JKO case, as the drift is time-indepedent. Also there may be some sign errors in their proof, but I haven't checked carefully.

This is an answer to a question older than my math career since undergraduate. Hope it helps someone at least!

A bootstrap argument is given in the proof of Lemma 10.7 of a paper of Mei-Montanari-Nguyen: https://web.stanford.edu/~montanar/RESEARCH/FILEPAP/mean_field.pdf

The authors only show $C^{1,2}$ derivatives exist classically under their set of assumptions, but it should be clear how to proceed with smoothness conditions to get infinite regularity. (Also $L^\infty_\text{loc}$ is nicer than $L^p$ since it's invariant under multiplication.)

I don't quite follow their step 5 and don't understand why equations 10.52 and 10.53 are correct. I think we should have, for example, for 10.52, $$ D_t [\varphi_1 \ast_2 G] = \Delta \varphi_1 \ast_2 G - \varphi_1 $$ by $(D_t - \Delta) G = \delta_{0,0}$. So in order to estimate $D_t^2 \rho$ one needs to estimate $D_t\varphi_1$, which again depends on the time-regularity of the drift. This is no-problem in the JKO case, as the drift is time-independent. Also there may be some sign errors in their proof, but I haven't checked carefully.

This is an answer to a question older than my math career since undergraduate. Hope it helps someone at least!

A bootstrap argument is given in the proof of Lemma 10.7 of a paper of Mei-Montanari-Nguyen: https://web.stanford.edu/~montanar/RESEARCH/FILEPAP/mean_field.pdf

The authors only show $C^{1,2}$ derivatives exist classically under their set of assumptions, but it should be clear how to proceed with smoothness conditions to get infinite regularity. (Also $L^\infty_\text{loc}$ is nicer than $L^p$ since it's invariant under multiplication.)

I don't quite follow their step 5 and don't understand why equations 10.52 and 10.53 are correct. I think we should have, for example, for 10.52, $$ D_t [\varphi_1 \ast_2 G] = \Delta \varphi_1 \ast_2 G - \varphi_1 $$ by $(D_t - \Delta) G = \delta_{0,0}$. So in order to estimate $D_t^2 \rho$ one needs to estimate $D_t\varphi_1$, while again depends on the time-regularity of the drift. This is no-problem in the JKO case, as the drift is time-indepedent. Also there may be some sign errors in their proof, but I haven't checked carefully.

This is an answer to a question older than my math career since undergraduate. Hope it helps someone at least!

A bootstrap argument is given in the proof of Lemma 10.7 of a paper of Mei-Montanari-Nguyen: https://web.stanford.edu/~montanar/RESEARCH/FILEPAP/mean_field.pdf

The authors only show $C^{1,2}$ derivatives exist classically under their set of assumptions, but it should be clear how to proceed with smoothness conditions to get infinite regularity. (Also $L^\infty_\text{loc}$ is nicer than $L^p$ since it's invariant under multiplication.)

I don't quite follow their step 5 and don't understand why equations 10.52 and 10.53 are correct. I think we should have, for example, for 10.52, $$ D_t [\varphi_1 \ast_2 G] = \Delta \varphi_1 \ast_2 G - \varphi_1 $$ by $(D_t - \Delta) G = \delta_{0,0}$. So in order to estimate $D_t^2 \rho$ one needs to estimate $D_t\varphi_1$, which again depends on the time-regularity of the drift. This is no-problem in the JKO case, as the drift is time-indepedent. Also there may be some sign errors in their proof, but I haven't checked carefully.

This is an answer to a question older than my math career since undergraduate. Hope it helps someone at least!