It can be shown that for $d=1$ the best upper bound on $\|f\|_\infty$ is given by $$\|f\|_\infty\le\sqrt{2L\|f\|_1}\,1(\|f\|_1\le L/2)+(L/2+\|f\|_1)\,1(\|f\|_1>L/2).$$ If there is a sufficient interest, I can later provide details on this.

One can see that, if $L=O(\|f\|_1)$, then the above bound is of the same order of magnitude as the bound $\|f\|_1+L$ given in a comment by Nate Eldredge. On the other hand, the above bound goes to $0$ (as it should) when e.g. $\|f\|_1\to0$ while $L\asymp1$.

We see that the optimal bound is rather complicated already for $d=1$.

The case of $d>1$ is much more complicated -- in particular, because it is hard to evaluate or even estimate the volume of the intersection of the hypercube and an arbitrary Euclidean ball -- cf. e.g. https://math.stackexchange.com/a/2008339/96609.

Anyhow, as Nate Eldredge pointed out, you conjectured bound cannot hold because it should increase with $L$. Also, the bound should of course depend on $\|f\|_1$. So, I think further help depends on whether you can tell us what kind of bound on $\|f\|_\infty$ will suffice for the purposes of your research.

It can be shown that for $d=1$ the best upper bound on $\|f\|_\infty$ is given by $$\|f\|_\infty\le\sqrt{2L\|f\|_1}\,1(\|f\|_1\le L/2)+(L/2+\|f\|_1)\,1(\|f\|_1>L/2).$$ If there is a sufficient interest, I can later provide details on this.

One can see that, if $L=O(\|f\|_1)$, then the above bound is of the same order of magnitude as the bound $\|f\|_1+L$ given in a comment by Nate Eldredge. On the other hand, the above bound goes to $0$ (as it should) when e.g. $\|f\|_1\to0$ while $L\asymp1$.

We see that the optimal bound is rather complicated already for $d=1$.

The case of $d>1$ is much more complicated -- in particular, because it is hard to evaluate or even estimate the volume of the intersection of the hypercube and an arbitrary Euclidean ball -- cf. e.g. https://math.stackexchange.com/a/2008339/96609.

Anyhow, as Nate Eldredge pointed out, your conjectured bound cannot hold because it should increase with $L$. Also, the bound should of course depend on $\|f\|_1$. So, I think further help depends on whether you can tell us what kind of bound on $\|f\|_\infty$ will suffice for the purposes of your research.

It can be shown that for $d=1$ the best upper bound on $\|f\|_\infty$ is given by $$\|f\|_\infty\le\sqrt{L\|f\|_1}\,1(\|f\|_1\le L/4)+(L/4+\|f\|_1)\,1(\|f\|_1>L/4).$$ If there is a sufficient interest, I can later provide details on this.

One can see that, if $L=O(\|f\|_1)$, then the above bound is of the same order of magnitude as the bound $\|f\|_1+L$ given in a comment by Nate Eldredge. On the other hand, the above bound goes to $0$ (as it should) when e.g. $\|f\|_1\to0$ while $L\asymp1$.

We see that the optimal bound is rather complicated already for $d=1$.

The case of $d>1$ is much more complicated -- in particular, because it is hard to evaluate or even estimate the volume of the intersection of the hypercube and an arbitrary Euclidean ball -- cf. e.g. https://math.stackexchange.com/a/2008339/96609.

Anyhow, as Nate Eldredge pointed out, you conjectured bound cannot hold because it should increase with $L$. Also, the bound should of course depend on $\|f\|_1$. So, I think further help depends on whether you can tell us what kind of bound on $\|f\|_\infty$ will suffice for the purposes of your research.

It can be shown that for $d=1$ the best upper bound on $\|f\|_\infty$ is given by $$\|f\|_\infty\le\sqrt{2L\|f\|_1}\,1(\|f\|_1\le L/2)+(L/2+\|f\|_1)\,1(\|f\|_1>L/2).$$ If there is a sufficient interest, I can later provide details on this.

One can see that, if $L=O(\|f\|_1)$, then the above bound is of the same order of magnitude as the bound $\|f\|_1+L$ given in a comment by Nate Eldredge. On the other hand, the above bound goes to $0$ (as it should) when e.g. $\|f\|_1\to0$ while $L\asymp1$.

We see that the optimal bound is rather complicated already for $d=1$.

The case of $d>1$ is much more complicated -- in particular, because it is hard to evaluate or even estimate the volume of the intersection of the hypercube and an arbitrary Euclidean ball -- cf. e.g. https://math.stackexchange.com/a/2008339/96609.

Anyhow, as Nate Eldredge pointed out, you conjectured bound cannot hold because it should increase with $L$. Also, the bound should of course depend on $\|f\|_1$. So, I think further help depends on whether you can tell us what kind of bound on $\|f\|_\infty$ will suffice for the purposes of your research.