= Solution
Put $r=\lceil\beta\rceil-1$ and $\alpha=\beta-r\in(0,1]$. The density Hölder class $\mathcal F(\beta,L)$ consists of nonnegative functions $f$ integrating to one, with derivatives through order $r$, such that
$$
|f^{(r)}(x)-f^{(r)}(y)|\leq L|x-y|^\alpha.
$$
This is the density version of the <Hölder class>. A <kernel for density estimation> is an integrable function $K$ with $\int K=1$. It has order $\ell$ when $\int u^jK(u)\,du=0$ for $1\leq j<\ell$ and $\int|u|^\ell|K(u)|\,du<\infty$.
Choose a bounded kernel of order at least $\lceil\beta\rceil$ with $\int|u|^\beta|K(u)|\,du<\infty$, and use the <kernel density estimator>
$$
\widehat f_h(x)=\frac1{nh}\sum_{i=1}^n
K\left(\frac{x-X_i}{h}\right).
$$
Taylor's theorem at $x$ and the vanishing kernel moments cancel every polynomial term below the remainder. Hence, for a constant depending only on $\beta$ and the fixed kernel,
$$
|\mathbb E_f\widehat f_h(x)-f(x)|
\leq C_\beta Lh^\beta.
$$
We also need a uniform density bound. The standard Hölder interpolation argument combines nonnegativity, $\int f=1$, and the Hölder constraint to give
$$
\lVert f\rVert_\infty\leq C_\beta L^{1/(\beta+1)}.
$$
Indeed, near a point where $f$ is close to its maximum $M$, Taylor's theorem and the derivative bounds implied by the Hölder constraint keep $f$ of order $M$ on an interval of length comparable to $(M/L)^{1/\beta}$; integrating over that interval gives $M^{1+1/\beta}\leq C_\beta L^{1/\beta}$.
Using this bound and independence,
$$
\begin{aligned}
\operatorname{Var}_f(\widehat f_h(x))
&\leq\frac1{nh^2}\mathbb E_f
K^2\left(\frac{x-X_1}{h}\right)\\
&\leq\frac{\lVert K\rVert_2^2\lVert f\rVert_\infty}{nh}
\leq\frac{C_\beta L^{1/(\beta+1)}}{nh}.
\end{aligned}
$$
Thus
$$
\sup_x\sup_{f\in\mathcal F(\beta,L)}
\mathbb E_f(\widehat f_h(x)-f(x))^2
\leq C_\beta\left(L^2h^{2\beta}
+\frac{L^{1/(\beta+1)}}{nh}\right).
$$
Choose
$$
h=L^{-1/(\beta+1)}n^{-1/(2\beta+1)}.
$$
Both terms then have order $L^{2/(\beta+1)}n^{-2\beta/(2\beta+1)}$. Since the infimum over all measurable estimators is no larger than the risk of this particular estimator,
$$
\inf_{\widehat f_n}\sup_x\sup_{f\in\mathcal F(\beta,L)}
\mathbb E_f(\widehat f_n(x)-f(x))^2
\leq C_\beta L^{2/(\beta+1)}n^{-2\beta/(2\beta+1)}.
$$
This is the <pointwise minimax rate for Hölder density estimation> upper bound.
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