Lebesgue space and weak Lebesgue space

Let $1\le p<\infty$. We define the weak Lebesgue space $wL^p(\mathbb{R}^d)$ as the set of all measurable functions $f$ on $\mathbb{R}^d$ such that \begin{equation} \|f\|_{wL^p}=\sup_{\gamma>0} \gamma (\{x\in \mathbb{R}^d : |f(x)|>\gamma \})^{1/p}<\infty. \end{equation} By Chebyshev inequality, we have $\|f\|_{wL^p}\le \|f\|_{L^p}$ for every $f\in L^p$.

My question is as follows: Let $1\le p, q <\infty$. Could we obtain some conditions for $p$ and $q$ such that \begin{equation} \|f\|_{L^p} \le C_1\|f\|_{wL^q} \end{equation} for every $f \in wL^q$ or \begin{equation} \|f\|_{wL^p} \le C_2\|f\|_{L^q} \end{equation} for every $f \in L^q$. Here, we denote by $C_1,C_2$ the positive constants that independent to $f$.

Edit: Maybe we may restrict the question to bounded subset $K$ of $\mathbb{R}^n$, that is, \begin{equation} \|f\|_{L^p(K)} \le C_1\|f\|_{wL^q(K)} \end{equation} for every $f \in wL^q$ or \begin{equation} \|f\|_{wL^p(K)} \le C_2\|f\|_{L^q(K)} \end{equation} for every $f \in L^q$.

On $\mathbb R^n$, the answer is negative: if $p\ne q$, there is a function in $L^q$ that is not in $wL^p$, e.g.,
$$f(x)=\frac{|x|^{-n/q}}{1+\log^2|x|}\tag{*}$$

On a subset of finite measure, Lebesgue spaces are nested: $L^p\subset L^q$ if $p\ge q$. Therefore, we have the inequalities $$\|f\|_{L^p(K)} \le C_1\|f\|_{wL^q(K)},\quad p< q\tag1$$ and (as a consequence of (1)), $$\|f\|_{wL^p(K)} \le C_2\|f\|_{L^q(K)},\quad p\le q\tag2$$ To prove (1): use Jensen's inequality, followed by Chebyshev's: $$\|f\|_{L^p(K)}\le |K|^{1/p-1/q}\|f\|_{L^q(K)}\le |K|^{1/p-1/q}\|f\|_{wL^q(K)}$$

For $p>q$ the inclusion still fails. Consider the same example (*), which is in $L^q$ on the unit ball $B$, but not in $L^p(B)$ for $p>q$.