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As is well known, Morrey spaces are widely used to investigate the local behavior of solutions to second order elliptic partial differential equations. Recall that the classical Morrey spaces $\mathcal{M}^{p,\lambda}(\mathbb{R}^{n})$ are defined by \begin{equation*} \mathcal{M}^{p,\lambda}(\mathbb{R}^{n}) = \left\{ f \in L^p_{\rm loc}(\mathbb{R}^{n}) : \left\| f\right\|_{\mathcal{M}^{p,\lambda}}: = \sup_{x \in \mathbb{R}^{n}, \; r>0 } r^{-\frac{\lambda}{p}} \|f\|_{L^p (B(x,r))} < \infty \right\}, \end{equation*} where $0 \le \lambda \le n,$ $1\le p < \infty$. Note that $\mathcal{M}^{p,0}(\mathbb{R}^{n})=L^{p}(\mathbb{R}^{n})$ and $\mathcal{M}^{p,n}(\mathbb{R}^{n})=L^{\infty}(\mathbb{R}^{n})$. If $\lambda<0$ or $\lambda>n$, then $\mathcal{M}^{p,\lambda }(\mathbb{R}^{n})={\Theta}$, where $\Theta$ is the set of all functions equivalent to $0$ on $\mathbb{R}^{n}$. Indeed, first let $\lambda<0$ and $f\in \mathcal{M}^{p,\lambda}(\mathbb{R}^{n})$, then we have for all $x\in\mathbb{R}^{n}$ and $r>0$ \begin{align*} \|f\|_{L^{p}(B(x,r))}\le r^{\frac{\lambda}{p}}\| f\|_{\mathcal{M}^{p,\lambda }}. \end{align*} Consequently we get $$ \|f\|_{L^{p}(\mathbb{R}^{n})}=\lim_{r\rightarrow \infty}\|f\|_{L^{p}(B(x,r))}=0\Rightarrow f(x)=0, \text{ for a.e. }x\in\mathbb{R}^{n}. $$ Now let $\lambda>n$ and $f\in \mathcal{M}^{p,\lambda}(\mathbb{R}^{n})$. Then, we have for all $x\in\mathbb{R}^{n}$ and $r>0$ \begin{align}\tag{*} \frac{\|f\chi_{B(x,r)}\|_{L^p(\mathbb{R}^{n})}}{\|\chi_{B(x,r)}\|_{L^p(\mathbb{R}^{n})}}\le v_n^{-\frac{1}{p}} r^{\frac{\lambda-n}{p}}\| f\|_{\mathcal{M}^{p,\lambda }}, \end{align} where $v_n$ is the volume of the unit ball in $\mathbb{R}^{n}$.

By the Lebesgue theorem on differentiation of integrals we know that for all $f\in L^p_{\rm loc}(\mathbb{R}^{n})$ \begin{align} \lim_{r\rightarrow 0} \frac{\|f\chi_{B(x,r)}\|_{L_p(\mathbb{R}^{n})}}{\|\chi_{B(x,r)}\|_{L^p(\mathbb{R}^{n})}}=|f(x)| \text{ for a.e. }x\in\mathbb{R}^{n}. \end{align} Using this fact in (*) we get $f(x)=0$ for a.e. $x\in\mathbb{R}^{n}$.

By $W\mathcal{M}^{p,\lambda}(\mathbb{R}^{n})$ we denote the weak Morrey space defined as the set of functions $f\in WL^{p}_{\rm loc}(\mathbb{R}^{n}) $ for which $$ \left\| f\right\|_{W\mathcal{M}^{p,\lambda }} := \sup_{x\in \mathbb{R}^{n}, \; r>0} r^{-\frac{\lambda}{p}} \|f\|_{WL^{p}(B(x,r))} <\infty, $$ where $WL^{p}(B(x,r))$ denotes the weak $L^p$-space of measurable functions $f$ for which \begin{align*} \begin{split} \|f\|_{WL^{p}(B(x,r))} & \equiv \|f \chi_{_{B(x,r)}}\|_{WL^{p}(\mathbb{R}^{n})} \\ &= \sup_{t>0}t \left|\left\{y\in B(x,r) : \,|f(y)| > t \right\} \right|^{1/{p}} %\label{ISAAC2} \\ &=\sup_{0<t\le |B(x,r)|} t^{1/{p}} \left( f \chi_{_{B(x,r)}} \right)^{*}(t) <\infty. \end{split} \end{align*} Here $g^{*}$ denotes the non-increasing rearrangement of the function $g$.

For the non-triviality of weak Morrey space where should $\lambda$ be? I think it is obvious that if $\lambda=0$ then $W\mathcal{M}^{p,0}(\mathbb{R}^{n})=WL^{p} (\mathbb{R}^{n})$ and if $\lambda<0$ then $W\mathcal{M}^{p,\lambda }(\mathbb{R}^{n})={\Theta}$. Also, for my opinion if $\lambda=n$ then \begin{align*} \begin{split} r^{-\frac{n}{p}} \|f\|_{WL^{p}(B(x,r))}&=r^{-\frac{n}{p}} \sup_{0<t\le |B(x,r)|} t^{1/{p}} \left( f \chi_{_{B(x,r)}} \right)^{*}(t)\\ &\leq v_{n}^{\frac{1}{p}}\left( f \chi_{_{B(x,r)}} \right)^{*}(0)\\ &\leq v_{n}^{\frac{1}{p}}\|f\|_{L^\infty(\mathbb{R}^{n})}. \end{split} \end{align*} So $L^\infty(\mathbb{R}^{n})\hookrightarrow W\mathcal{M}^{p,n}(\mathbb{R}^{n})$.

What do you think about the embedding $W\mathcal{M}^{p,n}(\mathbb{R}^{n})\hookrightarrow L^\infty(\mathbb{R}^{n})$ ? Is it true? Also what do you think about the other cases of $\lambda$ ? For which cases of $\lambda$ the weak Morrey spaces are trivial?

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