We give Trudinger-type inequalities for Riesz potentials of functions in Orlicz-Morrey spaces of an integral form over non-doubling metric measure spaces. Our results are new even for the doubling metric measure setting. In particular, our results improve and extend the previous results in Morrey spaces of an integral form in the Euclidean case.

Our aim in this paper is to establish a generalization of Sobolev’s inequality for Riesz potentials $I_{\unicode[STIX]{x1D6FC}(\,\cdot \,),\unicode[STIX]{x1D70F}}f$ of order $\unicode[STIX]{x1D6FC}(\,\cdot \,)$ with $f\in L^{\unicode[STIX]{x1D6F7},\unicode[STIX]{x1D705},\unicode[STIX]{x1D703}}(X)$ over bounded non-doubling metric measure spaces. As a corollary we obtain Sobolev’s inequality for double phase functionals with variable exponents.

Let $\mu $ be a nonnegative Radon measure on ${{\mathbb{R}}^{d}}$ that satisfies the growth condition that there exist constants ${{C}_{0}}\,>\,0$ and $n\,\in \,(0,\,d]$ such that for all $x\,\in \,{{\mathbb{R}}^{d}}$ and $r\,>\,0$, $\mu \left( B\left( x,\,r \right) \right)\,\le \,{{C}_{0}}{{r}^{n}}$, where $B(x,\,r)$ is the open ball centered at $x$ and having radius $r$. In this paper, the authors prove that if $f$ belongs to the $\text{BMO}$-type space $\text{RBMO(}\mu \text{)}$ of Tolsa, then the homogeneous maximal function ${{\dot{\mathcal{M}}}_{s}}\left( f \right)$ (when ${{\mathbb{R}}^{d}}$ is not an initial cube) and the inhomogeneous maximal function ${{\overset{{}}{\mathop{\mathcal{M}}}\,}_{s}}\left( f \right)$ (when ${{\mathbb{R}}^{d}}$ is an initial cube) associated with a given approximation of the identity $S $ of Tolsa are either infinite everywhere or finite almost everywhere, and in the latter case, ${{\dot{\mathcal{M}}}_{s}}$ and ${{\mathcal{M}}_{s}}$ are bounded from $\text{RBMO(}\mu \text{)}$ to the $\text{BLO}$-type space $\text{RBMO(}\mu \text{)}$. The authors also prove that the inhomogeneous maximal operator ${{\mathcal{M}}_{s}}$ is bounded from the local $\text{BMO}$-type space $\text{rbmo(}\mu \text{)}$ to the local $\text{BLO}$-type space $\text{rblo(}\mu \text{)}$.

Let $\mu$ be a positive Radon measure on $\mathbb{R}^d$ which satisfies $\mu(B(x,r))\le Cr^{n}$ for any $x\in\mathbb{R}^d$ and $r>0$ and some fixed constants $C>0$ and $n\in(0,d]$. In this paper, a new characterization of the space $\rbmo(\mu)$, which was introduced by Tolsa, is given. As an application, it is proved that the $L^p(\mu)$-boundedness with $p\in(1,\infty)$ of Calderón–Zygmund operators is equivalent to various endpoint estimates.

Let µ be Radon measure on Rd which may be non doubling. The only condition that µ must satisfy is the size condition µ(B(x, r)) ≤ Crn for some fixed n є (0, d). Recently, Tolsa introduced the spaces RBMO(µ) and Hatb1.∞ (µ), which, in some ways, play the role of the classical spaces BMO and H1 in case u is a doubling measure. In this paper, the author considers the local versions of the spaces RBMO(µ) and Hatb1.∞ (µ) in the sense of Goldberg and establishes the connections between the spaces RBMO(µ) and Hatb1.∞ (µ) with their local versions. An interpolation result of linear operators is also given.