We establish Trudinger-type inequalities for variable Riesz potentials $J_{\alpha (\cdot ), \tau }f$ of functions in Musielak–Orlicz spaces $L^{\Phi }(X)$ over bounded metric measure spaces X equipped with lower Ahlfors $Q(x)$-regular measures under conditions on $\Phi $ which are weaker than conditions in the previous paper (Houston J. Math. 48 (2022), no. 3, 479–497). We also deal with the case $\Phi $ is the double phase functional with variable exponents. As an application, Trudinger-type inequalities are discussed for Sobolev functions.

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 $J_{\unicode[STIX]{x1D6FC}(\cdot )}^{\unicode[STIX]{x1D70E}}f$ of functions $f$ in Musielak–Orlicz–Morrey spaces $L^{\unicode[STIX]{x1D6F7},\unicode[STIX]{x1D705}}(X)$. As a corollary we obtain Sobolev’s inequality for double phase functionals with variable exponents.

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.

Our aim in this paper is to deal with integrability of maximal functions for Herz–Morrey spaces on the unit ball with variable exponent $p_{1}(\cdot )$ approaching $1$ and for double phase functionals $\unicode[STIX]{x1D6F7}_{d}(x,t)=t^{p_{1}(x)}+a(x)t^{p_{2}}$, where $a(x)^{1/p_{2}}$ is nonnegative, bounded and Hölder continuous of order $\unicode[STIX]{x1D703}\in (0,1]$ and $1/p_{2}=1-\unicode[STIX]{x1D703}/N>0$. We also establish Sobolev type inequality for Riesz potentials on the unit ball.

Our aim in this paper is to deal with Sobolev inequalities for Riesz potentials of functions in Lebesgue spaces of variable exponents near Sobolev’s exponent over nondoubling metric measure spaces.

Our aim in this paper is to deal with Sobolev's embeddings for Sobolev–Orlicz functions with ∇u ∈ Lp(·) logLq(·)(Ω) for Ω ⊂ n. Here p and q are variable exponents satisfying natural continuity conditions. Also the case when p attains the value 1 in some parts of the domain is included in the results.