A Complete Characterization of the Generalized Multilinear Sobolev Inequality in Grand Product Lebesgue Spaces Defined on Non-homogeneous Spaces

Abstract

Necessary and sufficient condition governing the boundedness of the multilinear fractional integral operator $T_{\gamma , \mu }(\textbf{f})(x)= \int \nolimits _{X^m} \frac{ f_1(y_1) \dots f(y_m)}{\big (\rho (x, y_1)+ \cdots + \rho (x, y_m) \big )^{m-\gamma }}\, d\mu (\textbf{y})$ defined with respect to a measure $\mu$ from grand product space $\prod _{j=1}^m {\mathcal {L}}^{p_j), \theta }(X, \mu )$ to grand Lebesgue space $L^{q),\theta q/p} (X, \mu )$ is established, where $(X, \rho , \mu )$ is a quasi-metric measure space, $\frac{1}{p}= \sum _{j=1}^m \frac{1}{p_j}$, $p<q$ and $\theta >0$. In particular, we show that for that boundedness, the necessary and sufficient condition on $\mu$ is that it is upper Ahlfors $\beta _m$—regular for certain number $\beta _m$. To establish this result, first we derive the boundedness of $T_{\gamma , \mu }$ between more general grand spaces: $T_{\gamma , \mu }: \prod _{j=1}^m {\mathcal {L}}^{p_j), \Psi (\cdot )}(X, \mu )\mapsto L^{q), \Phi (\cdot )}(X, \mu )$. As a corollary, we have a complete characterization of the multilinear Sobolev inequality (i.e., when q is the Sobolev exponent of p) in these spaces. In this case criterion on $\mu$ is that it is upper Ahlfors 1– regular. Some weighted inequalities for the bilinear fractional integral operator $B_{\gamma }(f,g)(x) = \int \nolimits _{ {\mathbb {R}}^n} \frac{f(x-t)g(x+t) dt}{|t|^{n-\gamma } }$, $0<\gamma <n$, in the classical Lebesgue spaces are also discussed.