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Let$X$be a rearrangement-invariant Banach function space on$R$^{$n$}and let$V$^{1}$X$be the Sobolev space of functions whose gradient belongs to$X$. We give necessary and sufficient conditions on$X$under which$V$^{1}$X$is continuously embedded into BMO or into$L$_{∞}. In particular, we show that$L$_{$n, ∞$}is the largest rearrangement-invariant space$X$such that$V$^{1}$X$is continuously embedded into BMO and, similarly,$L$_{$n$, 1}is the largest rearrangement-invariant space$X$such that$V$^{1}$X$is continuously embedded into$L$_{∞}. We further show that$V$^{1}$X$is a subset of VMO if and only if every function from$X$has an absolutely continuous norm in$L$_{$n, ∞$}. A compact inclusion of$V$^{1}$X$into$C$^{0}is characterized as well.

No abstract uploaded!

No abstract uploaded!

The systematic investigation of contour integrals satisfying the system of partial differential equations associated with Appell's hypergeometric function$F$_{1}leads to new solutions of that system. Fundamental sets of solutions are given for the vicinity of all singular points of the system of partial differential equations. The transformation theory of the solutions reveals connections between the system under consideration and other hypergeometric systems of partial differential equations. Presently it is discovered that any hypergeometric system of partial differential equations of the second order (with two independent variables) which has only three linearly independent solutions can be transformed into the system of$F$_{1}or into a particular or limiting case of this system. There are also other hypergeometric systems (with four linearly independent solutions) the integration of which can be reduced to the integration of the system of$F$_{1}.