ABSTRACT

A complete characterization of a measure μ governing the boundedness of fractional integral operators defined on a quasi-metric measure space $(X,d,\mu )$ (non-homogeneous space) from one grand Lebesgue spaces $L_{\mu }^{p),{\theta }_1}\left(X\right)$ into another one $L_{\mu }^{q),{\theta }_2}\left(X\right)$ is established. As a corollary, we have a generalization of the Sobolev inequality for potentials with measure. An appropriate problem for grand Morrey spaces is studied. D. Adams trace inequality (i.e. $L_{\mu }^{p),{\theta }_1}\left(X\right)\mapsto L_{\nu }^{q),{\theta }_2}\left(X\right)$ boundedness) is also derived for these operators in grand Lebesgue spaces. In the case of Morrey spaces, we assume that the underlying sets of spaces might be of infinite measure. Under some additional conditions on a measure, we investigate the sharpness of the second parameter ${\theta }_2$ in the target space.

摘要

度量μ 的完整表征控制在拟度量度量空间上定义的分数积分算子的有界性$(X,d,\mu )$ （非齐次空间）来自一个大 Lebesgue 空间 ${升}_{\mu }^{磷),{\theta }_1}\left(X\right)$ 进入另一个 ${升}_{\mu }^{q),{\theta }_2}\left(X\right)$成立。作为推论，我们对带测度势的 Sobolev 不等式进行了推广。研究了大莫雷空间的一个适当问题。D. Adams 追踪不等式（即${升}_{\mu }^{磷),{\theta }_1}\left(X\right)\mapsto {升}_{\nu }^{q),{\theta }_2}\left(X\right)$有界）也在大 Lebesgue 空间中为这些算子推导出来。在莫雷空间的情况下，我们假设潜在的空间集可能是无限测度的。在测量的一些附加条件下，我们研究第二个参数的锐度${\theta }_2$ 在目标空间。