We analyze the general form of Caffarelli-Kohn-Nirenberg inequality. Due to a new introduced parameter, this inequality presents two distinguishable ranges. One of them, the inequality is shown to be the interpolation between Hardy and weighted Sobolev inequalities. The other range, which is no more an interpolation, the positive constant in the inequality is not necessarily bounded for all value of the parameters. In both cases, the precise value of the constants were given.

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关于Caffarelli-Kohn-Nirenberg不等式的精确标度

我们分析了Caffarelli-Kohn-Nirenberg不等式的一般形式。由于引入了新的参数，该不等式呈现了两个可区分的范围。其中之一，不等式显示为Hardy和加权Sobolev不等式之间的插值。另一个范围（不再是插值），不等式中的正常数不一定对所有参数值都具有边界。在这两种情况下，均给出了常数的精确值。