Abstract: The Gagliardo-Nirenberg-Sobolev inequalities are one of the most fundamental analysis tools in PDEs. Moreover, weighted norm Sobolev type inequalities, such as Hardy's inequalities and their extensions, have been extensively studied. Caffarelli, Kohn and Nirenberg consider such types of interpolation inequalities with isotropic weights given by a power of |x|. In this talk, I will discuss an extension of these inequalities with anisotropic weights. I will first have an overview about the development and related problems on Gagliardo-Nirenberg-Sobolev interpolation inequalities and Caffarelli-Kohn-Nirenberg inequalities. I will then describe a more general anisotropic version of CKN inequalities. I will briefly sketch the proof and discuss some applications of these inequalities. This is a joint work with Yanyan Li.
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