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Abstract

We establish new logarithmic Sobolev inequalities in the fractional Sobolev setting. Our approach relies on a interpolation inequality, which can be viewed as a fractional Caffarelli-Kohn-Nirenberg inequality without weights. We further relate the optimal constant in this interpolation inequality to a corresponding variational problem. These results extend classical logarithmic Sobolev inequalities to the nonlocal framework and provide new tools for analysis in fractional Sobolev spaces.