This paper continues the program that was initiated in [26] and continued in [27], where a high-dimensional limiting technique was developed and used to prove certain parabolic theorems from their elliptic counterparts. The articles [26] and [27] address the constant-coefficient and variable-coefficient settings, respectively. Here, we focus on fractional operators. As shown in [17, 48, 57], fractional operators may be associated with certain degenerate operators via extension problems, so we study the corresponding class of degenerate operators. Our high-dimensional limiting technique is demonstrated through new proofs of three theorems for degenerate parabolic equations. Specifically, we establish the monotonicity of Almgren-type, Weiss-type, and Alt-Caffarelli-Friedman-type functionals in the degenerate parabolic setting. Each new parabolic proof in this article is based on a (new) related elliptic theorem and a careful limiting argument that is reminiscent of those from [26] and [27]. Our proof of the degenerate parabolic Weiss-type monotonicity formula additionally uses an epiperimetric inequality for weakly a-harmonic functions, which we also prove. To the best of our knowledge, our Alt-Caffarelli-Friedman monotonicity result is new.
