Bakry–Émery, Hardy, and spectral gap estimates on manifolds with conical singularities

a version of the Bakry–Émery inequality

a novel Hardy inequality

a spectral gap estimate.

weighted spaces, e.g. $$M={{\mathbb {R}}}^n$$ M = R n with $$g=g^{Euclid}$$ g = g Euclid and $$m(dx)=|x|^\alpha d{\mathfrak {L}}^n(x)$$ m ( d x ) = | x | α d L n ( x ) for some $$\alpha \in {{\mathbb {R}}}$$ α ∈ R where the largest lower bound for Bakry–Émery Ricci tensor is given by $$ k(x)=-\frac{|\alpha |}{|x|^2}$$ k ( x ) = - | α | | x | 2 , and

Grushin-type spaces $$M={{\mathbb {R}}}^j \times _f {{\mathbb {R}}}^{n-j}$$ M = R j × f R n - j with $$f(y)=|y|^{-\alpha }$$ f ( y ) = | y | - α for suitable $$\alpha >0$$ α > 0 , either with Riemannian volume measure or with Lebesgue measure, which admit lower Ricci bounds of the form $$k(y,z)=-\frac{C}{|y|^2}$$ k ( y , z ) = - C | y | 2 .