Related papers: A note on the boundedness of Riesz transform for s…

In this paper, we give an affirmative answer to Gromov's conjecture ([3, Conjecture E]) by establishing an optimal Lipschitz lower bound for a class of smooth functions on orientable open $3$-manifolds with uniformly positive sectional…

We prove bounds for the truncated directional Hilbert transform in $L^p(\mathbb{R}^2)$ for any $1<p<\infty$ under a combination of a Lipschitz assumption and a lacunarity assumption. It is known that a lacunarity assumption alone is not…

We present a new proof of the dimensionless $L^p$ boundedness of the Riesz vector on manifolds with bounded geometry. Our proof has the significant advantage that it allows for a much stronger conclusion, namely that of a new dimensionless…

In this paper, a generalized Cauchy-Schwarz inequality for positive sesquilinear maps with values in noncommutative Lp-spaces for p > 1 are obtained. Bound estimates for their real and imaginary parts are also provided, and, as an…

Let $M^n$ be an $n$-dimensional Riemannian manifold with boundary $\partial M$. Assume that Ricci curvature is bounded from below by $(n-1)k$, for $k\in \RR$, we give a sharp estimate of the upper bound of $\rho(x)=\dis(x, \partial M)$, in…

It is shown that an elliptic scattering operator $A$ on a compact manifold with boundary with coefficients in the bounded operators of a bundle of Banach spaces of class (HT) and Pisier's property $(\alpha)$ has maximal regularity (up to a…

We prove that for any complete n-dimensional Riemannian manifold with nonnegative Ricci curvature, if the Nash inequality is satisfied, then it is diffeomorphic to $R^{n}$l.

Let $M$ be a closed connected spin manifold such that its spinor Dirac operator has non-vanishing (Rosenberg) index. We prove that for any Riemannian metric on $V = M \times [-1,1]$ with scalar curvature bounded below by $\sigma > 0$, the…

Assuming a lower bound on the Ricci curvature of a complete Riemannian manifold, for $p< 1/2$ we show the existence of bounds on the local $L^p$ norm of the Ricci curvature that depend only on the dimension and which improve with volume…

Let $L = \Delta + V$ be a Schr\"odinger operator with a non-negative potential $V$ on a complete Riemannian manifold $M$. We prove that the vertical Littlewood-Paley-Stein functional associated with $L$ is bounded on $L^p(M)$ {\it if and…

We prove that for a symmetric Markov semigroup, Ricci curvature bounded from below by a non-positive constant combined with a finite $L_\infty$-mixing time implies the modified log-Sobolev inequality. Such $L_\infty$-mixing time estimates…

In this paper, we investigate complete Riemannian manifolds satisfying the lower weighted Ricci curvature bound $\mathrm{Ric}_{N} \geq K$ with $K>0$ for the negative effective dimension $N<0$. We analyze two $1$-dimensional examples of…

We prove that metric measure spaces obtained as limits of closed Riemannian manifolds with Ricci curvature satisfying a uniform Kato bound are rectifiable. In the case of a non-collapsing assumption and a strong Kato bound, we additionally…

We show that any $L^\infty$ Riemannian metric $g$ on $\mathbb{R}^n$ that is smooth with nonnegative scalar curvature away from a singular set of finite $(n-\alpha)$-dimensional Minkowski content, for some $\alpha>2$, admits an approximation…

Comparison theorems are foundational to our understanding of the geometric features implied by various curvature constraints. This paper considers manifolds with a positive lower bound on either scalar, 2-Ricci, or Ricci curvature, and…

This article establishes a low-regularity Riemannian positive mass theorem for non-spin manifolds whose metrics are only $C^0 \cap W_{\mathrm{loc}}^{1,n}$ and smooth outside a compact set. The main theorem asserts that asymptotically flat…

We prove explicit $L^p$ bounds for second order Riesz transforms of the sub-Laplacian in the Lie groups $\mathbb H$, $\mathbb{SU}(2)$ and $\mathbb{SL}(2)$

We derive new, sharp lower bounds for certain curvature functionals on the space of Riemannian metrics of a smooth compact 4-manifold with a non-trivial Seiberg-Witten invariant. These allow one, for example, to exactly compute the infimum…

Let $L_k=-\Delta_k+V$ be the Dunk- Schr\"{o}dinger operators, where $\Delta_k=\sum_{j=1}^dT_j^2$ is the Dunkl Laplace operator associated to the dunkl operators $T_j$ on $\mathbb{R}^d$ and $V$ is a nonnegative potential function. In the…

Let $M$ be a connected complete noncompact $n$-dimensional Riemannian manifold with a base point $p \in M$ whose radial sectional curvature at $p$ is bounded from below by that of a noncompact surface of revolution which admits a finite…