Heat Equation and Convolution Inequalities

lVIULTIVARIATE PROBABILITY INEQUALITIES: CONVOLUTION THEORElVIS, COlVIPOSITION THEORElVlS, AND CONCENTRATION INEQUALITIES

Several important multivariate probability inequalities can be formulated in terms of multivariate convolutions of the form J!I (x )h(x (})dx, where usually !I = Ie is the indicator of a region C ~ R", h is a probability density on R , and (} is a translation parameter. Often fl and f2 possess convexity, monotonicity, and/or symmetry properties. More general multivariate compositions of the for...

Differential Harnack Inequalities on Riemannian Manifolds I : Linear Heat Equation

Abstract. In the first part of this paper, we get new Li-Yau type gradient estimates for positive solutions of heat equation on Riemmannian manifolds with Ricci(M) ≥ −k, k ∈ R. As applications, several parabolic Harnack inequalities are obtained and they lead to new estimates on heat kernels of manifolds with Ricci curvature bounded from below. In the second part, we establish a Perelman type L...

Middle Convolution and Heun’s Equation

The parameter q is called an accessory parameter. Although the local monodromy (local exponent) is independent of q, the global monodromy (e.g. the monodromy on the cycle enclosing two singularities) depends on q. Some properties of Heun’s equation are written in the books [21, 23], but an important feature related with the theory of finite-gap potential for the case γ, δ, ǫ, α−β ∈ Z+ 12 (see [...

Inequalities for Strongly Singular Convolution Operators

Best Constants for Two Non-convolution Inequalities

The norm of the operator which averages |f | in L p (R n) over balls of radius δ|x| centered at either 0 or x is obtained as a function of n, p and δ. Both inequalities proved are n-dimensional analogues of a classical inequality of Hardy in R 1. Finally, a lower bound for the operator norm of the Hardy-Littlewood maximal function on L p (R n) is given. A classical result of Hardy [HLP] states ...