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Let $M = mathbb R^m sharp mathcal R^n$ be a non-doubling manifold with two ends $mathbb R^m sharp mathcal R^n$, $m > n ge 3$. Let $Delta$ be the Laplace--Beltrami operator which is non-negative self-adjoint on $L^2(M)$. Then $Delta$ and its square root $sqrt{Delta}$ generate the semigroups $e^{-tDelta}$ and $e^{-tsqrt{Delta}}$ on $L^2(M)$, respectively. We give testing conditions for the two weight inequality for the Poisson semigroup $e^{-tsqrt{Delta}}$ to hold in this setting. In particular, we prove that for a measure $mu$ on $M_{+}:=Mtimes (0,infty)$ and $sigma$ on $M$: $$ |mathsf{P}_sigma(f)|_{L^2(M_{+};mu)} lesssim |f|_{L^2(M;sigma)}, $$ with $mathsf{P}_sigma(f)(x,t):= int_M mathsf{P}_t(x,y)f(y) ,dsigma(y)$ if and only if testing conditions hold for the Poisson semigroup and its adjoint. Further, the norm of the operator is shown to be equivalent to the best constant in these testing conditions.
For the maximal operator $ M $ on $ mathbb R ^{d}$, and $ 1< p , rho < infty $, there is a finite constant $ D = D _{p, rho }$ so that this holds. For all weights $ w, sigma $ on $ mathbb R ^{d}$, the operator $ M (sigma cdot )$ is bounded from $ L ^
We study the Rellich inequalities in the framework of equalities. We present equalities which imply the Rellich inequalities by dropping remainders. This provides a simple and direct understanding of the Rellich inequalities as well as the nonexistence of nontrivial extremisers.
One may define a trilinear convolution form on the sphere involving two functions on the sphere and a monotonic function on the interval $[-1,1]$. A symmetrization inequality of Baernstein and Taylor states that this form is maximized when the two fu
This paper is a companion to our prior paper arXiv:0705.4619 on the `Small Ball Inequality in All Dimensions. In it, we address a more restrictive inequality, and obtain a non-trivial, explicit bound, using a single essential estimate from our prior
This paper gives new two-weight bump conditions for the sparse operators related to iterated commutators of fractional integrals. As applications, the two-weight bounds for iterated commutators of fractional integrals under more general bump conditio