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Register a new userTwo Weight Inequalities for Positive Operators: Doubling Cubes
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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 ^{p} (sigma ) to L ^{p} (w)$ if and only the pair of weights $ (w, sigma )$ satisfy the two weight $ A _{p}$ condition, and this testing inequality holds: begin{equation*} int _{Q} M (sigma mathbf 1_{Q} ) ^{p} ; d w lesssim sigma ( Q), end{equation*} for all cubes $ Q$ for which there is a cube $ P supset Q$ satisfying $ sigma (P) < D sigma (Q)$, and $ ell P = rho ell Q$. This was recently proved by Kangwei Li and Eric Sawyer. We give a short proof, which is easily seen to hold for several closely related operators.
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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 conditions are obtained. Meanwhile, the necessity of two-weight bump conditions as well as the converse of Bloom type estimates for iterated commutators of fractional integrals are also given.
We prove in this note one weight norm inequalities for some positive Bergman-type operators.
It is shown that positivity in $(0,1)times (0,1)$ of Green function of positively defined fourth-order ordinary differential operator (with separated boundary conditions) is a criterium of sign-regularity of this operator.
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.
We address the optimal constants in the strong and the weak Stechkin inequalities, both in their discrete and continuous variants. These inequalities appear in the characterization of approximation spaces which arise from sparse approximation or have applications to interpolation theory. An elementary proof of a constant in the strong discrete Stechkin inequality given by Bennett is provided, and we improve the constants given by Levin and Stechkin and by Copson. Finally, the minimal constants in the weak discrete Stechkin inequalities and both continuous Stechkin inequalities are presented.
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