Two weight inequality for vector-valued positive dyadic operators by parallel stopping cubes

We study the vector-valued positive dyadic operator $$T_\lambda(f\sigma):=\sum_{Q\in\mathcal{D}} \lambda_Q \int_Q f \mathrm{d}\sigma 1_Q,$$ where the coefficients $\{\lambda_Q:C\to D\}_{Q\in\mathcal{D}}$ are positive operators from a Banach lattice $C$ to a Banach lattice $D$. We assume that the Banach lattices $C$ and $D^*$ each have the Hardy--Littlewood property. An example of a Banach lattice with the Hardy--Littlewood property is a Lebesgue space. In the two-weight case, we prove that the $L^p_C(\sigma)\to L^q_D(\omega)$ boundedness of the operator $T_\lambda( \cdot \sigma)$ is characterized by the direct and the dual $L^\infty$ testing conditions: $$\lVert 1_Q T_\lambda(1_Q f \sigma)\rVert_{L^q_D(\omega)}\lesssim \lVert f\rVert_{L^\infty_C(Q,\sigma)} \sigma(Q)^{1/p},$$ $$\lVert1_Q T^*_{\lambda}(1_Q g \omega)\rVert_{L^{p'}_{C^*}(\sigma)}\lesssim \lVert g\rVert_{L^\infty_{D^*}(Q,\omega)} \omega(Q)^{1/q'}.$$ Here $L^p_C(\sigma)$ and $L^q_D(\omega)$ denote the Lebesgue--Bochner spaces associated with exponents $1<p\leq q<\infty$, and locally finite Borel measures $\sigma$ and $\omega$. In the unweighted case, we show that the $L^p_C(\mu)\to L^p_D(\mu)$ boundedness of the operator $T_\lambda( \cdot \mu)$ is equivalent to the endpoint direct $L^\infty$ testing condition: $$\lVert1_Q T_\lambda(1_Q f \mu)\rVert_{L^1_D(\mu)}\lesssim \lVert f\rVert_{L^\infty_C(Q,\mu)} \mu(Q).$$ This condition is manifestly independent of the exponent $p$. By specializing this to particular cases, we recover some earlier results in a unified way.Comment: 32 pages. The main changes are: a) Banach lattice-valued functions are considered. It is assumed that the Banach lattices have the Hardy--Littlewood property. b) The unweighted norm inequality is characterized by an endpoint testing condition and some corollaries of this characterization are stated. c) Some questions about the borderline of the vector-valued testing conditions are pose

Initial treatment strategy and clinical outcomes in Finnish MS patients : a propensity-matched study

Background The optimal treatment strategy with disease-modifying therapies (DMTs) in relapsing-remitting multiple sclerosis (RRMS) remains uncertain. Objective To compare outcomes of initial treatment with infusion therapies and starting therapy with medium efficacy therapy in a propensity-matched cohort of Finnish RRMS patients. Methods A total of 154 RRMS patients initiating natalizumab, alemtuzumab, ocrelizumab or rituximab as first DMT (high efficacy DMT, heDMT group) and 1771 patients initially treated with injectable therapies, teriflunomide or dimethylfumarate and escalated based on disease activity (moderate efficacy DMT, meDMT group) were identified from the Finnish MS registry. Nearest neighbor propensity matching (1:1, caliper 0.1) was performed for age, sex, baseline Expanded Disability Status Scale (EDSS), annual relapse rate (ARR) one year prior DMT and time since MS symptom onset. Primary outcome was time to 6-month confirmed EDSS progression and the secondary outcome time to first relapse. Results In the propensity-matched group comparisons, the probability of 6-month confirmed disability progression (CDP) at 5 years after DMT start was 28.4% (95% CI 15.7-39.3) in the heDMT group (n = 66) and 47.0% (95% CI 33.1-58.1) in meDMT group (n = 66), p = 0.013. Probability of relapse at 5 years was 34.6% (95% CI 24.1-43.6) for heDMT (n = 105) and 47.2% (95% CI 36.6-56.1) for meDMT (n = 105), p = 0.019. Conclusions Initiating MS-therapy with heDMT significantly reduced the risk of 5-year disability progression and relapse compared to using meDMT as first DMT choice in propensity-matched groups of Finnish MS-patients.Peer reviewe

On Two-Weight Norm Inequalities for Positive Dyadic Operators

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