Endpoint Estimates Research Articles

Endpoint estimates for commutators with respect to the fractional integral operators on Orlicz–Morrey spaces

Endpoint estimates for commutators with respect to the fractional integral operators on Orlicz–Morrey spaces

Counterexamples and weak (1,1) estimates of wave operators for fourth-order Schrödinger operators in dimension three

This paper is dedicated to investigating the L^{p} -bounds of wave operators W_{\pm}(H,\Delta^{2}) associated with fourth-order Schrödinger operators H=\Delta^{2}+V on \mathbb{R}^{3} with real potentials satisfying |V(x)|\lesssim \langle x\rangle^{-\mu} for some \mu>0 . A recent work by Goldberg and Green (2021) has demonstrated that wave operators W_{\pm}(H,\Delta^{2}) are bounded on L^{p}(\mathbb{R}^{3}) for all 1<p<\infty under the condition that \mu>9 and zero is a regular point of H . In the paper, we aim to further establish endpoint estimates for W_{\pm}(H,\Delta^{2}) in two significant ways. First, we provide counterexamples to illustrate the unboundedness of W_{\pm}(H,\Delta^{2}) on the endpoint spaces L^{1}(\mathbb{R}^{3}) and L^{\infty}(\mathbb{R}^{3}) for non-zero compactly supported potentials V . Second, we establish weak (1,1) estimates for the wave operators W_{\pm}(H,\Delta^{2}) and their dual operators W_{\pm}(H,\Delta^{2})^{*} in the case where zero is a regular point and \mu>11 . These estimates depend critically on the singular integral theory of Calderón–Zygmund on a homogeneous space (X,d\omega) with a doubling measure d\omega .

Weighted mixed endpoint estimates of Fefferman-Stein type for commutators of singular integral operators

We deal with mixed weak estimates of Fefferman-Stein type for higher order commutators of Calderón-Zygmund operators with BMO symbol. The results obtained are Fefferman-Stein inequalities that include the estimates proved in [7] for the case of singular integral operators, as well as the classical weak endpoint estimate for commutators given in [26]. We also consider commutators of operators involving less regular kernels satisfying an LΦ–Hörmander condition. Particularly, the obtained results contain some previous estimates proved in [7] and [17].

Endpoint estimates for riesz transform on manifolds with ends

AbstractWe consider a class of non-doubling manifolds $$\mathcal {M}$$ M consisting of finite many “Euclidean” ends, where the Euclidean dimensions at infinity are not necessarily all the same. In [17], Hassell and Sikora proved that the Riesz transform on $$\mathcal {M}$$ M is of weak type (1, 1), bounded on $$L^{p}$$ L p if and only if $$1<p<n_*$$ 1 < p < n ∗ , where $$n_* = \min _k n_k$$ n ∗ = min k n k . In this note, we complete the picture by giving an endpoint estimate: Riesz transform is bounded on Lorentz space $$L^{n_*,1}$$ L n ∗ , 1 and unbounded from $$L^{n_*,p}\rightarrow L^{n_*,q}$$ L n ∗ , p → L n ∗ , q for all $$1<p<\infty $$ 1 < p < ∞ and $$p\le q\le \infty $$ p ≤ q ≤ ∞ .

Model-Based Bioequivalence Analysis to Assess and Predict the Relative Bioavailability of Valproic Acid Formulations.

Model-based bioequivalence (MBBE) encompasses the use of nonlinear mixed effect models supporting the estimation of pharmacokinetic endpoints to assess the relative bioavailability between multi-source drug products. This application emerges as a valuable alternative to the standard non-compartmental analysis (NCA) in bioequivalence (BE) studies in which dense sampling is not possible. In this work, we aimed to assess the application of MBBE compared to traditional methods in evaluating the relative bioavailability of two formulations with different drug release properties. Additionally, we sought to predict the performance of a modified-release formulation in a multiple-dose scenario, leveraging data from a single-dose study. MBBE analysis was implemented to estimate the BE endpoints (90%CI for the Test/Reference geometric mean ratio, T/R GMR) in area under the concentration-time curve (AUC) and maximum concentration (Cmax) using data from a single-dose, 2-period, 2-sequence BE study performed in 14 healthy subjects between a locally developed valproic acid extended-release formulation (Test) and the brand-name delayed-release formulation (Reference). Results were compared with the standard approach, revealing that MBBE analysis achieved higher discrimination between formulations for Cmax, addressing limitations of the experimental sampling design and highlighting an advantage for this model-based analysis even when rich data are available. Additionally, the bioequivalence outcome under the multiple-dose scenario was predicted through a simulation-based study for both total and unbound valproic acid concentrations, considering the impact of valproic acid saturable binding on BE conclusions. The MBBE analysis was superior to the NCA approach in detecting product-related differences, overcoming limitations in the study experimental design. Predictions for the multiple-dose scenario preclude that the extended-release properties of the Test formulation would persist at steady state, resulting in lower peak-to-trough fluctuation and bioequivalent performance in terms of the extent of drug absorption. Overall, these results should discourage unnecessary experimentation in healthy subjects.

Study of the Soaking Process of a ready-to-eat rice of Assam (Komal Chaul): A Mechanistic and a Machine Learning Based Approach for spectra-based Estimation of Endpoint

Study of the Soaking Process of a ready-to-eat rice of Assam (Komal Chaul): A Mechanistic and a Machine Learning Based Approach for spectra-based Estimation of Endpoint

Endpoint estimates of variation and oscillation operators associated with Zλ functions

This paper obtains weak-type estimates, limiting weak-type behaviors for variation operators associated with [Formula: see text] functions. Besides, we give a new characterization of Hardy space via the boundedness of variation operators associated with [Formula: see text] functions.

Endpoint estimates for harmonic analysis operators associated with Laguerre polynomial expansions

AbstractIn this paper, we give a criterion to prove boundedness results for several operators from the Hardy‐type space to and also from to the space of functions of bounded mean oscillation , with respect to the probability measure on when is a multi‐index in . We shall apply it to establish endpoint estimates for Riesz transforms, maximal operators, Littlewood–Paley functions, multipliers of the Laplace transform type, fractional integrals, and variation operators in the Laguerre setting.

Improved estimation of overall survival and progression-free survival for state transition modeling.

Aim: National Institute for Health and Care Excellence guidance (Technical Support Document 19) highlights a key challenge of state transition models (STMs) being their difficulty in achieving a satisfactory fit to the observed within-trial endpoints. Fitting poorly to data over the trial period can then have implications for long-term extrapolations. A novel estimation approach is defined in which the predicted overall survival (OS) and progression-free survival (PFS) extrapolations from anSTM are optimized to provide closer estimates of the within-trial endpoints. Materials & methods: An STM was fitted to the SQUIRE trial data in non-small-cell lung cancer (obtained from Project Data Sphere). Two methods were used: a standard approach whereby the maximum likelihood was utilized for the individual transitions and the best-fitting parametric model selected based on AIC/BIC, and a novel approach in which parameters were optimized by minimizing the area between the STM-predicted OS and PFS curves and the corresponding OS and PFS Kaplan-Meier curves. Sensitivity analyses were conducted to assess uncertainty. Results: The novel approach resulted in closer estimations to the OS and PFS Kaplan-Meier for all combinations of parametric distributions analyzed compared with the standard approach. Though the uncertainty associated with the novel approach was slightly larger, it provided better estimates to the restricted mean survival time in 10 of the 12 parametric distributions analyzed. Conclusion: A novel approach is defined which provides an alternative STM estimation method enabling improved fits to modeled endpoints, which can easily be extended to more complex model structures.

High-Throughput Transcriptomics Differentiates Toxic versus Non-Toxic Chemical Exposures Using a Rat Liver Model.

To address the challenge of limited throughput with traditional toxicity testing, a newly developed high-throughput transcriptomics (HTT) platform, together with a 5-day in vivo rat model, offers an alternative approach to estimate chemical exposures and provide reasonable estimates of toxicological endpoints. This study contains an HTT analysis of 18 environmental chemicals with known liver toxicity. They were evaluated using male Sprague Dawley rats exposed to various concentrations daily for five consecutive days via oral gavage, with data collected on the sixth day. Here, we further explored the 5-day rat model to identify potential gene signatures that can differentiate between toxic and non-toxic liver responses and provide us with a potential histopathological endpoint of chemical exposure. We identified a distinct gene expression pattern that differentiated non-hepatotoxic compounds from hepatotoxic compounds in a dose-dependent manner, and an analysis of the significantly altered common genes indicated that toxic chemicals predominantly upregulated most of the genes and several pathways in amino acid and lipid metabolism. Finally, our liver injury module analysis revealed that several liver-toxic compounds showed similarities in the key injury phenotypes of cellular inflammation and proliferation, indicating potential molecular initiating processes that may lead to a specific end-stage liver disease.

Corrigendum to “Improvements on Sawyer‐type estimates for generalized maximal functions”

AbstractThe purpose of this note is to correct a mixed estimate involving the generalized maximal function , when Φ is a Young function satisfying certain properties. The family considered include type functions. Although the obtained estimates turn out to be slightly different, they are still good extensions of mixed inequalities for the classical Hardy–Littlewood maximal operators corresponding to the power functions , with . They also allow us to derive the boundedness properties of between Lebesgue spaces by using interpolation techniques related to endpoint estimates of modular type.

A note on some Endpoint Estimates of Commutators of Fractional Integral Operators

A note on some Endpoint Estimates of Commutators of Fractional Integral Operators

Is CAR T a Drug or a Therapeutic Pathway? Systematic Review, (Re-)Estimation of Efficacy Endpoints and Metanalyses in Large B-Cell Lymphoma Real World Reports

In everyday clinical practice, the therapeutic pathway of autologous CAR T-cell starts from leukapheresis, but the drug is infused on average 4-7 weeks later. Between leukapheresis and CAR T infusion patients may die following disease progression or exit the therapeutic pathway due to comorbidities or other reasons, thereby introducing a potential bias in the estimation of survival benefit gained with the drug. Moreover, substantial differences may occur (e.g., timing of leukapheresis and enrollment, use of bridging chemotherapy, lymphodepleting regimens) between trials' designs and every-day clinical practice. All of this precludes a robust comparison among the different available CAR T products and the therapies which share the same indication. Considering analyses limited to treated patients with endpoints estimated from date of infusion should be avoided to prevent misleading treatment choices. We conceived a project involving researchers and cooperative groups worldwide which already published real world effectiveness data on per-protocol (PP, i.e. from infusion) or on intention-to-treat (ITT, i.e. from leukapheresis taking into account also subjects who were scheduled for but finally did not receive CAR T) populations with the aim to re-estimate efficacy endpoints in both modalities. This would offer the clinicians the possibility to draw more meaningful interpretations of the data according to these two different approaches (ALMA IDEA 2022 CUP:J33C22001420001). First a systematic review was conducted searching for real world studies on the effectiveness of anti-CD19 CAR T-cell products administered to large B-cell lymphoma patients in third line and beyond between 2017 (after FDA approval) and October 2022. For each study, we asked researchers about both infused and not infused patients, for overall response rate (ORR), complete response rate (CRR), median follow up, median survivals (event-free [EFS], progression-free [PFS] and overall [OS]) duration of response (DoR). ITT EFS, the new endpoint added for our project, was defined as the time the patients remained in the CAR T pathway from leukapheresis to exit due to any cause. When applicable, meta-regressions, metanalyses and pooled analyses were also performed. The results overview with the 10 selected studies is reported in Table 1 (Bachy et al. Nat Med. 2022; Bastos-Oreiro et al. Front Immunol. 2022; Ghafouri et al. Clin Lymphoma Myeloma Leuk. 2021; Iacoboni et al. Cancer Med. 2021; Kittai et al. Transplant Cell Ther. 2021; Kwon et al. Haematologica. 2023; Mian et al. Leuk Lymphoma. 2021; Nastoupil et al. J Clin Oncol. 2020; Pinnix et al. Blood Adv. 2020; Casadei et al. Cancers (Basel). 2021). Rates of not infused patients ranged from 3% to 29%. Meta-regressions did not indicate a statistically significant influence of the different ratio of infused/not infused patients among studies, nor the different lengths of the median follow up seem to bias project results. Metanalyses for ITT ORR and CRR resulted in 57% and 38% respectively while the PP ORR and CRR were 68% and 45% respectively. Although centers kept track of all patients scheduled for CAR T, around 50% did not record the exit date (decision of not infuse, due to any cause) for not-infused subjects, reflecting the inability to calculate the ITT EFS. Three studies reported a median PP PFS shorter than the ITT PFS, which may indicate a high mean time from leukapheresis to infusion, worthy of further investigation. For all studies but one median PP OS was higher than ITT OS: OS may be over-estimated when using a PP analysis. We should be using an ITT analysis, as reporting data in which we do not include the patients who underwent leukapheresis and were never treated could introduce a significant bias in results interpretation. Hematologists should understand which is the most clinical meaningful endpoint for this new therapeutic approach to reach effective results: have we to consider CAR T-cell therapy as a drug or a therapeutic pathway? The importance of considering this new treatment option as a pathway should translate into implementation of CAR T patient registries that include ITT as an endpoint to establish population-level effectiveness of these drugs. In addition, considering both PP and ITT analyses as well as infused and not infused patients provides a complete overview and leads to more information.

Trial in Progress: Patient-Reported Outcome Measures in the Prospective Observational Cohort Study of Patients at Risk for Chronic Graft-Versus-Host Disease in the United States (THRIVE)

Background and Significance: Allogeneic hematopoietic cell transplantation (alloHCT) is a curative immunotherapy for patients with hematologic disorders. However, graft-versus-host-disease (GVHD) is a major limitation to the success of alloHCT, with chronic GVHD (cGVHD) as the leading cause of morbidity and mortality. The pleomorphic nature of cGVHD makes the diagnosis and patient management challenging for clinicians; thus, patient-reported symptoms often guide in diagnosing and determining the severity of cGVHD. Our current understanding of cGVHD is largely based on data from registries, single-institution prospective studies, and the cGVHD Consortium with representation from tertiary care academic centers. Prospective studies including patient-reported outcomes and a focus on a more representative and generalizable sample of patients with cGVHD are needed to better understand cGVHD symptoms, disease burden, and practice patterns for management. This trial in progress focuses on broad patient-centered assessment of post-alloHCT medical and psychosocial complications and patient-reported caregiver involvement in a diverse and contemporary patient population. This study uses multiple organ- or system-directed patient-reported outcome measures (PROMs) to help identify cGVHD symptoms, severity, and impact on patient quality of life including financial impact. The primary objective of THRIVE is to assess the clinical course and burden of cGVHD, including diagnosis and severity recorded in electronic case report forms (eCRFs) and PROMs. Secondary objectives include measuring healthcare resource utilization and patient economic and employment impact associated with cGVHD management, as well as patient-reported organ-specific symptoms. Study Design and Methods: THRIVE is a prospective, longitudinal, observational study (NCT05919511) of alloHCT recipients in the United States that uses a hybrid model designed to work with HCT centers and a virtual platform. Eligible patients are ≥18 years old, received alloHCT 90-180 days before enrollment, and are able to complete patient-assessment questionnaires provided in their preferred language either alone or with minimal caregiver assistance. Up to 1500 alloHCT recipients will be recruited at HCT sites (with the expectation that ~500 will have cGVHD symptoms at enrollment; ~500 are estimated to develop cGVHD during the course of the study; ~500 are estimated not to develop cGVHD during the study; Figure). Patients have the option to participate on a virtual platform. Patients will be followed for 36 months from date of enrollment or until the earliest of study termination, withdrawal, or death. Data from 13 PROMs (Table) divided into 3 groups will be collected every 3 months using a decentralized virtual platform. To reduce survey burden, patients will be randomized into 1 of 6 sequence groups to capture 4-5 PROMs per month in year 1 and every 2 months in years 2 and 3. Clinical data, including medications and adverse events, will be collected and abstracted into eCRFs every 3 months. In general, reporting of results will be descriptive for variables of interest. Summary statistics for continuous variables and for categorical variables will be calculated for all patients as well as for subgroups of interest. Whenever applicable, 95% confidence intervals will be calculated for endpoint estimates. Univariate, multivariate, or repeated-measurement analyses will be performed to assess factors associated with evolution of clinical manifestations of and complications from cGVHD. Recruitment for this study is ongoing.

GRU-D-Weibull: A novel real-time individualized endpoint prediction

BackgroundIn the era of healthcare digital transformation, using electronic health record (EHR) data to generate various endpoint estimates for active monitoring is highly desirable in chronic disease management. However, traditional predictive modeling strategies leveraging well-curated data sets can have limited real-world implementation potential due to various data quality issues in EHR data. MethodsWe propose a novel predictive modeling approach, GRU-D-Weibull, which models Weibull distribution leveraging gated recurrent units with decay (GRU-D), for real-time individualized endpoint prediction and population level risk management using EHR data. ExperimentsWe systematically evaluated the performance and showcased the real-world implementability of the proposed approach through individual level endpoint prediction using a cohort of patients with chronic kidney disease stage 4 (CKD4). A total of 536 features including ICD/CPT codes, medications, lab tests, vital measurements, and demographics were retrieved for 6879 CKD4 patients. The performance metrics including C-index, L1-loss, Parkes' error, and predicted survival probability at time of event were compared between GRU-D-Weibull and other alternative approaches including accelerated failure time model (AFT), XGBoost based AFT (XGB(AFT)), random survival forest (RSF), and Nnet-survival. Both in-process and post-process calibrations were experimented on GRU-D-Weibull generated survival probabilities. ResultsGRU-D-Weibull demonstrated C-index of ~0.7 at index date, which increased to ~0.77 at 4.3 years of follow-up, comparable to that of RSF. GRU-D-Weibull achieved absolute L1-loss of ~1.1 years (sd≈0.95) at CKD4 index date, and a minimum of ~0.45 year (sd≈0.3) at 4 years of follow-up, comparing to second-ranked RSF of ~1.4 years (sd≈1.1) at index date and ~0.64 years (sd≈0.26) at 4 years. Both significantly outperform competing approaches. GRU-D-Weibull constrained predicted survival probability at time of event to smaller and more fixed range than competing models throughout follow-up. Significant correlations were observed between prediction error and missing proportions of all major categories of input features at index date (Corr ~0.1 to ~0.3), which faded away within 1 year after index date as more data became available. Through post training recalibration, we achieved a close alignment between the predicted and observed survival probabilities across multiple prediction horizons at different time points during follow-up. ConclusionGRU-D-Weibull shows advantages over competing methods in handling missingness commonly encountered in EHR data and providing both probability and point estimates for diverse prediction horizons during follow-up. The experiment highlights the potential of GRU-D-Weibull as a suitable candidate for individualized endpoint risk management, utilizing real-time clinical data to generate various endpoint estimates for monitoring. Additional research is warranted to evaluate the influence of different data quality aspects on prediction performance. Furthermore, collaboration with clinicians is essential to explore the integration of this approach into clinical workflows and evaluate its effects on decision-making processes and patient outcomes.

Products and commutators of martingales in H1 and BMO

Let f:=(fn)n∈Z+ and g:=(gn)n∈Z+ be two martingales related to the probability space (Ω,F,P) equipped with the filtration (Fn)n∈Z+. Assume that f is in the martingale Hardy space H1 and g is in its dual space, namely the martingale BMO. Then the semi-martingale f⋅g:=(fngn)n∈Z+ may be written as the sumf⋅g=G(f,g)+L(f,g). Here L(f,g):=(L(f,g)n)n∈Z+ with L(f,g)n:=∑k=0n(fk−fk−1)(gk−gk−1) for any n∈Z+, where f−1:=0=:g−1. The authors prove that L(f,g) is a process with bounded variation and limit in L1, while G(f,g) belongs to the martingale Hardy-Orlicz space Hlog associated with the Orlicz functionΦ(t):=tlog⁡(e+t),∀t∈[0,∞). The above bilinear decomposition L1+Hlog is sharp in the sense that, for particular martingales, the space L1+Hlog cannot be replaced by a smaller space having a larger dual. As an application, the authors characterize the largest subspace of H1, denoted by H1b with b∈BMO, such that the commutators [T,b] with classical sublinear operators T are bounded from H1b to L1. This endpoint boundedness of commutators allows the authors to give more applications. On the one hand, in the martingale setting, the authors obtain the endpoint estimates of commutators for both martingale transforms and martingale fractional integrals. On the other hand, in harmonic analysis, the authors establish the endpoint estimates of commutators both for the dyadic Hilbert transform beyond doubling measures and for the maximal operator of Cesàro means of Walsh–Fourier series.

Upper endpoint estimates and extrapolation for commutators

Upper endpoint estimates and extrapolation for commutators

Maximal regularity of parabolic equations associated with a discrete Laplacian

Let Δd be the discrete Laplacian defined on Zd by settingΔdf(n→)=∑j=1d−[f(n→+e→j)+f(n→−e→j)−2f(n→)],n→∈Zd, where {e→j:j=1,…,d} is the standard basis for Rd.In this paper, we prove weighted mixed norm estimates and end-point estimates for the maximal regularity of the discrete parabolic equation{ut+Δdu=f,t∈[0,T)u(0,⋅)=0, where T∈(0,∞).

Weighted norm inequalities for multilinear strongly singular Calderón–Zygmund operators on RD‐spaces

AbstractLet be an RD‐space, namely, a space of homogeneous type in the sense of Coifman and Weiss with the Borel measure μ satisfying the reverse doubling condition on . Based on this space, the authors define a multilinear strongly singular Calderón–Zygmund operator whose kernel does not need any size condition and has more singularities near the diagonal than that of a standard multilinear Calderón–Zygmund operator. For such an operator, we establish its boundedness on product of weighted Lebesgue spaces by means of the pointwise estimate for the sharp maximal function. In addition, the endpoint estimates of the type are also obtained. Moreover, we prove weighted boundedness results for multilinear commutators generated by multilinear strongly singular Calderón–Zygmund operators and BMO functions. These results contribute to the extension of multilinear strongly singular Calderón–Zygmund operator theory in the Euclidean case to the context of space of homogeneous type.

Blurring cluster randomized trials and observational studies: Two-Stage TMLE for subsampling, missingness, and few independent units.

Cluster randomized trials (CRTs) often enroll large numbers of participants; yet due to resource constraints, only a subset of participants may be selected for outcome assessment, and those sampled may not be representative of all cluster members. Missing data also present a challenge: if sampled individuals with measured outcomes are dissimilar from those with missing outcomes, unadjusted estimates of arm-specific endpoints and the intervention effect may be biased. Further, CRTs often enroll and randomize few clusters, limiting statistical power and raising concerns about finite sample performance. Motivated by SEARCH-TB, a CRT aimed at reducing incident tuberculosis infection, we demonstrate interlocking methods to handle these challenges. First, we extend Two-Stage targeted minimum loss-based estimation to account for three sources of missingness: (i) subsampling; (ii) measurement of baseline status among those sampled; and (iii) measurement of final status among those in the incidence cohort (persons known to be at risk at baseline). Second, we critically evaluate the assumptions under which subunits of the cluster can be considered the conditionally independent unit, improving precision and statistical power but also causing the CRT to behave like an observational study. Our application to SEARCH-TB highlights the real-world impact of different assumptions on measurement and dependence; estimates relying on unrealistic assumptions suggested the intervention increased the incidence of TB infection by 18% (risk ratio [RR]=1.18, 95% confidence interval [CI]: 0.85-1.63), while estimates accounting for the sampling scheme, missingness, and within community dependence found the intervention decreased the incident TB by 27% (RR=0.73, 95% CI: 0.57-0.92).