In this paper, a class of multiple fractional type weights A ( p,q) is defined aswhen there exists p i = 1 for some multilinear fractional type operators (e.g. fractional maximal operator, fractional integral, commutators of fractional integral operators) are obtained. As applications of these results, we give some weighted estimates for the above operators with rough homogeneous kernels when suitable conditions were assumed on the kernels. Weighted strong and L(log L) type endpoint estimates for commutators of multilinear fractional integral operators are also obtained. Similar results for multilinear Calderón-Zygmund singular integral can be found in A.K. Lerner et al. (in press) [12].

The stochastic gradient descent (SGD) algorithm has been widely used in statistical estimation for large-scale data due to its computational and memory efficiency. While most existing works focus on the convergence of the objective function or the error of the obtained solution, we investigate the problem of statistical inference of true model parameters based on SGD when the population loss function is strongly convex and satisfies certain smoothness conditions.Our main contributions are two-fold. First, in the fixed dimension setup, we propose two consistent estimators of the asymptotic covariance of the average iterate from SGD: (1) a plug-in estimator, and (2) a batch-means estimator, which is computationally more efficient and only uses the iterates from SGD. Both proposed estimators allow us to construct asymptotically exact confidence intervals and hypothesis tests.Second, for high-dimensional linear regression, using a variant of the SGD algorithm, we construct a debiased estimator of each regression coefficient that is asymptotically normal. This gives a one-pass algorithm for computing both the sparse regression coefficients and confidence intervals, which is computationally attractive and applicable to online data. MSC 2010 subject classifications: Primary 62J10, 62M02; secondary 60K35The batch-means estimator is a "weighted" sample covariance matrix that treats each batch-means as a sample.The idea of batch-means estimator can be traced to Markov Chain Monte Carlo (MCMC), where the batch-means method with equal batch size (see, e.g., Glynn and Iglehart (1990); Glynn and Whitt (1991); Damerdji (1991); Geyer (1992); Fishman (1996);Jones et al. (2006); Flegal and Jones (2010)) is widely used for variance estimation in a time-homogeneous Markov chain. The SGD iterates in (2) indeed form a Markov chain, as x i only depends on x i−1 . However, since the step size sequence η i is a diminishing sequence, it is a time-inhomogenous Markov chain. Moreover, the asymptotic behavior of SGD and MCMC are fundamentally different: while the former converges to the optimum, the latter travels ergodically inside the state space. As a consequence of these important differences, previous literature on batchmeans methods is not applicable to our analysis. To address this challenge, our new batch-means method constructs batches of increasing sizes. The sizes of batches are chosen to ensure that the correlation decays appropriately among far-apart batches, so that far-apart batch-means can be roughly treated as independent. In Theorem 4.3, we prove that the proposed batchmeans method is a consistent estimator of the asymptotic covariance. Further, we believe this new batch-means algorithm with increasing batch sizes is of independent interest since it can be used to estimate the covariance structure of other time-inhomogeneous Markov chains.As both the plug-in and the batch-means estimator provide asymptotically exact confidence intervals, each of them has its own advantages:1. The plug-in estimator has a faster convergence r...

Abstract. This is the first in a series of papers in which we investigate the resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds with applications to the restriction theorem, spectral multiplier results and Strichartz estimates. In this first paper, we construct the high energy resolvent on general non-trapping asymptotically hyperbolic manifolds, using semiclassical Lagrangian distributions and semiclassical intersecting Lagrangian distributions, together with the 0-calculus of Mazzeo-Melrose.Our results generalize recent work of Melrose, Sá Barreto and Vasy [23], which applies to metrics close to the exact hyperbolic metric. We note that there is an independent work by Y. Wang [30] which also constructs the high-energy resolvent.

We consider the Laplacian ∆ on an asymptotically hyperbolic manifold X, as defined by Mazzeo and Melrose [34]. We give pointwise bounds on the Schwartz kernel of the spectral measure for the operator (∆ − n 2 /4) 1/2 + on such manifolds, under the assumptions that X is nontrapping and there is no resonance at the bottom of the spectrum. This uses the construction of the resolvent given by Mazzeo and Melrose [34] (valid when the spectral parameter lies in a compact set), Melrose, Sá Barreto and Vasy [36] (high energy estimates for a perturbation of the hyperbolic metric) and the present authors [10] (see also [45]) in the general high-energy case.We give two applications of the spectral measure estimates. The first, following work due to Guillarmou and Sikora with the second author [19] in the asymptotically conic case, is a restriction theorem, that is, a L p (X) → L p (X) operator norm bound on the spectral measure. The second is a spectral multiplier result under the additional assumption that X has negative curvature everywhere, that is, a bound on functions F ((∆ − n 2 /4) 1/2 + ) of the square root of the Laplacian, in terms of norms of the function F . Compared to the asymptotically conic case, our spectral multiplier result is weaker, but the restriction estimate is stronger. In both cases, the difference can be traced to the exponential volume growth at infinity for asymptotically hyperbolic manifolds, as opposed to polynomial growth in the asymptotically conic setting.The pointwise bounds on the spectral measure established here will also be applied to Strichartz estimates in [8].