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Researcher profile

Ji Li

Ji Li contributes to research discovery and scholarly infrastructure.

ResearcherAffiliation not importedOpen to collaborate
math.CAmath.APmath.FAmath.CVDistributed, Parallel, and Cluster ComputingMachine Learningastro-ph.GAastro-ph.SRComputation and LanguageComputational GeometryComputer VisionCryptography and SecurityGraphicsmath.GTmath.OCphysics.atom-ph

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Published work

24 published item(s)

preprint2026arXiv

Covering Human Action Space for Computer Use: Data Synthesis and Benchmark

Computer-use agents (CUAs) automate on-screen work, as illustrated by GPT-5.4 and Claude. Yet their reliability on complex, low-frequency interactions is still poor, limiting user trust. Our analysis of failure cases from advanced models suggests a long-tail pattern in GUI operations, where a relatively small fraction of complex and diverse interactions accounts for a disproportionate share of task failures. We hypothesize that this issue largely stems from the scarcity of data for complex interactions. To address this problem, we propose a new benchmark CUActSpot for evaluating models' capabilities on complex interactions across five modalities: GUI, text, table, canvas, and natural image, as well as a variety of actions (click, drag, draw, etc.), covering a broader range of interaction types than prior click-centric benchmarks that focus mainly on GUI widgets. We also design a renderer-based data-synthesis pipeline: scenes are automatically generated for each modality, screenshots and element coordinates are recorded, and an LLM produces matching instructions and action traces. After training on this corpus, our Phi-Ground-Any-4B outperforms open-source models with fewer than 32B parameters. We will release our benchmark, data, code, and models at https://github.com/microsoft/Phi-Ground.git

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

$L^{p}$ estimates and weighted estimates of fractional maximal rough singular integrals on homogeneous groups

In this paper, we study the $L^{p}$ boundedness and $L^{p}(w)$ boundedness ($1<p<\infty$ and $w$ a Muckenhoupt $A_{p}$ weight) of fractional maximal singular integral operators $T_{Ω,α}^{\#}$ with homogeneous convolution kernel $Ω(x)$ on an arbitrary homogeneous group $\mathbb H$ of dimension $\mathbb{Q}$. We show that if $0<α<\mathbb{Q}$, $Ω\in L^{1}(Σ)$ and satisfies the cancellation condition of order $[α]$, then for any $1<p<\infty$, \begin{align*} \|T_{Ω,α}^{\#}f\|_{L^{p}(\mathbb{H})}\lesssim\|Ω\|_{L^{1}(Σ)}\|f\|_{L_α^{p}(\mathbb{H})}, \end{align*} where for the case $α=0$, the $L^p$ boundedness of rough singular integral operator and its maximal operator were studied by Tao (\cite{Tao}) and Sato (\cite{sato}), respectively. We also obtain a quantitative weighted bound for these operators. To be specific, if $0\leqα<\mathbb{Q}$ and $Ω$ satisfies the same cancellation condition but a stronger condition that $Ω\in L^{q}(Σ)$ for some $q>\mathbb{Q}/α$, then for any $1<p<\infty$ and $w\in A_{p}$, \begin{align*} \|T_{Ω,α}^{\#}f\|_{L^{p}(w)}\lesssim\|Ω\|_{L^{q}(Σ)}\{w\}_{A_p}(w)_{A_p}\|f\|_{L_α^{p}(w)},\ \ 1<p<\infty. \end{align*}

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preprint2022arXiv

A Secure and Efficient Federated Learning Framework for NLP

In this work, we consider the problem of designing secure and efficient federated learning (FL) frameworks. Existing solutions either involve a trusted aggregator or require heavyweight cryptographic primitives, which degrades performance significantly. Moreover, many existing secure FL designs work only under the restrictive assumption that none of the clients can be dropped out from the training protocol. To tackle these problems, we propose SEFL, a secure and efficient FL framework that (1) eliminates the need for the trusted entities; (2) achieves similar and even better model accuracy compared with existing FL designs; (3) is resilient to client dropouts. Through extensive experimental studies on natural language processing (NLP) tasks, we demonstrate that the SEFL achieves comparable accuracy compared to existing FL solutions, and the proposed pruning technique can improve runtime performance up to 13.7x.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Detection of DC electric forces with zeptonewton sensitivity by single-ion phonon laser

Detecting extremely small forces helps exploring new physics quantitatively. Here we demonstrate that the phonon laser made of a single trapped $^{40}$Ca$^{+}$ ion behaves as an exquisite sensor for small force measurement. We report our successful detection of small electric forces regarding the DC trapping potential with sensitivity of 2.41$\pm$0.49 zN/$\sqrt{\rm Hz}$, with the ion only under Doppler cooling, based on the injection-locking of the oscillation phase of the phonon laser in addition to the classical squeezing applied to suppress the measurement uncertainty. We anticipate that such a single-ion sensor would reach a much better force detection sensitivity in the future once the trapping system is further improved and the fluorescence collection efficiency is further enhanced.

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preprint2022arXiv

Endpoint weak Schatten class estimates and trace formula for commutators of Riesz transforms with multipliers on Heisenberg groups

Along the line of singular value estimates for commutators by Rochberg-Semmes, Lord-McDonald-Sukochev-Zanin and Fan-Lacey-Li, we establish the endpoint weak Schatten class estimate for commutators of Riesz transforms with multiplication operator $M_f$ on Heisenberg groups via homogeneous Sobolev norm of the symbol $f$. The new tool we exploit is the construction of a singular trace formula on Heisenberg groups, which, together with the use of double operator integrals, allows us to bypass the use of Fourier analysis and provides a solid foundation to investigate the singular values estimates for similar commutators in general stratified Lie groups.

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preprint2022arXiv

Schatten classes and commutator in the two weight setting, I. Hilbert transform

We characterize the Hilbert--Schmidt class membership of commutator with the Hilbert transform in the two weight setting. The characterization depends upon the symbol of the commutator being in a new weighted Besov space. This follows from a Schatten class $S_p$ result for dyadic paraproducts, where $1< p < \infty $. We discuss the difficulties in extending the dyadic result to the full range of Schatten classes for the Hilbert transform.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Singular integral operators, $T1$ theorem, Littlewood-Paley theory and Hardy spaces in Dunkl Setting

The purpose of this paper is to introduce a new class of singular integral operators in the Dunkl setting involving both the Euclidean metric and the Dunkl metric. Then we provide the $T1$ theorem, the criterion for the boundedness on $L^2$ for these operators. Applying this singular integral operator theory, we establish the Littlewood-Paley theory and the Dunkl-Hardy spaces. As applications, the boundedness of singular integral operators, particularly, the Dunkl-Rieze transforms, on the Dunkl-Hardy spaces is given. The $L^2$ theory and the singular integral operator theory play crucial roles. New tools developed in this paper include the weak-type discrete Calderón reproducing formulae, new test functions, and distributions, the Littlewood-Paley, the wavelet-type decomposition, and molecule characterizations of the Dunkl-Hardy space, Coifman&#39;s approximation to the identity and the decomposition of the identity operator on $L^2$, Meyer&#39;s commutation Lemma, and new almost orthogonal estimates in the Dunkl setting.

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preprint2022arXiv

The Schrödinger equation in $L^p$ spaces for operators with heat kernel satisfying Poisson type bounds

Let $L$ be a non-negative self-adjoint operator acting on $L^2(X)$ where $X$ is a space of homogeneous type with a dimension $n$. In this paper, we study sharp endpoint $L^p$-Sobolev estimates for the solution of the initial value problem for the Schrödinger equation, $i \partial_t u + L u=0 $ and show that for all $f\in L^p(X), 1<p<\infty,$ \begin{eqnarray*} \left\| e^{itL} (I+L)^{-{σn}} f\right\|_{p} \leq C(1+|t|)^{σn} \|f\|_{p}, \ \ \ t\in{\mathbb R}, \ \ \ σ\geq \big|{1\over 2}-{1\over p}\big|, \end{eqnarray*} where the semigroup $e^{-tL}$ generated by $L$ satisfies a Poisson type upper bound. This extends the previous result in \cite{CDLY1} in which the semigroup $e^{-tL}$ generated by $L$ satisfies the exponential decay.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

The Two Weight Inequality for Poisson Semigroup on Manifold with Ends

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 $Δ$ be the Laplace--Beltrami operator which is non-negative self-adjoint on $L^2(M)$. Then $Δ$ and its square root $\sqrtΔ$ generate the semigroups $e^{-tΔ}$ and $e^{-t\sqrtΔ}$ on $L^2(M)$, respectively. We give testing conditions for the two weight inequality for the Poisson semigroup $e^{-t\sqrtΔ}$ to hold in this setting. In particular, we prove that for a measure $μ$ on $M_{+}:=M\times (0,\infty)$ and $σ$ on $M$: $$ \|\mathsf{P}_σ(f)\|_{L^2(M_{+};μ)} \lesssim \|f\|_{L^2(M;σ)}, $$ with $\mathsf{P}_σ(f)(x,t):= \int_M \mathsf{P}_t(x,y)f(y) \,dσ(y)$ 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.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

Boundedness criterion for integral operators on the fractional Fock-Sobolev spaces

We provide a boundedness criterion for the integral operator $S_φ$ on the fractional Fock-Sobolev space $F^{s,2}(\mathbb C^n)$, $s\geq 0$, where $S_φ$ (introduced by Kehe Zhu) is given by \begin{eqnarray*} S_φF(z):= \int_{\mathbb{C}^n} F(w) e^{z \cdot\bar{w}} φ(z- \bar{w}) dλ(w) \end{eqnarray*} with $φ$ in the Fock space $F^2(\mathbb C^n)$ and $dλ(w): = π^{-n} e^{-|w|^2} dw$ the Gaussian measure on the complex space $\mathbb{C}^{n}$. This extends the recent result in Cao--Li--Shen--Wick--Yan. The main approach is to develop multipliers on the fractional Hermite-Sobolev space $W_H^{s,2}(\mathbb R^n)$.

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preprint2020arXiv

A Boundedness Criterion for Singular Integral Operators of convolution type on the Fock Space

We show that for an entire function $φ$ belonging to the Fock space ${\mathscr F}^2(\mathbb{C}^n)$ on the complex Euclidean space $\mathbb{C}^n$, the integral operator \begin{eqnarray*} S_φF(z)=\int_{\mathbb{C}^n} F(w) e^{z \cdot\bar{w}} φ(z- \bar{w})\,dλ(w), \ \ \ \ \ z\in \mathbb{C}^n, \end{eqnarray*} is bounded on ${\mathscr F}^2(\mathbb{C}^n)$ if and only if there exists a function $m\in L^{\infty}(\mathbb{R}^n)$ such that $$ φ(z)=\int_{\mathbb{R}^n} m(x)e^{-2\left(x-\frac{i}{2} z \right)\cdot \left(x-\frac{i}{2} z \right)} dx, \ \ \ \ \ \ z\in \mathbb{C}^n. $$ Here $dλ(w)= π^{-n}e^{-\left\vert w\right\vert^2}dw$ is the Gaussian measure on $\mathbb C^n$. With this characterization we are able to obtain some fundamental results including the normaility, the algebraic property, spectrum and compactness of this operator $S_φ$. Moreover, we obtain the reducing subspaces of $S_φ$. In particular, in the case $n=1$, we give a complete solution to an open problem proposed by K. Zhu for the Fock space ${\mathscr F}^2(\mathbb{C})$ on the complex plane ${\mathbb C}$ (Integr. Equ. Oper. Theory {\bf 81} (2015), 451--454).

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

A two weight inequality for Calderón-Zygmund operators on spaces of homogeneous type with applications

Let $(X,d,μ)$ be a space of homogeneous type in the sense of Coifman and Weiss, i.e. $d$ is a quasi metric on $X$ and $μ$ is a positive measure satisfying the doubling condition. Suppose that $u$ and $v$ are two locally finite positive Borel measures on $(X,d,μ)$. Subject to the pair of weights satisfying a side condition, we characterize the boundedness of a Calderón--Zygmund operator $T$ from $L^{2}(u)$ to $L^{2}(v)$ in terms of the $A_{2}$ condition and two testing conditions. For every cube $B\subset X$, we have the following testing conditions, with $\mathbf{1}_{B}$ taken as the indicator of $B$ \begin{equation*} \Vert T(u\mathbf{1}_{B})\Vert _{L^{2}(B, v)}\leq \mathcal{T}\Vert 1_{B}\Vert _{L^{2}(u)}, \end{equation*} \begin{equation*} \Vert T^{\ast }(v\mathbf{1}_{B})\Vert _{L^{2}(B, u)}\leq \mathcal{T}\Vert 1_{B}\Vert _{L^{2}(v)}. \end{equation*} The proof uses stopping cubes and corona decompositions originating in work of Nazarov, Treil and Volberg, along with the pivotal side condition.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

FTRANS: Energy-Efficient Acceleration of Transformers using FPGA

In natural language processing (NLP), the &#34;Transformer&#34; architecture was proposed as the first transduction model replying entirely on self-attention mechanisms without using sequence-aligned recurrent neural networks (RNNs) or convolution, and it achieved significant improvements for sequence to sequence tasks. The introduced intensive computation and storage of these pre-trained language representations has impeded their popularity into computation and memory-constrained devices. The field-programmable gate array (FPGA) is widely used to accelerate deep learning algorithms for its high parallelism and low latency. However, the trained models are still too large to accommodate to an FPGA fabric. In this paper, we propose an efficient acceleration framework, Ftrans, for transformer-based large scale language representations. Our framework includes enhanced block-circulant matrix (BCM)-based weight representation to enable model compression on large-scale language representations at the algorithm level with few accuracy degradation, and an acceleration design at the architecture level. Experimental results show that our proposed framework significantly reduces the model size of NLP models by up to 16 times. Our FPGA design achieves 27.07x and 81x improvement in performance and energy efficiency compared to CPU, and up to 8.80x improvement in energy efficiency compared to GPU.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

H2O-Cloud: A Resource and Quality of Service-Aware Task Scheduling Framework for Warehouse-Scale Data Centers -- A Hierarchical Hybrid DRL (Deep Reinforcement Learning) based Approach

Cloud computing has attracted both end-users and Cloud Service Providers (CSPs) in recent years. Improving resource utilization rate (RUtR), such as CPU and memory usages on servers, while maintaining Quality-of-Service (QoS) is one key challenge faced by CSPs with warehouse-scale data centers. Prior works proposed various algorithms to reduce energy cost or to improve RUtR, which either lack the fine-grained task scheduling capabilities, or fail to take a comprehensive system model into consideration. This article presents H2O-Cloud, a Hierarchical and Hybrid Online task scheduling framework for warehouse-scale CSPs, to improve resource usage effectiveness while maintaining QoS. H2O-Cloud is highly scalable and considers comprehensive information such as various workload scenarios, cloud platform configurations, user request information and dynamic pricing model. The hierarchy and hybridity of the framework, combined with its deep reinforcement learning (DRL) engines, enable H2O-Cloud to efficiently start on-the-go scheduling and learning in an unpredictable environment without pre-training. Our experiments confirm the high efficiency of the proposed H2O-Cloud when compared to baseline approaches, in terms of energy and cost while maintaining QoS. Compared with a state-of-the-art DRL-based algorithm, H2O-Cloud achieves up to 201.17% energy cost efficiency improvement, 47.88% energy efficiency improvement and 551.76% reward rate improvement.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Mapping the Galactic disk with the LAMOST and Gaia Red clump sample: I: precise distances, masses, ages and 3D velocities of $\sim$ 140000 red clump stars

We present a sample of $\sim$ 140,000 primary red clump (RC) stars of spectral signal-to-noise ratios higher than 20 from the LAMOST Galactic spectroscopic surveys, selected based on their positions in the metallicity-dependent effective temperature--surface gravity and color--metallicity diagrams, supervised by high-quality $Kepler$ asteroseismology data. The stellar masses and ages of those stars are further determined from the LAMOST spectra, using the Kernel Principal Component Analysis method, trained with thousands of RCs in the LAMOST-$Kepler$ fields with accurate asteroseismic mass measurements. The purity and completeness of our primary RC sample are generally higher than 80 per cent. For the mass and age, a variety of tests show typical uncertainties of 15 and 30 per cent, respectively. Using over ten thousand primary RCs with accurate distance measurements from the parallaxes of Gaia DR2, we re-calibrate the $K_{\rm s}$ absolute magnitudes of primary RCs by, for the first time, considering both the metallicity and age dependencies. With the the new calibration, distances are derived for all the primary RCs, with a typical uncertainty of 5--10 per cent, even better than the values yielded by the Gaia parallax measurements for stars beyond 3--4 kpc. The sample covers a significant volume of the Galactic disk of $4 \leq R \leq 16$ kpc, $|Z| \leq 5$ kpc, and $-20 \leq ϕ\leq 50^{\circ}$. Stellar atmospheric parameters, line-of-sight velocities and elemental abundances derived from the LAMOST spectra and proper motions of Gaia DR2 are also provided for the sample stars. Finally, the selection function of the sample is carefully evaluated in the color-magnitude plane for different sky areas. The sample is publicly available.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Maximal function, Littlewood--Paley theory, Riesz transform and atomic decomposition in the multi-parameter flag setting

In this paper, we develop via real variable methods various characterisations of the Hardy spaces in the multi-parameter flag setting. These characterisations include those via, the non-tangential and radial maximal function, the Littlewood--Paley square function and area integral, Riesz transforms and the atomic decomposition in the multi-parameter flag setting. The novel ingredients in this paper include (1) establishing appropriate discrete Calderón reproducing formulae in the flag setting and a version of the Plancherel--Pólya inequalities for flag quadratic forms; (2) introducing the maximal function and area function via flag Poisson kernels and flag version of harmonic functions; (3) developing an atomic decomposition via the finite speed propagation and area function in terms of flag heat semigroups. As a consequence of these real variable methods, we obtain the full characterisations of the multi-parameter Hardy space with the flag structure.

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preprint2020arXiv

Orbital Stability of smooth solitary waves for the Degasperis-Procesi Equation

The Degasperis-Procesi equation is the integrable Camassa-Holm-type model which is an asymptotic approximation for the unidirectional propagation of shallow water waves. This work establishes the orbital stability of localized smooth solitary waves to the Desgasperis-Procesi (DP) equation on the real line. %extending our previous work on their spectral stability \cite{LLW}. The main difficulty stems from the fact that the translation symmetry for the DP equation gives rise to a conserved quantity equivalent to the $L^2$-norm, which by itself can not bound the higher-order nonlinear terms in the Lagrangian. The remedy is to observe that, given a sufficiently smooth initial condition satisfying a measurable constraint, the $L^\infty$ orbital norm of the perturbation is bounded above by a function of its $L^2$ orbital norm, yielding the orbital stability in the $L^2\cap L^\infty$ space.

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preprint2020arXiv

Quantitative weighted bounds for Calderón commutator with rough kernel

We consider weighted $L^p(w)$ boundedness ($1<p<\infty $ and $w$ a Muckenhoupt $A_p$ weight) of the Calderón commutator $\mathcal C_Ω$ associated with rough homogeneous kernel, under the condition $Ω\in L^q(\mathbb S^{n-1})$ for $q_0<q\leq\infty$ with $q_0$ a fixed constant depending on $w$. Comparing to the previous related known results (assuming $Ω\in L^\infty(\mathbb S^{n-1})$), our result for $Ω\in L^q(\mathbb S^{n-1})$ with $q$ in the range $(q_0,\infty)$ is new. We also obtain a quantitative weighted bound for this $\mathcal C_Ω$ on $L^p(w)$, which is the best known quantitative result for this class of operators.

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preprint2020arXiv

Sharp endpoint $L^p$ estimates for Schrödinger groups

Let $L$ be a non-negative self-adjoint operator acting on $L^2(X)$ where $X$ is a space of homogeneous type with a dimension $n$. Suppose that the heat operator $e^{-tL}$ satisfies the generalized Gaussian $(p_0, p&#39;_0)$-estimates of order $m$ for some $1\leq p_0 < 2$. In this paper we prove {\it sharp} endpoint $L^p$-Sobolev bound for the Schrödinger group $e^{itL}$, that is for every $p\in (p_0, p&#39;_0)$ there exists a constant $C=C(n,p)>0$ independent of $t$ such that \begin{eqnarray*} \left\| (I+L)^{-{s}}e^{itL} f\right\|_{p} \leq C(1+|t|)^{s}\|f\|_{p}, \ \ \ t\in{\mathbb R}, \ \ \ s\geq n\big|{1\over 2}-{1\over p}\big|. \end{eqnarray*} As a consequence, the above estimate holds for all $1<p<\infty$ when the heat kernel of $L$ satisfies a Gaussian upper bound. This extends classical results due to Feffermann and Stein, and Miyachi for the Laplacian on the Euclidean spaces ${\mathbb R}^n$. We also give an application to obtain an endpoint estimate for $L^p$-boundedness of the Riesz means of the solutions of the Schrödinger equations.

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preprint2020arXiv

Zygmund type and flag type maximal functions, and sparse operators

We prove that the maximal functions associated with a Zygmund dilation dyadic structure in three-dimensional Euclidean space, and with the flag dyadic structure in two-dimensional Euclidean space, cannot be bounded by multiparameter sparse operators associated with the corresponding dyadic grid. We also obtain supplementary results about the absence of sparse domination for the strong dyadic maximal function.

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preprint2019arXiv

Solving Phase Retrieval via Graph Projection Splitting

Phase retrieval with prior information can be cast as a nonsmooth and nonconvex optimization problem. We solve the problem by graph projection splitting (GPS), where the two proximity subproblems and the graph projection step can be solved efficiently. With slight modification, we also propose a robust graph projection splitting (RGPS) method to stabilize the iteration for noisy measurements. Contrary to intuition, RGPS outperforms GPS with fewer iterations to locate a satisfying solution even for noiseless case. Based on the connection between GPS and Douglas-Rachford iteration, under mild conditions on the sampling vectors, we analyze the fixed point sets and provide the local convergence of GPS and RGPS applied to noiseless phase retrieval without prior information. For noisy case, we provide the error bound of the reconstruction. Compared to other existing methods, thanks for the splitting approach, GPS and RGPS can efficiently solve phase retrieval with prior information regularization for general sampling vectors which are not necessarily isometric. For Gaussian phase retrieval, compared to existing gradient flow approaches, numerical results show that GPS and RGPS are much less sensitive to the initialization. Thus they markedly improve the phase transition in noiseless case and reconstruction in the presence of noise respectively. GPS shows sharpest phase transition among existing methods including RGPS, while it needs more iterations than RGPS when the number of measurement is large enough. RGPS outperforms GPS in terms of stability for noisy measurements. When applying RGPS to more general non-Gaussian measurements with prior information, such as support, sparsity and TV minimization, RGPS either outperforms state-of-the-art solvers or can be combined with state-of-the-art solvers to improve their reconstruction quality.

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preprint2018arXiv

A Complete Real-Variable Theory of Hardy Spaces on Spaces of Homogeneous Type

Let $(X,d,μ)$ be a space of homogeneous type, with the upper dimension $ω$, in the sense of R. R. Coifman and G. Weiss. Assume that $η$ is the smoothness index of the wavelets on $X$ constructed by P. Auscher and T. Hytönen. In this article, when $p\in(ω/(ω+η),1]$, for the atomic Hardy spaces $H_{\mathrm{cw}}^p(X)$ introduced by Coifman and Weiss, the authors establish their various real-variable characterizations, respectively, in terms of the grand maximal function, the radial maximal function, the non-tangential maximal functions, the various Littlewood-Paley functions and wavelet functions. This completely answers the question of R. R. Coifman and G. Weiss by showing that no any additional (geometrical) condition is necessary to guarantee the radial maximal function characterization of $H_{\mathrm{cw}}^1(X)$ and even of $H_{\mathrm{cw}}^p(X)$ with $p$ as above. As applications, the authors obtain the finite atomic characterizations of $H^p_{\mathrm{cw}}(X)$, which further induce some criteria for the boundedness of sublinear operators on $H^p_{\mathrm{cw}}(X)$. Compared with the known results, the novelty of this article is that $μ$ is not assumed to satisfy the reverse doubling condition and $d$ is only a quasi-metric, moreover, the range $p\in(ω/(ω+η),1]$ is natural and optimal.

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