The Soft X-ray Imager(SXI)on the SMILE Mission

The Soft X-ray Imager(SXI)is part of the scientific payload of the Solar wind Magnetosphere Ionosphere Link Explorer(SMILE)mission.SMILE is a joint science mission between the European Space Agency(ESA)and the Chinese Academy of Sciences(CAS)and is due for launch in 2025.SXI is a compact X-ray telescope with a wide field-of-view(FOV)capable of encompassing large portions of Earth’s magnetosphere from the vantage point of the SMILE orbit.SXI is sensitive to the soft X-rays produced by the Solar Wind Charge eXchange(SWCX)process produced when heavy ions of solar wind origin interact with neutral particles in Earth’s exosphere.SWCX provides a mechanism for boundary detection within the magnetosphere,such as the position of Earth’s magnetopause,because the solar wind heavy ions have a very low density in regions of closed magnetic field lines.The sensitivity of the SXI is such that it can potentially track movements of the magnetopause on timescales of a few minutes and the orbit of SMILE will enable such movements to be tracked for segments lasting many hours.SXI is led by the University of Leicester in the United Kingdom(UK)with collaborating organisations on hardware,software and science support within the UK,Europe,China and the United States.

Finding the magnetopause location using soft X-ray observations and a statistical inverse method

Variability in the location and shape of the dayside magnetopause is attributed to magnetic reconnection,a fundamental process that enables the transfer of mass,energy,and momentum from the solar wind into the magnetosphere.The spatial and temporal properties of the magnetopause,under varying solar and magnetospheric conditions,remain largely unknown because empirical studies using in-situ observations are challenging to interpret.Global wide field-of-view(FOV)imaging is the only means to simultaneously observe the spatial distribution of the plasma properties over the vast dayside magnetospheric region and,subsequently,quantify the energy transport from the interplanetary medium into the terrestrial magnetosphere.Two upcoming missions,ESA/CAS SMILE and NASA’s LEXI will provide wide-field imagery of the dayside magnetosheath in soft X-rays,an emission generated by charge exchange interactions between high charge-state heavy ions of solar wind origin and exospheric neutral atoms.High-cadence two-dimensional observations of the magnetosheath will allow the estimation of dynamic properties of its inner boundary,the magnetopause,and enable studies of its response to changes in the solar wind dynamic pressure and interplanetary magnetic field orientation.This work introduces a statistically-based estimation approach based on inverse theory to estimate the spatial distribution of magnetosheath soft X-ray emissivities and,with this,identify the location of the magnetopause over the Sun−Earth line.To do so,we simulate the magnetosheath structure using the MHD-based OpenGGCM model and generate synthetic soft X-ray images using LEXI’s orbit and attitude information.Our results show that 3-D estimations using the described statistically-based technique are robust against Poisson-distributed shot noise inherent to soft X-ray images.Also,our proposed methodology shows that the accuracy of both three-dimensional(3-D)estimation and the magnetopause standoff distance calculation highly depends on the observational point.

Estimating the subsolar magnetopause position from soft X-ray images using a low-pass image filter

The Lunar Environment heliospheric X-ray Imager(LEXI)and Solar wind Magnetosphere Ionosphere Link Explorer(SMILE)missions will image the Earth’s dayside magneto pause and cusps in soft X-rays after their respective launches in the near future,to specify glo bal magnetic reconnection modes for varying solar wind conditions.To suppo rt the success of these scientific missions,it is critical to develop techniques that extract the magnetopause locations from the observed soft X-ray images.In this research,we introduce a new geometric equation that calculates the subsolar magnetopause position(RS)from a satellite position,the look direction of the instrument,and the angle at which the X-ray emission is maximized.Two assumptions are used in this method:(1)The look direction where soft X-ray emissions are maximized lies tangent to the magnetopause,and(2)the magnetopause surface near the subsolar point is almost spherical and thus RSis nea rly equal to the radius of the magneto pause curvature.We create synthetic soft X-ray images by using the Open Geospace General Circulation Model(OpenGGCM)global magnetohydrodynamic model,the galactic background,the instrument point spread function,and Poisson noise.We then apply the fast Fourier transform and Gaussian low-pass filte rs to the synthetic images to re move noise and obtain accurate look angles for the soft X-ray pea ks.From the filte red images,we calculate RS and its accuracy for different LEXI locations,look directions,and solar wind densities by using the OpenGGCM subsolar magnetopause location as ground truth.Our method estimates RS with an accuracy of<0.3 RE when the solar wind density exceeds>10 cm-3.The accuracy improves for greater solar wind densities and during southward interplanetary magnetic fields.The method ca ptures the magnetopause motion during southwa rd interplaneta ry magnetic field turnings.Consequently,the technique will enable quantitative analysis of the magnetopause motion and help reveal the dayside reconnection modes for dynamic solar wind conditions.This technique will suppo rt the LEXI and SMILE missions in achieving their scientific o bjectives.

SMILE soft X-ray Imager flight model CCD370 pre-flight device characterisation

Throughout the SMILE mission the satellite will be bombarded by radiation which gradually damages the focal plane devices and degrades their performance.In order to understand the changes of the CCD370s within the soft X-ray Imager,an initial characterisation of the devices has been carried out to give a baseline performance level.Three CCDs have been characterised,the two flight devices and the flight spa re.This has been carried out at the Open University in a bespo ke cleanroom measure ment facility.The results show that there is a cluster of bright pixels in the flight spa re which increases in size with tempe rature.However at the nominal ope rating tempe rature(-120℃) it is within the procure ment specifications.Overall,the devices meet the specifications when ope rating at -120℃ in 6 × 6 binned frame transfer science mode.The se rial charge transfer inefficiency degrades with temperature in full frame mode.However any charge losses are recovered when binning/frame transfer is implemented.

Hybrid-Vlasov simulation of soft X-ray emissions at the Earth’s dayside magnetospheric boundaries

Solar wind charge exchange produces emissions in the soft X-ray energy range which can enable the study of near-Earth space regions such as the magnetopause,the magnetosheath and the polar cusps by remote sensing techniques.The Solar wind Magnetosphere Ionosphere Link Explorer(SMILE)and Lunar Environment heliospheric X-ray Imager(LEXI)missions aim to obtain soft Xray images of near-Earth space thanks to their Soft X-ray Imager(SXI)instruments.While earlier modeling works have already simulated soft X-ray images as might be obtained by SMILE SXI during its mission,the numerical models used so far are all based on the magnetohydrodynamics description of the space plasma.To investigate the possible signatures of ion-kinetic-scale processes in soft Xray images,we use for the first time a global hybrid-Vlasov simulation of the geospace from the Vlasiator model.The simulation is driven by fast and tenuous solar wind conditions and purely southward interplanetary magnetic field.We first produce global X-ray images of the dayside near-Earth space by placing a virtual imaging satellite at two different locations,providing meridional and equatorial views.We then analyze regional features present in the images and show that they correspond to signatures in soft X-ray emissions of mirrormode wave structures in the magnetosheath and flux transfer events(FTEs)at the magnetopause.Our results suggest that,although the time scales associated with the motion of those transient phenomena will likely be significantly smaller than the integration time of the SMILE and LEXI imagers,mirror-mode structures and FTEs can cumulatively produce detectable signatures in the soft X-ray images.For instance,a local increase by 30%in the proton density at the dayside magnetopause resulting from the transit of multiple FTEs leads to a 12%enhancement in the line-of-sight-and time-integrated soft X-ray emissivity originating from this region.Likewise,a proton density increase by 14%in the magnetosheath associated with mirror-mode structures can result in an enhancement in the soft X-ray signal by 4%.These are likely conservative estimates,given that the solar wind conditions used in the Vlasiator run can be expected to generate weaker soft X-ray emissions than the more common denser solar wind.These results will contribute to the preparatory work for the SMILE and LEXI missions by providing the community with quantitative estimates of the effects of small-scale,transient phenomena occurring on the dayside.

Path Planning and Tracking Control for Parking via Soft Actor-Critic Under Non-Ideal Scenarios

Parking in a small parking lot within limited space poses a difficult task. It often leads to deviations between the final parking posture and the target posture. These deviations can lead to partial occupancy of adjacent parking lots, which poses a safety threat to vehicles parked in these parking lots. However, previous studies have not addressed this issue. In this paper, we aim to evaluate the impact of parking deviation of existing vehicles next to the target parking lot(PDEVNTPL) on the automatic ego vehicle(AEV) parking, in terms of safety, comfort, accuracy, and efficiency of parking. A segmented parking training framework(SPTF) based on soft actor-critic(SAC) is proposed to improve parking performance. In the proposed method, the SAC algorithm incorporates strategy entropy into the objective function, to enable the AEV to learn parking strategies based on a more comprehensive understanding of the environment. Additionally, the SPTF simplifies complex parking tasks to maintain the high performance of deep reinforcement learning(DRL). The experimental results reveal that the PDEVNTPL has a detrimental influence on the AEV parking in terms of safety, accuracy, and comfort, leading to reductions of more than 27%, 54%, and 26%respectively. However, the SAC-based SPTF effectively mitigates this impact, resulting in a considerable increase in the parking success rate from 71% to 93%. Furthermore, the heading angle deviation is significantly reduced from 2.25 degrees to 0.43degrees.

Two methods for separating the magnetospheric solar wind charge exchange soft X-ray emission from the diffuse X-ray background

Solar wind charge exchange(SWCX)is the process of solar wind high-valence ions exchanging charges with neutral components and generating soft X-rays.Recently,detecting the SWCX emission from the magnetosphere is proposed as a new technique to study the magnetosphere using panoramic soft X-ray imaging.To better prepare for the data analysis of upcoming magnetospheric soft X-ray imaging missions,this paper compares the magnetospheric SWCX emission obtained by two methods in an XMM-Newton observation,during which the solar wind changed dramatically.The two methods differ in the data used to fit the diffuse X-ray background(DXB)parameters in spectral analysis.The method adding data from the ROSAT All-Sky Survey(RASS)is called the RASS method.The method using the quiet observation data is called the Quiet method,where quiet observations usually refer to observations made by the same satellite with the same target but under weaker solar wind conditions.Results show that the spectral compositions of magnetospheric SWCX emission obtained by the two methods are very similar,and the changes in intensity over time are highly consistent,although the intensity obtained by the RASS method is about 2.68±0.56 keV cm^(-2)s^(-1)sr^(-1)higher than that obtained by the Quiet method.Since the DXB intensity obtained by the RASS method is about 2.84±0.74 keV cm^(-2)s^(-1)sr^(-1)lower than that obtained by the Quiet method,and the linear correlation coefficient between the difference of SWCX and DXB obtained by the two methods in diffe rent energy band is close to-1,the diffe rences in magnetospheric SWCX can be fully attributed to the diffe rences in the fitted DXB.The difference between the two methods is most significant when the energy is less than 0.7 keV,which is also the main energy band of SWCX emission.In addition,the difference between the two methods is not related to the SWCX intensity and,to some extent,to solar wind conditions,because SWCX intensity typically va ries with the solar wind.In summary,both methods are robust and reliable,and should be considered based on the best available options.

Simulation of the SMILE Soft X-ray Imager response to a southward interplanetary magnetic field turning

The Solar wind Magnetosphere Ionosphere Link Explorer(SMILE)Soft X-ray Imager(SXI)will shine a spotlight on magnetopause dynamics during magnetic reconnection.We simulate an event with a southward interplanetary magnetic field turning and produce SXI count maps with a 5-minute integration time.By making assumptions about the magnetopause shape,we find the magnetopause standoff distance from the count maps and compare it with the one obtained directly from the magnetohydrodynamic(MHD)simulation.The root mean square deviations between the reconstructed and MHD standoff distances do not exceed 0.2 RE(Earth radius)and the maximal difference equals 0.24 RE during the 25-minute interval around the southward turning.

Direct 4D printing of functionally graded hydrogel networks for biodegradable,untethered,and multimorphic soft robots

Recent advances in functionally graded additive manufacturing(FGAM)technology have enabled the seamless hybridization of multiple functionalities in a single structure.Soft robotics can become one of the largest beneficiaries of these advances,through the design of a facile four-dimensional(4D)FGAM process that can grant an intelligent stimuli-responsive mechanical functionality to the printed objects.Herein,we present a simple binder jetting approach for the 4D printing of functionally graded porous multi-materials(FGMM)by introducing rationally designed graded multiphase feeder beds.Compositionally graded cross-linking agents gradually form stable porous network structures within aqueous polymer particles,enabling programmable hygroscopic deformation without complex mechanical designs.Furthermore,a systematic bed design incorporating additional functional agents enables a multi-stimuli-responsive and untethered soft robot with stark stimulus selectivity.The biodegradability of the proposed 4D-printed soft robot further ensures the sustainability of our approach,with immediate degradation rates of 96.6%within 72 h.The proposed 4D printing concept for FGMMs can create new opportunities for intelligent and sustainable additive manufacturing in soft robotics.

Experimental Study on Vortex-Induced Vibration of Rough Risers Coupling with Interference Effect in Tandem Arrangement

In order to study the response law of vortex-induced vibration(VIV)of marine risers under the combined action of roughness and interference effects,and to reveal the coupling mechanism of roughness and interference effects on the riser,a VIV experiment of rough risers in tandem arrangement was conducted in a wave−current combined flume.The experiment characterized the risers’roughness by arranging different specifications of attachments on the surface of the risers.Three rough risers with different roughness and smooth risers were arranged in tandem arrangement,with the rough risers arranged downstream.The experimental results indicate that the suppression of the attachments on the downstream risers’vibration are more significant both in the CF and IL directions as the reduced velocity increases.For the downstream riser,the amplitude response of rough riser is more significantly weakened compared with the smooth one at high reduced velocity.For the upstream risers,changes in the roughness and spacing ratio have an impact on their‘lock-in’region.When the roughness of downstream risers is relatively large(0.1300)and the spacing between risers is small(S/D=4.0),the reduced velocity range of‘lock-in’region in the CF direction of upstream risers is obviously expanded,and the displacement in the‘lock-in’region is severer.

Shear behavior and off-fault damage of saw-cut smooth and tension-induced rough joints in granite

The damage of rock joints or fractures upon shear includes the surface damage occurring at the contact asperities and the damage beneath the shear surface within the host rock.The latter is commonly known as off-fault damage and has been much less investigated than the surface damage.The main contribution of this study is to compare the results of direct shear tests conducted on saw-cut planar joints and tension-induced rough granite joints under normal stresses ranging from 1 MPa to 50 MPa.The shear-induced off-fault damages are quantified and compared with the optical microscope observation.Our results clearly show that the planar joints slip stably under all the normal stresses except under 50 MPa,where some local fractures and regular stick-slip occur towards the end of the test.Both post-peak stress drop and stick-slip occur for all the rough joints.The residual shear strength envelopes for the rough joints and the peak shear strength envelope for the planar joints almost overlap.The root mean square(RMS)of asperity height for the rough joints decreases while it increases for the planar joint after shear,and a larger normal stress usually leads to a more significant decrease or increase in RMS.Besides,the extent of off-fault damage(or damage zone)increases with normal stress for both planar and rough joints,and it is restricted to a very thin layer with limited micro-cracks beneath the planar joint surface.In comparison,the thickness of the damage zone for the rough joints is about an order of magnitude larger than that of the planar joints,and the coalesced micro-cracks are generally inclined to the shear direction with acute angles.The findings obtained in this study contribute to a better understanding on the frictional behavior and damage characteristics of rock joints or fractures with different roughness.

Experimental Study on Vortex-Induced Vibration of Rough Risers with Coupling Interference Effect Under Side-by-Side Arrangement

A vortex-induced vibration(VIV)experiment of rough risers with coupling interference effect under a side-by-side arrangement was carried out in a wave-current combined flume.The roughness of the riser was characterized by arranging different specifications of surface attachments on the surface of the riser.Rough risers with three different roughnesses were arranged side by side with smooth risers to explore the VIV response of the riser under the combined action of roughness and interference effect,and to reveal the coupling mechanism between roughness and interference effect.The experimental results show that,compared with that of a smooth riser,the VIV of a rough riser under the coupling interference effect has a wider"lock-in"region,and the displacement decreases more significantly at a high reduced velocity,which is more likely to excite higher-order modes and frequency responses.In addition,the displacement response and frequency response of the smooth riser are not significantly affected by wake interference from the rough riser,which is caused by the decrease of the wake region due to the delay of the boundary layer separation point of the rough riser.

Estimating the yield stress of soft materials via laser-induced breakdown spectroscopy

Taking three typical soft samples prepared respectively by loose packings of 77-,95-,and 109-μm copper grains as examples,we perform an experiment to investigate the energy-dependent laser-induced breakdown spectroscopy(LIBS)of soft materials.We discovered a reversal phenomenon in the trend of energy dependence of plasma emission intensity:increasing initially and then decreasing separated by a well-defined critical energy.The trend reversal is attributed to the laser-induced recoil pressure at the critical energy just matching the sample's yield strength.As a result,a one-to-one correspondence can be well established between the samples'yield stress and the critical energy that is easily obtainable from LIBS measurements.This allows us to propose an innovative method for estimating the yield stress of soft materials via LIBS with attractive advantages including in-situ remote detection,real-time data collection,and minimal destructive to sample.

MXenes for Bioinspired Soft Actuators:Advancements in Angle-Independent Structural Colors and Beyond

Soft actuators have garnered substantial attention in current years in view of their potential appliances in diverse domains like robotics,biomedical devices,and biomimetic systems.These actuators mimic the natural movements of living organisms,aiming to attain enhanced flexibility,adaptability,and versatility.On the other hand,angle-independent structural color has been achieved through innovative design strategies and engineering approaches.By carefully controlling the size,shape,and arrangement of nanostructures,researchers have been able to create materials exhibiting consistent colors regardless of the viewing angle.One promising class of materials that holds great potential for bioinspired soft actuators is MXenes in view of their exceptional mechanical,electrical,and optical properties.The integration of MXenes for bioinspired soft actuators with angle-independent structural color offers exciting possibilities.Overcoming material compatibility issues,improving color reproducibility,scalability,durability,power supply efficiency,and cost-effectiveness will play vital roles in advancing these technologies.This perspective appraises the development of bioinspired MXene-centered soft actuators with angleindependent structural color in soft robotics.

Complex Decision Modeling Framework with Fairly Operators and Quaternion Numbers under Intuitionistic Fuzzy Rough Context

The main goal of informal computing is to overcome the limitations of hypersensitivity to defects and uncertainty while maintaining a balance between high accuracy,accessibility,and cost-effectiveness.This paper investigates the potential applications of intuitionistic fuzzy sets(IFS)with rough sets in the context of sparse data.When it comes to capture uncertain information emanating fromboth upper and lower approximations,these intuitionistic fuzzy rough numbers(IFRNs)are superior to intuitionistic fuzzy sets and pythagorean fuzzy sets,respectively.We use rough sets in conjunction with IFSs to develop several fairly aggregation operators and analyze their underlying properties.We present numerous impartial laws that incorporate the idea of proportionate dispersion in order to ensure that the membership and non-membership activities of IFRNs are treated equally within these principles.These operations lead to the development of the intuitionistic fuzzy rough weighted fairly aggregation operator(IFRWFA)and intuitionistic fuzzy rough ordered weighted fairly aggregation operator(IFRFOWA).These operators successfully adjust to membership and non-membership categories with fairness and subtlety.We highlight the unique qualities of these suggested aggregation operators and investigate their use in the multiattribute decision-making field.We use the intuitionistic fuzzy rough environment’s architecture to create a novel strategy in situation involving several decision-makers and non-weighted data.Additionally,we developed a novel technique by combining the IFSs with quaternion numbers.We establish a unique connection between alternatives and qualities by using intuitionistic fuzzy quaternion numbers(IFQNs).With the help of this framework,we can simulate uncertainty in real-world situations and address a number of decision-making problems.Using the examples we have released,we offer a sophisticated and systematically constructed illustrative scenario that is intricately woven with the complexity ofmedical evaluation in order to thoroughly assess the relevance and efficacy of the suggested methodology.

Mechanical properties and acoustic emission characteristics of soft rock with different water contents under dynamic disturbance

Uniaxial compression tests and cyclic loading acoustic emission tests were conducted on 20%,40%,60%,80%,dry and saturated muddy sandstone by using a creep impact loading system to investigate the mechanical properties and acoustic emission characteristics of soft rocks with different water contents under dynamic disturbance.The mechanical properties and acoustic emission characteristics of muddy sandstones at different water contents were analysed.Results of experimental studies show that water is a key factor in the mechanical properties of rocks,softening them,increasing their porosity,reducing their brittleness and increasing their plasticity.Under uniaxial compression,the macroscopic damage characteristics of the muddy sandstone change from mono-bevel shear damage and‘X’type conjugate bevel shear damage to a roadway bottom-drum type damage as the water content increases.Dynamic perturbation has a strengthening effect on the mechanical properties of samples with 60%and less water content,and a weakening effect on samples with 80%and more water content,but the weakening effect is not obvious.Macroscopic damage characteristics of dry samples remain unchanged,water samples from shear damage and tensile–shear composite damage gradually transformed into cleavage damage,until saturation transformation monoclinic shear damage.The evolution of acoustic emission energy and event number is mainly divided into four stages:loading stage(I),dynamic loading stage(II),yield failure stage(III),and post-peak stage(IV),the acoustic emission characteristics of the stages were different for different water contents.The characteristic value of acoustic emission key point frequency gradually decreases,and the damage degree of the specimen increases,corresponding to low water content—high main frequency—low damage and high water content—low main frequency—high damage.

Mechanism of high-preload support based on the NPR anchor cable in layered soft rock tunnels

The control of large deformation problems in layered soft rock tunnels needs to solve urgently.The roof problem is particularly severe among the deformation issues in tunnels.This study first analyzes the asymmetric deformation modes in layered soft rock tunnels with large deformations.Subsequently,we construct a mechanical model under ideal conditions for controlling the roof of layered soft rock tunnels through high preload with the support of NPR anchor cables.The prominent roles of long and short NPR anchor cables in the support system are also analyzed.The results indicate the significance of high preload in controlling the roof of layered soft rock tunnels.The short NPR anchor cables effectively improve the integrity of the stratified soft rock layers,while the long NPR anchor cables effectively mobilize the self-bearing capacity of deep-stable rock layers.Finally,the high-preload support method with NPR anchor cables is validated to have a good effect on controlling large deformations in layered soft rock tunnels through field monitoring data.

Roughness characterization and shearing dislocation failure for rock-backfill interface

Shearing dislocation is a common failure type for rock–backfill interfaces because of backfill sedimentation and rock strata movement in backfill mining goaf.This paper designed a test method for rock–backfill shearing dislocation.Using digital image techno-logy and three-dimensional(3D)laser morphology scanning techniques,a set of 3D models with rough joint surfaces was established.Further,the mechanical behavior of rock–backfill shearing dislocation was investigated using a direct shear test.The effects of interface roughness on the shear–displacement curve and failure characteristics of rock–backfill specimens were considered.The 3D fractal dimen-sion,profile line joint roughness coefficient(JRC),profile line two-dimensional fractal dimension,and the surface curvature of the frac-tures were obtained.The correlation characterization of surface roughness was then analyzed,and the shear strength could be measured and calculated using JRC.The results showed the following:there were three failure threshold value points in rock–backfill shearing dis-location:30%–50%displacement before the peak,70%–90%displacement before the peak,and 100%displacement before the peak to post-peak,which could be a sign for rock–backfill shearing dislocation failure.The surface JRC could be used to judge the rock–backfill shearing dislocation failure,including post-peak sliding,uniform variations,and gradient change,corresponding to rock–backfill disloca-tion failure on the field site.The research reveals the damage mechanism for rock–backfill complexes based on the free joint surface,fills the gap of existing shearing theoretical systems for isomerism complexes,and provides a theoretical basis for the prevention and control of possible disasters in backfill mining.

Advanced Design of Soft Robots with Artificial Intelligence

In recent years,breakthrough has been made in the field of artificial intelligence(AI),which has also revolutionized the industry of robotics.Soft robots featured with high-level safety,less weight,lower power consumption have always been one of the research hotspots.Recently,multifunctional sensors for perception of soft robotics have been rapidly developed,while more algorithms and models of machine learning with high accuracy have been optimized and proposed.Designs of soft robots with AI have also been advanced ranging from multimodal sensing,human-machine interaction to effective actuation in robotic systems.Nonethe-less,comprehensive reviews concerning the new developments and strategies for the ingenious design of the soft robotic systems equipped with AI are rare.Here,the new development is systematically reviewed in the field of soft robots with AI.First,background and mechanisms of soft robotic systems are briefed,after which development focused on how to endow the soft robots with AI,including the aspects of feeling,thought and reaction,is illustrated.Next,applications of soft robots with AI are systematically summarized and discussed together with advanced strategies proposed for performance enhancement.Design thoughts for future intelligent soft robotics are pointed out.Finally,some perspectives are put forward.

Maximal operators of pseudo-differential operators with rough symbols

Consider a pseudo-differential operator T_(a)f(x)=∫_(R^(n))e^(ix,ζ)a(x,ζ)f(ζ)dζwhere the symbol a is in the rough Hormander class L^(∞)S_(ρ)^(m)with m∈R andρ∈[0,1].In this note,when 1≤p≤2,if n(ρ-1)/p and a∈L^(∞)S_(ρ)^(m),then for any f∈S(R^(n))and x∈R^(n),we prove that M(T_(a)f)(x)≤C(M(|f|^(p))(x))^(1/p) where M is the Hardy-Littlewood maximal operator.Our theorem improves the known results and the bound on m is sharp,in the sense that n(ρ-1)/p can not be replaced by a larger constant.