Nanoscience is a booming field incorporating some of the most fundamental questions concerning structure, function, and applications. The cutting-edge research in nanoscience requires access to advanced techniques and instrumentation capable of approaching these unanswered questions. Over the past few decades, atomic force microscopy (AFM) has been developed as a powerful platform, which enables in situ characterization of topological structures, local physical properties, and even manipulating samples at nanometer scale. Currently, an imaging mode called PeakForce Tapping (PFT) has attracted more and more attention due to its advantages of nondestructive characterization, high-resolution imaging, and concurrent quantitative property mapping. In this review, the origin, principle, and advantages of PFT on nanoscience are introduced in detail. Three typical applications of this technique, including high-resolution imaging of soft samples in liquid environment, quantitative nanomechanical property mapping, and electrical/electrochemical property measurement will be reviewed comprehensively. The future trends of PFT technique development will be discussed as well.

Abstract. We investigate the weighted bounds for multilinear maximal functions and Calderón-Zygmund operators from, where 1 < p1, · · · , pm < ∞ with 1/p1+· · ·+1/pm = 1/p and w is a multiple A P weight. We prove the sharp bound for the multilinear maximal function for all such p1, . . . , pm and prove the sharp bound for m-linear Calderón-Zymund operators when p ≥ 1.

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We study the asymptotic properties of Gabor frame operators defined by the Riemannian sums of inverse windowed Fourier transforms. When the analysis and the synthesis window functions are the same, we give necessary and sufficient conditions for the Riemannian sums to be convergent as the sampling density tends to infinity. Moreover, we show that Gabor frame operators converge to the identity operator in operator norm whenever they are generated with locally Riemann integrable window functions in the Wiener space.

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Abstract. We prove a Hörmander type multiplier theorem for multilinear Fourier multipiers with multiple weights. We also give weighted estimates for their commutators with vector BM O functions.