We give sharp estimates for the fractional maximal function in terms of Hausdorff capacity. At the same time we identify the real interpolation spaces between$L$_{1}and the Morrey space $\mathcal{L}^{1,\lambda}$ . The result can be viewed as an analogue of the Hardy–Littlewood maximal theorem for the fractional maximal function.

Different from conventional spaceborne or airborne synthetic aperture radar (SAR) with optimal aperture length, an imaging radar with highly suboptimal aperture length acquires the data in short bursts by a geometry spreading over a large range. A polarlike or pseudopolar format grid is introduced to sample data close to the resolution, which presents the design of a separable kernel for efficient FFT implementation. The proposed imaging algorithm formulates the reflectivity image of the target scene as an interpolation-free double image series expansion with two usual approximation-induced phase error terms being taken into account, whereby more generalized application scenarios with high frequency, large bandwidth or larger aperture length for imaging a target scene located within either the far-field or the near-field of the radar aperture are processable with high accuracy. In addition, convergence

Speech signals are often produced or received in the presence of noise, which is known to degrade the performance of a speech recognition system. In this paper, a perception-and PDE-based nonlinear transformation was developed to process spoken words in noisy environment. Our goal is to distinguish essential speech features and suppress noise so that the processed words are better recognized by a computer software. The nonlinear transformation was made on the spectrogram (short-term Fourier spectra) of speech signals, which reveals the signal energy distribution in time and frequency. The transformation reduces noise through time adaptation (reducing temporally slowly varying portions of spectra) and enhances spectral peaks (formants) by evolving a focusing quadratic fourth-order PDE. Short-term spectra of speech signals were initially divided into three (low, mid and high) frequency bands based

We prove that all maximal-tb positive Legendrian torus links (n,m) in the standard contact 3-sphere, except for (2,m), (3,3), (3,4) and (3,5), admit infinitely many Lagrangian fillings in the standard symplectic 4-ball. This is proven by constructing infinite order Lagrangian concordances that induce faithful actions of the modular group PSL(2,Z) and the mapping class group M0,4 into the coordinate rings of algebraic varieties associated to Legendrian links. In particular, our results imply that there exist Lagrangian concordance monoids with subgroups of exponential-growth, and yield Stein surfaces homotopic to a 2-sphere with infinitely many distinct exact Lagrangian surfaces of higher-genus. We also show that there exist infinitely many satellite and hyperbolic knots with Legendrian representatives admitting infinitely many exact Lagrangian fillings.

This is the second in a series of papers on a new equivariant cohomology that takes values in a vertex algebra. In an earlier paper, the first two authors gave a construction of the cohomology functor on the category of O(sg) algebras. The new cohomology theory can be viewed as a kind of "chiralization'' of the classical equivariant cohomology, the latter being defined on the category of $G^*$ algebras a la H. Cartan. In this paper, we further develop the chiral theory by first extending it to allow a much larger class of algebras which we call sg[t] algebras. In the geometrical setting, our principal example of an O(sg) algebra is the chiral de Rham complex Q(M) of a G manifold M. There is an interesting subalgebra of Q(M) which does not admit a full O(sg) algebra structure but retains the structure of an sg[t] algebra, enough for us to define its chiral equivariant cohomology. The latter then turns out to have many surprising features that allows us to delineate a number of interesting geometric aspects of the G manifold M, sometimes in ways that are quite different from the classical theory.