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Many geometric and analytic properties of sets hinge on the properties of elliptic measure, notoriously missing for sets of higher co-dimension. The aim of this manuscript is to develop a version of elliptic theory, associated to a linear PDE, which ultimately yields a notion analogous to that of the harmonic measure, for sets of codimension higher than 1. To this end, we turn to degenerate elliptic equations. Let In another article to appear, we will prove that when

Many smoothness spaces in harmonic analysis are decomposition spaces. In this paper we ask: Given two such spaces, is there an embedding between the two? A decomposition space We establish readily verifiable criteria which ensure the existence of a continuous inclusion (“an embedding”) In a nutshell, in order to apply the embedding results presented in this article, no knowledge of Fourier analysis is required; instead, one only has to study the geometric properties of the involved coverings, so that one can decide the finiteness of certain sequence space norms defined in terms of the coverings. These sufficient criteria are quite sharp: For almost arbitrary coverings and certain ranges of We also prove a The resulting embedding theory is illustrated by applications to

This volume is dedicated to the memory of Björn Jawerth. It contains original research contributions and surveys in several of the areas of mathematics to which Björn made important contributions. Those areas include harmonic analysis, image processing, and functional analysis, which are of course interrelated in many significant and productive ways.Among the contributors are some of the world's leading experts in these areas. With its combination of research papers and surveys, this book may become an important reference and research tool.This book should be of interest to advanced graduate students and professional researchers in the areas of functional analysis, harmonic analysis, image processing, and approximation theory. It combines articles presenting new research with insightful surveys written by foremost experts.

The goal of this article is to define, explore and analyze two new families of meromorphically harmonic functions by applying the concept of a certain generalized convolution q -operator along with the idea of convolution. We investigate convolution properties and sufficiency criteria for these families of meromorphically harmonic functions. Some of the interesting consequences of our investigation are also included.

Accurate corrections for ionospheric total electron content (TEC) and early warning information are crucial for global navigation satellite system (GNSS) applications under the influence of space weather. In this study, we propose to use a new machine learning model—the Prophet model, to predict the global ionospheric TEC by establishing a short-term ionospheric prediction model. We use 15th-order spherical harmonic coefficients provided by the Center for Orbit Determination in Europe (CODE) as the training data set. Historical spherical harmonic coefficient data from 7 days, 15 days, and 30 days are used as the training set to model and predict 256 spherical harmonic coefficients. We use the predicted coefficients to generate a global ionospheric TEC forecast map based on the spherical harmonic function model and select a year with low solar activity (63.4 < F10.7 < 81.8) and a year with the high solar activity (79.5 < F10.7 < 255.0) to carry out a sliding 2-day forecast experiment. Meanwhile, we verify the model performance by comparing the forecasting results with the CODE forecast product (COPG) and final product (CODG). The results show that we obtain the best predictions by using 15 days of historical data as the training set. Compared with the results of CODE’S 1-Day (C1PG) and CODE’S 2-Day (C2PG). The number of days with RMSE better than COPG on the first and second day of the low-solar-activity year is 151 and 158 days, respectively. This statistic for high-solar-activity year is 183 days and 135 days.

Suppose α , β ∈ R ∖ Z − such that α + β > − 1 and 1 ≤ p ≤ ∞ . Let u = P α , β [ f ] be an ( α , β ) -harmonic function on D , the unit disc of C , with the boundary f being absolutely continuous and f ˙ ∈ L p ( 0 , 2 π ) , where f ˙ ( e i θ ) : = d d θ f ( e i θ ) . In this paper, we investigate the membership of the partial derivatives ∂ z u and ∂ z ‾ u in the space H G p ( D ) , the generalized Hardy space. We prove, if α + β > 0 , then both ∂ z u and ∂ z ‾ u are in H G p ( D ) . For α + β < 0 , we show if ∂ z u or ∂ z ‾ u ∈ H G 1 ( D ) then u = 0 or u is a polyharmonic function.

The spectral theory on the S -spectrum was introduced to give an appropriate mathematical setting to quaternionic quantum mechanics, but it was soon realized that there were different applications of this theory, for example, to fractional heat diffusion and to the spectral theory for the Dirac operator on manifolds. In this seminal paper we introduce the harmonic functional calculus based on the S -spectrum and on an integral representation of axially harmonic functions. This calculus can be seen as a bridge between harmonic analysis and the spectral theory. The resolvent operator of the harmonic functional calculus is the commutative version of the pseudo S -resolvent operator. This new calculus also appears, in a natural way, in the product rule for the F -functional calculus.

In this article we study two classical potential-theoretic problems in convex geometry. The first problem is an inequality of Brunn-Minkowski type for a nonlinear capacity, In the first part of this article, we prove the Brunn-Minkowski inequality for this capacity: In the second part of this article we study a Minkowski problem for a certain measure associated with a compact convex set

This book gives a comprehensive and self-contained introduction to the theory of symmetric Markov processes and symmetric quasi-regular Dirichlet forms. In a detailed and accessible manner, Zhen-Qing Chen and Masatoshi Fukushima cover the essential elements and applications of the theory of symmetric Markov processes, including recurrence/transience criteria, probabilistic potential theory, additive functional theory, and time change theory. The authors develop the theory in a general framework of symmetric quasi-regular Dirichlet forms in a unified manner with that of regular Dirichlet forms, emphasizing the role of extended Dirichlet spaces and the rich interplay between the probabilistic and analytic aspects of the theory. Chen and Fukushima then address the latest advances in the theory, presented here for the first time in any book. Topics include the characterization of time-changed Markov processes in terms of Douglas integrals and a systematic account of reflected Dirichlet spaces, and the important roles such advances play in the boundary theory of symmetric Markov processes. This volume is an ideal resource for researchers and practitioners, and can also serve as a textbook for advanced graduate students. It includes examples, appendixes, and exercises with solutions.

Ionospheric delay errors significantly reduce the positioning accuracy of global navigation satellite systems (GNSSs), whereas precise ionospheric modeling can effectively mitigate this issue. The ionosphere exhibits large-scale symmetry, and spherical harmonics (SHs) can effectively describe this property due to their rotational symmetry on the sphere. However, mathematical fitting models such as spherical harmonic functions and polynomial models encounter boundary inaccuracies caused by edge effects. To address this problem, we developed a spherical harmonic–radial basis function (SH-RBF) hybrid method based on the integration of spherical harmonics and radial basis function interpolation techniques. This method leverages the global symmetry of spherical harmonics and utilizes the local adaptability of radial basis functions to correct regional distortions. Validation using European GNSS data during both geomagnetically quiet and active periods, in comparison with the CODE global ionospheric map (GIM), demonstrates that the modeling accuracy of spherical harmonics surpasses that of POLY during geomagnetically quiet periods. Compared to spherical harmonics, SH-RBF improves overall modeling accuracy by 8.87–27.27% and enhances accuracy in edge regions by 34.16–83.91%. During geomagnetically active periods, the SH-RBF method also achieves notable improvements. This study confirms that SH-RBF is a reliable technique for significantly reducing edge effects in regional ionospheric modeling.