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"Harmonics"
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Elliptic Theory for Sets with Higher Co-dimensional Boundaries
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
eBook
Multi-Parameter Hardy Spaces Theory and Endpoint Estimates for Multi-Parameter Singular Integrals
The main purpose of this paper is to establish the theory of the multi-parameter Hardy spaces
More precisely, Street (2014) studied the
eBook
Maximal Functions, Littlewood–Paley Theory, Riesz Transforms 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.
eBook
Functional Analysis, Harmonic Analysis, and Image Processing
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.
eBook
The Riesz Transform of Codimension Smaller Than One and the Wolff Energy
Fix $d\\geq 2$, and $s\\in (d-1,d)$. The authors characterize the non-negative locally finite non-atomic Borel measures $\\mu $ in $\\mathbb R^d$ for which the associated $s$-Riesz transform is bounded in $L^2(\\mu )$ in terms of the Wolff energy. This extends the range of $s$ in which the Mateu-Prat-Verdera characterization of measures with bounded $s$-Riesz transform is known. As an application, the authors give a metric characterization of the removable sets for locally Lipschitz continuous solutions of the fractional Laplacian operator $(-\\Delta )^\\alpha /2$, $\\alpha \\in (1,2)$, in terms of a well-known capacity from non-linear potential theory. This result contrasts sharply with removability results for Lipschitz harmonic functions.
eBook
Dyadic-probabilistic methods in bilinear analysis
We demonstrate and develop dyadic–probabilistic methods in connection with non-homogeneous bilinear operators, namely singular
integrals and square functions. We develop the full non-homogeneous theory of bilinear singular integrals using a modern point of view.
The main result is a new global
While proving our bilinear results we also advance and
refine the linear theory of Calderón–Zygmund operators by improving techniques and results. For example, we simplify and make more
efficient some non-homogeneous summing arguments appearing in
eBook
Frames and Harmonic Analysis
This volume contains the proceedings of the AMS Special Sessions on Frames, Wavelets and Gabor Systems and Frames, Harmonic Analysis, and Operator Theory, held from April 16-17, 2016, at North Dakota State University in Fargo, North Dakota. The papers appearing in this volume cover frame theory and applications in three specific contexts: frame constructions and applications, Fourier and harmonic analysis, and wavelet theory.
eBook
Grid integration and a power quality assessment of a wave‐energy park
This study presents a step toward the grid connection of a wave‐energy park through an electric power conversion system (EPCS) developed and installed for the wave‐energy harvesting in Lysekil, Sweden. The EPCS comprises a rectifier, a DC bus, and an inverter followed by a harmonic filter (HF). The higher‐ and lower‐order harmonics injected by the inverter in a power quality context are investigated. The lower‐order voltage harmonics partially distort the voltage‐source inverter output grid current. A phase‐locked loop‐based (PLL) grid‐phase tracking is used to attenuate the lower‐order harmonics by reflecting the grid harmonics in the inverter output. An expression for the grid‐current harmonics as a function of the grid‐voltage harmonics has been derived and implemented. A mathematical model is derived to obtain a transfer function for the PLL, and finally, proportional–integral gains are tuned for stable system operation. An HF for mitigating the higher‐order harmonics has been implemented. The total harmonic distortion is evaluated experimentally, and the results fulfil the grid‐code requirements at various frequencies and harmonic orders.
Journal Article
Embeddings of Decomposition Spaces
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
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