# On $$(\phi ,\psi )$$ ( ϕ , ψ ) -Inframonogenic Functions in Clifford Analysis

27 Jul 2021-pp 1-17

TL;DR: In this paper, a new integral representation formula for inframonogenic functions was derived for multidimensional Ahlfors-Beurling transforms closely connected to the use of two different orthogonal basis.

Abstract: Solutions of the sandwich equation $${^\phi \!\underline{\partial }}[f]{^\psi \!\underline{\partial }}=0$$
, where $${^\phi \!\underline{\partial }}$$
stands for the Dirac operator with respect to a structural set $$\phi $$
, are referred to as $$(\phi ,\psi )$$
-inframonogenic functions and capture the standard inframonogenic ones as special case. We derive a new integral representation formula for such functions as well as for multidimensional Ahlfors–Beurling transforms closely connected to the use of two different orthogonal basis in $${{\mathbb {R}}}^m$$
. Moreover, we also establish sufficient conditions for the solvability of a jump problem for the system $${^\phi \!\underline{\partial }}[f]{^\psi \!\underline{\partial }}=0$$
in domains with fractal boundary.

##### Citations

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TL;DR: In this article, the relationship between elliptic partial differential equations and harmonic Clifford algebra valued functions has been clarified, showing that the harmonic functions do not share the good structure and properties of the harmonic ones.

Abstract: In recent works, arbitrary structural sets in the non-commutative Clifford analysis context have been used to introduce non-trivial generalizations of harmonic Clifford algebra valued functions in $${\mathbb R}^m$$
. Being defined as the solutions of elliptic (genera-lly non-strongly elliptic) partial differential equations, $$(\varphi ,\psi )$$
-inframonogenic and $$(\varphi ,\psi )$$
-harmonic functions do not share the good structure and properties of the harmonic ones. The aim of this paper it to show and clarified the relationship between these classes of functions.

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TL;DR: In this article , the Dirichlet problem in the ball for the so-called inframonogenic functions, i.e., the solutions of the sandwich equation ∂x_f∂x _=0, where ∂ x_ stands for the Dirac operator in Rm.

##### References

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01 Feb 1971

TL;DR: Stein's seminal work Real Analysis as mentioned in this paper is considered the most influential mathematics text in the last thirty-five years and has been widely used as a reference for many applications in the field of analysis.

Abstract: Singular integrals are among the most interesting and important objects of study in analysis, one of the three main branches of mathematics. They deal with real and complex numbers and their functions. In this book, Princeton professor Elias Stein, a leading mathematical innovator as well as a gifted expositor, produced what has been called the most influential mathematics text in the last thirty-five years. One reason for its success as a text is its almost legendary presentation: Stein takes arcane material, previously understood only by specialists, and makes it accessible even to beginning graduate students. Readers have reflected that when you read this book, not only do you see that the greats of the past have done exciting work, but you also feel inspired that you can master the subject and contribute to it yourself. Singular integrals were known to only a few specialists when Stein's book was first published. Over time, however, the book has inspired a whole generation of researchers to apply its methods to a broad range of problems in many disciplines, including engineering, biology, and finance. Stein has received numerous awards for his research, including the Wolf Prize of Israel, the Steele Prize, and the National Medal of Science. He has published eight books with Princeton, including Real Analysis in 2005.

9,595 citations

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16 Nov 2007

TL;DR: In this paper, the authors propose integration and integral theorems for series expansions and local behavior. But they do not discuss the relationship between series expansion and local behaviour.Numbers.

Abstract: Numbers.- Functions.- Integration and integral theorems.- Series expansions and local behavior.

336 citations

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TL;DR: In this paper, the authors extend the validity of (1) to much more general boundaries while still using the ordinary Lebesgue integral; this is the topic of the present paper.

Abstract: v is the 1-vectorfield “dual” to ω: if ω = ∑ (−1)fi dx1 ∧ · · · ∧ ∧ dxi ∧ · · · ∧ dxn, then v = (f1, . . . , fn).) There has been considerable effort in the literature (e.g. [JK], [M], [P]) to extend this formula to permit integrands of less regularity by generalizing the Lebesgue integral. On the other hand, invariably the situations in which (1) holds require fairly strong hypotheses on the boundary ∂Ω, e.g. that it should have sigmafinite (n−1)-measure, or that the gradient of the characteristic function of Ω be a vector valued measure with finite total variation [F], [P]. However there is a natural way to expand the validity of (1) to much more general boundaries while still using the ordinary Lebesgue integral; this is the topic of the present paper. For the case of Lipschitz forms, the results of this paper follow readily from Whitney’s theory of flat chains [W2]. However his approach to the Gauss-Green theorem is not widely appreciated because he focused on chains and cochains, where effectively (1) is used to define the exterior derivative. In [HN] we extend Whitney’s method to treat the more general Holder case. (Only the case n = 2 is discussed

107 citations

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TL;DR: In this paper, the design of Log Periodic Fractal Koch Antennas (LPFKA) is proposed for UHF band applications, and the procedure to design the LPFKA with three different numbers of iterations to reduce the antenna size is discussed.

Abstract: In this paper, the design of Log Periodic Fractal Koch Antennas (LPFKA) is proposed for Ultra High Frequency (UHF) band applications. The procedure to design the LPFKA with three different numbers of iterations in order to reduce the antenna size is discussed. The Computer Simulation Technology (CST) software has been used to analyze the performances of the designed antennas such as return loss, radiation patterns, current distribution and gain. The antennas have been fabricated using FR4 laminate board with wet etching technique. Using fractal Koch technique, the size of the antenna can be reduced up to 27% when the series iteration is applied to the antennas without degrading the overall performances. Both simulated and measured results are compared, analyzed and presented in this paper.

92 citations