By adding to the title and its anisotropy of this project the non-local or (singular) integral operators = my response to an ask of feedback to this paper, see below (paper put in reference to my own project)= (+ -via google translation, French text below) it is quite interesting = the anisotropy is here xi-cartesian constant (and the study comes out of the 1d to speak of the true multidimensional) and one can imagine more without difficulties The localized varying declinations of this anisotropy with Cartesian bases and ellipsoids/paraboloids and weights and functions, and your localized exponents ai, varying in x all of them in their own way, in centers, angles and Rotations, values etc, for your operators and various weights and functions on which your operators apply, variabilities more or less slow, fast, (ir /) regular, adapted, local, nice or not etc. It is quite speaking when the function is the derivative or the gradient of another or with studies in spaces of besov, sobolev or all functional spaces Es,p,q, with s not zero, s being the index (Integral) of derivation (instead of the spaces with s = 0, Lp, Lq etc) with s, p and p variables in x = we approach geometric anisotropy, foliations and associated irregularities which can eg to account for very natural situations in mathematical physics such as vortex patches and many others. (+-via google traduction, texte francais plus bas) c'est tout a fait interessant = l'anisotropie est ici xi-cartesienne constante (et l'etude sort du 1d pour parler du vrai multidimensionnel) et on peut imaginer de plus sans trop de difficultés, les declinaisons variables localisées de cette anisotropie avec des bases cartesiennes et des ellipsoïdes/paraboloides et des poids et fonctions et vos exposents 'ai', tous localisés, variant en x tous chacun d'eux a sa facon, en centres, angles et en rotations, valeurs etc, pour vos operateurs et poids et fonctions diverses sur lesquels s'appliquent vos operateurs, variabilités plus ou moins lentes, rapides, (ir/)regulieres, adaptées, locales, gentilles ou pas etc. c'est assez parlant quand la fonction est la derivée ou le gradient d'une autre ou avec des etudes dans des espaces de besov, de sobolev ou tous espaces fonctionnels Es,p,q, avec s pas nul, s etant l'indice (integral) de derivation (au lieu des espaces a s=0, Lp, Lq etc) avec donc s, p et p variables en x = on s'approche de l'anisotropie geometrique, les feuilletages et foliations et des irregularités associées qui peuvent eg bien rendre compte de situations tres naturelles en physique mathematique comme les vortex patches et beaucoup d'autres.
(paper and project "Weighted Anisotropic Morrey Spaces Estimates for Anisotropic Maximal Operators" and "weighted anisotropic Morrey spaces...." by Ferit Gürbüz= I pronounce my self on its/their subject and not on the novelty that its author brings to it). the rg-profile of the author has 2 (slightly different) papers with same title, but one of them has a "full text" with the title ending with: "...AND 0 -ORDER ANISOTROPIC PSEUDO-DIFFERENTIAL OPERATORS WITH SMOOTH SYMBOLS" with an additional math paragraph.
Dear Prof. Serfatti,
You have raised a very importanrt issue which affects our newly discovered Dbranes String FUNCTOR ALGEBRA CALCULUS using Mathematical Physics and Quantitative Finance experiments and is based on the Analysis provided in proving the P vs. NP problem of Millenium Maths problems (Mallick, Hamburger, Mallick (2016)). However, so far we have been able to formulate only the Fundamental Theorem in plain language which is stated on our www.econometricsociety.org/Soumitra K. Mallick website. The point is that we have not studied convergence properties of the FAC Integrals so I cannot tell you FAC has particular group homology properties over FAC algebras. This is a long drawn process of development of the Mathematical Fields which if you are interested in developing some feel free to communicate. My son who is in this with me will be specialising in Pharmaceutical Engineering Science (now undergraduate) where he studies Analysis as part of his course. So he may take more part in expanding this Maths also. Thanks for your question.
Prof. Dr. S.K.Mallick ForMemEPS, ForMemReS, MES, MAICTE, QC
for S.K.Mallick, N. Hamburger, S.Mallick
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