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Öğe A general approach to weighted L p rellich type inequalities related to greiner operator(American Institute of Mathematical Sciences, 2019) Kömbe, İsmail; Yener, AbdullahIn this paper we exhibit some sufficient conditions that imply general weighted L p Rellich type inequality related to Greiner operator without assuming a priori symmetric hypotheses on the weights. More precisely, we prove that given two nonnegative functions a and b, if there exists a positive supersolution ? of the Greiner operator ? ? such that ? ? (a|? ? ?| p-2 ? ? ?)?b? p-1 almost everywhere in R 2n+1 ; then a and b satisfy a weighted L p Rellich type inequality. Here, p > 1 and ? ? = ? n j=1 (x 2 j +y 2 j ) is the sub-elliptic operator generated by the Greiner vector fields x j {equation presented} where (z,l)=(x,y,l)? R 2n+1 =R n ×R n ×R,|Z|={equation presented} and k ? 1. The method we use is quite practical and constructive to obtain both known and new weighted Rellich type inequalities. On the other hand, we also establish a sharp weighted L p Rellich type inequality that connects first to second order derivatives and several improved versions of two-weight L p Rellich type inequalities associated to the Greiner operator ? ? on smooth bounded domains ? in R 2n+1 . © 2019 American Institute of Mathematical Sciences. All Rights Reserved.Öğe A general approach to weighted Rellich type inequalities on Carnot groups(Springer-Verlag Wien, 2018) Goldstein, Jerome A.; Kömbe, İsmail; Yener, AbdullahWe give a simple sufficient criterion on a pair of nonnegative weight functions a and b on a Carnot group G, so that the following general weighted Lp Rellich type inequality?Ga|?Gu|pdx??Gb|u|pdxholds for every u?C0?(G) and p> 1. It is worthwhile to notice that our method easily derives previously known weighted Rellich type inequalities with a sharp constant in a more adequate fashion and also enables us to obtain new ones. We also present a sharp Lp Rellich type inequality that connects first to second order derivatives and some new two-weight Rellich type inequalities with remainders on bounded domains ? in G via a differential inequality and the improved two-weight Hardy inequality in Goldstein et al. (Discret Contin Dyn Syst 37:2009–2021, 2017). © 2017, Springer-Verlag Wien.Öğe General weighted hardy type inequalities related to Baouendi-Grushin operators(Taylor and Francis Ltd., 2018) Kömbe, İsmail; Yener, AbdullahIn this paper, we derive a sufficient condition on a pair of nonnegative weight functions ? and w in ?m+k so that the general weighted Hardy type inequality with a remainder term (Formula Presented) is the sub-elliptic gradient. It is worth emphasizing here that our unifying method may be readily used to recover most of the previously known sharp weighted Hardy and Heisenberg-Pauli-Weyl type inequalities as well as to construct other new inequalities with an explicit constant. Furthermore, we also obtain new results on two-weight Lp Hardy type inequalities with remainder terms on smooth bounded domains ? in ?m+k via a non-linear partial differential inequality. © 2017 Informa UK Limited, trading as Taylor & Francis Group.Öğe General weighted Hardy-type inequalities related to greiner operators(Rocky Mountain Mathematics Consortium, 2018) Yener, AbdullahIn this article, we present a general method that can be used to deduce weighted Hardy-type inequalities from a particular non-linear partial differential inequality in a relatively simple and unified way on the sub-Riemannian manifold R 2 n +1 = R n ×R n ×R, defined by the Greiner vector fields ? X j = ?xj + 2ky j |z| 2 k? 2 ? ?l , ? Y j = ?yj ? 2kx j |z| 2 k? 2 ? ?l , j = 1, . . ., n, where z = x + iy ? C n , l ? R, k ? 1. Our method allows us to improve, extend, and unify many previously obtained sharp weighted Hardy-type inequalities as well as to yield new ones. These cases are illustrated by giving many concrete examples, including radial, logarithmic, hyperbolic and non-radial weights. Furthermore, we introduce a new technique for constructing two-weight L p Hardy-type inequalities with remainder terms on smooth bounded domains ? in R 2 n +1 . We also give several applications leading to various weighted Hardy inequalities with remainder terms. Copyright © 2018 Rocky Mountain Mathematics Consortium.Öğe Greiner operatörüyle ilişkilendirilmiş hızlı difüzyon denklemi ve bazı integral eşitsizlikleri(İstanbul Ticaret Üniversitesi, 2024) Utku, Ahmet Uğur; Yener, Abdullah[Abstract Not Available]Öğe Heisenberg grubunda hardy, rellich eşitsizlikleri ve bu eşitsizliklerin bazı uygulamaları(İstanbul Ticaret Üniversitesi, 2017) Yener, Abdullah; Kömbe, İsmailBu tezde; ilk olarak, sırasıyla ağırlıklı p—alt-Laplace ve ağırlıklı p—biharmonik doğrusal olmayan kısmi diferansiyel eşitsizliklerinden yola çıkılarak, Heisenberg grubunda genel ağırlıklı Lp Hardy ve Lp Rellich eşitsizlikleri ispatlanmıştır. Burada kullanılan metodlar, üzerinde hem bilinen hem de yeni ağırlıklı Hardy, Rellich ve Heisenberg-Pauli-Weyl tipi eşitsizlikler elde etme adına oldukça pratik ve üretkendir. Hn’de veya Hn’nin bazı alt bölgelerinde çeşitli ağırlık fonksiyonlarına sahip Hardy ve Rellich tipi eşitsizlikler elde etmek için, sırasıyla ağırlıklı p—alt-Laplace ve ağırlıklı p—biharmonik esitsizliklerini sağlayan uygun fonksiyonları belirlemek yeterlidir. Bu durum, tezin uygulama kısımlarında birçok somut örnek vererek gösterilmiştir. Daha sonra, Heisenberg grubunda, uygun bir fonksiyonun ikinci mertebeden türevi ile birinci mertebeden türevi arasında bir ilişki kuran en iyi sabitli Lp Rellich- II tipi bir eşitsizlik elde edilmiş ve bu eşitsizlikten faydalanılarak ikinci mertebeden Heisenberg-Pauli-Weyl tipi bir eşitsizliğin de geçerli olduğu gösterilmiştir. Ayrıca, en iyi sabite sahip ağırlıklı Lp Rellich-II eşitsizliğinin ispatında kullanılan tekniğe benzer bir teknikle, Rellich-Hardy-Poincare tipi yeni bir eşitsizlik bulunmuştur. Son olarak, içindeki düzgün sınırlı bir Q bölgesinde geliştirilmiş iki-ağırlıklı genel Lp Hardy ve Lp Rellich tipi eşitsizlikler üzerine bazı yeni sonuçlar elde edilmiştir. Bu tipten Hardy ve Rellich eşitsizliklerinin ispatındaki temel dayanak noktalardan biri bazı doğrusal olmayan kısmi diferansiyel eşitsizliklerin varlığı olmuştur. Bu diferansiyel eşitsizliklerin çözümlerinden yola çıkılarak; östel, logaritmik ve radyal tipli çok çeşitli ağırlık fonksiyonlarına sahip geliştirilmiş Lp Hardy ve Lp Rellich eşitsizliklerine bazı somut örnekler verilmiştir.Öğe Heisenberg grubunda ısı denklemi(İstanbul Ticaret Üniversitesi, 2013) Yener, Abdullah; Kömbe, İsmailBu çalışmada,u_{t}-u_{xx}=f(x,t)ısı denkleminindeki R ve Rdeki temel özellikleri verildikten sonra, H Heisenberg grubunda a(w)>0, V(w)L_{loc}¹() potansiyelleri ile verilmiş doğrusal olmayan parabolik probleminin ne zaman pozitif çözümünün olmadığı ispatlanmıştır. Anahtar Sözcükler: Isı denklemi, başlangıç değer problemi, başlangıç sınır değer problemi, temel çözüm, Fourier dönüşümü, Heisenberg grubu, singüler potan- siyel, doğrusal olmayan parabolik denklem.Öğe Several Hardy-type inequalities with weights related to Baouendi–Grushin operators(Tübitak, 2018) Yener, AbdullahIn this paper we shall prove several weighted Lp Hardy-type inequalities associated to the Baouendi–Grushintype operators.Öğe A unified approach to weighted Hardy type inequalities on Carnot groups(Southwest Missouri State University, 2017) Goldstein, J.A.; Kömbe, İsmail; Yener, AbdullahWe find a simple sufficient criterion on a pair of nonnegative weight functions V (x) and W (x) on a Carnot group G; so that the general weighted Lp Hardy type inequality (Equation presentted) is valid for any ? ? C? 0 (G) and p > 1: It is worth noting here that our unifying method may be readily used both to recover most of the previously known weighted Hardy and Heisenberg-Pauli-Weyl type inequalities as well as to construct other new inequalities with an explicit best constant on G: We also present some new results on two-weight Lp Hardy type inequalities with remainder terms on a bounded domain ? in G via a differential inequality.Öğe Weighted Hardy and Rellich type inequalities on Riemannian manifolds(Wiley-VCH Verlag, 2016) Kömbe, İsmail; Yener, AbdullahIn this paper we present new results on two-weight Hardy, Hardy-Poincaré and Rellich type inequalities with remainder terms on a complete noncompact Riemannian Manifold M. The method we use is flexible enough to obtain more weighted Hardy type inequalities. Our results improve and include many previously known results as special cases. © 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.Öğe WEIGHTED HARDY TYPE INEQUALITIES ON THE HEISENBERG GROUP H-n(Element, 2016) Yener, AbdullahIn the present article, we provide a sufficient condition on a pair of nonnegative weight functions V and W on the Heisenberg group H-n, so that we establish a general L-p Hardy type inequality involving these weights with a remainder term. The method we use here is practical enough to get more weighted Hardy type inequalities. We also obtain new results on two-weight Hardy and Hardy-Poincare type inequalities with remainder terms on Hn. Our findings improve and include many previously known results in special cases.Öğe Weighted hardy type inequalities with Robin boundary conditions(American Institute of Mathematical Sciences, 2024) Kömbe, İsmail; Yener, AbdullahIn this paper, we establish a general weighted Hardy type inequality for the p?Laplace operator with Robin boundary condition. We provide various concrete examples to illustrate our results for different weights. Furthermore, we present some Heisenberg-Pauli-Weyl type inequalities with boundary terms on balls centred at the origin with radius R in RnÖğe Weighted rellich type inequalities related to baouendi-grushin operators(American Mathematical Society, 2017) Kömbe, İsmail; Yener, AbdullahWe find a simple sufficient criterion on a pair of nonnegative weight functions a (x, y) and b (x, y) in ?m+k so that the general weighted Lp Rellich type inequality (Formula presented) holds for all u ? C0?(?m+k). Here ?? = ?x + |x|2??y is the Baouendi-Grushin operator, ? > 0, m, k ? 1 and p > 1. It is important to point out here that our approach is constructive in the sense that it allows us to retrieve already established weighted sharp Rellich type inequalities as well as to get other new results with an explicit constant on ?m+k. We also obtain a sharp Lp Rellich type inequality that connects first to second order derivatives and several new two-weight Rellich type inequalities with remainder terms on smooth bounded domains ? in ?m+k via a nonlinear differential inequality. © 2017 American Mathematical Society.
