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Maximal and Potential Operators in Weighted Lebesgue Spaces With Non- Standard Growth

Thesis Info

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Author

Sarwar, Muhammad

Program

PhD

Institute

Government College University

City

Lahore

Province

Punjab

Country

Pakistan

Thesis Completing Year

2006

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/handle/123456789/1478

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676726623212

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Two–weight criteria of various type for one–sided maximal functions and one–sided potentials are established in variable exponent Lebesgue spaces. Among other re- sults we derive the Hardy–Littlewood, Fefferman–Stein and trace inequalities in these spaces. Weighted estimates for Hardy–type, maximal, potential and singular opera- tors defined by means of a quasi–metric and a doubling measure are derived in Lp(x) spaces. In some cases examples of weights guaranteeing the appropriate weighted estimates are given.
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آئو وطن آباد کریں

آئو وطن آباد کر یں
نحمدہ ونصلی علی رسولہ الکریم امّا بعد فاعوذ بااللہ من الشیطن الرجیم
بسم اللہ الرحمن الرحیم
معزز اسا تذہ کرام اور میرے ہم مکتب ساتھیو!
آج مجھے جس موضوع پر اظہار خیال کرنا ہے وہ ہے:’’آؤوطن آباد کر یں‘‘
صدرِذی وقار!
وطن کی محبت ایمان سے ہے، وطن سے والہانہ عقیدت ایمان کا حصہ ہے، وطن ہے تو ہم ہیں ، وطن ہے تو ہمارا وجود قائم ہے، وطن کی خوشبو سے ہمارے دماغ معطر رہتے ہیں،وطن کے صحرودر یا ہمارا سرمایہ ہیں، وطن کے شجر وحجر ہمارا اثاثہ ہیں۔
محترم صدر!
اگر یہ الفاظ ہم دِل کی اتھاہ گہرائیوں سے کہتے ہیں، تو ہم قابلِ فخر ہیں، ہماری حیات کی ساعتیں قابلِ صد مبارکباد ہیں، ہمارے وطن کے بارے میں تصورات یقینا صائب ہیں، ہماری محبت واقعی وطن کے لیے حقیقی ہے، ہمارا خیال اپنی سرزمین کے لیے واقعی طلسماتی اور کرشماتی ہے۔
صدرِمحترم!
وطن سے محبت اور وطن کی آبادکاری دماغ کے سوچنے کانام ہیں، وطن کی تعمیر صرف زبان کے اظہار کا نام نہیں ،وطن کے گلشن کی تز ئین صرف جسم کی حرکات کا نام نہیں ، وطن سے محبت اور پیار صرف قول و قرار کانام نہیں۔
معزز سامعین!
وطن سے محبت کرنی ہے تو وطن کے افراد سے محبت کرنا ہوگی ، وطن کے در و دیوار سے محبت کرنا ہوگی ، وطن کے نقصان کو اپنا نقصان سمجھنا ہوگا ، وطن کے مفاد کو اپنے مفادات پر ترجیح دینا ہوگی ، وطن کی تعمیر میں لاثانی اور مثالی کردار ادا کرنا ہو گا کیوں کہ وطن ہی ہماری آن ہے، وطن سے ہماری شان ہے وطن ہے تو ہم ہیں وطن نہیں ہے تو ہم بھی نہیں ہیں کیونکہ یہی وطن ہی تو ہماری شناخت ہے۔
اقوام کے وجود...

سیاسی اضطرابی صورتحال کا جائزہ اور اس کا حل: سیرۃ نبویﷺ کی روشنی میں

Pakistan is an Ideological country, there are only two ideological states in the world. First one was Madinah (Saudi Arabia) and the second one is Pakistan. Political situation in our country is worst. There is no any stability in political situation. From its existence Pakistan is facing different problems and issues. In this research article, will try to draw reasons and will table solutions to get political stability in Pakistan.

Mechanistic Studies of Thermal Behavior of Certain Polymeric Systems With Additives

The present work deals with the pyrolytical studies of nine systems, i.e., two polymers [poly(methyl methacrylate) and poly(vinyl acetate)] and one copolymer [(poly(S-co-MMA)] blended with three inorganic additives, namely, aluminum tribromide, phosphorus tribromide and tin tetrachloride. Each polymeric/copolymeric substance is cast in the form of film (from common solvent) with each additive to get binary system such as PMMA-AlBr3, P(S-co- MMA)-AlBr3, PVAc-AlBr3, etc. The interest is based on the retardance of flammability, thermal stability, alteration in degradation mechanism, etc. of organic materials in the presence of inorganic species. The first chapter details the historical use of polymers, their wear and tear under the influence of different conditions, i.e., UV, g-radiations, oxygen, biological agents, heat, etc. Major emphasis is laid on the effects of heat and the reactions involved during the deterioration. The role of additives in imparting stability or otherwise is briefly described. Literature review is given in second chapter which includes the summary of the work published by various scientists on the subject of thermal degradation of polymers/copolymers alone as well as in the presence of additives over a long span of time. The polymers/copolymer selected for current investigations get major share of the review. In the third chapter, aims and objectives of the present research are included. Furthermore, the incentives and motivation for the work conducted are presented along with the plans and strategies devised. Experimental portion, chapter 4, consists of purification procedures for solvents, the preparatory methods for polymers, copolymers and additives, sample preparation techniques for flammability test, set-up designed for the pyrolysis of the systems explored and the instrumental techniques employed for the characterizations. All results gleaned from instrumentation, viz., TG-DTG-DTA, IR, GC-MS, flammability, about the binary systems are listed in chapter 5. Tabulated findings are available including activation energies and order of reactions. Chapter 6 discusses the mutual thermal behavior of PMMA and AlBr3, PBr3, SnCl4 in each other‟s presence. The three systems start degrading at lower temperatures when compared with T0 (temperature corresponding to first weight-loss) of PMMA. In the case of PMMA-AlBr3, T0 is lower than the T0 of the additive. Despite early destabilizations, stabilization zones (in terms of temperature) and other factors promoting stability are identified. A number of new products have evolved in all cases which provide reasons to believe that chemical interaction has taken place and the path of polymer degradation appears modified. PMMA-PBr3 blend reveals that phosphorus is incorporated in the backbone of PMMA during the course of degradation. Phosphorus also interacts with carbonyl oxygens of PMMA pendent groups in a similar fashion as is observed for Al (from AlBr3) and Sn (from SnCl4), however, it cannot be termed as „co-ordination‟ or „complexation‟ as observed for Sn in PMMA-SnCl4 system. Not only the positively charged parts of the additives, i.e., Al, P and Sn interact with the pendent groups and backbone of the polymer but the negatively charged parts (Br and Cl) also form bonds after the partial or complete removal of pendent groups. In other instances, they also replace the methyls attached to backbone carbons. The flammability parameter indicates the effectiveness of all the additives which markedly lower the burning rate of PMMA. Degradation mechanisms are proposed on the basis of gathered data and products identified. Monomer is not the major product in the studies undertaken presently. The same chapter, (6), explains the thermal behavior of P(S-co-MMA) in the presence of additives already named in the last paragraph. The blends [P(S-co-MMA)-AlBr3, P(S-co- MMA)-PBr3 and P(S-co-MMA)-SnCl4] exhibit a low-temperature degradation which is thought to be linked to the decomposition of non-copolymeric part of the blends. This destabilization is followed by stabilization of either component of the systems (general observation). The release of free radicals by the disintegration of „free‟ or „attached‟ additives initiates as well as inhibits depolymerization. The „co-ordination‟ or „complex‟ type structures become less appreciable in the present case when this feature is compared with the PMMA blends revealing the lower number of pendent (-COOCH3) groups available in copolymer. New products are noticed in all blends which furnish the basis for chemical influence of the constituents on each other‟s pyrolysis. Phosphorus appears attached to the degrading copolymer for P(S-co-MMA)-PBr3. Benzene ring, substituted with halogens, evidence the types of reactions that occur as the degradation processes progress. The pyrolysis of P(S-co-MMA)-SnCl4 unveils the formation of bisubstituted benzene which means chlorine is more reactive than bromine (no such instances are observed with P(S-co- MMA)-AlBr3 and P(S-co-MMA)-PBr3). These reactions are believed to proceed free radically. The test of flammability proves that all the additives are equally effective in case of copolymer‟s burning.