In this paper, we prove a weighted atom inequality for Hermite expansions and want to obtain similar type of inequalities for 1-dimensional Hermite expansions of weighted Hardy's inequalities.

In 19th-century Britain, women as compared with men, did not have equal professional and educational opportunities; therefore, they were often outside the center of politics and the economy, and were regarded as being subordinate in the household. This Woman Question had been fervently discussed in the press since the 1880s. And it was Ouida who ¡§christened¡¨ the ¡§New Woman¡¨ for this new class of women in 1894 from Sarah Grand¡¦s essay ¡§The New Aspect of the Woman Question.¡¨ The ¡§New Woman¡¨ was abhorred by traditional British society and condemned as a temptress who, so it observed, tended to satisfy her own sexual needs, regardless of social mores and household responsibilities. In contrast to ¡§the good angel in the house¡¨ or ¡§the proper lady,¡¨ the New Woman was accused of neglecting her female virtues of a selfless housewife, wife, mother, and daughter. Traditionally, women were economically dependent, and that was the reason why the grace of a Victorian proper lady lay in her submission to social morality, but not in her assertion of individualism. Since she was only considered part of the family instead of herself, it was important for a woman to be a virgin before marriage in order to guarantee her chastity and loyalty to her husband. According to Mona Caird, the ideal of virginity was worshipped in order to reinforce the idea that women¡¦s virginity belonged to their husbands-to-be instead of to themselves. This ideology satisfied the male possessive attitudes toward their wives. In her essay, ¡§Marriage,¡¨ Caird related virginity to marriage as a historically situated institution, which was temporary and challengeable. So once the gender barrier between the male master and the female housekeeper was questioned and even dismissed, women could claim the same freedom of choice, both of their life and of their bodies. With the intensification of capitalism and the individual economic unit, women seemed to represent men¡¦s property. It was in this skewed relationship between the two sexes that the feminine ideal was erected and institutionalized in order to maintain masculine domination. From a man¡¦s point of view, a modest woman, or a proper lady, should repress her sexual desire so that she would seem to have no desire for men at all. The basic reason for women to suppress their desire was because the Victorians regarded human desire as solely masculine; in other words, they saw women as the objects of desire, not the agents from which the desire originated. Women¡¦s subordinate position constrained their individualism. In this way, women became selfless and functional. The loss of virginity out of wedlock was the primary cause of a woman¡¦s tragedy. This happened to two of Hardy¡¦s ¡§New Woman¡¨ heroines, Tess d¡¦Urbervilles and Sue Bridehead. The two heroines were chosen to represent the New Women because both of them, to a large extent, acted against the decorum of a traditional proper lady. Tess of the d¡¦Urbervilles dealt with the relationship between men and women and the problems in conventional marriage. Presenting the emancipation of the heroine Sue, Jude the Obscure dealt with the inequality women suffered in marriage and the possible solution of ¡§free union¡¨ (couples living together out of wedlock) as an alternative to conventional marriage. ¡§Free union,¡¨ as this novel suggested, was a means of accommodating sexual relations with fairness to both sexes and with a more permanent relationship than traditional marriage. Hence Tess and Jude the Obscure were designated as the New Woman novels whose protagonists--Tess and Sue--illustrated the predicament encountered by the New Woman on her way to emancipation. In order to penetrate into the male psychology toward virginity in the two novels, I will delineate the different attitudes toward female virginity through the seventeenth to the nineteenth century in England to provide a historical perspective. In the seventeenth and eighteenth centuries the legal status of women remained family-oriented, which was based on the precepts of Roman law. The Puritans deplored adultery and further stressed the importance of virginity. Compared with the Puritans, the Evangelicals further idealized female nature as morally more self-restrained than men¡¦s. On a larger social scale, the religious ideals of the Evangelicals helped reinforce the social hierarchy. Because if the poor and women embraced those ideals largely to gain spiritual salvation, their governors could thereby control their labor and productivity by means of propagandizing morality. In the Victorian society, women tended to obey the rigid law and conformed to their narrowly defined domestic roles. To this dogma much had been contributed by the Puritans and the Evangelicals, from whose doctrines the ideal of a dutiful wife arose. The entire social machinery helped promote this censorship on female sexuality and individualism, while consolidating male authority simultaneously. Women, realizing that they could hardly alter the male-centered society, considered the best policy as submission and conformity to the traditional values so as to gain social acknowledgement. Hardy¡¦s presentation of Tess aroused severe criticism, because he refused to condemn her for her misbehavior. Instead of making Tess a completely helpless victim, Hardy endowed her with her own sense of strength to protect herself against the misfortunes. Sue practiced the New Woman ideal of ¡§free union,¡¨ and she herself was a well-learned student of philosophy. Sue felt herself doomed as a fallen woman, with the loss of all her extra-marital children murdered by the legitimate child of her lover. That was the reason why she considered it a compromise with the society to return to her legal husband. New Woman heroines became the scapegoats of the social machinery, in which patriarchal value was the center and women¡¦s rights were cast aside. Hardy reflected this feminist issue on the two different New Woman heroines, especially their relationship with the men around them. Readers might feel ambivalent toward both Tess and Sue, who were far from evil but were degraded. Finally, both of them were executed, Tess physically while Sue mentally. Tess paid her price for love and justice, but Sue gave up love and justice for shelter. In the thesis, I will, first of all, discuss the differences between the so-called New Woman and the proper lady in the context of Hardy¡¦s two novels, Tess and Jude. I will discuss Tess¡¦ and Sue¡¦s desires, centering on their suppressed sexuality. In the first chapter, ¡§¡¦New Women¡¦ in the Victorian Era,¡¨ while examining the function of a proper lady, I will also delineate the origin and exhibit the traits of the so-called New Woman. In the second chapter, ¡§Tess¡¦ Subjectivity and Revenge,¡¨ I shall re-evaluate Tess¡¦ tragedy by focusing on her sense of responsibility and her New Woman subjectivity. In addition, Tess¡¦ murder of Alec will be interpreted as a New Woman¡¦s revenge on male chauvinists. In the third chapter, ¡§Sue¡¦s Experiments of ¡¥Free Union¡¦ and the Final Defeat,¡¨ I will argue that Sue epitomizes a typical New Woman in her advocacy of ¡§free union¡¨ and her subsequent defeat by traditional society. In the conclusion, in addition to my personal feedback on the New Woman issue, I will make comparison between Tess and Sue as the unconventional New Woman heroines and discuss Hardy¡¦s intention in portraying the two heroines the way they are.

Cette thèse est composée de deux parties indépendantes. La première partie concerne l'étude des opérateurs bilinéaires. On consacre les deux premiers chapitres à détailler les arguments d'une décomposition "temps-fréquence" afin d'obtenir des estimations localisées sur ces opérateurs. En utilisant ces estimations hors-diagonales, nous obtenons principalement les continuités de ces opérateurs bilinéaires sur les espaces de Lebesgue et les espaces de Sobolev. Nous finissons ce deuxième chapitre par l'étude d'un calcul pseudo-différentiel bilinéaire. Le troisième chapitre porte sur une étude géométrique de ces opérateurs bilinéaires. Afin de compléter ce travail, nous étudions dans le quatrième chapitre différents résultats divers tels qu'une généralisation des résultats pour des variables multi-dimensionnelles. La deuxième partie porte sur la notion d'espace de Hardy. On y définit une construction abstraite de nouveaux espaces de Hardy. Puis en comparant avec les espaces de Hardy déjà connus et utilisés, nous essayons de dégager les conditions minimales pour conserver les propriétés essentielles de ces espaces. Nous obtenons donc un critère pour obtenir la continuité $H^1-L^1$ de certains opérateurs. Nous nous intéressons ensuite à l'étude des espaces intermédiaires par interpolation entre ces espaces $H^1$ obtenus et les espaces de Lebesgue. Nous finissons ensuite par appliquer ces résultats abstraits pour répondre au problème de régularité maximale $L^p$ sur les équations d'évolution.

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Thesis (Ph.D.)--University of Melbourne, Dept. of Mathematics and Statistics, 2005. / Typescript (photocopy). Includes bibliographical references and index (leaves 143-151).

Zugl.: Göttingen, Univ., Diss., 2008

Thesis (M.A.)--Villanova University, 2008. / English Dept. Includes bibliographical references.