Rozprawy doktorskie: „Hardy's inequalities”

1

Irvine, William Thomas Mark. "Hardy's thought experiment, Bell's inequalities and entanglement from photonic crystals". Thesis, University of Oxford, 2004. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.442452.

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Tidblom, Jesper. "Improved Lp Hardy Inequalities". Doctoral thesis, Stockholm : Department of Mathematics, Stockholm University, 2005. http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-615.

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Wedestig, Anna. "Weighted inequalities of Hardy-type and their limiting inequalities /". Luleå, 2003. http://epubl.luth.se/1402-1544/2003/17.

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Johansson, Maria. "Hardy and Carleman type inequalities /". Luleå, 2004. http://epubl.luth.se/1402-1757/2004/81.

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5

Handley, G. D. "Hilbert and Hardy type inequalities /". Connect to thesis, 2005. http://eprints.unimelb.edu.au/archive/00000818.

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Johansson, Maria. "Carleman type inequalities and Hardy type inequalities for monotone functions /". Luleå : Luleå University of Technology, 2007. http://epubl.ltu.se/1402-1544/2007/53/.

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7

Routin, Eddy. "Local Tb theorems and Hardy type inequalities". Phd thesis, Université Paris Sud - Paris XI, 2011. http://tel.archives-ouvertes.fr/tel-00656023.

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In this thesis, we study local Tb theorems for singular integral operators in the setting of spaces of homogeneous type. We give a direct proof of the local Tb theorem with L^2 integrability on the pseudo- accretive system. Our argument relies on the Beylkin-Coifman-Rokhlin algorithm applied in adapted Haar wavelet basis and some stopping time results. Motivated by questions of S. Hofmann, we extend it to the case when the integrability conditions are lower than 2, with an additional weak boundedness type hypothesis, which incorporates some Hardy type inequalities. We study the possibility of relaxing the support conditions on the pseudo-accretive system to a slight enlargement of the dyadic cubes. We also give a result in the case when, for practical reasons, hypotheses on the pseudo-accretive system are made on balls rather than dyadic cubes. Finally we study the particular case of perfect dyadic operators for which the proof gets much simpler. Our argument gives us the opportunity to study Hardy type inequalities. The latter are well known in the Euclidean setting, but seem to have been overlooked in spaces of homogeneous type. We prove that they hold without restriction in the dyadic setting. In the more general case of a ball B and its corona 2B\B, they can be obtained from some geometric conditions relative to the distribution of points in the homogeneous space. For example, we prove that some relative layer decay property suffices. We also prove that this property is implied by the monotone geodesic property of Tessera. Finally, we give some explicit examples and counterexamples in the complex plane to illustrate the relationship between the geometry of the homogeneous space and the validity of the Hardy type inequalities.

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8

Abuelela, Waleed Mostafa Kamal Abdelfatah. "Hardy type inequalities for non-convex domains". Thesis, University of Birmingham, 2010. http://etheses.bham.ac.uk//id/eprint/1268/.

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9

Chen, Tieling. "Weak and strong inequalities for Hardy type operators". Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2001. http://www.collectionscanada.ca/obj/s4/f2/dsk3/ftp04/NQ58204.pdf.

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Okpoti, Christopher Adjei. "Weight characterizations of Hardy and Carleman type inequalities /". Luleå : Department of Mathematics, Luleå University of Technology, 2006. http://epubl.ltu.se/1402-1544/2006/36/.

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11

Ushakova, Elena P. "Norm inequalities of Hardy and Pólya-Knopp types /". Luleå : Department of Mathematics, Luleå University of Technology, 2006. http://epubl.ltu.se/1402-1544/2006/53/.

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Tidblom, Jesper. "Improved Lp Hardy Inequalities". Doctoral thesis, Stockholm University, Department of Mathematics, 2005. http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-615.

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Paper 1 : A geometrical version of Hardy's inequality for W_0^{1,p}(D).

The aim of this article is to prove a Hardy-type inequality, concerning functions in W_0^{1,p}(D) for some domain D in R^n, involving the volume of D and the distance to the boundary of D. The inequality is a generalization of a previously proved inequality by M. and T. Hoffmann-Ostenhof and A. Laptev, which dealt with the special case p=2.

Paper 2 : A Hardy inequality in the Half-space.

Here we prove a Hardy-type inequality in the half-space which generalize an inequality originally proved by V. Maz'ya to the so-called L^p case. This inequality had previously been conjectured by the mentioned author. We will also improve the constant appearing in front of the reminder term in the original inequality (which is the first improved Hardy inequality appearing in the litterature).

Paper 3 : Hardy type inequalities for Many-Particle systems.

In this article we prove some results about the constants appearing in Hardy inequalities related to many particle systems. We show that the problem of estimating the best constants there is related to some interesting questions from Geometrical combinatorics. The asymptotical behaviour, when the number of particles approaches infinity, of a certain quantity directly related to this, is also investigated.

Paper 4 : Various results in the theory of Hardy inequalities and personal thoughts.

In this article we give some further results concerning improved Hardy inequalities in Half-spaces and other conic domains. Also, some examples of applications of improved Hardy inequalities in the theory of viscous incompressible flow will be given.

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13

Frank, Rupert L. "Hardy-Lieb-Thirring inequalities for eigenvalues of Schrödinger operators". Doctoral thesis, KTH, Matematik (Avd.), 2007. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-4344.

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This thesis is devoted to quantitative questions about the discrete spectrum of Schrödinger-type operators. In Paper I we show that the Lieb-Thirring inequalities on moments of negative eigen¬values remain true, with possibly different constants, when the critical Hardy weight is subtracted from the Laplace operator. In Paper II we prove that the one-dimensional analog of this inequality holds even for the critical value of the moment parameter. In Paper III we establish Hardy-Lieb-Thirring inequalities for fractional powers of the Laplace operator and, in particular, relativistic Schrödinger operators. We do so by first establishing Hardy-Sobolev inequalities for such operators. We also allow for the inclu¬sion of magnetic fields. As an application, in Paper IV we give a proof of stability of relativistic matter with magnetic fields up to the critical value of the nuclear charge. In Paper V we derive inequalities for moments of the real part and the modulus of the eigen¬values of Schrödinger operators with complex-valued potentials.
QC 20100708

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14

Frank, Rupert L. "Hardy-Lieb-Thirring inequalities for eigenvalues of Schrödinger operators /". Stockholm : Institutionen för matematik, Kungliga Tekniska högskolan, 2007. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-4344.

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15

Okpoti, Christopher Adjei. "Weight characterizations of discrete Hardy and Carleman type inequalities /". Luleå : Luleå University of Technology, 2005. http://epubl.luth.se/1402-1757/2005/45.

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16

Prokhorov, Dmitry V. "Weighted inequalities involving Riemann-Liouville and Hardy-type operators /". Luleå, 2003. http://epubl.luth.se/1402-1544/2003/38.

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17

Arendarenko, Larissa. "Some new Hardy-type Inequalities for integral operators with kernels". Licentiate thesis, Luleå tekniska universitet, Matematiska vetenskaper, 2011. http://urn.kb.se/resolve?urn=urn:nbn:se:ltu:diva-26661.

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This Licentiate thesis deals with the theory of Hardy-type inequalities in anew situation, namely when the classical Hardy operator is replaced by amore general operator with kernel. The kernels we consider belong to thenew classes O+ n and O-n , n = 0; 1; :::, which are wider than co-called Oinarovclass of kernels.The thesis consists of three papers (papers A, B and C), an appendix topaper A and an introduction, which gives an overview to this specific fieldof functional analysis and also serves to put the papers in this thesis into amore general frame.In paper A some new Hardy-type inequalities for the case with Hardy-Volterra integral operators involved are proved and discussed. The case 1

Godkänd; 2011; 20111114 (larare); LICENTIATSEMINARIUM Ämnesområde: Matematik/Mathematics Examinator: Professor Lars-Erik Persson, Institutionen för teknikvetenskap och matematik, Luleå tekniska universitet Diskutant: Professor Massimo Lanza de Cristoforis, Dipartamento di Matematica, Universita degli Studi di Padova, Italy Tid: Tisdag den 20 december 2011 kl 10.00 Plats: D2214-15, Luleå tekniska universitet

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18

Nassyrova, Maria. "Weighted inequalities involving Hardy-type and limiting geometric mean operators /". Luleå, 2002. http://epubl.luth.se/1402-1544/2002/03/index.html.

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19

Sababheh, Mohammad Suboh. "Constructions of bounded functions related to two-sided Hardy inequalities". Thesis, McGill University, 2006. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=102160.

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We investigate inequalities that can be viewed as generalizations of Hardy's inequality about the Fourier coefficients of a function analytic on the circle. The proof of the Littlewood conjecture opened a wide door in front of questions regarding possible generalizations of Hardy's inequality. The proof of the Littlewood conjecture was based on some constructions of bounded functions having particular properties.
In 1993, I. Klemes investigated one of the constructions (we shall call it the algebraic construction) and proved what is called a mixed norm generalization of Hardy's inequality. It turns out that we can work with the same construction and examine more properties of it in order to get more results.
The objectives of the thesis are to give more detailed properties of the algebraic construction and to use these properties in order to prove various versions of two-sided Hardy inequalities.

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20

D'Ambrosio, Lorenzo. "Hardy Inequalities and Liouville Type Theorems Associated to Degenerate Operators". Doctoral thesis, SISSA, 2002. http://hdl.handle.net/20.500.11767/4170.

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21

Aermark, Lior Alexandra. "Hardy and spectral inequalities for a class of partial differential operators". Doctoral thesis, Stockholms universitet, Matematiska institutionen, 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-97067.

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This thesis is devoted to the study of Hardy and spectral inequalities for the Heisenberg and the Grushin operators. It consists of five chapters. In chapter 1 we present basic notions and summarize the main results of the thesis. In chapters 2-4 we deal with different types of Hardy inequalities for Laplace and Grushin operators with magnetic and non-magnetic fields. It was shown in an article by Laptev and Weidl that for some magnetic forms in two dimensions, the Hardy inequality holds in its classical form. More precisely, by considering the Aharonov-Bohm magnetic potential, we can improve the constant in the respective Hardy inequality. In chapter 2 we establish an Lp - Hardy inequality related to Laplacians with magnetic fields with Aharonov-Bohm vector potentials. In chapter 3 we introduce a suitable notion of a vector field for the Grushin sub-elliptic operator G and obtain an improvement of the Hardy inequality, which was previously obtained in the paper of N. Garofallo and E. Lanconelli. In chapter 4 we find an Lp version of the Hardy inequality obtained in chapter 2. Finally in chapter 5 we aim to find the CLR and Lieb-Thirringbninequalities for harmonic Grushin-type operators. As the Grushin operator is non-elliptic, these inequalities will not take their classical form.

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22

Sloane, Craig Andrew. "Hardy-Sobolev-Maz'ya inequalities for fractional integrals on halfspaces and convex domains". Diss., Georgia Institute of Technology, 2011. http://hdl.handle.net/1853/41125.

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This thesis will present new results involving Hardy and Hardy-Sobolev-Maz'ya inequalities for fractional integrals. There are two key ingredients to many of these results. The first is the conformal transformation between the upper halfspace and the unit ball. The second is the pseudosymmetric halfspace rearrangement, which is a type of rearrangment on the upper halfspace based on Carlen and Loss' concept of competing symmetries along with certain geometric considerations from the conformal transformation. After reducing to one dimension, we can use the conformal transformation to prove a sharp Hardy inequality for general domains, as well as an improved fractional Hardy inequality over convex domains. Most importantly, the sharp constant is the same as that for the halfspace. Two new Hardy-Sobolev-Maz'ya inequalities will also be established. The first will be a weighted inequality that has a strong relationship with the pseudosymmetric halfspace rearrangement. Then, the psuedosymmetric halfspace rearrangement will play a key part in proving the existence of the standard Hardy-Sobolev-Maz'ya inequality on the halfspace, as well as some results involving the existence of minimizers for that inequality.

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23

Kalybay, Aigerim A. "A new development of Nikol'skii - Lizorkin and Hardy type inequalities with applications /". Luleå : Luleå University of Technology, 2006. http://epubl.ltu.se/1402-1544/2006/21/.

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24

Ruszkowski, Bartosch [Verfasser], i Timo [Akademischer Betreuer] Weidl. "Spectral and Hardy inequalities for the Heisenberg Laplacian / Bartosch Ruszkowski ; Betreuer: Timo Weidl". Stuttgart : Universitätsbibliothek der Universität Stuttgart, 2017. http://d-nb.info/113065706X/34.

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25

Albuquerque, Nacib André Gurgel e. "Hardy-Littlewood/Bohnenblust-Hille multilinear inequalities and Peano curves on topological vector spaces". Universidade Federal da Paraíba, 2014. http://tede.biblioteca.ufpb.br:8080/handle/tede/7448.

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This work is divided in two subjects. The first concerns about the Bohnenblust-Hille and Hardy- Littlewood multilinear inequalities. We obtain optimal and definitive generalizations for both inequalities. Moreover, the approach presented provides much simpler and straightforward proofs than the previous one known, and we are able to show that in most cases the exponents involved are optimal. The technique used is a combination of probabilistic tools and of an interpolative approach; this former technique is also employed in this thesis to improve the constants for vector-valued Bohnenblust-Hille type inequalities. The second subject has as starting point the existence of Peano spaces, that is, Haurdor spaces that are continuous image of the unit interval. From the point of view of lineability we analyze the set of continuous surjections from an arbitrary euclidean spaces on topological spaces that are, in some natural sense, covered by Peano spaces, and we conclude that large algebras are found within the families studied. We provide several optimal and definitive result on euclidean spaces, and, moreover, an optimal lineability result on those special topological vector spaces.
Este trabalho édividido em dois temas. O primeiro diz respeito às desigualdades multilineares de Bohnenblust-Hille e Hardy-Littlewood. Obtemos generalizações ótimas e definitivas para ambas desigualdades. Mais ainda, a abordagem apresentada fornece demonstrações mais simples e diretas do que as conhecidas anteriormente, além de sermos capazes de mostrar que os expoentes envolvidos são ótimos em varias situações. A técnica utilizada combina ferramentas probabilísticas e interpolativas; esta ultima e ainda usada para melhorar as estimativas das versões vetoriais da desigualdade de Bohnenblust-Hille. O segundo tema possui como ponto de partida a existência de espaços de Peano, ou seja, os espaços de Hausdor que são imagem contínua do intervalo unitário. Sob o ponto de vista da lineabilidade, analisamos o conjunto das sobrejecoes contínuas de um espaço euclidiano arbitrário em um espaço topológico que, de certa forma, e coberto por espaços de Peano, e concluímos que grandes álgebras são encontradas nas famílias estudadas. Fornecemos vários resultados ótimos e definitivos em espaços euclidianos, e, mais ainda, um resultado de lineabilidade ótimo naqueles espaços vetoriais topológicos especiais.

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26

Portmann, Fabian. "Spectral Inequalities and Their Applications in Quantum Mechanics". Doctoral thesis, KTH, Matematik (Avd.), 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-145210.

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The work presented in this thesis revolves around spectral inequalities and their applications in quantum mechanics. In Paper A, the ground state energy of an atom confined to two dimensions is analyzed in the limit when the charge of the nucleus Z becomes very large. The main result is a two-term asymptotic expansion of the ground state energy in terms of Z. Paper B deals with Hardy inequalities for the kinetic energy of a particle in the presence of an external magnetic field. If the magnetic field has a non-trivial radial component, we show that Hardy’s classical lower bound can be improved by an extra term depending on the magnetic field. In Paper C we study interacting Bose gases and prove Lieb-Thirring type estimates for several types of interaction potentials, such as the hard-sphere interaction in three dimensions, the hard-disk interaction in two dimensions as well as homogeneous potentials.

QC 20140520

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27

Moradifam, Amir. "Hardy-Rellich inequalities and the critical dimension of fourth order nonlinear elliptic eigenvalue problems". Thesis, University of British Columbia, 2010. http://hdl.handle.net/2429/27775.

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This thesis consists of three parts and manuscripts of seven research papers studying improved Hardy and Hardy-Rellich inequalities, nonlinear eigenvalue problems, and simultaneous preconditioning and symmetrization of linear systems. In the first part that consists of three research papers we study improved Hardy and Hardy-Rellich inequalities. In sections 2 and 3, we give necessary and sufficient conditions on a pair of positive radial functions V and W on a ball B of radius R in Rn, n ≥ 1, so that the following inequalities hold for all u \in C_{0}^{\infty}(B): \begin{equation*} \label{one} \hbox{$\int_{B}V(x)|\nabla u²dx \geq \int_{B} W(x)u²dx,} \end{equation*} \begin{equation*} \label{two} \hbox{$\int_{B}V(x)|\Delta u|²dx \geq \int_{B} W(x)|\nabla %%@ u|^²dx+(n-1)\int_{B}(\frac{V(x)}{|x|²}-\frac{V_r(|x|)}{|x|})|\nabla u|²dx. \end{equation*} This characterization makes a very useful connection between Hardy-type inequalities and the oscillatory behavior of certain ordinary differential equations. This allows us to improve, extend, and unify many results about Hardy and Hardy-Rellich type inequalities. In section 4, with a similar approach, we present various classes of Hardy-Rellich inequalities on H²\cap H¹₀ The second part of the thesis studies the regularity of the extremal solution of fourth order semilinear equations. In sections 5 and 6 we study the extremal solution u_{\lambda^*}$ of the semilinear biharmonic equation $\Delta² u=\frac{\lambda}{(1-u)², which models a simple Micro-Electromechanical System (MEMS) device on a ball B\subset R^N, under Dirichlet or Navier boundary conditions. We show that u* is regular provided N ≤ 8 while u_{\lambda^*} is singular for N ≥ 9. In section 7, by a rigorous mathematical proof, we show that the extremal solutions of the bilaplacian with exponential nonlinearity is singular in dimensions N ≥ 13. In the third part, motivated by the theory of self-duality we propose new templates for solving non-symmetric linear systems. Our approach is efficient when dealing with certain ill-conditioned and highly non-symmetric systems.

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28

Abylayeva, Akbota. "Inequalities for some classes of Hardy type operators and compactness in weighted Lebesgue spaces". Doctoral thesis, Luleå tekniska universitet, Institutionen för teknikvetenskap och matematik, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:ltu:diva-59667.

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This PhD thesis is devoted to investigate weighted differential Hardy inequalities and Hardy-type inequalities with the kernel when the kernel has an integrable singularity, and also the additivity of the estimate of a Hardy type operator with a kernel.The thesis consists of seven papers (Papers 1, 2, 3, 4, 5, 6, 7) and an introduction where a review on the subject of the thesis is given. In Paper 1 weighted differential Hardy type inequalities are investigated on the set of compactly supported smooth functions, where necessary and sufficient conditions on the weight functions are established for which this inequality and two-sided estimates for the best constant hold. In Papers 2, 3, 4 a more general class of $\alpha$ -order fractional integrationoperators are considered including the well-known classical Weyl, Riemann-Liouville, Erdelyi-Kober and Hadamard operators. Here 0 < $\alpha$ < 1. In Papers 2 and 3 the boundedness and compactness of two classes of such operators are investigated namely of Weyl and Riemann-Liouville type, respectively, in weighted Lebesgue spaces for 1 < p ≤ q < 1 and 0 < q < p < ∞. As applications some new results for the fractional integration operators of Weyl, Riemann-Liouville, Erdelyi-Kober and Hadamard are given and discussed.In Paper 4 the Riemann-Liouville type operator with variable upper limit is considered. The main results are proved by using a localization method equipped with the upper limit function and the kernel of the operator. In Papers 5 and 6 the Hardy operator with kernel is considered, where the kernel has a logarithmic singularity. The criteria of the boundedness and compactness of the operator in weighted Lebesgue spaces are given for 1 < p ≤ q < ∞ and 0 < q < p < ∞, respectively. In Paper 7 we investigated the weighted additive estimates for integral operators K+ and K¯ defined by K+ ƒ(x) := ∫ k(x,s) ƒ(s)ds, K¯ ƒ(x) := ∫ k(x,s)ƒ(s)ds. It is assumed that the kernel k of the operators K+and K- belongs to the general Oinarov class. We derived the criteria for the validity of these addittive estimates when 1 ≤ p≤ q < ∞

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Araújo, Gustavo da Silva. "Some classical inequalities, summability of multilinear operators and strange functions". Universidade Federal da Paraíba, 2016. http://tede.biblioteca.ufpb.br:8080/handle/tede/9310.

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This work is divided into three parts. In the first part, we investigate the behavior of the constants of the Bohnenblust–Hille and Hardy–Littlewood polynomial and multilinear inequalities. In the second part, we show an optimal spaceability result for a set of non-multiple summing forms on `p and we also generalize a result related to cotype (from 2010) as highlighted by G. Botelho, C. Michels, and D. Pellegrino. Moreover, we prove new coincidence results for the class of absolutely and multiple summing multilinear operators (in particular, we show that the well-known Defant–Voigt theorem is optimal). Still in the second part, we show a generalization of the Bohnenblust–Hille and Hardy–Littlewood multilinear inequalities and we present a new class of summing multilinear operators, which recovers the class of absolutely and multiple summing operators. In the third part, it is proved the existence of large algebraic structures inside, among others, the family of Lebesgue measurable functions that are surjective in a strong sense, the family of non-constant di↵erentiable real functions vanishing on dense sets, and the family of noncontinuous separately continuous real functions.
Este trabalho est´a dividido em trˆes partes. Na primeira parte, investigamos o comportamento das constantes das desigualdades polinomial e multilinear de Bohnenblust–Hille e Hardy–Littlewood. Na segunda parte, mostramos um resultado ´otimo de espa¸cabilidade para o complementar de uma classe de operadores m´ultiplo somantes em `p e tamb´em generalizamos um resultado relacionado a cotipo (de 2010) devido a G. Botelho, C. Michels e D. Pellegrino. Al´em disso, provamos novos resultados de coincidˆencia para as classes de operadores multilineares absolutamente e m´ultiplo somantes (em particular, mostramos que o famoso teorema de Defant–Voigt ´e ´otimo). Ainda na segunda parte, mostramos uma generaliza¸c˜ao das desigualdades multilineares de Bohnenblust–Hille e Hardy–Littlewood e apresentamos uma nova classe de operadores multilineares somantes, a qual recupera as classes dos operadores multilineares absolutamente e m´ultiplo somantes. Na terceira parte, provamos a existˆencia de grandes estruturas alg´ebricas dentro de certos conjuntos, como, por exemplo, a fam´ılia das fun¸c˜oes mensur´aveis `a Lebesgue que s˜ao sobrejetivas em um sentido forte, a fam´ılia das fun¸c˜oes reais n˜ao constantes e diferenci´aveis que se anulam em um conjunto denso e a fam´ılia das fun¸c˜oes reais n˜ao cont´ınuas e separadamente cont´ınuas.

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30

Ekholm, Tomas. "Schrödinger Operators in Waveguides". Doctoral thesis, KTH, Mathematics (Dept.), 2005. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-410.

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In this thesis, which consists of four papers, we study the discrete spectrum of Schrödinger operators in waveguides. In these domains the quadratic form of the Dirichlet Laplacian operator does not satisfy any Hardy inequality. If we include an attractive electric potential in the model or curve the domain, then bound states will always occur with energy below the bottom of the essential spectrum. We prove that a magnetic field stabilises the threshold of the essential spectrum against small perturbations. We deduce this fact from a magnetic Hardy inequality, which has many interesting applications in itself.

In Paper I we prove the magnetic Hardy inequality in a two-dimensional waveguide. As an application, we establish that when a magnetic field is present, a small local deformation or a small local bending of the waveguide will not create bound states below the essential spectrum.

In Paper II we study the Dirichlet Laplacian operator in a three-dimensional waveguide, whose cross-section is not rotationally invariant. We prove that if the waveguide is locally twisted, then the lower edge of the spectrum becomes stable. We deduce this from a Hardy inequality.

In Paper III we consider the magnetic Schrödinger operator in a three-dimensional waveguide with circular cross-section. If we include an attractive potential, eigenvalues may occur below the bottom of the essential spectrum. We prove a magnetic Lieb-Thirring inequality for these eigenvalues. In the same paper we give a lower bound on the ground state of the magnetic Schrödinger operator in a disc. This lower bound is used to prove a Hardy inequality for the magnetic Schrödinger operator in the original waveguide setting.

In Paper IV we again study the two-dimensional waveguide. It is known that if the boundary condition is changed locally from Dirichlet to magnetic Neumann, then without a magnetic field bound states will occur with energies below the essential spectrum. We however prove that in the presence of a magnetic field, there is a critical minimal length of the magnetic Neumann boundary condition above which the system exhibits bound states below the threshold of the essential spectrum. We also give explicit bounds on the critical length from above and below.

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31

Zghal, Mohamed Khalil. "Inégalités de type Trudinger-Moser et applications". Thesis, Paris Est, 2016. http://www.theses.fr/2016PESC1077/document.

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Cette thèse porte sur quelques inégalités de type Trudinger-Moser et leurs applications à l'étude des injections de Sobolev qu'elles induisent dans les espaces d'Orlicz et à l'analyse d'équations aux dérivées partielles non linéaires à croissance exponentielle.Le travail qu'on présente ici se compose de trois parties. La première partie est consacrée à la description du défaut de compacité de l'injection de Sobolev 4D dans l'espace d'Orlicz dansle cadre radial.L'objectif de la deuxième partie est double. D'abord, on caractérise le défaut de compacité de l'injection de Sobolev 2D dans les différentes classes d'espaces d'Orlicz. Ensuite, on étudiel'équation de Klein-Gordon semi-linéaire avec non linéarité exponentielle, où la norme d'Orlicz joue un rôle crucial. En particulier, on aborde les questions d'existence globale, de complétude asymptotique et d'étude qualitative.Dans la troisième partie, on établit des inégalités optimales de type Adams, en étroite relation avec les inégalités de Hardy, puis on fournit une description du défaut de compacité des injections de Sobolev qu'elles induisent
This thesis focuses on some Trudinger-Moser type inequalities and their applications to the study of Sobolev embeddings they induce into the Orlicz spaces, and the investigation of nonlinear partial differential equations with exponential growth.The work presented here includes three parts. The first part is devoted to the description of the lack of compactness of the 4D Sobolev embedding into the Orlicz space in the radialframework.The aim of the second part is twofold. Firstly, we characterize the lack of compactness of the 2D Sobolev embedding into the different classes of Orlicz spaces. Secondly, we undertakethe study of the nonlinear Klein-Gordon equation with exponential growth, where the Orlicz norm plays a crucial role. In particular, issues of global existence, scattering and qualitativestudy are investigated.In the third part, we establish sharp Adams-type inequalities invoking Hardy inequalities, then we give a description of the lack of compactness of the Sobolev embeddings they induce

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32

Kumar, Rakesh. "Hardy's inequalities for Grushin operator and Hermite multipliers on Modulation spaces". Thesis, 2021. https://etd.iisc.ac.in/handle/2005/5582.

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This thesis consists of two broad themes. First one revolves around the Hardy's inequalities for the fractional power of Grushin operator $\G$ which is chased via two different approaches. In the first approach, we first prove Hardy's inequality for the generalized sublaplacian defined on $\R\times\R^+$, then using the spherical harmonics, applying Hardy's inequality for individual components, we derive Hardy's inequality for Grushin operator. The techniques used for deriving Hardy's inequality for generalized sublaplacian are in parallel with the ones used in \cite{thangaveluroncal}. We first find Cowling-Haagerup type of formula for the fractional generalised sublaplacian and then using the modified heat kernel, we find integral representations of fractional generalized sublaplacian. Then we derive Hardy's inequality for generalized sublaplacian. In the second approach, we start with an extension problem for Grushin, with initial condition $f\in L^p(\R^{n+1})$. We derive a solution $u(\cdot,\rho)$ to that extension problem and show that solution goes to $f$ in $L^p(\R^{n+1})$ as the extension variable $\rho$ goes to $0$. Further $-\rho^{1-2s}\partial_\rho u $ goes to $B_s\G_s f$ in $L^p(\R^{n+1})$ as $\rho$ goes to $0$, thereby giving us an another way of defining fractional powers of Grushin operator $\G_s$. We also derive trace Hardy inequality for the Grushin operator with the help of extension problem. Finally we prove $L^p$-$L^q$ inequality for fractional Grushin operator, thereby deriving Hardy-Littlewood-Sobolov inequality for the Grushin operator.\\ Second theme consists of Hermite multipliers on modulation spaces $M^{p,q}(\R^n)$. We find a relation between sublaplacian multipliers $m(\tilde{\L})$ on polarised Heisenberg group $\Hb^n_{pol}$ and Hermite multipliers $m(\H)$ on modulation spaces $M^{p,q}(\R^n)$, thereby deriving the conditions on the multipliers $m$ to be Hermite multipliers on modulation spaces. We believe that the conditions on multipliers that we have found are more strict than required. We improve the results for the case the modulation spaces $M^{p,q}(\R^n)$ have $p=q$ by finding a relation between the boundedness of Hermite multipliers on $M^{p,p}$ and the boundedness of Fourier multipliers on torus $\T^n$. We also derive the conditions for boundedness of the solution of wave equation related to Hermite and the solution of Schr\"odinger equation related to Hermite on modulation spaces.
CSIR

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33

Lin, Shu-Huey, i 林淑惠. "Hardy-Littlewood type inequalities for Laguerre series". Thesis, 1995. http://ndltd.ncl.edu.tw/handle/69719741984032117283.

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34

JIAN, SHOU-PING, i 簡守平. "On hardy type inequalities in two variables". Thesis, 1992. http://ndltd.ncl.edu.tw/handle/43298069225292959158.

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35

Skrzypczak, Iwona. "Hardy–type inequalities and nonlinear eigenvalue problems". Doctoral thesis, 2013.

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Naszym celem jest wprowadzenie nowej metody konstruowania nier´owno´sci typu Hardy’ego. Konstruujemy je znaj¸ac rozwi¸azania u zagadnie´n p oraz A– harmonicznych. Wyprowadzamy nier´owno´sci typu Caccioppoli dla u. Jako wniosek z nich otrzymujemy wa˙zone nier´owno´sci typu Hardy’ego dla funkcji Lipschitzowskich o zwartym no´sniku.

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36

Su, Hung-Wei, i 蘇弘偉. "A study of Hardy''s inequalities in the weighted Hardy spaces". Thesis, 2006. http://ndltd.ncl.edu.tw/handle/46380868668375971128.

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碩士
國立中山大學
應用數學系研究所
94
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.

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37

"The Best constant for a general Sobolev-Hardy inequality". Chinese University of Hong Kong, 1991. http://library.cuhk.edu.hk/record=b5886942.

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by Chu Chiu Wing.
Thesis (M.Phil.)--Chinese University of Hong Kong, 1991.
Bibliography: leaves 31-32.
Introduction
Chapter Section 1. --- A Minimization Problem
Chapter Section 2. --- Radial Symmetry of The Solution
Chapter Section 3. --- Proof of The Main Theorem
References

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38

Zeng, Guan-Cheng, i 曾冠逞. "On Hardy-Hilbert Type Inequalities and Stability of Cauchy Additive Mappings". Thesis, 2008. http://ndltd.ncl.edu.tw/handle/80965927858906631726.

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碩士
國立中央大學
數學研究所
96
This thesis is concerned with two subjects of research; Hardy-Hilbert type inequalities and the stability of Cauchy additive mappings. The following are done : 1) to extend B. Yang''s result on the norm of a bounded self- adjoint integral operator T : L2 (0,∞) → L2 (0,∞) and its applications to Hardy-Hilbert type integral inequalities from the space L2 (0,∞) to the space Lp (0,∞) with p > 1 ; 2) to generalize Rassias''s theorem on the stability of Cauchy additive mappings ; 3) to give a correct proof of Park et al''s theorem in [6]; 4) to approximate the odd part of a certain vector mapping by a unique group homomorphism and ring homomorphism, respectively.

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39

Pappalardo, Francesco. "Weighted Multipolar Hardy Inequalities in R^N and Kolmogorov Type Operators". Tesi di dottorato, 2018. http://www.fedoa.unina.it/12587/1/pappalardo_francesco_31.pdf.

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The main purpose of the thesis, which describes the topics I was involved and the results achieved so far, is to introduce the multipolar weighted Hardy inequalities in R^N in the context of the study of Kolmogorov type operators perturbed by singular potentials and of the related evolution problems. The thesis describes, in the first part (Chapter 1), the reference results we can find in literature about the behaviour of the operators with inverse square potentials in the unipolar and multipolar case (existence and nonexistence of positive solutions to evolution problems with Schrodinger and Kolmogorov type operators and positivity of the quadratic form associated with Schrodinger operators). Furthermore we recall the Hardy inequalities in the case of Lebesgue measure and in the weighted case. In the second part (Chapters 2 and 3) we report our results about Kolmogorov type operators and weighted Hardy inequalities.

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40

Pasteczka, Paweł. "Analytic methods in inequalities concerning means". Doctoral thesis, 2015.

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41

BORDONI, SARA. "Nonlinear elliptic problems in the Heisenberg group". Doctoral thesis, 2018. http://hdl.handle.net/2158/1121183.

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The aim of this Ph.D. thesis is to present new results concerning the study of nonlinear elliptic problems in the context of the Heisenberg group. We deal with different problems, but the common thread consists in extending to a more general setting, the Heisenberg group, results proved in the Euclidean case. This generalization process in the Heisenberg framework implies a series of technical difficulties, that force the use of new key theorems.

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Bathory, Michal. "Konjugovaná funkce". Master's thesis, 2016. http://www.nusl.cz/ntk/nusl-347544.

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Using interpolation methods, new results on the boundedness of quasilinear joint weak type operators on Lorentz-Karamata (LK) spaces are established. LK spaces generalize many function spaces introduced before in literature, for example, the generalized Lorentz- Zygmund spaces, the Zygmund spaces, the Lorentz spaces and, of course, the Lebesgue spaces. The focus is mainly on the limiting cases of interpolation, where the spaces involved are, in certain sense, very close to the endpoint spaces. The results contain both necessary and sufficient conditions for the boundedness of the given operator on LK spaces. The complete characterization of embeddings of LK spaces is also included and the optimality of achieved results is then discussed. Finally, we apply our results to the conjugate function operator, which is known to be bounded on $L_p$ only if $1

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