Sharif Digital Repository

HAedbbergshtasrpraovecdtthefolowinginequalityforany1pnof;nandpIsuchft(xa)tCkfkpnp(Mf(x)1np .8Tfh2erLepe(xRisnt)saCindependent fHwuhhenedecrSbteioeobrnIgo'losefvinffie.nqieusqautlhiatleyityhReefloipsrsztihpneodtReenaeltiisinazglpwooftitefhntpaiarnlod,bil.Meem.fsrdeelnatoitnegtuhseinHgatrhdeyf-uLnitcttlieowno.ofodrmexaaxmimpalel SwohbeorelevCiniseqinudaeliptyencdaenntbeofexft,racactnkeIdbefrdofmekdnuHnspeepddbferroCgm'ksfHinepedqbuearlgit'ys.iHneeqrueawlietyw.iAllgseontehraeliczleasHsiecda-l berg'sinequalitytoOrliczspaces.
 

In this thesis, it has been introduced Hardy inequality and it’s extends. The fractional Hardy inequality for Ω ⊆ Rn and f ∈ C ∞ (Ω) is:1f (x) − f ( y)2∫α +n2dx dy ≥ kn,α ∫f (x)dx( )αΩ×Ωx − yΩ M α x In this thesis we are going to introduce the fractional and derivative forms of Hardy inequality then the Hardy inequality will be proved to fractional form on Euclidean and Hyperbolic domains, and finally we will get right into Hardy inequality with remainder.
 

In this thesis, we present some extension of Burton and Kirk fixed point theorem for the sum of two nonlinear operators, which one of them is compact and the other one is strict contraction; and we investigate the existence of fixed point where these two operators don’t need to be weakly continuous. Next, we will check the necessity of some conditions, like being strict contraction or compactness, in some specific Banach spaces, such as reflective spaces or Banach spaces equipped with uniformly convex norm.Finally, we analysis the existence of the solution for integral equations in L1 space by using the expressed theorems  

The so called T(b) theorem for boundedness of generalized CalderonZygmund singular integrals is investigated. This theorem provides a criterion, in which boundedness of the operator is equivalent to a special condition for the image of some function under action of that operator.Then we discuss some applications of the theorem in analysis and partial differential equations  

For a Banach space E and a compact metric space (X, d), a function F : X → E is a Lipschitz function if there exists k > 0 such that ∥F (x) − F (y)∥ ≤ kd(x, y) for all x, y ∈ X.The smallest such k is called the Lipschitz constant L(F ) for F . The space Lip(X, E) of all Lipschitz functions from X to E is a Banach space under the norm defined by ∥F ∥ = max{L(F ), ∥F ∥∞}, where ∥F ∥∞ = sup{∥F (x)∥ : x ∈ X}.Recent results characterizing isometries on these vector-valued Lipschitz spaces require the Banach space E to be strictly convex. We inves- tigate the nature of the extreme points of the dual ball for Lip(X, E) and use the information to describe the surjective isometries on Lip(X, E) under... 

Let C(S) and C(T) denote the sup-normed Banach spaces of real or complex-valued continuous functions on the compact Hausdorff spaces S and T, respectively. A linear map H:C(T)→C(S) is called separating if, for x,y∈C(T), xy≡0 implies HxHy≡0. In the second chapter of this thesis, we will show that any continuous separating map is a continuous multiple of a composition map. Moreover, it will be proved that any linear separating isomorphism of C(T) onto C(S) is continuous. We will also define separating and biseparating maps on the rings of continuous functions equipped with the compact-open topology. In addition, vector-valued separating maps will be investigated. For example, assume that C(T)... 

In this thesis we consider dynamical aspects of fractals. More precisely, answering questions like how heat diffuses on fractals and how does a material with fractal structure vibrates? To give an answer to these questions we need a PDE theory on fractals. Since fractals do not have smooth structures, defining differential operators like Laplacian is not possible from a classical viewpoint of analysis, to overcome such a difficulty we also need a theory of analysis on fractals. So as a good instance of analysis on fractals we first define Laplacian on Sierpinsky gasket and we try to extend the concept on other finitely ramified self-similar fractals. We also construct Dirichlet forms,... 

Monge’s problem was solved by Brenier in 1990. In general, the problem remained unresolved for a long time. some of its cases were solved by assumptions, but the general case and its analytical solution for the first time by Ian Bernier The French mathematician was introduced. He wrote his famous article Solved this nonlinear problem with technical assumptions. With the help of convex analysis and fundamental theorems in functions with vector values, he proved the existence and unity of this nonlinear problem. Monge’s problem, also known as optimal transport, suggests whether a stable­sized mapping can be done by having two probabilistic spaces and one cost function. (Inverted image of any... 

One of the important questions in mathematics is generalized. In case of metric spaces, one of the questions is: Can we extend notions of Riemannian geometry to arbitrary metric spaces smoothly? Ricci curvature is one of the most important concepts in differential geometry. Ricci curvature is defined for the Riemannian manifold and has many applications in mathematics and physics like Einstein’s equation in relativity theory. There is a good notion for a metric space having “sectional curvature bounded below by K” or “sectional curvature bounded above by K”, due to Alexandrov. Can we define the concept of Ricci curvature in metric space? A motivation for this question comes from Gromov’s... 

In this thesis, we introduce and prove the inequalities and theorems of Smulian which are about derivatives of convex functions and optimization of linear functionals. Then we show that some old ideas of Smulian can be used to give another proof of theorems of Bourgain. The first theorem of Bourgain associates to any bounded, convex, closed and dentable subset like D of a Banach space X a G-delta subset of the dual space and the second theorem of Bourgain investigates the optimization in Banach spaces. Finally, we characterize subsets of Banach spaces having the Radon-Nikodym property by means of optimization results.
 

The development of intrinsic theories for area-minimization problems was motivated in the 1950 by the diffichlty to prove, existence for the Plateau problem for surfaces in Euclidean spaces of dimension higher than two. After the pioneering work of R. Caccioppoli and E. De Giorgi on set with finite perimeter, W. H. Fleming and H. Federer developed the theory of currents, which leads to existence results for the Plateau problem for oriented surfaces of any dimension and codimension. The aim of this paper is to develop an existence of the Federer-Fleming theory to spaces without a differentiable structure, and virtually to any complete metric space. The starting point of our research has been... 

In this thesis, vve generali :ed the notion of (upper) gradient to arbitrary func- tions on a. metric measure space a.nd under some conditions -vve study deriva- t.ives of Lipschitz functions and their differential properties. Finally, we consider metric measure spaces that satisfy certa.in additiona.l conditions and give more detailed study of these notions  

Suppose $X$ is a locally compcat Hausdorf‌f space. We denote by $C_0(X)$ the Banach space of all continuous functions real or complex-valued def‌ined on $X$ which vanish at inf‌inity, equipped with the supremum norm. Suppose linear subspace $A$ of $C_0(X)$ is strongly separating, that is, for all pair of distinct points $x_1,x_2 \in X$, there exists $f \in A$ such that $|f(x_1)|\neq |f(x_2)|$. In this thesis, we show that Shilov boundary of $A$ exists and is closure Choquet boundary of $A$. Furthermore, \ we show that a linear isometry $T$ of $A$ onto such a subspace $B$ of $C_0(Y)$ induces a homeomorphism $h$ between two certain subspaces of the Shilov boundaries of $B$ and $A$, sending the... 

We prove that if an n-dimensional complete minimal submanifold M in hyperbolic space has sufficiently small total scalar curvature then M has only one end. We also prove that for such M there exist no nontrivial L^2 harmonic 1-forms on M  

The Kakeya needle problem was stated by a Japanese mathematician, Kakeya, in 1917 as follows: \What is the least area of a planar shape in which a needle of unit length can be turned 180 degrees?" Later, this problem was closely related to some problems in Fourier analysis and harmonic analysis, theory of partial dierential equations, theory of arithmetic combinatorics, and analytic number theory. In this thesis, after stating some elementary considerations in chapter one, we describe the Kakeya needle problem and Besicovitch's solution in chapter two. In chapter three there will be some discussions about the Fourier multiplier operator and in particular, disk multiplier operator, and we... 

The main purpose of this thesis is to establish the relationship between Orlicz-Hardy inequalities on a bounded Euclidean domain to certain fatness conditions on the comple-ment. Suppose Rn is a bounded domain and let d(x) := dist(x; @). We consider integral Hardy inequalities 8u 2 C1
0 ( ) Z ju(x)jd(x) C Z (jru(x)j)dxand norm Hardy inequalities8u 2 C1
0 () ju(x)jd(x) Cjkru(x)kj where : [0;1) ! [0;1) is a Young function and jk kj is representation of Luxemburg norm. Also the concept of fatness is introduced by concept of capacity  

In this thesis, we are going to study Sobolev classes of weakly differentiable mappings, specially Orlicz-Sobolev class, between compact Riemannian manifolds without boundary. These mappings posses less regularity than the mappings in the borderline case. Two major themes we investigate are smooth approximation of these mappings and the integrability of the Jacobian determinant. We impose no topological restrictions on manifolds in the approximation issue. We characterize classes of weakly differentiable mappings satisfying the approximation property and claim the Orlicz-Sobolev class is in these classes. The importance of our approach is that we are able to find tiny sets on which Sobolev...