Functional Inequalities in Stratified Lie groups with Sobolev, Besov, Lorentz and Morrey spaces

The study of Sobolev inequalities can be divided in two cases: p = 1 and 1 < p < +$\infty$. In the case p = 1 we study here a relaxed version of refined Sobolev inequalities. When p > 1, using as base space classical Lorentz spaces associated to a weight from the Arino-Muckenhoupt class Bp, we will study Gagliardo-Nirenberg inequalities. As a by-product we will also consider Morrey-Sobolev inequalities. These arguments can be generalized to many different frameworks, in particular the proofs are given in the setting of stratified Lie groups

Some new results concerning Lorentz sequence spaces and Schur multipliers : characterization of some new Banach spaces of infinite matrices

This Licentiate thesis consists of an introduction and three papers, which deal with some new spaces of infinite matrices and Lorentz sequence spaces.In the introduction we give an overview of the area that serves as a frame for the rest of the thesis. In particular, a short description of Schur multipliers is given.In Paper 1 we prove that the space of all bounded operators on $\ell^2$ is contained in the space of all Schur multipliers on $B_w(\ell^2)$, where $B_w(\ell^2)$ is the space of linear (unbounded) operators on $\ell^2$ which map decreasing sequences from $\ell^2$ into sequences from $\ell^2$.In Paper 2 using a special kind of Schur multipliers and G. Bennett's factorization technique we characterize the upper triangular positive matrices from $B_w(\ell^p)$, $1In Paper 3 we consider the Lorentz spaces$\ell^{p,q}$in the range$1$$\|x\|_{p,q}=\left(\sum_{n=1}^\infty (x^*)^q n^{\frac{q}{p}-1}\right)^\frac{1}{q}$$is only a quasi-norm. In particular, we derive the optimal constant in the triangle inequality for this quasi-norm, which leads us to consider the following decomposition norm:$$\|x\|_{(p,q)}=\inf\{\sum_k \|x^{(k)}\|_{p,q}\},$$where the infimum is taken over all finite representations $x=\sum_k x^{(k)}$.Godkänd; 2009; 20090423 (ancmar); LICENTIATSEMINARIUM Ämnesområde: Matematik/Mathematics Examinator: Professor Lars-Erik Persson, Luleå tekniska universitet Tid: Onsdag den 3 juni 2009 kl 10.15 Plats: D 2214, Luleå tekniska universitet</p

Some new results concerning Banach spaces of infinite matrices and Lorentz sequence spaces

This PhD thesis consists of an introduction and five papers, Which deal with some new spaces of infinite matrices and Lorentz sequence spaces.In the introduction we give an overview of the area that serves as a frame for the rest of the thesis. In particular, a short description of Schur multipliers is given.In Paper 1 we prove that the space of all bounded operators on \ ell ^ 2 is contained in the space of all Schur multipliers on $B_w (\ ell ^ 2)$, where $B_w (\ ell ^ 2)$ is the space of linear (unbounded) operators on $\ ell ^ 2$ which map sequences decreasing from \ ell ^ 2 into sequences from $\ ell ^ 2$.In Paper 2 using a special kind of Schur multipliers and G. Bennett 's actually Riza tion technique we characterize the upper triangular matrices positive from $B_w (\ ell ^ p)$, $1 In Paper 3 we consider the Lorentz spaces$ \ ell ^ (p, q) $in the range$ 1 \ [\ | x \ | _ (p, q) = \ left (\ sum_ (n = 1) ^ \ infty (x_n ^ *) ^ qn ^ (\ frac (q) (p) -1) \ right) ^ \ frac (1) (q)\]is only a quasi-norm. In particular, we derive the optimal constant in the triangle inequality for this quasi-norm, Which leads us to consider the following decomposition rule:\ [\ | x \ | _ ((p, q)) = \ inf \ (\ sum_k \ | x ^ ((k)) \ | _ (p, q) \);\]Where the Infimum is taken over all finite representation $x = \ sum_k x ^ ((k))$.In Paper 4 we denotes by $B_p (\ ell ^ 2)$ the Besov-Schatten space of all upper triangular matrices $A$ such that\ [\ | A \ | _ (B_p (\ ell ^ 2)) = \ left [\ int_0 ^ 1 (1-r ^ 2) ^ (2p) \ | A''(r) \ | _ (C_p) ^ pd \ lambda (r) \ right] ^ \ frac (1) (p) &lt;\ infty.\]and we prove a natural relationship between the Bergman projection and the Besov-Schatten spaces.In Paper 5, given a matrix $A$ satisfying that\ [Ax \ in \ ell ^ p \ text (for every) x \ in \ ell ^ p \ text (with) | x_k | \ searrow 0\]For 1 \ leq p <\ infty , we show that\ [A \ in B_w (\ ell ^ p) \ text (if and only if) \ sup_ (n \ in \ mathbb N ^ *) \ left (\ frac (1) (n) \ sum_ (k = 1) ^ n | a_k | ^ p \ right) ^ \ frac (1) (p) &lt;\ infty\]For $A = a_0$ given by $a = (a_k) _ (k \ in \ mathbb (N))$. We prove that there exist some classes of operators either Belonging to $B_w (\ ell ^ p)$ or to the space of all Schur multipliers on $B_w (\ ell ^ p)$.Godkänd; 2010; 20100127 (ancmar); DISPUTATION Ämnesområde: Matematik/Mathematics Opponent: Professor Vladimir Stepanov, Peoples Friendship University, Moskva, Ryssland Ordförande: Professor Lars-Erik Persson, Luleå tekniska universitet Tid: Fredag den 19 mars 2010, kl 13.00 Plats: D 2214-15, Luleå tekniska universitet</p

On a class of linear operators on lp and its Schur multipliers

In this paper we consider the spaces , , of infinite matrices defined by the norm . We consider the Schur product of matrices and prove that is not closed under this product. Moreover, we prove that linear and bounded operators on are Schur multipliers on , a result which is not obvious, since is not a Schur algebra. Most of the results are sharp in the sense that they are given via necessary and sufficient conditions. ( ) p w B 1p A 1 ( ) =1 =1 1 0 := sup p p p jk k B j k w x p xk A a x ( ) p w B p ( ) p w B ( ) p w BValiderad; 2016; Nivå 2; 20160510 (andbra

Some new characterizations of Bloch type spaces of infinite matrices via Schur multipliers

We consider the innite matrix version B(D, ℓ2) of the Bloch space. In this paperwe complement the results in the recent book [18] by deriving some new characteriza-tions of B(D, ℓ2) and related spaces and duals via Schur multipliers. As applicationswe nd the largest solid subspace of B(D, ℓ2) and its \conjugate" space I(ℓ2) consid-ered in [18] and [19].Validerad; 2015; Nivå 2; 20151013 (andbra