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Register a new userOptimality of the rearrangement inequality with applications to Lorentz-type sequence spaces
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We characterize the sequences $(w_i)_{i=1}^infty$ of non-negative numbers for which [ sum_{i=1}^infty a_i w_i quad text{ is of the same order as } quad sup_n sum_{i=1}^n a_i w_{1+n-i} ] when $(a_i)_{i=1}^infty$ runs over all non-increasing sequences of non-negative numbers. As a by-product of our work we settle a problem raised in [F. Albiac, Jose L. Ansorena and B. Wallis; arXiv:1703.07772[math.FA]] and prove that Garling sequences spaces have no symmetric basis.
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We are proving a Bernstein type inequality in the shift-invariant spaces of $L_2(R)$.
Interpolation inequalities in Triebel-Lizorkin-Lorentz spaces and Besov-Lorentz spaces are studied for both inhomogeneous and homogeneous cases. First we establish interpolation inequalities under quite general assumptions on the parameters of the function spaces. Several results on necessary conditions are also provided. Next, utilizing the interpolation inequalities together with some embedding results, we prove Gagliardo-Nirenberg inequalities for fractional derivatives in Lorentz spaces, which do hold even for the limiting case when one of the parameters is equal to 1 or $infty$.
We find necessary and sufficient conditions on a family $mathcal{R} = (r_i)_{i in I}$ in a Boolean algebra $mathcal{B}$ under which there exists a unique positive probability measure $mu$ on $mathcal{B}$ such that $mu ( bigcap_{k=1}^n theta_k r_{i_k} ) = 2^{-n}$ for all finite collections of distinct indices $i_1, ldots, i_n in I$ and all collections of signs $theta_1, ldots, theta_n in {-1,1}$, where the product $theta x$ of a sign $theta$ by an element $x in mathcal{B}$ is defined by setting $1 x = x$ and $-1 x = - x = mathbf{1} setminus x$. Such a family we call a complete Rademacher family. We prove that Dedekind $sigma$-complete Boolean algebras admitting complete Rademacher systems of the same cardinality are isomorphic. As a consequence, we obtain that a Dedekind $sigma$-complete Boolean algebra is homogeneous measurable if and only if it admits a complete Rademacher family. This new way to define a measure on a Boolean algebra allows us to define classical systems on an arbitrary Riesz space, such as Rademacher and Haar. We define a complete Rademacher system of any cardinality and a countable complete Haar system on an element $e > 0$ of a vector lattice $E$ in such a way that if $e$ is an order unit of $E$ then the corresponding systems become complete for the entire $E$. We prove that if $E$ is Dedekind complete then any complete Haar system on $e$ is an order Schauder basis for the ideal $A_e$ generated by $e$. Finally, we develop a theory of integration in a Riesz space of elements of the band $B_e$ generated by a fixed $e > 0$ with respect to the measure on the Boolean algebra $mathfrak{F}_e$ of fragments of $e$ generated by a complete Rademacher family on $mathfrak{F}_e$. Much space is devoted to examples showing that our way of thinking is sharp (e.g., we show the essentiality of each of the condition in the definition of a Rademacher family).
The Kahane--Salem--Zygmund inequality is a probabilistic result that guarantees the existence of special matrices with entries $1$ and $-1$ generating unimodular $m$-linear forms $A_{m,n}:ell_{p_{1}}^{n}times cdotstimesell_{p_{m}}^{n}longrightarrowmathbb{R}$ (or $mathbb{C}$) with relatively small norms. The optimal asymptotic estimates for the smallest possible norms of $A_{m,n}$ when $left{ p_{1},...,p_{m}right} subsetlbrack2,infty]$ and when $left{ p_{1},...,p_{m}right} subsetlbrack1,2)$ are well-known and in this paper we obtain the optimal asymptotic estimates for the remaining case: $left{ p_{1},...,p_{m}right} $ intercepts both $[2,infty]$ and $[1,2)$. In particular we prove that a conjecture posed by Albuquerque and Rezende is false and, using a special type of matrices that dates back to the works of Toeplitz, we also answer a problem posed by the same authors.
We revisit and comment on the Harnack type determinantal inequality for contractive matrices obtained by Tung in the nineteen sixtieth and give an extension of the inequality involving multiple positive semidefinite matrices.
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