Let $\Phi \left(t\right)={ʃ}_0^{t}a\left(s\right)ds$ and $\Psi \left(t\right)={ʃ}_0^{t}b\left(s\right)ds$, where a(s) is a positive continuous function such that ${ʃ}_1^{\infty }a\left(s\right)/sds=\infty$ and b(s) is quasi-increasing and $li{m}_{s\to \infty }b\left(s\right)=\infty$. Then the following statements for the Hardy-Littlewood maximal function Mf(x) are equivalent: (j) there exist positive constants $c_1$ and $s_0$ such that ${ʃ}_1^{s}a\left(t\right)/tdt\ge c_{1}b\left(c_{1}s\right)$ for all $s\ge s_0$; (jj) there exist positive constants $c_2$ and $c_3$ such that ${ʃ}_0^{2\pi }\Psi (\left(c_2\right)/{\left(\right|⨍|}_{}\left)\right|⨍\left(x\right)\left|\right)dx\le c_3+c_3{ʃ}_0^{2\pi }{\Phi (1/(\left|⨍\right|}_{}\left)\right)Mf\left(x\right)dx$ for all $⨍\in L^1\left(\right)$.

We define an operator α on ℂ³ ⊗ ℂ³ associated with the quantum group $U_q\left(2\right)$, which satisfies the Yang-Baxter equation and a cubic equation (α² - 1)(α + q²) = 0. This operator can be extended to a family of operators $h_j:=I_j\otimes \alpha \otimes I_{n-2-j}$ on ${\left(ℂ³\right)}^{\otimes n}$ with 0 ≤ j ≤ n - 2. These operators generate the cubic Hecke algebra ${\mathscr{H}}_{q,n}\left(2\right)$ associated with the quantum group $U_q\left(2\right)$. The purpose of this note is to present the construction.

By using D. Preiss' approach to a construction from a paper by J. Matoušek and E. Matoušková, and some results of E. Matoušková, we prove that we can decompose a separable Banach space with modulus of convexity of power type p as a union of a ball small set (in a rather strong symmetric sense) and a set which is Aronszajn null. This improves an earlier unpublished result of E. Matoušková. As a corollary, in each separable Banach space with modulus of convexity of power type p, there exists a closed...

A definition of seminorm in the Sobolev space Ws,p (Γ) on a smooth compact manifold Gamma without boundary, using a localization procedure without partition of unity.

The classical theorem of Borsuk and Ulam [2] says that for any continuous mapping $f:S^k\to {ℝ}^k$ there exists a point $x\in S^k$ such that f(-x) = f(x). In this note a discrete version of the antipodal theorem is proved in which $S^k$ is replaced by the set of vertices of a high-dimensional cube equipped with Hamming’s metric. In place of equality we obtain some optimal estimates of $in{f}_x\left|\right|f\left(x\right)-f(-x)\left|\right|$ which were previously known (as far as the author knows) only for f linear (cf. [1]).

We transfer a renorming method of transfer, due to G. Godefroy, from weakly compactly generated Banach spaces to Vašák, i.e., weakly K-countably determined Banach spaces. Thus we obtain a new construction of a locally uniformly rotund norm on a Vašák space. A further cultivation of this method yields the new result that every dual Vašák space admits a dual locally uniformly rotund norm.

Sea T un operador lineal acotado e inyectivo de un espacio de Banach X en un espacio de Hilbert H con rango denso y sea {xn} ⊂ X una sucesión tal que {Txn} es ortogonal. Se estudian propiedades de {Txn} dependientes de propiedades de {xn}. También se estudia la ""situación opuesta"", es decir, la acción de un operador T : H → X sobre sucesiones ortogonales.

A modification of the Nikolskij extension theorem for functions from Sobolev spaces $H^k\left(\Omega \right)$ is presented. This modification requires the boundary $\partial \Omega$ to be only Lipschitz continuous for an arbitrary $k\in \mathbb{N}$; however, it is restricted to the case of two-dimensional bounded domains.