We show in two dimensions that if $Kf={\int }_{ℝ₊²}k(x,y)f\left(y\right)dy$, $k(x,y)=\left(e^{i{x}^a·y^b}\right){/\left(\right|x-y|}^{\eta })$, p = 4/(2+η), a ≥ b ≥ 1̅ = (1,1), $v_p\left(y\right)=y^{(p/p^{\text{'}})(1̅-b/a)}$, then ${\left|\right|Kf\left|\right|}_p{\le C\left|\right|f\left|\right|}_{p,v_p}$ if η + α₁ + α₂ < 2, ${\alpha }_j=1-b_j/a_j$, j = 1,2. Our methods apply in all dimensions and also for more general kernels.

For an absolutely continuous contraction T on a Hilbert space 𝓗, it is shown that the factorization of various classes of L¹ functions f by vectors x and y in 𝓗, in the sense that ⟨Tⁿx,y⟩ = f̂(-n) for n ≥ 0, implies the existence of invariant subspaces for T, or in some cases for rational functions of T. One of the main tools employed is the operator-valued Poisson kernel. Finally, a link is established between L¹ factorizations and the moment sequences studied in the Atzmon-Godefroy method, from...

Let E be a Riesz space. By defining the spaces $L{¹}_E$ and $L_E^{\infty }$ of E, we prove that the center $Z\left(L{¹}_E\right)$ of $L{¹}_E$ is $L_E^{\infty }$ and show that the injectivity of the Arens homomorphism m: Z(E)” → Z(E˜) is equivalent to the equality $L{¹}_E=Z{\left(E\right)}^{\text{'}}$. Finally, we also give some representation of an order continuous Banach lattice E with a weak unit and of the order dual E˜ of E in $L{¹}_E$ which are different from the representations appearing in the literature.

Let $P_t$ be the Markov semigroup generated by a weighted Laplace operator on a Riemannian manifold, with μ an invariant probability measure. If the curvature associated with the generator is bounded below, then the exponential convergence of $P_t$ in L¹(μ) implies its hypercontractivity. Consequently, under this curvature condition L¹-convergence is a property stronger than hypercontractivity but weaker than ultracontractivity. Two examples are presented to show that in general, however, L¹-convergence...

Let Ω be an open subset of ${ℝ}^d$ with 0 ∈ Ω. Furthermore, let $H_{\Omega }=-{\sum }_{i,j=1}^d{\partial }_i{c}_{ij}{\partial }_j$ be a second-order partial differential operator with domain $C_c^{\infty }\left(\Omega \right)$ where the coefficients $c_{ij}\in W_{loc}^{1,\infty }\left(\Omega ̅\right)$ are real, $c_{ij}=c_{ji}$ and the coefficient matrix $C=\left(c_{ij}\right)$ satisfies bounds 0 < C(x) ≤ c(|x|)I for all x ∈ Ω. If ${\int }_0^{\infty }ds{s}^{d/2}{e}^{-\lambda \mu \left(s\right)²}<\infty$ for some λ > 0 where $\mu \left(s\right)={\int }_0^{s}dtc{\left(t\right)}^{-1/2}$ then we establish that $H_{\Omega }$ is L₁-unique, i.e. it has a unique L₁-extension which generates a continuous semigroup, if and only if it is Markov unique, i.e. it has a unique L₂-extension which generates a submarkovian semigroup. Moreover...

Let m be a Radon measure on C without atoms. In this paper we prove that if the Cauchy transform is bounded in L2(m), then all 1-dimensional Calderón-Zygmund operators associated to odd and sufficiently smooth kernels are also bounded in L2(m).

Sea H un espacio de Hilbert complejo, separable y de dimensión infinita. Denotaremos por L(H) al álgebra de todos los operadores acotados en H. Carl Pearcy en 1977 introdujo el concepto de figura espectral de un operador T en L(H) [13]. Sin lugar a dudas hay dos resultados que hacen de la figura espectral de un operador un concepto importante. El primero se debe a Brown, Douglas y Fillmore:"Dos operadores esencialmente normales son débilmente equivalentes si y sólo si tienen la misma figura espectral".El...

Dans son livre [H. Stein, Ann. of Math. Studies, 63, Princeton Univ. Press, (1970)] E. Stein associe à tout opérateur de Sturm-Liouville la $g$-fonction de Littlewood-Paley et conjecture que, pour tout $p$ dans l’intervalle $]1,\infty [$, il existe deux constantes $C_p$ et $D_p$ telles que :$$C_p\parallel f{\parallel }_p\le \parallel g\left(f\right){\parallel }_p\le D_p\parallel f{\parallel }_p.$$On démontre ces inégalités pour une classe d’opérateurs différentiels singuliers sur $]0,\infty [$ et on énonce alors un résultat sur les multiplicateurs concernant ces opérateurs.