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In this paper we consider the $X_s$ spaces that lie between $H^1(R^n)$ and $L^1(R^n)$. We discuss the interpolation properties of these spaces, and the behavior of maximal functions and singular integrals acting on them.
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By using the notion of contraction of Lie groups, we transfer $L^p-L^2$ estimates for joint spectral projectors from the unit complex sphere $sfera$ in ${{mathbb{C}}}^{n+1}$ to the reduced Heisenberg group $h^{n}$. In particular, we deduce some estimates recently obtained by H. Koch and F. Ricci on $h^n$. As a consequence, we prove, in the spirit of Sogges work, a discrete restriction theorem for the sub-Laplacian $L$ on $h^n$.
This paper deals with the inequalities devoted to the comparison between the norm of a function on a compact hypergroup and the norm of its Fourier coefficients. We prove the classical Paley inequality in the setting of compact hypergroups which further gives the Hardy-Littlewood and Hausdorff-Young-Paley (Pitt) inequalities in the noncommutative context. We establish Hormanders $L^p$-$L^q$ Fourier multiplier theorem on compact hypergroups for $1<p leq 2 leq q<infty$ as an application of Hausdorff-Young-Paley inequality. We examine our results for the hypergroups constructed from the conjugacy classes of compact Lie groups and for a class of countable compact hypergroups.
We investigate an infinite, linear system of ordinary differential equations that models the evolution of fragmenting clusters. We assume that each cluster is composed of identical units (monomers) and we allow mass to be lost, gained or conserved during each fragmentation event. By formulating the initial-value problem for the system as an abstract Cauchy problem (ACP), posed in an appropriate weighted $ell^1$ space, and then applying perturbation results from the theory of operator semigroups, we prove the existence and uniqueness of physically relevant, classical solutions for a wide class of initial cluster distributions. Additionally, we establish that it is always possible to identify a weighted $ell^1$ space on which the fragmentation semigroup is analytic, which immediately implies that the corresponding ACP is well posed for any initial distribution belonging to this particular space. We also investigate the asymptotic behaviour of solutions, and show that, under appropriate restrictions on the fragmentation coefficients, solutions display the expected long-term behaviour of converging to a purely monomeric steady state. Moreover, when the fragmentation semigroup is analytic, solutions are shown to decay to this steady state at an explicitly defined exponential rate.
If alpha and beta are countable ordinals such that beta eq 0, denote by tilde{T}_{alpha,beta} the completion of $c_{00}$ with respect to the implicitly defined norm ||x|| = max{||x||_{c_{0}}, 1/2 sup sum_{i=1}^{j}||E_{i}x||}, where the supremum is taken over all finite subsets E_{1},...,E_{j} of $mathbb{N}$ such that $E_{1}<...<E_{j}$ and {min E_{1},...,min E_{j}} in S_beta. It is shown that the Bourgain $ell^{1}$-index of tilde{T}_{alpha,beta} is omega^{alpha+beta.omega}. In particular, if alpha =omega^{alpha_{1}}. m_{1}+...+omega^{alpha_{n}}. m_{n} in Cantor normal form and alpha_{n} is not a limit ordinal, then there exists a Banach space whose ell^{1}-index is omega^{alpha}.
In this paper, we give the definability of bilinear singular and fractional integral operators on Morrey-Banach space, as well as their commutators and we prove the boundedness of such operators on Morrey-Banach spaces. Moreover, the necessary condition for BMO via the bounedness of bilinear commutators on Morrey-Banach space is also given. As a application of our main results, we get the necessary conditions for BMO via the bounedness of bilinear integral operators on weighted Morrey space and Morrey space with variable exponents. Finally, we obtain the boundedness of bilinear C-Z operator on Morrey space with variable exponents.
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