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Register a new userHausdorff-Young inequality for Orlicz spaces on compact homogeneous manifolds
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We prove the classical Hausdorff-Young inequality for Orlicz spaces on compact homogeneous manifolds.
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Let $H$ be a compact subgroup of a locally compact group $G$ and let $m$ be the normalized $G$-invariant measure on homogeneous space $G/H$ associated with Weils formula. Let $varphi$ be a Young function satisfying $Delta_2$-condition. We introduce the notion of left module action of $L^1(G/H, m)$ on the Orlicz spaces $L^varphi(G/H, m).$ We also introduce a Banach left $L^1(G/H, m)$-submodule of $L^varphi(G/H, m).$
In this paper, for a locally compact commutative hypergroup $K$ and for a pair $(Phi_1, Phi_2)$ of Young functions satisfying sequence condition, we give a necessary condition in terms of aperiodic elements of the center of $K,$ for the convolution $fast g$ to exist a.e., where $f$ and $g$ are arbitrary elements of Orlicz spaces $L^{Phi_1}(K)$ and $L^{Phi_2}(K)$, respectively. As an application, we present some equivalent conditions for compactness of a compactly generated locally compact abelian group. Moreover, we also characterize compact convolution operators from $L^1_w(K)$ into $L^Phi_w(K)$ for a weight $w$ on a locally compact hypergroup $K$.
Given a compact (Hausdorff) group $G$ and a closed subgroup $H$ of $G,$ in this paper we present symbolic criteria for pseudo-differential operators on compact homogeneous space $G/H$ characterizing the Schatten-von Neumann classes $S_r(L^2(G/H))$ for all $0<r leq infty.$ We go on to provide a symbolic characterization for $r$-nuclear, $0< r leq 1,$ pseudo-differential operators on $L^{p}(G/H)$-space with applications to adjoint, product and trace formulae. The criteria here are given in terms of the concept of matrix-valued symbols defined on noncommutative analogue of phase space $G/H times widehat{G/H}.$ Finally, we present applications of aforementioned results in the context of heat kernels.
In this paper we extend classical Titchmarsh theorems on the Fourier transform of H$ddot{text{o}}$lder-Lipschitz functions to the setting of harmonic $NA$ groups, which relate smoothness properties of functions to the growth and integrability of their Fourier transform. We prove a Fourier multiplier theorem for $L^2$-H$ddot{text{o}}$lder-Lipschitz spaces on Harmonic $NA$ groups. We also derive conditions and a characterisation of Dini-Lipschitz classes on Harmonic $NA$ groups in terms of the behaviour of their Fourier transform. Then, we shift our attention to the spherical analysis on Harmonic $NA$ group. Since the spherical analysis on these groups fits well in the setting of Jacobi analysis we prefer to work in the Jacobi setting. We prove $L^p$-$L^q$ boundedness of Fourier multipliers by extending a classical theorem of H$ddot{text{o}}$rmander to the Jacobi analysis setting. On the way to accomplish this classical result we prove Paley-type inequality and Hausdorff-Young-Paley inequality. We also establish $L^p$-$L^q$ boundedness of spectral multipliers of the Jacobi Laplacian.
We consider a class of compact homogeneous CR manifolds, that we call $mathfrak n$-reductive, which includes the orbits of minimal dimension of a compact Lie group $K_0$ in an algebraic homogeneous variety of its complexification $K$. For these manifolds we define canonical equivariant fibrations onto complex flag manifolds. The simplest example is the Hopf fibration $S^3tomathbb{CP}^1$. In general these fibrations are not $CR$ submersions, however they satisfy a weaker condition that we introduce here, namely they are CR-deployments.
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