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We completely characterize those positive Borel measures $mu$ on the unit ball $mathbb{B}_ n$ such that the Carleson embedding from Hardy spaces $H^p$ into the tent-type spaces $T^q_ s(mu)$ is bounded, for all possible values of $0<p,q,s<infty$.
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In this note we give several characterisations of weights for two-weight Hardy inequalities to hold on general metric measure spaces possessing polar decompositions. Since there may be no differentiable structure on such spaces, the inequalities are given in the integral form in the spirit of Hardys original inequality. We give examples obtaining new weighted Hardy inequalities on $mathbb R^n$, on homogeneous groups, on hyperbolic spaces, and on Cartan-Hadamard manifolds.
In this note, we obtain a full characterization of radial Carleson measures for the Hilbert-Hardy space on tube domains over symmetric cones. For large derivatives, we also obtain a full characterization of the measures for which the corresponding embedding operator is continuous. Restricting to the case of light cones of dimension three, we prove that by freezing one or two variables, the problem of embedding derivatives of the Hilbert-Hardy space into Lebesgue spaces reduces to the characterization of Carleson measures for Hilbert-Bergman spaces of the upper-half plane or the product of two upper-half planes.
Let $mathcal{D}$ be the class of radial weights on the unit disk which satisfy both forward and reverse doubling conditions. Let $g$ be an analytic function on the unit disk $mathbb{D}$. We characterize bounded and compact Volterra type integration operators [ J_{g}(f)(z)=int_{0}^{z}f(lambda)g(lambda)dlambda ] between weighted Bergman spaces $L_{a}^{p}(omega )$ induced by $mathcal{D}$ weights and Hardy spaces $H^{q}$ for $0<p,q<infty$.
In this paper we characterize off-diagonal Carleson embeddings for both Hardy-Orlicz spaces and Bergman-Orlicz spaces of the upper-half plane. We use these results to obtain embedding relations and pointwise multipliers between these spaces.
Let $1leq p,q < infty$ and $1leq r leq infty$. We show that the direct sum of mixed norm Hardy spaces $big(sum_n H^p_n(H^q_n)big)_r$ and the sum of their dual spaces $big(sum_n H^p_n(H^q_n)^*big)_r$ are both primary. We do so by using Bourgains localization method and solving the finite dimensional factorization problem. In particular, we obtain that the spaces $big(sum_{nin mathbb N} H_n^1(H_n^s)big)_r$, $big(sum_{nin mathbb N} H_n^s(H_n^1)big)_r$, as well as $big(sum_{nin mathbb N} BMO_n(H_n^s)big)_r$ and $big(sum_{nin mathbb N} H^s_n(BMO_n)big)_r$, $1 < s < infty$, $1leq r leq infty$, are all primary.
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