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Register a new userCharacterization of parabolic Hardy spaces by Littlewood-Paley functions
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Shuichi Sato
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We consider Littlewood-Paley functions associated with non-isotropic dilations. We prove that they can be used to characterize the parabolic Hardy spaces of Calder{o}n-Torchinsky.
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We establish a characterization of the Hardy spaces on the homogeneous groups in terms of the Littlewood-Paley functions. The proof is based on vector-valued inequalities shown by applying the Peetre maximal function.
We consider certain Littlewood-Paley operators and prove characterization of some function spaces in terms of those operators. When treating weighted Lebesgue spaces, a generalization to weighted spaces will be made for Hormanders theorem on the invertibility of homogeneous Fourier multipliers. Also, applications to the theory of Sobolev spaces will be given.
Let ${mathbb{P}_t}_{t>0}$ be the classical Poisson semigroup on $mathbb{R}^d$ and $G^{mathbb{P}}$ the associated Littlewood-Paley $g$-function operator: $$G^{mathbb{P}}(f)=Big(int_0^infty t|frac{partial}{partial t} mathbb{P}_t(f)|^2dtBig)^{frac12}.$$ The classical Littlewood-Paley $g$-function inequality asserts that for any $1<p<infty$ there exist two positive constants $mathsf{L}^{mathbb{P}}_{t, p}$ and $mathsf{L}^{mathbb{P}}_{c, p}$ such that $$ big(mathsf{L}^{mathbb{P}}_{t, p}big)^{-1}big|fbig|_{p}le big|G^{mathbb{P}}(f)big|_{p} le mathsf{L}^{mathbb{P}}_{c,p}big|fbig|_{p},,quad fin L_p(mathbb{R}^d). $$ We determine the optimal orders of magnitude on $p$ of these constants as $pto1$ and $ptoinfty$. We also consider similar problems for more general test functions in place of the Poisson kernel. The corresponding problem on the Littlewood-Paley dyadic square function inequality is investigated too. Let $Delta$ be the partition of $mathbb{R}^d$ into dyadic rectangles and $S_R$ the partial sum operator associated to $R$. The dyadic Littlewood-Paley square function of $f$ is $$S^Delta(f)=Big(sum_{RinDelta} |S_R(f)|^2Big)^{frac12}.$$ For $1<p<infty$ there exist two positive constants $mathsf{L}^{Delta}_{c,p, d}$ and $ mathsf{L}^{Delta}_{t,p, d}$ such that $$ big(mathsf{L}^{Delta}_{t,p, d}big)^{-1}big|fbig|_{p}le big|S^Delta(f)big|_{p}le mathsf{L}^{Delta}_{c,p, d}big|fbig|_{p},quad fin L_p(mathbb{R}^d). $$ We show that $$mathsf{L}^{Delta}_{t,p, d}approx_d (mathsf{L}^{Delta}_{t,p, 1})^d;text{ and }; mathsf{L}^{Delta}_{c,p, d}approx_d (mathsf{L}^{Delta}_{c,p, 1})^d.$$ All the previous results can be equally formulated for the $d$-torus $mathbb{T}^d$. We prove a de Leeuw type transference principle in the vector-valued setting.
This paper obtains new characterizations of weighted Hardy spaces and certain weighted $BMO$ type spaces via the boundedness of variation operators associated with approximate identities and their commutators, respectively.
Using the Fourier analysis techniques on hyperbolic spaces and Greens function estimates, we confirm in this paper the conjecture given by the same authors in [43]. Namely, we prove that the sharp constant in the $frac{n-1}{2}$-th order Hardy-Sobolev-Mazya inequality in the upper half space of dimension $n$ coincides with the best $frac{n-1}{2}$-th order Sobolev constant when $n$ is odd and $ngeq9$ (See Theorem 1.6). We will also establish a lower bound of the coefficient of the Hardy term for the $k-$th order Hardy-Sobolev-Mazya inequality in upper half space in the remaining cases of dimension $n$ and $k$-th order derivatives (see Theorem 1.7). Precise expressions and optimal bounds for Greens functions of the operator $ -Delta_{mathbb{H}}-frac{(n-1)^{2}}{4}$ on the hyperbolic space $mathbb{B}^n$ and operators of the product form are given, where $frac{(n-1)^{2}}{4}$ is the spectral gap for the Laplacian $-Delta_{mathbb{H}}$ on $mathbb{B}^n$. Finally, we give the precise expression and optimal pointwise bound of Greens function of the Paneitz and GJMS operators on hyperbolic space, which are of their independent interest (see Theorem 1.10).
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