اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدLittlewood-Paley equivalence and homogeneous Fourier multipliers
79
0
0.0
(
0
)
تأليف
Shuichi Sato
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
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.
قيم البحث
اقرأ أيضاً
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 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.
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 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 furt
her 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 prove endpoint-type sparse bounds for Walsh-Fourier Marcinkiewicz multipliers and Littlewood-Paley square functions. These results are motivated by conjectures of Lerner in the Fourier setting. As a corollary, we obtain novel quantitative weighted
norm inequalities for these operators. Among these, we establish the sharp growth rate of the $L^p$ weighted operator norm in terms of the $A_p$ characteristic in the full range $1<p<infty$ for Walsh-Littlewood-Paley square functions, and a restricted range for Marcinkiewicz multipliers. Zygmunds $L{(log L)^{{frac12}}}$ inequality is the core of our lacunary multi-frequency projection proof. We use the Walsh setting to avoid extra complications in the arguments.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
