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Shunchao Long

Publications and source records attributed to Shunchao Long.

8 recordsLinked to original sources
arXiv2021

On estimate of operator for $0<p<\infty $

Operators such as Carleson operator are known to be bounded on $L^p$ for all $1<p<\infty$, but not from $L^1$ to weak-$L^1$ and from $H^p$ to $L^p$ for each $0<p\leq 1$, the object of this article is to give a estimate for all $0<p<\infty$. For the weights $w$ satisfying the doubling condition of order $q$ with $0<q<p$ and the reverse Hölder condition, by using some new functions spaces, we prove that: $\bullet$ some sublinear operators are bounded from some subspaces of $L^p_w$ to $L^p_w$ and to themselves for all $0<p< \infty$; in particular, these imply the endpoint estimates from $H^p_w$ to $L^p_w$ and from $H^p_w$ to itself for all $0<p\leq 1$; these results are applied to many operators, such as Hardy-Littlewood maximal operator, singular integral operators with rough kernels, Calderón commutators, Carleson operator, the polynomial Carleson operator, et al, and give the endpoint versions of classical theorems such as Carleson-Hunt theorem and a conjecture of Stein; $\bullet$ $H^p_w$ with $0<p\leq 1$ is characterized by blocks without vanishing moment conditions; $\bullet$ $H^p_w$ with $0<p\leq 1$ is characterized by a convolution maximal function with a non-smooth kernel.

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arXiv2021

Weighted estimates of commutators for $0<p<\infty$

We establish weighted inequalities for $BMO$ commutators of sublinear operators for all $0<p<\infty$. For weights $w$ satisfying the doubling condition of order $q$ with $0<q<p$ and the reverse Hölder condition, we prove that $\bullet$ commutators $T_b$, which are bounded on $L^p$ with $1<p<\infty$, are bounded from some subspaces of $L^p_w$ to $L^p_w$ and to themselves for all $0<p<\infty$, these are applied to the commutators of singular integral operators and Hardy-Littelwood maximal operator, et.al, which are known to fail to be bounded from $H^1$ to $L^1$ and whose estimate has been open problems for $0<p$ enough small; $\bullet$ commutators $T_b$, whose associated operators $T$ are bounded on $L^p$ with $1<p<\infty$, are bounded from some subspaces of $L^p_w$ to $L^p_w$ and from some subspaces of $L^p_w$ to others for all $0<p<\infty$, these are applied to the commutators of maximal operators such as singular integral maximal operators, Carleson operator and the polynomial Carleson operator, et.al, the estimate of these commutators has been open problems for each $0<p<\infty$; $\bullet$ in particular, these imply that the commutators above are bounded from $H^p_w$ to $L^p_w$ and to itself for all $0<p\leq 1$.

math.CA↗
arXiv2019

Littlewood-Paley Characterization for Musielak-Orlicz-Hardy Spaces Associated with Operators

Let $X$ be a space of homogeneous type. Assume that $L$ is an non-negative second-order self-adjoint operator on $L^2\left(X\right)$ with (heart) kernel associated to the semigroup $e^{ - tL}$ that satisfies the Gaussian upper bound. In this paper, the authors introduce a new characterization of the Musielak-Orlicz-Hardy Space $H_{\varphi, L}\left(X\right)$ associated with $L$ in terms of the Lusin area function where $\varphi$ is a growth function. Further, the authors prove that the Musielak-Orlicz-Hardy Space $H_{L,G,\varphi}\left(X\right)$ associated with $L$ in terms of the Littlewood-Paley function is coincide with $H_{\varphi, L}\left(X\right)$ and their norms are equivalent.

math.CA↗
arXiv2014

Differentiation of Integrals

No functions class for general measurable sets classes are known whose functions have the property of differentiability of integrals associated to such sets classes. In this paper,we give some subspaces of $L^s$ with $1<s<\infty$, whose functions are proven to have the differentiability of integrals associated to measurable sets classes in ${\bf R}^n $, this gives an answer to a question stated by Stein in his book Harmonic Analysis. We give also a example of some functions in these classes on ${\bf R}^2 $, which is continuous nowhere.

math.CA↗
arXiv2011

Estimates at or beyond endpoint in harmonic analysis: Bochner-Riesz means and spherical means

We introduce some new functions spaces to investigate some problems at or beyond endpoint. First, we prove that Bochner-Riesz means $B_R^λ$ are bounded from some subspaces of $L^p_{|x|^α}$ to $L^p_{|x|^α}$ for $ \frac{n-1}{2(n+1)}<λ\leq \frac{n-1}{2}, 0 < p\leq p'_λ=\frac{2n}{n+1+2λ}, n(\frac{p}{p_λ}-1)< α<n(\frac{p}{p'_λ}-1)$, and $0<R<\infty,$ and so are the maximal Bochner-Riesz means $B_*^λ$ for $ \frac{n-1}{2}\leq λ< \infty, 0 < p\leq 1$ and $-n< α<n(p-1)$. From these we obtain the $L^p_{|x|^α}$-norm convergent property of $B_R^λ$ for these $λ,p,$ and $α$. Second, let $n\geq 3,$ we prove that the maximal spherical means are bounded from some subspaces of $L^p_{|x|^α}$ to $L^p_{|x|^α}$ for $0<p\leq \frac{n}{n-1}$ and $ -n(1-\frac{p}{2})<α<n(p-1)-n$. We also obtain a $L^p_{|x|^α}$-norm convergent property of the spherical means for such $p$ and $α$. Finally, we prove that some new types of $|x|^α$-weighted estimates hold at or beyond endpoint for many operators, such as Hardy-Littlewood maximal operator, some maximal and truncated singular integral operators, the maximal Carleson operator, etc. The new estimates can be regarded as some substitutes for the $(H^p,H^p)$ and $(H^p,L^p)$ estimates for the operators which fail to be of types $(H^p,H^p)$ and $(H^p,L^p)$.

math.CA↗
arXiv2011

Convergence of Fourier series at or beyond endpoint

We consider several problems at or beyond endpoint in harmonic analysis. The solutions of these problems are related to the estimates of some classes of sublinear operators. To do this, we introduce some new functions spaces $RL^{p,s}_{|x|^α}({\bf R}^n)$ and $\dot{R}L^{p,s}_{|x|^α}({\bf R}^n)$, which play an analogue role with the classical Hardy spaces $H^p({\bf R}^n)$. These spaces are subspaces of $L^p_{|x|^α}({\bf R}^n)$ with $1<s<\infty, 0<p\leq s$ and $-n<α<n(p-1)$, and $\dot{R}L^{p,s}_{|x|^α}({\bf R}^n) \supset L^s({\bf R}^n)$ when $ -n<α<n(p/s-1)$. We prove the following results. First, $μ_α$-a.e. convergence and ${L}^{p}_{|x|^α}({\bf R})$ -norm convergence of Fourier series hold for all functions in $ RL^{p,s}_{|x|^α}({\bf R})$ and $ \dot{R}L^{p,s}_{|x|^α}({\bf R})$ with $1<s<\infty, 0<p\leq s$ and $-1<α<p-1$, where $μ_α(x)=|x|^α$; Second, many sublinear operators initially defined for the functions in $L^p({\bf R}^n)$ with $1<p<\infty$, such as Calderón-Zygmund operators, C.Fefferman's singular multiplier operator, R.Fefferman's singular integral operator, the Bochner-Riesz means at the critical index, certain oscillatory singular integral operators, and so on, admit extensions which map $RL^{p,s}_{|x|^α}({\bf R}^n)$ and $\dot{R}L^{p,s}_{|x|^α}({\bf R}^n)$ into $L^p_{|x|^α}({\bf R}^n)$ with $1<s<\infty, 0<p\leq s$ and $-n<α<n(p-1)$; Final, Hardy-Littlewood maximal operator is bounded from $RL^{p,s}_{|x|^α}({\bf R}^n)$ (or $\dot{R}L^{p,s}_{|x|^α}({\bf R}^n)$) to ${L}^{p}_{|x|^α}({\bf R}^n)$ for $ 1<s<\infty$ and $0<p\leq s$ if and only if $-n<α<n(p-1)$.

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