L p Bounds for the parabolic singular integral operator

Abstract

Let 1 < p <∞ and n ≥ 2. The authors establish the L p (ℝ n+1 ) boundedness for a class of parabolic singular integral operators with rough kernels. MR(2000) Subject Classification: 42B20; 42B25.

L bounds for rough parabolic maximal operators

“…Precisely, they proved that is bounded for any under the condition that . Recently, a considerable amount of research has been done to obtain the boundedness of the operator , the readers are refereed (for instance to [3] , [4] and the references therein).…”

Rough singular integrals and maximal operators with mixed homogeneity along compound curves

“…Recently, Chen, Ding and Fan [10] extended further the condition to the case ∈ H 1 S n−1 . Moreover, it follows from Chen, Wang and Yu's work [11] that T is bounded on L p (R n ) for 2β/(2β − 1) < p < 2β provided that ∈ F β S n−1 for some β > 1, where…”

“…Recently, Chen, Ding and Fan extended further the condition to the case . Moreover, it follows from Chen, Wang and Yu's work that T is bounded on for provided that for some , where …”

Mixed radial‐angular integrability for rough maximal singular integrals and Marcinkiewicz integrals with mixed homogeneity

“…Later on, Chen, Ding, and Fan [3] extended further the condition to the case: $$\Omega \in H^1(S^{n-1})$$, where $$H^1(S^{n-1})$$ denotes the Hardy space defined on $$S^{n-1}$$. Moreover, it follows from Chen, Wang, and Yu's work [4] that $$T_{\Omega }$$ is bounded on $$L^p(\mathbb {R}^n)$$ for $$2\beta /(2\beta -1)&lt;p&lt;2\beta$$ provided that $$\Omega \in \mathcal {F}_{\beta }(S^{n-1})$$ for some $$\beta &gt;1$$. Here, $$\mathcal {F}_{\beta }(S^{n-1})$$ denotes the set of all Ω, which are integrable over $$S^{n-1}$$ and satisfies …”

“…\end{equation*}$$This class of functions was introduced by Grafakos and Stefanov [9] and it was showed that $$$$\begin{equation*} \mathcal {F}_{\beta _2}{\big(S^{n-1}\big)}\subsetneqq \mathcal {F}_{\beta _1}{\big(S^{n-1}\big)},\quad \forall \,\, 1&lt;\beta _1&lt;\beta _2, \end{equation*}$$$$$$$$\begin{equation*} \bigcap \limits _{\beta &gt;1}\mathcal {F}_{{\beta }}{\big(S^{n-1}\big)}\nsubseteq H^1{\big(S^{n-1}\big)}\nsubseteq \bigcup \limits _{\beta &gt;1} \mathcal {F}_{\beta }{\big(S^{n-1}\big)}, \end{equation*}$$$$and $$$$\begin{equation*} \bigcup \limits _{q&gt;1}L^q{\big(S^{n-1}\big)}\subsetneqq \mathcal {F}_{\beta }{\big(S^{n-1}\big)},\quad \forall \:\,\beta &gt;1. \end{equation*}$$$$Very recently, the authors in [11] further generalized the result of [4] to the mixed radial‐angular spaces …”