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[L<sup>p</sup>] ESTIMATES FOR A ROUGH MAXIMAL OPERATOR ON PRODUCT SPACES
AL-QASSEM HUSSAIN MOHAMMED Korean Mathematical Society 2005 대한수학회지 Vol.42 No.3
We establish appropriate $L^p$ estimates for a class of maximal operators $S_{\Omega}^{(\gamma)}$ on the product space $R^n\;\times\;R^m\;when\;\Omega$ lacks regularity and $1\;\le\;\gamma\;\le\;2.\;Also,\;when\;\gamma\;=\;2$, we prove the $L^p\;(2\;{\le}\;P\;<\;\infty)\;boundedness\;of\;S_{\Omega}^{(\gamma)}\;whenever\;\Omega$ is a function in a certain block space $B_q^{(0,0)}(S^{n-1}\;\times\;S^{m-1})$ (for some q > 1). Moreover, we show that the condition $\Omega\;{\in}\;B_q^{(0,0)}(S^{n-1}\;\times\;S^{m-1})$ is nearly optimal in the sense that the operator $S_{\Omega}^{(2)}$ may fail to be bounded on $L^2$ if the condition $\Omega\;{\in}\;B_q^{(0,0)}(S^{n-1}\;\times\;S^{m-1})$ is replaced by the weaker conditions $\Omega\;{\in}\;B_q^{(0,\varepsilon)}(S^{n-1}\;\times\;S^{m-1})\;for\;any\;-1\;<\;\varepsilon\;<\;0.$
Weighted L<sup>P</sup> Estimates for a Rough Maximal Operator
Al-Qassem, H.M. Department of Mathematics 2005 Kyungpook mathematical journal Vol.45 No.2
This paper is concerned with studying the weighted $L^P$ boundedness of a class of maximal operators related to homogeneous singular integrals with rough kernels. We obtain appropriate weighted $L^P$ bounds for such maximal operators. Our results are extensions and improvements of the main theorems in [2] and [5].
Weighted L<sup>p</sup> Boundedness for the Function of Marcinkiewicz
Al-Qassem, Hussain M. Department of Mathematics 2006 Kyungpook mathematical journal Vol.46 No.1
In this paper, we prove a weighted norm inequality for the Marcinkiewicz integral operator $\mathcal{M}_{{\Omega},h}$ when $h$ satisfies a mild regularity condition and ${\Omega}$ belongs to $L(log L)^{1l2}(S^{n-1})$, $n{\geq}2$. We also prove the weighted $L^p$ boundedness for a class of Marcinkiewicz integral operators $\mathcal{M}^*_{{\Omega},h,{\lambda}}$ and $\mathcal{M}_{{\Omega},h,S}$ related to the Littlewood-Paley $g^*_{\lambda}$-function and the area integral S, respectively.
L^(p) Boundedness for Singular Integral Operators with L(log^(+) L)² Kernels on Product Spaces
AL-QASSEM, HUSSAIN,ALI, MOHAMMED 대한수학회 2006 Kyungpook mathematical journal Vol.46 No.3
In this paper, we study the L^(P) mapping properties of singular integral operators related to homogeneous mappings on product spaces with kernels which belong to L(log^(+) L)^(2). Our results extend as well as improve some known results on singular integrals.
Weighted L^(p) Boundedness for the Function of Marcinkiewicz
AL-QASSEM, HUSSAIN M. 대한수학회 2006 Kyungpook mathematical journal Vol.46 No.1
In this paper, we prove a weighted norm inequality for the Marcinkiewicz integral operator M_(Ω,h) when h satisfies a mild regularity condition and Ω belongs to L(log L)^(1/2)(S^(n-1)), n ≥ 2. We also prove the weighted L^(P) boundedness for a class of Marcinkiewicz integral operators M^(*)_(Ω,h,λ) and M_(Ω,h,s) related to the Littlewood-Paley g^(*)_(λ)-function and the area integral S, respectively.
WEIGHTED ESTIMATES FOR ROUGH PARAMETRIC MARCINKIEWICZ INTEGRALS
Al-Qassem, Hussain Mohammed Korean Mathematical Society 2007 대한수학회지 Vol.44 No.6
We establish a weighted norm inequality for a class of rough parametric Marcinkiewicz integral operators $\mathcal{M}^{\rho}_{\Omega}$. As an application of this inequality, we obtain weighted $L^p$ inequalities for a class of parametric Marcinkiewicz integral operators $\mathcal{M}^{*,\rho}_{\Omega,\lambda}\;and\;\mathcal{M}^{\rho}_{\Omega,S}$ related to the Littlewood-Paley $g^*_{\lambda}-function$ and the area integral S, respectively.
A van der Corput type lemma for oscillatory integrals with H\"older amplitudes and its applications
Hussain Al-Qassem,Leslie Cheng 대한수학회 2021 대한수학회지 Vol.58 No.2
We prove a decay estimate for oscillatory integrals with \linebreak H\"older amplitudes and polynomial phases. The estimate allows us to answer certain questions concerning the uniform boundedness of oscillatory singular integrals on various spaces.
Lp ESTIMATES FOR A ROUGH MAXIMALOPERATOR ON PRODUCT SPACES
Hussain Mohammed Al-Qassem 대한수학회 2005 대한수학회지 Vol.42 No.3
We establish appropriate Lp estimates for a class of maximal operators S(°) on the product space Rn £ Rm when lacks regularity and 1 · ° · 2. Also, when ° = 2; we prove the Lp (2 · p < 1) boundedness of S(2)whenever is a function in a certain block space B(0;0) q (Sn¡1 £ Sm¡1) (for some q > 1). Moreover, we show that the condition 2 B(0;0) q (Sn¡1 £ Sm¡1) is nearly optimal in the sense that the operator S(2) may fail to be bounded on L2 if the condition 2 B(0;0)q (Sn¡1 £ Sm¡1) is replaced by the weaker conditions 2 B(0;") q (Sn¡1£Sm¡1) for any ¡1 < " < 0