# Ahmad Al-SalmanYarmouk University | YU · Department of Mathematics

Ahmad Al-Salman

PhD

## About

64

Publications

2,410

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652

Citations

Citations since 2017

## Publications

Publications (64)

In this paper, we introduce a class of singular Radon transforms on Rn with kernels supported in a subvariety in Rn×Rn determined by a polynomial mapping from Rn×Rn into Rn. The class of considered operators is related to the composition of homogeneous singular integral operators. We prove that the operators are bounded on Lp provided that the kern...

We introduce a class of singular integral operators on product domains along twisted surfaces. We prove that the operators are bounded on Lp provided that the kernels satisfy weak conditions.

Marcinkiewicz integral operators on product domains defined by translates determined by twisted surfaces are introduced. Maximal functions along twisted surfaces are also introduced. Conditions on the underlined surfaces implying that the corresponding Marcinkiewicz integral operators map \begin{document}$ L^{p}\rightarrow L^{p} $\end{document} for...

We consider the general second order difference equation $x_{n+1}=F(x_n,x_{n-1})$ in which $F$ is continuous and of mixed monotonicity in its arguments. In equations with negative terms, a persistent set can be a proper subset of the positive orthant, which motivates studying global stability with respect to compact invariant domains. In this paper...

We prove Lp estimates of a class of parametric Marcinkiewicz integral operators when their kernels satisfy only the \(L^1(\mathbb{S}^{n-1})\) integrability condition. The obtained Lp estimates resolve a problem left open in previous work. Our argument is based on duality technique and direct estimation of operators. As a consequence of our result,...

In this note, we obtain sharp L^ p estimates of parametric Marcinkiewicz integral operators. Our result resolves a long standing open problem. Also, we present a class of parametric Marcinkiewicz integral operators that are bounded provided that their kernels belong to the sole space L^ 1( S^{ n−1}) .

In this paper we prove Lp estimates of Marcinkiewicz integral operators with kernels in the Hardy space and supported on general subvarieties. The considered subvarieties are of the type that caries partially the polynomial behavior as well as the behavior of convex functions. Results obtained in this paper improve as well as generalize known resul...

In this paper, we study the L
p
mapping properties of certain class of maximal oscillatory singular integral operators. We prove a general theorem for a class of maximal functions along surfaces. As a consequence of such theorem, we establish the L
p
boundedness of various maximal oscillatory singular integrals provided that their kernels belong to...

In this paper, we study the L p mapping properties of certain class of maximal oscillatory singular integral operators. We prove a general theorem for a class of maximal functions along surfaces. As a consequence of such theorem, we establish the L p boundedness of various maximal oscillatory singular integrals provided that their kernels belong to...

In this paper, we prove Lp estimates of a class of parabolic maximal functions provided that their kernels are in Lq . Using the obtained estimates, we prove the boundedness of the maximal functions under very weak conditions on the kernel. In particular, we prove the Lp -boundedness of our maximal functions when their kernels are in Llog L1/2 (Sⁿ⁻...

In this paper, we prove Lp bounds for singular integrals with rough kernels associated to certain surfaces. Our results extend as well as improve previously obtained results.

We introduce a class of integral operators related to parametric Marcinkiewicz operators. We present a multiplier formula characterizing the L2 boundedness of such class of operators. Also, we prove L p-β (inhomogeneous Sobolev space)→Lp estimates provided that the kernels are in L(logL)(Sn-1). In fact, we show that the global parts of the introduc...

In this paper, we characterize periodic solutions of p-periodic difference equations. We classify the periods into multiples of p and nonmultiples of p. We show that the elements of the set of multiples of p follow the well-known Sharkovsky’s ordering multiplied by p. On the other hand, we show that the elements of the set Γp of nonmultiples of p a...

Integrals and sums involving special functions are in constant demand in
applied mathematics. Rather than refer to a handbook of integrals or to a
computer algebra system, we present a do-it-yourself systematic approach
that shows how the evaluation of such integrals and sums can be made as
simple as possible. Illustrating our method, we present se...

In this paper, parabolic Marcinkiewicz integral operators along surfaces on the product domain ℝ
n
× ℝ
m
(n,m ⩾ 2) are introduced. L
p
bounds of such operators are obtained under weak conditions on the kernels.
KeywordsParabolic Marcinkiewicz integral operators-parabolic maximal functions-Fourier transform-rough kernels
MR(2000) Subject Classific...

In this paper, we derive a multiplier formula for Hörmander's parametric Marcinkiewicz integral operator. As a consequence, we prove the optimality of the kernel size condition as well as the optimality of the Block size condition.

The author establishes the Lp boundedness for a class of maximal functions related to singular integrals associated to surfaces of revolution on product domains with rough kernels in L(logL)(Sn−1).

In this paper, we study the L
p
boundedness for a class of maximal functions along surfaces in ℝ
n
× ℝ
m
of the form
$
\{ (\varphi _1 (|u|)u',\varphi _2 (|v|)v'):(u,v) \in \mathbb{R}^n \times \mathbb{R}^m \} .
$
\{ (\varphi _1 (|u|)u',\varphi _2 (|v|)v'):(u,v) \in \mathbb{R}^n \times \mathbb{R}^m \} .
We prove that such maximal functions are bo...

In this paper, we establish the Lp boundedness of a class of maximal functions with rough kernels supported by subvarieties. Moreover, several LP estimates of singular integrals, Marcinkiewicz integrals, and parametric Marcinkiewicz integrals will be studied.

In this paper, we establish the L p boundedness of a class of max-imal functions with rough kernels supported by subvarieties. Moreover, sev-eral L p estimates of singular integrals, Marcinkiewicz integrals, and paramet-ric Marcinkiewicz integrals will be studied.

The author studies the L
p
mapping properties of a class of maximal functions that are related to oscillatory singular integral operators. L
p
estimates, as well as the corresponding weighted estimates of such maximal functions, are obtained. Moreover, several applications
of our results are highlighted.

The diagonal Δ of Rn is Chebeshev with respect to the p-norm for every p ∈ (1, ∞) but not for p = 1. As a result, the median is multi-valued, since the median of a data set {a1, ··· ,an} can be thought of as the number(s) μ for which the point (μ, ··· ,μ) is a point on Δ that best approximates the point (a1, ··· ,an) with respect to the � 1 -norm....

In this paper, we study the L p boundedness of a class of parametric Marcinkiewicz integral operators with rough kernels in L(log + L)(S n−1). Our result in this paper solves an open problem left by the authors of ([6]).

We establish the L p boundedness of a class of fractional integral operators with rough kernels involving Bessel functions of the first kind. Our result generalizes Calderón-Zygmund’s result [A. P. Calderón and A. Zygmund, Am. J. Math. 78, 289–309 (1956; Zbl 0072.11501)].

In this paper, we study the Lp boundedness of a class of parametric Marcinkiewicz integral operators with rough kernels in L(log + L)(Sn-1). Our result in this paper solves an open problem left by the authors of ([6]).

In this paper, we study the Lp mapping properties of a class of singular integral operators with rough kernels belonging to certain block spaces. We prove that our operators are bounded on Lp provided that their kernels satisfy a size condition much weaker than that for the classical Calderón-Zygmund singular integral operators. Moreover, we presen...

This paper is concerned with singular integral operators on product domains with rough kernels in L(logL)2. We prove, among other things, L p bounds (1 < p < ∞) for such singular integral operators as well as for their corresponding maximal truncated singular integrals. We also establish the optimality of our condition in the sense that the space L...

The author establishes the L p boundedness for a class of maximal functions related to singular integrals associated to surfaces of revolution with rough kernels in L(logL) 1/2 (S n-1 ).

We prove a weighted norm inequality for Marcinkiewicz integral operators with rough kernels in L(log + L) 1 2 (S n-1 ). The L(log + L) 1 2 (S n-1 )-size condition is known to be the most desirable size condition for the L 2 boundedness to hold. We also establish the weighted L p boundedness of the corresponding Marcinkiewicz integral operators that...

We establish L p estimates for certain class of maximal functions with kernels in L q (S n-1 ). As a consequence of such L p estimates, we obtain the L p boundedness of our maximal functions when their kernels are in L(logL) 1/2 (S n-1 ) or in the block space B q 0,-1/2 (S n-1 ), q>1. Several applications of our results are also presented.

We consider Marcinkiewicz integral operators associated to subvarieties determined by flat surfaces with kernels in the Hardy space H 1 (S n-1 ). We establish L p bounds of these operators under weak convexity conditions on these surfaces. Moreover, we establish L p bounds for corresponding Marcinkiewicz integral operators that are related to the a...

In this paper, we study the LP boundedness of certain maximal operators on product domains with rough kernels in L (log L). We prove that our operators are bounded on L P for all 2 <= p <= infinity. Moreover, we show that our condition on the kernel is optimal in the sense that the space L (log L) cannot be replaced by L (log L)(r) for any r < 1. O...

Our point of departure is the following $L^{p}$ boundedness result form [St], Theorem 1.1, ... Recently, the results in Theorem 1.1 were improved by Fan, Guo, and Pan in [FGP] who showed that the $L^{p}$ boundedness of $T_{\Phi}$ and $M_{\Phi}$ continues to hold if the condition $\Omega \in \bb{C}^{1}(Sn-1)$ is replaced by the weaker condition $\Om...

We prove the L p boundedness of the Marcinkiewicz integral operators μΩ on ℝ n1 × ... × ℝ nk under the condition that Ω∈L(log L) k/2(struck S sign n1-1 × ... × struck S sign nk-1). The exponent k/2 is the best possible. This answers an open question posed in [?].

In this paper, we study Marcinkiewicz integral operators with rough kernels supported by surfaces given by flat curves. Under convexity assumptions on our surfaces, we establish an Lp boundedness result of such operators. Moreover, we obtain the Lp boundedness of the corresponding Marcinkiewicz integral operators that are related to area integral a...

In this paper, we study the L-p mapping properties of maximal functions with rough kernels that are related to certain class of singular integral operators. We prove that our maximal functions are bounded on LP provided that their kernels are in L (log L)(1/2)(Sn-1). Moreover, we present an example showing that our size condition on the kernel is o...

In this paper, we study the L p mapping properties of singular integral operators related to homogeneous mappings with kernels belonging to certain block spaces. An example is presented to show that our condition on the kernel is nearly optimal.

In this paper, we study a classes of oscillatory singular integral operators of nonconvolution type with phases more general than polynomials. We prove that such operators are bounded on Lp provided their kernels satisfy a very weak condition. In addition, we also study the related truncated oscillatory singular integral operators. Moreover, we pre...

In this paper we establish the Lp boundedness of a class of singular integrals with rough kernels associated to polynomial mappings.

In this paper, we study singular integrals on product domains with kernels in L(log L) 2 (S n−1 ×S m−1) supported by surfaces of revolutions. We prove that our operators are bounded on L p under certain convexity assumption on the surfaces. Also, in this paper we prove that the convexity assumption is not necessary for the L 2 boundedness to hold....

We study the mapping properties of singular integral operators defined by mappings of finite type. We prove that such singular integral operators are bounded on the Lebesgue spaces under the condition
that the singular kernels are allowed to be in certain block spaces.

We study the Lp mapping properties of a class of Marcinkiewicz integral operators on product domains with rough kernels supported by subvarieties.

In this paper, we study the L p mapping properties of parametric Marcinkiewicz operators introduced by Hörmander. We prove the L p bound-edness of these operators under very weak size conditions on their kernels. Our result substantially improves the result in [10].

In this paper, we present a systematic treatment of Marcinkiewicz integrals with block space function kernels and prove the L p boundedness of several classes of Marcinkiewicz integrals along surfaces of revolution. The results in this paper extend as well as improve previously known results.

This paper is primarily concerned with proving the Lp boundedness of Marcinkiewicz integral operators with kernels belonging to certain block spaces. We also show the optimality of our condition on the kernel for the L2 boundedness of the Marcinkiewicz integral.

In this paper, we study certain classes of oscillatory singular integral operators with kernels in L logL(Sn 1) which is known to be the most desirable size condition for the Lp boundedness to hold. We prove that such operators are bounded on Lp. Our results extend and improve previously known results. Variations of our approach in this paper can b...

We study singular integral operators along subvarieties determined by flat curves and kernels in the Hardy space H
1 (S
n−1). We prove that these operators are bounded on L
p
for all p ∈ (1, ∞). Our results extend previously known results.

In this paper we study integral operators of Marcinkiewicz type. We formulate a general method which allows us to obtain the Lp boundedness of several classes of integral operators of Marcinkiewicz type. Our results extend as well as improve previously known results on Marcinkiewicz integral operators.

A systematic treatment is given of several classes of singular integrals. Their
$L^p$
boundedness is proved when their kernels are given by functions
$\Omega$
in
$L\log L({\bf S}^{n-1})$
.

We study the Marcinkiewicz integral operator MÃ°ÂÂ’Â«f(x)=(Ã¢ÂˆÂ«Ã¢ÂˆÂ’Ã¢ÂˆÂžÃ¢ÂˆÂž|Ã¢ÂˆÂ«|y|Ã¢Â‰Â¤2tf(xÃ¢ÂˆÂ’Ã°ÂÂ’Â«(y))(ÃŽÂ©(y)/|y|nÃ¢ÂˆÂ’1)dy|2dt/22t)1/2, where Ã°ÂÂ’Â« is a polynomial mapping from Ã¢Â„Ân into Ã¢Â„Âd and ÃŽÂ© is a homogeneous function of degree zero on Ã¢Â„Ân with mean value zero over the unit sphere SnÃ¢ÂˆÂ’1. We prove an...

In this paper, we study the Lpmapping properties of singular integral operators with kernels belonging to certain block spaces. These operators have singularities along sets of the form {x = Φ (|y|)y′} where Φ satisfies certain growth conditions. Our results improve as well as extend previously known results on singular integrals.

Thesis (Ph.D.)--Cornell Univ., Feb. 1965.

In this paper, we study the $L^p$ mapping properties of maximal functions with rough kernels that are related to certain class of singular integral operators. We prove that our maximal functions are bounded on $L^p$ provided that their kernels are in $L(\log L)^{1/2}(\mathbb{S}^{n-1})$. Moreover, we present an example showing that our size conditio...

## Projects

Project (1)