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In this paper, we prove weighted norm inequalities for vector-valued multilinear singular integrals with nonsmooth kernels and commutators on generalized weighted Morrey space.

Some new nonlinear weakly singular integral inequalities of Gronwall–Bellman–Pachpatte type are given. The estimations of unknown functions are obtained by analysis techniques. These estimates are very significant tools in the study of differential-integral equations.

For a prime number p, let $\mathbb{Q}_{p}$ be the field of p-adic numbers. In this paper, we establish the boundedness of a class of p-adic singular integral operators on the p-adic generalized Morrey spaces. We also consider the corresponding boundedness for the commutators generalized by the p-adic singular integral operators and p-adic Lipschitz functions or p-adic generalized Campanato functions.

In this paper, we study methods of solution for some kinds of convolution type singular integral equations with Cauchy kernel. By means of the classical boundary value problems for analytic functions and of the theory of complex analysis, we deal with the necessary and sufficient conditions of solvability and obtain the general solutions and the conditions of solvability for such equations. All cases as regards the index of the coefficients in the equations are considered in detail. Especially, we discuss some properties of the solutions at the nodes. This paper will be of great significance for the study of improving and developing complex analysis, integral equation and boundary value problems for analytic functions (that is, Riemann–Hilbert problems). Therefore, the classical theory of integral equations is extended.

In this paper, we study the strongly singular integrals \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{n, \beta, \gamma}f(x)=\mathrm{p.v.} \int_{-1}^{1}f\bigl(x-\Gamma_{\theta}(t) \bigr)\frac {e^{-2\pi i \vert t \vert ^{-\beta}}}{t \vert t \vert ^{\gamma}}\,dt $$\end{document}Tn,β,γf(x)=p.v.∫−11f(x−Γθ(t))e−2πi|t|−βt|t|γdt along homogeneous curves \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Gamma_{\theta}(t)$\end{document}Γθ(t). We prove that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T_{n, \beta, \gamma}$\end{document}Tn,β,γ is bounded on the α-modulation spaces, including the inhomogeneous Besov spaces and the classical modulation spaces.

In this paper, we establish some new retarded nonlinear Volterra-Fredholm type integral inequalities with maxima in two independent variables, and we present the applications to research the boundedness of solutions to retarded nonlinear Volterra-Fredholm type integral equations.

In this paper we consider a singular integral operator and a parametric Marcinkiewicz integral operator with rough kernel. These operators have singularity along sets of the form curves $\{x=P(\varphi (|y|))y'\}$ , where P is a real polynomial satisfying $P(0)=0$ and φ satisfies certain smooth conditions. Under the conditions that $\varOmega \in H^{1} (\mathbf{S}^{n-1})$ and $h\in \Delta _{\gamma }(\mathbf{R}_{+})$ for some $\gamma >1$ , we prove that the above operators are bounded on the Lebesgue space $L^{2}( \mathbf{R}^{n})$ . Moreover, the $L^{2}$ -bounds of the maximal functions related to the above integrals are also established. Particularly, the bounds are independent of the coefficients of the polynomial P. In addition, we also present certain Hardy type inequalities related to these operators.

The focus of the present study is to prove some new Pólya-Szegö type integral inequalities involving the generalized Riemann-Liouville k-fractional integral operator. These inequalities are used then to establish some fractional integral inequalities of Chebyshev type.

In this article, we study some classes of singular integral equations of convolution type with Cauchy kernels in the class of exponentially increasing functions. Such equations are transformed into Riemann boundary value problems on either a straight line or two parallel straight lines by Fourier transformation. We propose one method different from the classical one for the study of such problems and obtain the general solutions and the conditions of solvability. Thus, the result in this paper improves the theory of integral equations and the classical boundary value problems for analytic functions.

The purpose of the present paper is to establish some new retarded weakly singular integral inequalities of Gronwall-Bellman type for discontinuous functions, which generalize some known weakly singular and impulsive integral inequalities. The inequalities given here can be used in the analysis of the qualitative properties of certain classes of singular differential equations and singular impulsive equations.

The main purpose of this paper is to establish the weighted \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(L^{p},L^{q})$\end{document}(Lp,Lq) inequalities of the oscillation and variation operators for the multilinear Calderón-Zygmund singular integral with a Lipschitz function.

In this paper, we present the $L^{p}$ mapping properties of multiple singular integrals related to homogeneous mappings with rough kernels given by the radial function $h\in\Delta_{\gamma}$ (or $h\in U_{\gamma}$ ) for some $\gamma>1$ (or $\gamma\geq1$ ) and the sphere function $\Omega\in L(\log^{+}L)^{2} (S^{m-1}\times S^{n-1})$ (or $\Omega\in L(\log^{+}L)^{2/\gamma'} (S^{m-1}\times S^{n-1})$ ). In addition, the $L^{p}$ bounds for the related maximal operators are also given. Our main results extend and improve some known ones.

In this article, we first define a kind of generalized singular integral operator and discuss its properties. Then we propose a kind of boundary value problem for an inhomogeneous partial differential system in $R^{4}$ . Finally, the integral representation of the solution to a boundary value problem for the inhomogeneous partial differential system is obtained using the above singular integral operator.

In this article we give some generalized singular number inequalities for products and sums of τ-measurable operators. Some related arithmetic-geometric mean and Heinz mean inequalities for a generalized singular number of τ-measurable operators are proved.

This paper is devoted to investigating the boundedness of the oscillation and variation operators for the commutators generated by Calderon-Zygmund singular integrals with Lipschitz functions in the weighted Lebesgue spaces and the endpoint spaces in dimension 1. Certain criterions of boundedness are given. As applications, the weighted $(L^{p}, L^{q})$ -estimates for the oscillation and variation operators on the iterated commutators of Hilbert transform and Hermitian Riesz transform, the $(L^{p}, \dot{\wedge}_{(\beta -1/p)})$ -bounds as well as the endpoint estimates for the oscillation and variation operators of the corresponding first order commutators are established.

By applying the method of weight functions and the technique of real analysis, a multidimensional Hilbert-type integral inequality with multi-parameters and the best possible constant factor related to the gamma function is given. The equivalent forms and the reverses are obtained. We also consider the operator expressions and a few particular results related to the kernels of non-homogeneous and homogeneous.

In the present paper, we consider the boundedness of the rough singular integral operator \(T_{\Omega,h,\phi}\) along a surface \(\Gamma=\{x=\phi(|y|)y/|y|\}\) on the Triebel-Lizorkin space \(\dot{F}_{p,q}^{\alpha}({\mathbb{R}^{n}})\) for \(\Omega\in H^{1}({{S}^{n-1}})\) and Ω belonging to some class \(W\mathcal{F}_{\alpha}({{S}^{n-1}})\), which relates to the Grafakos-Stefanov class.

By applying the way of real and functional analysis and estimating the weight functions, we build some lemmas and deduce some Hilbert-type and Hilbert-Hardy-type integral inequalities with the best possible constant factors. The equivalent forms, the reverses and the operator expressions are all considered. The composition formula of two Hilbert-Hardy-type integral operators and some examples are given.

In this paper, the authors establish the boundedness of the strongly singular Calderon-Zygmund operator on weighted Morrey spaces. Moreover, the boundedness of the commutator generated by the strongly singular Calderon-Zygmund operator and the weighted BMO function on weighted Morrey spaces is also obtained. MSC:42B20, 42B25, 42B35.

In this paper, some sharp maximal function inequalities for the commutators related to certain generalized fractional singular integral operators are proved. As an application, we obtain the boundedness of the commutators on Lebesgue, Morrey and Triebel-Lizorkin spaces. MSC:42B20, 42B25.

In this paper, we establish the weighted sharp maximal function inequalities for a multilinear operator associated to a singular integral operator with general kernels. As an application, we obtain the boundedness of the operator on weighted Lebesgue and Morrey spaces. MSC:42B20, 42B25.

Vector-valued inequalities are considered for the commutator of the singular integral with rough kernel. The results obtained in this paper are substantial improvement and extension of some known results.

In this paper, we establish the weighted sharp maximal function inequalities for the multilinear singular integral operators. As an application, we obtain the boundedness of the multilinear operators on weighted Lebesgue and Morrey spaces. MSC:42B20, 42B25.

In this paper we study the iterated commutators for multilinear singular integrals on weighted Morrey spaces. A strong type estimate and a weak endpoint estimate for the commutators are obtained. In the last section we present a problem for the multilinear Fourier multiplier with limited smooth condition. MSC: 42B20; 42B25

In the article, the authors introduce a new notion, ‘α-preinvex function’, establish an integral identity for the newly introduced function, and find some Hermite-Hadamard type integral inequalities for a function of which the power of the absolute value of the first derivative is α-preinvex.

In this paper, we show the sharp maximal function estimates for the Toeplitz type operators related to the strongly singular integral operators. As an application, we obtain the boundedness of the operators on weighted Lebesgue and Triebel-Lizorkin spaces.

In this paper, we establish a sharp inequality for some multilinear singular integral operators with non-smooth kernels. As an application, we obtain the weighted -norm inequality and -type inequality for the multilinear operators. MSC:42B20, 42B25.

In this paper, the inequality of boundedness for the multilinear fractional singular integral operators associated to the weighted Lipschitz functions is estimated. The operators include the Calderón-Zygmund singular integral operator and the fractional integral operator. MSC:42B20, 42B25.

Let Tj,1 and Tj,2 be singular integrals with non-smooth kernels, which are associated with an approximation of identity or ±I (I is the identity operator). Denote the Toeplitz-type operator by Tb = N=1 Tj,1MbTj,2 ,w hereMbf (x )= b(x)f (x). In this paper, the estimates of the Toeplitz operator Tb(f ) related to singular integral operators with non-smooth kernels and b ∈ BMO(R n ) in weighted Morrey spaces is established. MSC: 47B35

With a more concise condition for the ordered parameter groups, some nonlinear weakly singular integral inequalities of Wendroff type, which generalize some existing results, are established. Furthermore, application examples in the boundedness and uniqueness of the solution of a singular partial integral equation are given.

This paper is devoted to studying the singular integrals and Marcinkiewicz integrals with mixed homogeneity along surfaces, which contain many classical surfaces as model examples, on the product domains R m × R n (m, n ≥ 2). Under rather weak size conditions of the kernels, the L p (R m × R n )-boundedness for such operators is established. These results essentially extend certain previous results. MSC: 42B20; 42B25

In this article we discuss some new generalized nonlinear Gronwall-Bellman-Type integral inequalities with two variables, which include a non-constant term outside the integrals. We use our result to deal with the estimate on the solutions of partial differential equations with the initial and boundary conditions. Mathematics Subject Classification 2000: 26D10; 26D15; 26D20; 34A40.

In this article, fractional integro-differential inequalities with singular coefficients have been considered. The bounds obtained for investigating the behavior of the solution of a class of singular nonlinear fractional differential equations has been used, some applications are provided.

We establish the Poincaré-type inequalities for the composition of the Laplace-Beltrami operator and the Green's operator applied to the solutions of the non-homogeneous A-harmonic equation in the John domain. We also obtain some estimates for the integrals of the composite operator with a singular density.

Let be a nondoubling measure on . A class of commutators associated with multilinear fractional integrals and RBMO functions are introduced and shown to be bounded on product of Lebesgue spaces with .

In this paper we have introduced an integral general operator. For this general operator which is a generalization of more known integral operators we have demonstrated some univalence properties.

We study some special nonlinear integral inequalities and the corresponding integral equations in measure spaces. They are significant generalizations of Bihari type integral inequalities and Volterra and Fredholm type integral equations. The kernels of the integral operators are determined by concave functions. Explicit upper bounds are given for the solutions of the integral inequalities. The integral equations are investigated with regard to the existence of a minimal and a maximal solution, extension of the solutions, and the generation of the solutions by successive approximations.

Some new weakly singular integral inequalities of Wendroff type are established, which generalized some known weakly singular inequalities for functions in two variables and can be used in the analysis of various problems in the theory of certain classes of integral equations and evolution equations. Application examples are also given.

We define an integral operator denoted by , and we give sufficient conditions such that is univalent.

The purpose of this paper is to prove several results in approximation by complex Picard, Poisson-Cauchy, and Gauss-Weierstrass singular integrals with Jackson-type rate, having the quality of preservation of some properties in geometric function theory, like the preservation of coefficients' bounds, positive real part, bounded turn, starlikeness, and convexity. Also, some sufficient conditions for starlikeness and univalence of analytic functions are preserved.

We provide explicit formulas in terms of the special function gamma for the best constants in nontensorial multilinear extensions of some classical integral inequalities due to Hilbert, Hardy, and Hardy-Littlewood-Polya.