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Let $R_{1,2}$ be scalar Riesz transforms on $\mathbb{R}^2$. We prove that the $L^p$ norms of $k$-th powers of the operator $R_2+iR_1$ behave exactly as $|k|^{1-2/p}p$, uniformly in $k\in\mathbb{Z}\backslash\{0\}$, $p\geq2$. This gives a complete asymptotic answer to a question suggested by Iwaniec and Martin in 1996. The main novelty are the lower estimates, of which we give three different proofs. We also conjecture the exact value of $\|(R_2+iR_1)^k\|_p$. Furthermore, we establish the sharp behaviour of weak $(1,1)$ constants of $(R_2+iR_1)^k$ and an $L^\infty$ to $BMO$ estimate that is sharp up to a logarithmic factor., 32 pages

In this paper, we study the rough singular integral operator T Ω f ( x ) = p.v. ∫ R n f ( x − y ) Ω ( y ′ ) | y | n d y , \begin{equation*} T_\Omega f(x)=\text {p.v.}\int _{\mathbb {R}^n}f(x-y)\frac {\Omega (y’)}{|y|^n}dy, \end{equation*} and the corresponding maximal singular integral operator T Ω ∗ f ( x ) = sup ε > 0 | ∫ | y | ≥ ε f ( x − y ) Ω ( y ′ ) | y | n d y | , \begin{equation*} T^*_\Omega f(x)=\sup _{\varepsilon >0}\Big |\int _{|y|\geq \varepsilon }f(x-y)\frac {\Omega (y’)}{|y|^n}dy\Big |, \end{equation*} where the kernel Ω ∈ H 1 ( S n − 1 ) \Omega \in H^1(\mathrm {S}^{n-1}) has zero mean value and n ≥ 2 n\geq 2 . We prove that T Ω T_\Omega and T Ω ∗ T^*_\Omega are bounded on the mixed radial-angular spaces L | x | p L θ p ~ ( R n ) L_{|x|}^pL_{\theta }^{\tilde {p}}(\mathbb {R}^n) for some suitable indexes 1 > p , p ~ > ∞ 1>p, \tilde {p}>\infty . The corresponding vector-valued versions are also established.

Let $T$ be a linear operator that, for some $p_1\in(1,\infty)$, is bounded on $L^{p_1}(\tilde w)$ for all $\tilde w\in A_{p_1}(\mathbb R^d)$ and in addition compact on $L^{p_1}(w_1)$ for some $w_1\in A_{p_1}(\mathbb R^d)$. Then $T$ is bounded and compact on $L^p(w)$ for all $p\in(1,\infty)$ and all $w\in A_p(\mathbb R^d)$. This "compact version" of Rubio de Francia's celebrated weighted extrapolation theorem follows from a combination of results in the interpolation and extrapolation theory of weighted spaces on the one hand, and of compact operators on abstract spaces on the other hand. Moreover, generalizations of this extrapolation of compactness are obtained for operators that are bounded from one space to a different one ("off-diagonal estimates") or only in a limited range of the $L^p$ scale. As applications, we easily recover several recent results on the weighted compactness of commutators of singular integral operators, fractional integrals and pseudo-differential operators, and obtain new results about the weighted compactness of commutators of Bochner-Riesz multipliers., V4: 34 pages; final version, incorporated referee comments, to appear in Rev. Mat. Iberoam. (2022)

We prove a theorem on weighted pointwise convergence of \ multidimensional integral operators with radial kernels to generating function of several variables, which are in general non-integrable in $n$-dimensional Euclidean space $E_{n}$ in the sense of Lebesgue$.$ Main result holds at almost every point of $E_{n}.$

Let $\mathbb{H}$ be the first Heisenberg group, and let $k \in C^{\infty}(\mathbb{H} \, \setminus \, \{0\})$ be a kernel which is either odd or horizontally odd, and satisfies $$|\nabla_{\mathbb{H}}^{n}k(p)| \leq C_{n}\|p\|^{-1 - n}, \qquad p \in \mathbb{H} \, \setminus \, \{0\}, \, n \geq 0.$$ The simplest examples include certain Riesz-type kernels first considered by Chousionis and Mattila, and the horizontally odd kernel $k(p) = \nabla_{\mathbb{H}} \log \|p\|$. We prove that convolution with $k$, as above, yields an $L^{2}$-bounded operator on regular curves in $\mathbb{H}$. This extends a theorem of G. David to the Heisenberg group. As a corollary of our main result, we infer that all $3$-dimensional horizontally odd kernels yield $L^{2}$ bounded operators on Lipschitz flags in $\mathbb{H}$. This was known earlier for only one specific operator, the $3$-dimensional Riesz transform. Finally, our technique yields new results on certain non-negative kernels, introduced by Chousionis and Li., Comment: 78 pages. v4: main result extended to non-homogeneous kernels. New application to Lipschitz flags

In this paper, the authors study the boundedness properties of a class of multilinear strongly singular integral operator with generalized kernels on product of weighted Lebesgue spaces and product of variable exponent Lebesgue spaces, respectively. Moreover, the types $ L^{\infty}\times \dots \times L^{\infty}\rightarrow BMO $ and $ BMO\times \dots \times BMO\rightarrow BMO $ endpoint estimates are also obtained.

In this paper, starting with a relatively simple observation that the variational estimates of the commutators of the standard Calder\'on-Zygmund operators with the BMO functions can be deduced from the weighted variational estimates of the standard Calder\'on-Zygmund operators themselves, we establish similar variational estimates for the commutators of the BMO functions with rough singular integrals which do not admit any weighted variational estimates. The proof involves many Littlewood-Paley type inequalities with commutators as well as Bony decomposition and related para-product estimates., Comment: 27 pages

In this paper, we develop a method for solving planar nonstationary problems on the influence of surface pressure on an elastic-porous half-space. Using surface influence functions, we obtain an integral relation, which expresses the desired displacements with the surface pressure. A numericalanalytical algorithm is constructed. For calculating singular integrals, we derive special quadrature formulas based on the canonical regularization of integrands.

In this paper, we formulate mathematical foundations of applications of boundary integral equations with strongly singular integrals understood in the sense of finite Hadamard value to numerical solution of certain boundary-value problems. We describe numerical schemes for solving boundary strongly singular equations based on quadrature formulas and the collocation method. Also, we make references to known results on the mathematical justification of the numerical methods described in the paper.

By using an analog of the δc -model, we reduce the problem of stressed state and limit equilibrium of an inhomogeneous shell of revolution weakened by an internal crack of any configuration with plastic strains developed on the continuation of the crack in the form of a narrow strip to an elastic problem. The indicated elastic problem is then reduced to a system of singular integral equations with unknown limits of integration and discontinuous functions on the right-hand sides. We propose an algorithm for the numerical solution of these systems with regard for the conditions of plasticity of thin shells and the conditions of boundedness for stresses. The effects of loading, geometric parameters, and mechanical characteristics on the crack-opening displacement and the sizes of plastic zones are investigated for cylindrical and spherical shells made of functionally graded materials and containing internal parabolic cracks.

The work in this paper is concerned with application of a very simple numerical technique called line element method to solve singular integral equations with weakly singular kernel and hypersingular kernel. Of the integral equations considered here are first kind Abel integral equation and integral equation with log kernel and hypersingular integral equations of first and second kind. In this method, the range of integration as well as the interval of definition of the integral equation are discretised into finite number of small subintervals and the unknown function satisfying the integral equation is assumed to be constant in each small subinterval. This reduces the integral equations to a system of algebraic equations which is then solved to obtain the unknown function in each subinterval. The method is illustrated with examples. It is observed that a very accurate result is obtained by applying this method. The error analysis for this method is also given.

In this paper, we consider the maximal operator related to the Laplace-Bessel differential operator ( B B -maximal operator) on L p ( ⋅ ) , γ ( R k , + n ) {L}_{p\left(\cdot ),\gamma }\left({{\mathbb{R}}}_{k,+}^{n}) variable exponent Lebesgue spaces. We will give a necessary condition for the boundedness of the B B -maximal operator on variable exponent Lebesgue spaces. Moreover, we will obtain that the B B -maximal operator is not bounded on L p ( ⋅ ) , γ ( R k , + n ) {L}_{p\left(\cdot ),\gamma }\left({{\mathbb{R}}}_{k,+}^{n}) variable exponent Lebesgue spaces in the case of p − = 1 {p}_{-}=1 . We will also prove the boundedness of the fractional maximal function associated with the Laplace-Bessel differential operator (fractional B B -maximal function) on L p ( ⋅ ) , γ ( R k , + n ) {L}_{p\left(\cdot ),\gamma }\left({{\mathbb{R}}}_{k,+}^{n}) variable exponent Lebesgue spaces.

Problems of factorization of matrix functions are closely connected with the solution of matrix Riemann boundary value problems and with the solution of vector singular integral equations. In this article, we study functional operators with orientation-reversing shift reflection on the real axes. We introduce the concept of multiplicative representation of functional operators with shift and its partial indices. Based on the classical notion of matrix factorization, the correctness of the definitions is shown. A theorem on the relationship between factorization of functional operators with reflection and factorization of the corresponding matrix functions is proven.

In this paper, we examine the unique solvability of the Gevrey problem for a mixed parabolic equation with singular coefficients in a band. We prove the existence and uniqueness of a solution to the problem stated. The uniqueness of a solution is proved by the method of energy integrals and the existence by methods of the theory of Volterra, Fredholm, and singular integral equations.

Jabbarov [1] obtained the exact value of the exponent of convergence of the singular integral in Tarry’s problem for homogeneous polynomials of degree . We extend this result to the case of polynomials of degree .

The classical composite midpoint rectangle rule for computing Cauchy principal value integrals on an interval is studied. By using a piecewise constant interpolant to approximate the density function, an extended error expansion and its corresponding superconvergence results are obtained. The superconvergence phenomenon shows that the convergence rate of the midpoint rectangle rule is higher than that of the general Riemann integral when the singular point coincides with some priori known points. Finally, several numerical examples are presented to demonstrate the accuracy and effectiveness of the theoretical analysis. This research is meaningful to improve the accuracy of the collocation method for singular integrals.

We consider weighted norm inequalities for multilinear multipliers whose symbols satisfy a product-type Hörmander condition. Our approach is to consider a more general family of multilinear singular integral operators associated to a family of smooth kernels that satisfy an L r L^r -Schwartz regularity condition. We give conditions for these operators to satisfy weighted Hardy space estimates and derive our results for multipliers as a special case. As an additional application, we use Rubio de Francia extrapolation to prove multilinear estimates on the variable exponent Hardy spaces.

If $$I\subset {\mathbb {R}}$$ is a bounded interval, we prove the boundedness of Calderon singular operator and of Hardy-Littlewood Maximal operator in the generalized weighted Grand Lebesgue spaces $$L_p^{p),{\delta }}(I)$$ , $$1

In this article, solvability of infinite system of nonlinear singular integral equations of two variables in Banach sequence spaces $$c_{0}$$ and $$\ell _{1}$$ is investigated. For this purpose, Hausdorff measure of noncompactness (in short, Hausdorff MNC) and Meir–Keeler condensing operators are employed. The guarantee of our results are given by some examples. Also to approach an approximation of semi-analytic solution, we introduce a coupled modified homotopy perturbation and Adomian decomposition method. Thus, an iterative algorithm is constructed to find the above solution. The numerical results show that the produced sequence to approximate the solution has a high accuracy.

We study a system of nonlinear singular integral equations with a sum-difference kernel on the positive half-line. Such a system occurs (in various representations) in many branches of mathematical physics and applied mathematics. In particular, one encounters a system of equations with a kernel representing a Gaussian distribution and with a power nonlinearity in the dynamic theory of p-adic open-closed strings; in the case when the nonlinearity has a certain exponential structure, such a system occurs in mathematical biology, namely, in the theory of the spatio-temporal distribution of an epidemic. We prove constructive theorems on the existence of nonnegative nontrivial continuous and bounded solutions. We study the uniqueness and the asymptotic behavior of constructed solutions at infinity. In conclusion, we give concrete applied examples of the mentioned equations that satisfy all assumptions of the proved theorems.

Let $$x=(x_1,x_2)$$ with $$x_1,x_2 \in \mathbb {R}^n$$ and let $$K(x)={\Omega \big ({x}/{|x|}\big )}{\big |x\big |^{-2n}}$$ , where $$\Omega \in L^{\infty }(\mathbb {S}^{2n-1})$$ and satisfies $$\int _{\mathbb {S}^{2n-1}}\Omega =0$$ . We show that the maximal truncated bilinear singular integrals with rough kernel $$K(x_1,x_2)$$ satisfy a sparse bound by (p, p, p)-averages for all $$p>1$$ . As consequences, we obtain some quantitative weighted estimates for these rough singular integrals. A pointwise sparse domination for commutators of bilinear rough singular integrals were also established, which can be used to establish some weighted inqualities.

We prove a theorem on the equivalence of a scalar Riemann boundary value problem with a shift-reflection on the real axis and a matrix Riemann boundary value problem without the shift. A relationship between the boundary value problem in question and a boundary value problem with orientation-preserving shift-rotation on the unit circle is established. The operator identities constructed by the present authors in their preceding papers and permitting one to eliminate the shift-reflection from the integral equation corresponding to the boundary value problem are the main research tool.

In the present paper, a characterization of the Favard classes for the sampling Kantorovich operators based upon bandlimited kernels has been established. In order to achieve the above result, a wide preliminary study has been necessary. First, suitable high order asymptotic type theorems in$$L^p$$Lp-setting,$$1 \le p \le +\infty $$1≤p≤+∞, have been proved. Then, the regularization properties of the sampling Kantorovich operators have been investigated. Here, we show how the regularity of the kernel influences the operator itself; this has been shown for bandlimited kernels, or more in general for kernels in Sobolev spaces, satisfying a Strang-Fix type condition of order$$r \in \mathbb {N}^+$$r∈N+. Further, for the order of approximation of the sampling Kantorovich operators, quantitative estimates based on the$$L^p$$Lpmodulus of smoothness of orderrhave been established. As a consequence, the qualitative order of approximation is also derived assumingfin suitable Lipschitz and generalized Lipschitz classes. Moreover, an inverse theorem of approximation has been stated, allowing to obtain a full characterization of the Lipschitz and of the generalized Lipschitz classes in terms of convergence of the above sampling type series. These approximation results have been proved for not necessarily bandlimited kernels. From the above mentioned characterization, it remains uncovered the saturation case that, however, can be treated by a totally different approach assuming that the kernel is bandlimited. Indeed, since sampling Kantorovich (discrete) operators based upon bandlimited kernels can be viewed as double-singular integrals, exploiting the properties of the convolution in Fourier Analysis, we become able to get the desired result obtaining a complete overview of the approximation properties in$$L^p(\mathbb {R})$$Lp(R),$$1 \le p \le +\infty $$1≤p≤+∞, for the sampling Kantorovich operators.

We study Fourier multiplier operators associated with symbols $\xi\mapsto \exp(i\lambda\phi(\xi/|\xi|))$, where $\lambda$ is a real number and $\phi$ is a real-valued $C^\infty$ function on the standard unit sphere $\mathbb{S}^{n-1}\subset\mathbb{R}^n$. For $1, Comment: 25 pages; v2: references added, submitted for publication; v3: another reference added, accepted for publication

In this work, for a class of nonlinear weakly singular integral equations defined in the space of quadratically summable functions in a circle, the rationale for the general projection method is given. Using the obtained general results, the convergence of the well-known Galerkin method is proved.

A linear integro-differential equation of the first order given on a closed curve located on the complex plane is studied. The coefficients of the equation have a special structure. The equation contains a singular integral, which can be understood as the main value by Cauchy, and a hypersingular integral which can be understood as the end part by Hadamard. The analytical continuation method is applied. The equation is reduced to a sequential solution of the Riemann boundary value problem and two linear differential equations. The Riemann problem is solved in the class of analytic functions with special points. Differential equations are solved in the class of analytical functions on the complex plane. The conditions for the solvability of the original equation are explicitly given. The solution of the equation when these conditions are fulfilled is also given explicitly. Examples are considered. A non-obvious special case is analyzed.

We investigate a class of nonlinear singular integral equations with two monotone nonlinearities on the whole line. These equations have both theoretical and practical interest. We prove the existence theorem of a solution and the uniqueness theorem in a certain class of continuous and odd on $$\mathbb {R}\setminus\{0\}$$ functions. The results of the work summarize some previously obtained research results in this direction. We find the limits of the solution of the equation at $$\pm\infty$$ . For obtained solution we investigate some its properties. We also establish the integral asymptotic for the constructed solutions. The results are illustrated by examples of the equations under consideration.

In this attempt, we introduce a new technique to solve main generalized Abel’s integral equations and generalized weakly singular Volterra integral equations analytically. This technique is based on the Adomian decomposition method, Laplace transform method, andΨ-Riemann–Liouville fractional integrals. Finally, some examples are proposed and they illustrate the rapidness of our new technical method.

The purpose of this paper is for the first time to provide a careful and accessible exposition of the study of the existence of positive solutions of semilinear Neumann problems for diffusive logistic equations with discontinuous coefficients, which models population dynamics in environments with spatial heterogeneity. A biological interpretation of our main result is that when the environment has an impassable boundary and is on the average unfavorable, then high diffusion rates have the same effect (that is, the ultimate extinction of the population) as they always have when the boundary is deadly; but if the boundary is impassable and the environment is on the average neutral or favorable, then the population can persist, no matter what its rate of diffusion. The approach here is based on explicit representation formulas for the solutions of the Neumann problem and also on the $$L^{p}$$ boundedness of Calderon–Zygmund singular integral operators appearing in those representation formulas. That is why we consider the case where the space dimension is greater than 2. Moreover, we make use of an $$L^{p}$$ variant of an estimate for the Green operator of the Neumann problem introduced in the study of Feller semigroups.

We study singular modes of volume singular integral equations describing problems of scattering of electromagnetic wave in anisotropic dielectric structures. An explicit form of these modes is obtained for a certain class of anisotropic media. Example of real media in which singular modes can exist are considered.

We obtain a two weight local Tb theorem for any elliptic and gradient elliptic fractional singular integral operator T on the real line, and any pair of locally finite positive Borel measures on the line. This includes the Hilbert transform and in a sense improves on the T1 theorem by the authors and M. Lacey., 120 pages, 3 figures, 56 pages of appendices. We clarify arguments, mostly in appendix B, especially regarding the backward Poisson testing condition. We thank the referee for a thorough and insightful report. Main results unchanged

In this paper the jump formulas for the double layer potential and other singular integrals are proved for arbitrary rectifiable sets, by defining suitable non-tangential limits. The arguments are quite straightforward and only require some Calder\'on-Zygmund techniques., Comment: Correction of a few typos in this version

A new scale of weighted grand mixed norm Lebesgue spaces L w 1 p 1 , φ 1 ⁢ ( L w 2 p 2 , φ 2 ) {L_{w_{1}}^{p_{1},\varphi_{1}}(L_{w_{2}}^{p_{2},\varphi_{2}})} is introduced and the boundedness criteria for multiple singular operators are established.

We study the problem of limit equilibrium of a long cylindrical shell with longitudinal crack subjected to the action of a load that varies with time according to the exponential law. For the case of symmetric loading of the crack, we construct a system of singular integral equations. We also study the influence of the rate of changes in the load on the value of the force intensity factor in the vicinity of the crack ends.

For a mixed elliptic-hyperbolic type equation in a mixed domain, when the elliptic part is a sector of the circle, and the hyperbolic part consists of two characteristic triangles, the Bitsadze–Samarskii type nonlocal problem is investigated. A feature of such problems is that the boundary conditions in the hyperbolic parts of the boundary are determined by a first-order differential operator and pointwise link the values of the partial derivatives of the desired solution on the characteristics with the partial derivatives on arbitrary monotone curves lying inside the characteristic triangles of the equation. To solve the problem, the methods of the theory of partial differential equations and the theory of singular integral equations as well as the methods of energy integrals and complex analysis were used, with the help of which the existence and uniqueness theorem for the solution of the investigated problem is proved. Also, a method of reducing the investigated problem to an equivalent system of singular integral equations is shown, also, a method for solving this system is proposed, which allows to obtain a solution to the problem in explicit form.

This paper is devoted to studying singular integrals and maximal operators with rough kernels in the mixed homogeneity setting. Assuming the kernels satisfy certain rather weak sized conditions, the boundedness for such operators on the mixed radial-angular spaces are established. Meanwhile, the corresponding vector-valued versions are also obtained. All of the results are new even in the isotropic setting.

In this work, we present a modified linear approximation for solving the first and the second kind Abel–Volterra integral equations. This approximation was used by the author to approximate a weakly singular integral on the curve. Noting that this new technique gives a good approximation of these solutions compared with several methods in several numerical examples.

The monograph consists of three parts. Part I is presented here. In this monograph, we develop a new approach (mainly based on papers of the author). Many results are published for the first time here. Chapter 1 is introductory. It provides the necessary background from functional analysis (for completeness). In this monograph, we mostly use weighted HOlder spaces; they are considered in Chap. 2. Chapter 3 plays the key role: in weighted HOlder spaces, we consider estimates of integral operators with homogeneous difference kernels, covering potential-type integrals and singular integrals as well as Cauchy-type integrals and double layer potentials. In Chap. 4, similar estimates in weighted Lebesgue spaces are proved. Integrals with homogeneous difference kernels will play an important role in Part III of the monograph, which will be devoted to elliptic boundary-value problems. They naturally arise in integral representations of solutions of first-order elliptic systems in terms of fundamental matrices or their parametrices. The investigation of boundary-value problems for second-order and higher-order elliptic equations or systems is reduced to first-order elliptic systems.

In this paper we analyze the convergence of the following type of series $$\begin{eqnarray}T_{N}^{{\mathcal{L}}}f(x)=\mathop{\sum }_{j=N_{1}}^{N_{2}}v_{j}\big(e^{-a_{j+1}{\mathcal{L}}}f(x)-e^{-a_{j}{\mathcal{L}}}f(x)\big),\quad x\in \mathbb{R}^{n},\end{eqnarray}$$ where ${\{e^{-t{\mathcal{L}}}\}}_{t>0}$ is the heat semigroup of the operator ${\mathcal{L}}=-\unicode[STIX]{x1D6E5}+V$ with $\unicode[STIX]{x1D6E5}$ being the classical laplacian, the nonnegative potential $V$ belonging to the reverse Hölder class $RH_{q}$ with $q>n/2$ and $n\geqslant 3$, $N=(N_{1},N_{2})\in \mathbb{Z}^{2}$ with $N_{1}, ${\{v_{j}\}}_{j\in \mathbb{Z}}$ is a bounded real sequences, and ${\{a_{j}\}}_{j\in \mathbb{Z}}$ is an increasing real sequence.Our analysis will consist in the boundedness, in $L^{p}(\mathbb{R}^{n})$ and in $BMO(\mathbb{R}^{n})$, of the operators $T_{N}^{{\mathcal{L}}}$ and its maximal operator $T^{\ast }f(x)=\sup _{N}T_{N}^{{\mathcal{L}}}f(x)$.It is also shown that the local size of the maximal differential transform operators (with $V=0$) is the same with the order of a singular integral for functions $f$ having local support. Moreover, if ${\{v_{j}\}}_{j\in \mathbb{Z}}\in \ell ^{p}(\mathbb{Z})$, we get an intermediate size between the local size of singular integrals and Hardy–Littlewood maximal operator.

It is known that the compactness of the commutators of point-wise multiplication with bilinear homogeneous Calderon–Zygmund operators acting on product of Lebesgue spaces is characterized by the multiplying function being in the space CMO. This space is the closure in BMO of its subspace of smooth functions with compact support. It is shown in this work that for bilinear Calderon–Zygmund operators arising from smooth (inhomogeneous) bilinear Fourier multipliers or bilinear pseudodifferential operators, one can actually consider multiplying functions in a new subspace of BMO larger than CMO.

Let $\Omega$ be homogeneous of degree zero and have mean value zero on the unit sphere ${S}^{n-1}$, $T_{\Omega}$ be the convolution singular integral operator with kernel $\frac{\Omega(x)}{|x|^n}$. For $b\in{\rm BMO}(\mathbb{R}^n)$, let $T_{\Omega,\,b}$ be the commutator of $T_{\Omega}$. In this paper, by establishing suitable sparse dominations, the authors establish some weak type endpoint estimates of $L\log L$ type for $T_{\Omega,\,b}$ when $\Omega\in L^q(S^{n-1})$ for some $q\in (1,\,\infty]$.

In this paper, we introduce the notion of Krasnoselskii and Dugundji-Granas condensing operators in Banach spaces. In order to pave the way for a study the solvability of some classes of singular integral equations in the Banach algebra C[a,b], we provide some results for the existence of fixed points for such condensing operators. An example is presented to show the applicability of the results.

This paper is devoted to the study of dual singular integral equations with convolution kernels in the case of non-normal type. Via using the Fourier transforms, we transform such equations into Riemann boundary value problems. To solve the equation, we establish the regularity theory of solvability. The general solutions and the solvable conditions of the equation are obtained. Especially, we investigate the asymptotic property of solutions at nodes. This paper will have a significant meaning for the study of improving and developing complex analysis, integral equations and Riemann boundary value problems.

We prove sharp weak type weighted estimates for a class of sparse operators that includes majorants of standard $\alpha$-fractional singular integrals, fractional integral operators, Marcinkiewicz integral operators, and square functions. These bounds are knows to be sharp in many cases, and our main new result is the optimal bound $$[w]_{A_{p,q}}^{\frac{1}{q}}[w^{q}]_{A_{\infty}}^{\frac{1}{2}-\frac{1}{p}}\lesssim[w]_{A_{p,q}}^{\frac{1}{2}-\frac{\alpha}{d}}$$ for proper conditions which satisfy that three index $p$, $q$ and $\alpha$ ensure weak type norm of fractional square functions on $L^{q}(w^{q})$ with $p>2$.

In the paper, some weighted maximal inequalities for the Toeplitz operator related to the singular integral transform with variable CalderȮn-Zygmund kernel are proved. As an application, the boundedness of the operator on weighted Lebesgue space are obtained.