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In this paper, we study higher-order Riesz transforms associated with the inverse Gaussian measure given byπn/2ex2dxonℝn. We establishLpℝn,ex2dx-boundedness properties and obtain representations as principal values singular integrals for the higher-order Riesz transforms. New characterizations of the Banach spaces having the UMD property by means of the Riesz transforms and imaginary powers of the operator involved in the inverse Gaussian setting are given.

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

This paper is devoted to investigating the boundedness, continuity and compactness for variation operators of singular integrals and their commutators on Morrey spaces and Besov spaces. More precisely, we establish the boundedness for the variation operators of singular integrals with rough kernels Ω ∈ Lq (S n−1) (q > 1) and their commutators on Morrey spaces as well as the compactness for the above commutators on Lebesgue spaces and Morrey spaces. In addition, we present a criterion on the boundedness and continuity for a class of variation operators of singular integrals and their commutators on Besov spaces. As applications, we obtain the boundedness and continuity for the variation operators of Hilbert transform, Hermit Riesz transform, Riesz transforms and rough singular integrals as well as their commutators on Besov spaces.

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.

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 aim of this paper is to tackle the numerical calculation of the so-called finite-part integrals, including Cauchy principal value integrals and any supersingular integral. The numerical procedure consists of replacing the density function f ( t ) by the polynomial that is the best fit in a least-squares sense for the values f ( z k ) , where { z k } is a set of distinct points located on an ellipse that surrounds the interval [ − 1 , 1 ] . This method works in all mentioned cases, it does not depend on derivatives of f ( t ) , and provides very accurate results when f is smooth enough. Some examples show the performance of this approach.

Let $$(X, d, \mu )$$ be a space of homogeneous type with a metric d and a doubling measure $$\mu $$ . Assume that $$\rho $$ is a critical function on X which has an associated class of weights containing the Muckenhoupt weights as a proper subset. In this paper, we prove the quantitative weighted estimates for certain singular integrals corresponding to the new class of weights. It is important to note that the assumptions on the kernels of these singular integrals do not have any regularity conditions. Our applications include the spectral multipliers and the Riesz transforms associated to Schrodinger operators in various settings, ranging from the magnetic Schrodinger operators in Euclidean spaces to the Laguerre operators.

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.

In this paper, we consider the following non-linear equations in unbounded domains Ω with exterior Dirichlet condition: { ( − Δ ) p s u ( x ) = f ( u ( x ) ) , x ∈ Ω , u ( x ) > 0 , x ∈ Ω , u ( x ) = 0 , x ∈ R n ∖ Ω , where ( − Δ ) p s is the fractional p-Laplacian defined as (0.1) ( − Δ ) p s u ( x ) = C n , s , p P . V . ∫ R n | u ( x ) − u ( y ) | p − 2 [ u ( x ) − u ( y ) ] | x − y | n + s p d y with 0 s 1 and p ≥ 2 . We first establish a maximum principle in unbounded domains involving the fractional p-Laplacian by estimating the singular integral in (0.1) along a sequence of approximate maximum points. Then, we obtain the asymptotic behavior of solutions far away from the boundary. Finally, we develop a sliding method for the fractional p-Laplacians and apply it to derive the monotonicity and uniqueness of solutions. There have been similar results for the classical Laplacian [3] and for the fractional Laplacian [39] , which are linear operators. Unfortunately, many approaches there no longer work for the fully non-linear fractional p-Laplacian here. To circumvent these difficulties, we introduce several new ideas, which enable us not only to deal with non-linear non-local equations, but also to remarkably weaken the conditions on f ( ⋅ ) and on the domain Ω. We believe that the new methods developed in our paper can be widely applied to many problems in unbounded domains involving non-linear non-local operators.

This research and survey article deals exclusively with the study of the approximation of generalized multivariate Gauss-Weierstrass singular integrals to the identity-unit operator. Here we study quantitatively most of their approximation properties. The multivariate generalized Gauss-Weierstrass operators are not in general positive linear operators. In particular we study the rate of convergence of these operators to the unit operator, as well as the related simultaneous approximation. These are given via Jackson type inequalities and by the use of multivariate high order modulus of smoothness of the high order partial derivatives of the involved function. Also we study the global smoothness preservation properties of these operators. These multivariate inequalities are nearly sharp and in one case the inequality is attained, that is sharp. Furthermore we give asymptotic expansions of Voronovskaya type for the error of multivariate approximation. The above properties are studied with respect to Lp norm, 1 ≤ p ≤ ∞.

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.

We prove the boundedness for a class of multi-sublinear singular integral operators on the product of central Morrey spaces with variable exponents. Based on this result, we obtain the boundedness for the multilinear singular integral operators and two kinds of multilinear singular integral commutators on the above spaces.

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

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.

The purpose of this paper is to provide necessary and sufficient conditions of the boundedness for singular integrals on the local Hardy space and its dual. Particularly the singular integrals considered in this paper include the pseudo-differential operators $T_{\sigma }f(x)=\int \limits \sigma (x \xi )e^{2\pi ix\xi }\hat {f}(\xi )d\xi $ with $\sigma \in S_{1 0}^{0}$ . As a consequence our results give another proof of the boundedness of the pseudo-differential operators on the local Hardy space (Goldberg Duke Math. J. 46(1) 27–42 1979).

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

In this note, we propose a new class of orthogonal polynomials (named Bachok–Hasham polynomials H 1 n k ( x ) ), which is an extension of the Chebyshev polynomials. Eigenfunctions and corresponding eigenvalues are found for the homogeneous second kind of Logarithmic Singular Integral Equations (LogSIEs). For non-homogeneous LogSIEs truncated series of the first kind Bachok–Hasham polynomials are used to find approximate solution. It is found that first kind of Bachok–Hasham polynomials ( H 1 n k ( x ) ) are orthogonal with weight w k ( x ) = x k - 1 1 - x 2 k , where k is positive odd integer. Properties of first kind of Bachok–Hasham polynomials are also proved. Finally, two examples are presented to show the validity and accuracy of the proposed method.

The main purpose of this paper is to establish a Hardy space theory associated with a new multi-parameter structure and characterize this Hardy space as the intersection of flag Hardy spaces. The key idea used here is to identify the new kernels as sums of two flag kernels.

This paper explores the limiting weak-type behaviors of certain classical operators in harmonic analysis including maximal operators, singular and fractional integral operators and maximal truncated singular integrals as well as the general convolution operators with weak-type Young’s inequalities. Some optimal limiting weak-type behaviors are given, which essentially improve and extend the previous results. As applications, several characterizations on the weak-type endpoint boundedness for fractional integral operators and maximal operators with homogeneous kernels, and the weak-type Young’s inequality for the general convolution operator are also obtained.

In this paper, we investigate the behavior of the bounds of the composition for rough singular integral operators on the weighted space. More precisely, we obtain the quantitative weighted bounds of the composite operator for two singular integral operators with rough homogeneous kernels on $$L^p({\mathbb {R}}^d,\,w)$$ , $$p\in (1,\,\infty )$$ , which is smaller than the product of the quantitative weighted bounds for these two rough singular integral operators. Moreover, at the endpoint $$p=1$$ , the $$L\log L$$ weighted weak-type bound is also obtained, which has interests of its own in the theory of rough singular integral even in the unweighted case.

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 μ be a non-negative Radon measure on Rd which only satisfies the following growth condition that there exists a positive constant C such that μ(B(x,r)) ≤ Crn for all x∈ Rd, r > 0 and some fixed n ∈ (0,d]. This paper is interested in the properties of the iterated commutators of multilinear singular integral operators on Morrey spaces .Precisely speaking, we show that the iterated commutators generated by multilinear singular integrals operators are bounded from to where (Regular Bounded Mean Oscillation space) and 1 qj ≤ pj ∞ with 1/p = 1/p1 + ... + 1/pm and 1/q = 1/q1+ ... + 1/qm.

The properties of the class of functions of generalized bounded variation are studied. The “anomaly” feature of this class is revealed. There is the notation of absolute continuity with respect to $$((p_n), \phi )$$ and it’s connection with the ordinary absolute continuity is investigated. The problems of approximation by Steklov’s functions and singular integrals are studied.

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.

In this paper we provide some quantitative mixed weak-type estimates assuming conditions that imply that $$uv\in A_{\infty }$$ for Calderon–Zygmund operators, rough singular integrals and commutators. The main novelty of this paper lies in the fact that we rely upon sparse domination results, pushing an approach to endpoint estimates that was introduced in Domingo-Salazar et al. (Bull Lond Math Soc 48(1):63–73, 2016) and extended in Lerner et al. (Adv Math 319:153–181, 2017) and Li et al. (J Geom Anal, 2018).

Let Ω ∈ Ls(Sn−1) (s > 1) be a homogeneous function of degree zero and b be a BMO function or Lipschitz function. In this paper, the authors obtain some boundedness of the Calderon-Zygmund singular integral operator TΩ and its commutator [b, TΩ] on Herz-Morrey spaces with variable exponent.

Under some pretty much weaker size conditions assumed on the integral kernels both on the unit sphere and in the radial directions, the L p {L^{p}} boundedness was given for the rough singular integrals defined by translates of a real-analytic submanifold in ℝ n {\mathbb{R}^{n}} . Certain L p {L^{p}} estimates for the corresponding maximal rough singular integrals were also established.

In this paper, we prove the sharp weighted bound for certain singular integrals which have non-smooth kernels and do not belong to the class of standard Calderon–Zygmund operators. Our assumptions are weaker than those known in literature, since in particular we do not assume the Cotlar type inequality condition. Applications include sharp weighted estimates for the Riesz transforms associated to the Dirichlet Laplacians on open connected domains, the Riesz transforms associated to the Schrodinger operators with real potentials on the Euclidean spaces, the Riesz transforms associated to the degenerate Schrodinger operators and the Riesz transforms associated to the Schrodinger operators with inverse square potentials.

We prove l p ( Z d ) bounds, for p ∈ ( 1 , ∞ ) , of discrete maximal functions corresponding to averaging operators and truncated singular integrals of Radon type, and their applications to pointwise ergodic theory. Our new approach is based on a unified analysis of both types of operators, and also yields an extension to the vector-valued form of these results.

In this paper, we establish the extrapolation theory for the amalgam spaces and the Hardy-amalgam spaces. By using the extrapolation theory, we obtain the mapping properties for the Calderón-Zygmund operators and its commutator, the Carleson operators and establish the Rubio de Francia inequalities for Littlewood-Paley functions of arbitrary intervals to the amalgam spaces. We also obtain the boundedness of the Calder{ó}n-Zygmund operators and the intrinsic square function on the Hardy-amalgam spaces.

Let X0,X1,…,Xq(q&lt, N) be real vector fields, which are left invariant on homogeneous group G, provided that X0 is homogeneous of degree two and X1,…,Xq are homogeneous of degree one. We consider the following nondivergence degenerate operator with drift L=∑i,j=1qaij(x)XiXj+a0(x)X0, where the coefficients aij(x), a0(x) belonging to vanishing mean oscillation space are bounded measurable functions. Furthermore, aij(x) satisfies the uniform ellipticity condition on Rq and a0(x)≠0. We obtain the local weighted Sobolev–Morrey estimates by applying the boundedness of commutators and interpolation inequalities on weighted Morrey spaces.

Let $M$ be a non-doubling parabolic manifold with ends and $L$ a non-negative self-adjoint operator on $L^{2}(M)$ which satisfies a suitable heat kernel upper bound named the upper bound of Gaussian type. These operators include the Schrodinger operators $L = \Delta + V$ where $\Delta$ is the Laplace–Beltrami operator and $V$ is an arbitrary non-negative potential. This paper will investigate the behaviour of the Poisson semi-group kernels of $L$ together with its time derivatives and then apply them to obtain the weak type $(1, 1)$ estimate of the functional calculus of Laplace transform type of $\sqrt{L}$ which is defined by $\mathfrak{M}(\sqrt{L}) f(x) := \int_{0}^{\infty} \bigl[\sqrt{L} e^{-t \sqrt{L}} f(x)\bigr] m(t) dt$ where $m(t)$ is a bounded function on $[0, \infty)$. In the setting of our study, both doubling condition of the measure on $M$ and the smoothness of the operators' kernels are missing. The purely imaginary power $L^{is}$, $s \in \mathbb{R}$, is a special case of our result and an example of weak type $(1, 1)$ estimates of a singular integral with non-smooth kernels on non-doubling spaces.

For oscillatory singular integrals with polynomial phases and Holder class kernels, we establish their uniform boundedness on $$L^p$$ spaces as well as a sharp logarithmic bound on the Hardy space $$H^1$$ . These results improve the ones in (Pan in Forum Math 31: 535–542, 2019) by removing the restriction that the phase polynomials be quadratic.

This work develops a new method, based on the use of Gustafson's integrals and on the evaluation of singular integrals, allowing one to establish the unitarity of the separation of variables transform for infinite-dimensional representations of rank one quantum integrable models. We examine in detail the case of the $\mathrm{SL}(2,\mathbb R)$ spin chains.

In this paper, we study the mixed radial-angular integrability for maximal operators related to a class of rough singular integrals along “polynomial curves". Under assuming that the kernel satisfies certain rather weak size condition on sphere, we obtain the boundedness of such maximal operators on the mixed radial-angular Lebesgue spaces. Meanwhile, on the corresponding vector-valued versions are also established.

Weighted inequalities with power-type weights for operators of harmonic analysis, such as maximal and singular integral operators, and commutators of singular integrals in grand variable exponent Lebesgue spaces are derived. The spaces and operators are defined on quasi-metric measure spaces with doubling condition (spaces of homogeneous type). The proof of the result regarding the Hardy–Littlewood maximal operator is based on the appropriate sharp weighted norm estimates with power-type weights. To obtain the results for singular integrals and commutators we prove appropriate weighted extrapolation statement in grand variable exponent Lebesgue spaces. The extrapolation theorem deals with a family of pairs of functions (f, g). One of the consequences of the latter result is the weighted extrapolation for sublinear operators S acting in these spaces. As one of the applications of the main results we present weighted norm estimates for the Hardy–Littlewood maximal function, Cauchy singular integral operator, and its commutators in grand variable exponent Lebesgue spaces defined on rectifiable regular curves.

Utilising some recent ideas from our bilinear bi-parameter theory, we give an efficient proof of a two-weight Bloom-type inequality for iterated commutators of linear bi-parameter singular integrals. We prove that if $T$ is a bi-parameter singular integral satisfying the assumptions of the bi-parameter representation theorem, then $$\begin{equation*} \| [b_k,\cdots[b_2, [b_1, T]]\cdots]\|_{L^p(\mu) \to L^p(\lambda)} \lesssim_{[\mu]_{A_p}, [\lambda]_{A_p}} \prod_{i=1}^k\|b_i\|_{\operatorname{bmo}(\nu^{\theta_i})}, \end{equation*}$$where $p \in (1,\infty )$, $\theta _i \in [0,1]$, $\sum _{i=1}^k\theta _i=1$, $\mu , \lambda \in A_p$, $\nu := \mu ^{1/p}\lambda ^{-1/p}$. Here $A_p$ stands for the bi-parameter weights in ${\mathbb{R}}^n \times{\mathbb{R}}^m$, and $\operatorname{bmo}(\nu )$ is a suitable weighted little BMO space. We also simplify the proof of the known 1st order case.

Motivated by the asymptotic estimates and Hasse principle for multidimensional Waring's problem via the circle method, we prove for the first time that the corresponding singular series is bounded below by an absolute positive constant without any nonsingular local solubility assumption. The number of variables we need is near-optimal. By proving a more general uniform density result over certain complete discrete valuation rings with finite residue fields, we also establish uniform lower bounds for both singular series and singular integral in F q [ t ] . We thus obtain asymptotic formulas and the Hasse principle for multidimensional Waring's problem in F q [ t ] via a variant of the circle method.

Though multi-parameter Hardy space theory has been well developed in the past half century, not much has been studied for a local Hardy space theory in the multi-parameter settings. Such multi-parameter local Hardy spaces can play an important role in studying the boundedness of multi-parameter pseudo-differential operators, multi-parameter singular integrals of non-convolution type, and applications to partial differential equations, etc. By establishing a bi-parameter local reproducing formula, bi-parameter local Hardy space h p ( R n 1 × R n 2 ) is introduced in this paper. This space coincides with L p ( R n 1 + n 2 ) when p > 1 . While p ≤ 1 , this space is substantially different from the classical bi-parameter Hardy spaces H p ( R n 1 × R n 2 ) . We will establish its atomic decomposition of the bi-parameter local Hardy spaces and as an application, we prove the boundedness from h p ( R n 1 × R n 2 ) to L p ( R n 1 + n 2 ) of the bi-parameter singular integral operators in inhomogeneous Journe class for p near 1. For the simplicity, we have chosen to present all the results in the bi-parameter setting. Nevertheless, all of them hold for arbitrary number of parameters. The multi-parameter local theory developed in this paper can serve as a model case for similar theory in other multi-parameter settings.

The first author is supported by NSF Grant DMS-1362425 and research funds from the Dean of the College of Arts & Sciences, the University of Alabama. The second author is supported by the Simons Foundation. We establish the boundedness of the multilinear Calderon{Zygmund operators from a product of weighted Hardy spaces into a weighted Hardy or Lebesgue space. Our results generalize to the weighted setting results obtained by Grafakos and Kalton [18] and recent work by the third author, Grafakos, Nakamura, and Sawano [20]. As part of our proof we provide a finite atomic decomposition theorem for weighted Hardy spaces, which is interesting in its own right. As a consequence of our weighted results, we prove the corresponding estimates on variable Hardy spaces. Our main tool is a multilinear extrapolation theorem that generalizes a result of the first author and Naibo [10].