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This paper extends the Rubio de Francia extrapolation method to the grand Morrey spaces on Euclidean spaces. By using this extended extrapolation method, we obtain the boundedness of the rough singular integral operators, the spherical maximal functions and the maximal Bochner-Riesz operators on the grand Morrey spaces on Euclidean spaces. [ABSTRACT FROM AUTHOR]

In this paper, we prove the two-weighted boundedness of maximal operator, singular integral operators and their commutators in weighted global Morrey-type spaces GMp,θ,ω,φ(Rn). Furthermore, we give weighted global Morrey-type a priori estimates and a priori estimates for non-divergent elliptic equations in GMp,θ,ω,φ(Rn) spaces as applications. [ABSTRACT FROM AUTHOR]

We consider a class of hypoelliptic operators of the following type L = ∑ i , j = 1 p 0 a ij ∂ x i x j 2 + ∑ i , j = 1 N b ij x i ∂ x j - ∂ t , where (a ij) , (b ij) are constant matrices and (a ij) is symmetric positive definite on R p 0 (p 0 ≤ N) . We obtain generalized Hölder estimates for L on R N + 1 by establishing several estimates of singular integrals in generalized Morrey spaces. [ABSTRACT FROM AUTHOR]

This paper concerns the study singular integrals along twisted surfaces of the form We prove Lp bounds for the corresponding operators when the surfaces are defined by mappings more general than polynomials and convex functions, provided that the kernels are in L(logL)2(Sn-1x Sm-1) . [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to develop the theory of product Hardy spaces built on Banach lattices on R n × R m . First we introduce new product Hardy spaces H X (R n × R m) associated with ball quasi-Banach function spaces X (R n × R m) via applying the Littlewood–Paley–Stein theory. Then we establish a decomposition theorem for H X (R n × R m) in terms of the discrete Calderón’s identity. Moreover, we explore some useful and general extrapolation theorems of Rubio de Francia on X (R n × R m) and give some applications to boundedness of operators. Finally, we conclude that the bi-parameter singular integral operators T ~ are bounded from H X (R n × R m) to itself and bounded from H X (R n × R m) to X (R n × R m) via extrapolation. The main results obtained in this paper have a wide range of generality. Especially, we can apply these results to many concrete examples of ball quasi-Banach function spaces, including product Herz spaces, weighted product Morrey spaces and product Orlicz spaces. [ABSTRACT FROM AUTHOR]

We prove the weak type (1,1) estimate for maximal function of the truncated rough Hilbert transform considered in Paluszy'nski (ASNSCS 910:679-704, 2019), Paluszy'nski (CM 164(2):305-325, 2021). [ABSTRACT FROM AUTHOR]

We obtain the mapping properties of the strongly singular Calderón–Zygmund operators on Hardy spaces associated with ball quasi-Banach function spaces. We established this result by using the idea from extrapolation originated from Rubio de Francia. As applications of this result, we present the mapping properties of the strongly singular Calderón–Zygmund operators to the Hardy Orlicz-slice spaces, the Hardy local Morrey spaces with variable exponents and the Herz–Hardy spaces with variable exponents. [ABSTRACT FROM AUTHOR]

Motivated by the composition of singular integral operators, we introduce a class of singular integral operators in the Euclidean space R n with kernels supported in surfaces in R n × R n . The surfaces are allowed to be very flat at the origin in R n × R n . The boundedness of the operators on L p is proved under the assumption that the kernels are rough in L (log L) 2 (S n - 1 × S n - 1) . [ABSTRACT FROM AUTHOR]

In this article we study the boundedness of classical singular integral operator over local fields with rough kernel. We are relaxing the smoothness condition on kernel by block spaces and found the boundedness of truncated singular integral operator on different function spaces such as Lebesgue spaces, Besov spaces and Triebel–Lizorkin spaces. Unlike Euclidean space, our bound of truncated singular integral operator depends on the constant which leads the unboundedness of singular integral operator over local fields. [ABSTRACT FROM AUTHOR]

Singular integral operators with piecewise continuous matrix coefficients defined on segments of the real axis in weighted Lebesgue and Holder spaces are considered. In contrast to the classical case, the singular Cauchy operator together with the noncompact integral operators of a special form whose kernels are approximately homogeneous of degree –1 are among these operators. Similar operators arise in many applications. The Fredholm property criterion is established for these operators as well as a formula for the index. Examples of singular integral equations arising in the study of boundary value problems for forward-backward parabolic equations are displayed. [ABSTRACT FROM AUTHOR]

In this paper, we extend the two-weighted norm inequalities of singular integral operators to local Morrey spaces. This gives the mapping properties of singular integral operators from weighted local Morrey spaces to other local Morrey spaces with different weight. As application of this result, we extend the Córdoba- Fefferman inequalities to weighted local Morrey spaces. [ABSTRACT FROM AUTHOR]

In this article, we first give a modified Schwarz–Pompeiu formula in a general sector ring with angle ϑ = π α , α ≥ 1 / 2 by proper conformal mappings, and obtain the solution of the Schwarz problem for the Cauchy–Riemann equation in explicit forms. Furthermore, we construct some integral operators and discuss their properties in detail. Finally, by virtue of these operators, Schwarz problems for an inhomogeneous polyanalytic equation and for a generalized polyanalytic equation are investigated, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we study the L p boundedness and L p (w) boundedness ( 1 < p < ∞ and w a Muckenhoupt A p weight) of fractional maximal singular integral operators T Ω , α # with homogeneous convolution kernel Ω (x) on an arbitrary homogeneous group H of dimension Q . We show that if 0 < α < Q , Ω ∈ L 1 (Σ) and satisfies the cancellation condition of order [ α ] , then for any 1 < p < ∞ , ‖ T Ω , α # f ‖ L p (H) ≲ ‖ Ω ‖ L 1 (Σ) ‖ f ‖ L α p (H) . where for the case α = 0 , the L p boundedness of rough singular integral operator and its maximal operator were studied by Tao (Indiana Univ Math J 48:1547–1584, 1999) and Sato (J Math Anal Appl 400:311–330, 2013), respectively. We also obtain a quantitative weighted bound for these operators. To be specific, if 0 ≤ α < Q and Ω satisfies the same cancellation condition but a stronger condition that Ω ∈ L q (Σ) for some q > Q / α , then for any 1 < p < ∞ and w ∈ A p , ‖ T Ω , α # f ‖ L p (w) ≲ ‖ Ω ‖ L q (Σ) { w } A p (w) A p ‖ f ‖ L α p (w) , 1 < p < ∞. [ABSTRACT FROM AUTHOR]

In this paper, we establish effective necessary and sufficient conditions for the Noetherian property of two-dimensional singular integral equations of the Mikhlin–Calderón–Zygmund type in Lebesgue spaces with a weight and provide a formula for calculating the index. [ABSTRACT FROM AUTHOR]

Let μ be a Radon measure on Rd . We define and study conical energies Eμ,p(x, v,α), which quantify the portion of μ lying in the cone with vertex x € Rd , direction V € G (d,d - n), and aperture α €(0,1).We use these energies to characterize rectifiability and the big pieces of Lipschitz graphs property. Furthermore, if we assume that μ has polynomial growth, we give a sufficient condition for L2 ( μ ) boundedness of singular integral operators with smooth odd kernels of convolution type. [ABSTRACT FROM AUTHOR]

We present a general approach to sparse domination based on single-scale L p -improving as a key assumption. The results are formulated in the setting of metric spaces of homogeneous type and avoid completely the use of dyadic-probabilistic techniques as well as of Christ–Hytönen–Kairema cubes. Among the applications of our general principle, we recover sparse domination of Dini-continuous Calderón–Zygmund kernels on spaces of homogeneous type, we prove a family of sparse bounds for maximal functions associated to convolutions with measures exhibiting Fourier decay, and we deduce sparse estimates for Radon transforms along polynomial submanifolds of R n . [ABSTRACT FROM AUTHOR]

Singular integral operators with piecewise continuous matrix coefficients are considered on a piecewise smooth curve in weighted Lebesgue spaces. In contrast to the classical case, the operators have generalized Cauchy kernels arising as a parametrix of first-order elliptic systems in the plane. A Fredholmness criterion and an index formula for these operators are obtained in weighted Lebesgue spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study the rough singular integral operator \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 \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 \Omega \in H^1(\mathrm {S}^{n-1}) has zero mean value and n\geq 2. We prove that T_\Omega and T^*_\Omega are bounded on the mixed radial-angular spaces L_{|x|}^pL_{\theta }^{\tilde {p}}(\mathbb {R}^n) for some suitable indexes 1

Let G = (RN,◦,δλ) be a homogeneous group, Q be the homogeneous dimension of G, X0,X1,...,Xm be left invariant real vector fields on G and satisfy H ¨ormander's rank condition on RN. Assume that X1,...,Xm (m ≤ N - 1) are homogeneous of degree one and X0 is homogeneous of degree two with respect to the family of dilations (δλ)λ>0. Consider the following hypoelliptic operator with drift on G ... where (aij) is a constant matrix satisfying the elliptic condition in Rm and a0 ≠ 0. In this paper, for this class of operators we obtain generalized weighted Sobolev-Morrey estimates by establishing boundedness of a large class of sublinear operators Tα, α ∈ [0,Q) generated by Calder 'on-Zygmund operators (α = 0) and generated by fractional integral operator (α > 0) on generalized weighted Morrey spaces and proving interpolation results in generalized weighted Sobolev-Morrey spaces on G. [ABSTRACT FROM AUTHOR]

In this paper, by the aid of the Poisson integral, we establish a Carleson type characterization of Q type spaces Q K p (R n) (n ≥ 1) with weight K. As an application, we prove that convolution singular integral operators are bounded on Q K p (R n) for n ≥ 2 . [ABSTRACT FROM AUTHOR]

In a recent article (2021) we introduced and studied conical energies. We used them to prove three results: a characterization of rectifiable measures, a characterization of sets with big pieces of Lipschitz graphs, and a sufficient condition for boundedness of nice singular integral operators. In this note we give two examples related to sharpness of these results. One of them is due to Joyce and Morters (2000), the other is new and could be of independent interest as an example of a relatively ugly set containing big pieces of Lipschitz graphs. [ABSTRACT FROM AUTHOR]

Let Ω be a domain which belongs to a class of bounded weakly pseudoconvex domains of finite type in C n , let d λ be the Monge–Ampère boundary measure on b Ω and ϱ ≥ 0 be a non-decreasing function. The aim of this paper is to establish the characterizations of boundedness and compactness for the commutator operators of Cauchy–Fantappiè type integrals with L 1 (b Ω , d λ) functions on the generalized Morrey spaces L ϱ p (b Ω , d λ) , with p ∈ (1 , ∞) . [ABSTRACT FROM AUTHOR]

We give two-weighted norm estimates for higher order commutator of classical operators such as singular integral and fractional type operators, between weighted L p and certain spaces that include Lipschitz, BMO and Morrey spaces. We also give the optimal parameters involved with these results, where the optimality is understood in the sense that the parameters defining the corresponding spaces belong to certain region out of which the classes of weights are satisfied by trivial weights. We also exhibit pairs of non-trivial weights in the optimal region satisfying the conditions required. Finally, we exhibit an extrapolation result that allows us to obtain boundedness results of the type described above in the variable setting and for a great variety of operators, by starting from analogous inequalities in the classical context. In order to get this result we prove a Calderón–Scott type inequality with weights that connects adequately the spaces involved. [ABSTRACT FROM AUTHOR]

In this paper we consider local "complementary" generalized Morrey spaces with variable exponent p(x) and a general function ω(r) defining a Morrey-type norm. We prove the boundedness of the Hardy-Littlewood maximal operator and Calderón-Zygmund singular operators with standard kernel in such spaces in case of unbounded sets Ω ⊂ ℝn. Also we prove the boundedness of commutators of Hardy-Littlewood maximal operator and Calderón-Zygmund singular integral operators. [ABSTRACT FROM AUTHOR]

The main purpose of this article is to present a proof of the T(1) Theorem that uses a continuous version of the Cotlar-Knapp-Stein lemma, due to A. P. Calderón and R. Vaillancourt.

In this paper, the authors define the weak Herz spaces and the weak Herz-type Hardy spaces with variable exponent. As applications, the authors establish the boundedness for a class of singular integral operators including some critical cases. [ABSTRACT FROM AUTHOR]

We introduce a class of Morrey-type spaces M p , q , Ω λ , which includes the classical Morrey spaces and discuss their properties. We prove a Marcinkiewicz-type interpolation theorem for such spaces. This theorem is then applied to obtaining an analogue of O'Neil's inequality for convolutions and to proving the boundedness in the introduced Morrey-type spaces of the Riesz potential and singular integral operators. [ABSTRACT FROM AUTHOR]

We prove L p boundedness in A ∞ weighted spaces for operators defined by almost-orthogonal expansions indexed over the dyadic cubes. The constituent functions in the almost-orthogonal families satisfy weak decay, smoothness, and cancellation conditions. We prove that these expansions are stable (with respect to the L p operator norm) when the constituent functions suffer small dilation and translation errors. [ABSTRACT FROM AUTHOR]

We obtain the boundedness from a product of Lebesgue or Hardy spaces into Hardy spaces under suitable cancellation conditions for a large class of multilinear operators that includes the Coifman–Meyer class, sums of products of linear Calderón–Zygmund operators and combinations of these two types. [ABSTRACT FROM AUTHOR]

In 2004, Nakai defined generalized Orlicz-Morrey spaces of the first kind. This is an extension of the 1994 paper by Nakai, where Nakai showed that singular integral operators are bounded from generalized Morrey spaces to weak generalized Morrey spaces. Since generalized Orlicz- Morrey spaces are shown to be embedded into generalized Morrey spaces, it makes sense to restrict them from generalized Morrey spaces to generalized Orlicz-Morrey spaces. Via the weak boundedness, the boundedness property of singular integral operators acting on generalized Orlicz-Morrey spaces of the first kind is investigated in this paper. As a byproduct, the vector-valued boundedness of the Hardy-Littlewood maximal operator is obtained for generalized Orlicz-Morrey spaces of the first kind. Our results contain the weak type boundedness of operators as well as the one of strong type. The strong type boundedness is covered by the work of Nakai in 2006, while the contributions in this paper are the weak type boundedness and the vector-valued extension of the boundedness of the Hardy-Littlewood maximal operator. [ABSTRACT FROM AUTHOR]

Abstract In this paper we provide an extension of a theorem of David and Semmes [3] to general non-atomic measures. The result provides a geometric characterization of the non-atomic measures for which a certain class of square function operators, or singular integral operators, are bounded in L 2. The description is given in terms of a modification of Jones' β -coefficients. [ABSTRACT FROM AUTHOR]

In this article we generalize the singular integral operator theory on weighted tent spaces to spaces of homogeneous type. This generalization of operator theory is in the spirit of C. Fefferman and Stein since we use some auxiliary functionals on tent spaces which play roles similar to the Fefferman-Stein sharp and box maximal functions in the Lebesgue space setting. Our contribution in this operator theory is twofold: for singular integral operators (including maximal regularity operators) on tent spaces pointwise Carleson type estimates are proved and this recovers known results; on the underlying space no extra geometrical conditions are needed and this could be useful for future applications to parabolic problems in rough settings. [ABSTRACT FROM AUTHOR]

The main concern of this paper is to study the boundedness of singular integrals related to the Monge-Ampère equation established by Caffarelli and Gutiérrez. They obtained the L2 boundedness. Since then the Lp,1

Abstract 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 $R^{4}$ . 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.