Calderon-Zygmund Singular Operators Research Articles

Generalization of the notion of grand Lebesgue space

We introduce families of weighted grand Lebesgue spaces which generalize weighted grand Lebesgue spaces (known also as Iwaniec-Sbordone spaces). The generalization admits a possibility of expanding usual (weighted) Lebesgue spaces to grand spaces by various ways by means of additional functional parameter. For such generalized grand spaces we prove a theorem on the boundedness of linear operators under the information of their boundedness in ordinary weighted Lebesgue spaces. By means of this theorem we prove boundedness of the Hardy-Littlewood maximal operator and the Calderon-Zygmund singular operators in the weighted grand spaces.

Variable Exponent Herz Spaces

We introduce a new type of variable exponent function spaces Ḣp(·),q(·),α(·)(\({\mathbb{R}^n}\)) and Hp(·),q(·),α(·)(\({\mathbb{R}^n}\)) of Herz type, homogeneous and non-homogeneous versions, where all the three parameters are variable, and give comparison of continual and discrete approaches to their definition. Under the only assumption that the exponents p, q and α are subject to the log-decay condition at infinity, we prove that sublinear operators, satisfying the size condition known for singular integrals and bounded in Lp(·)(\({\mathbb{R}^n}\)), are also bounded in the nonhomogeneous version of the introduced spaces, which includes the case maximal and Calderon-Zygmund singular operators.

Maximal, potential and singular operators in vanishing generalized Morrey spaces

We introduce vanishing generalized Morrey spaces $${V\mathcal{L}^{p,\varphi}_\Pi (\Omega), \Omega \subseteq \mathbb{R}^n}$$ with a general function $${\varphi(x, r)}$$ defining the Morrey-type norm. Here $${\Pi \subseteq \Omega}$$ is an arbitrary subset in Ω including the extremal cases $${\Pi = \{x_0\}, x_0 \in \Omega}$$ and Â? = Ω, which allows to unify vanishing local and global Morrey spaces. In the spaces $${V\mathcal{L}^{p,\varphi}_\Pi (\mathbb{R}^n)}$$ we prove the boundedness of a class of sublinear singular operators, which includes Hardy-Littlewood maximal operator and Calderon-Zygmund singular operators with standard kernel. We also prove a Sobolev-Spanne type $${V\mathcal{L}^{p,\varphi}_\Pi (\mathbb{R}^n) \rightarrow V\mathcal{L}^{q,\varphi^\frac{q}{p}}_\Pi (\mathbb{R}^n)}$$ -theorem for the potential operator I Â? . The conditions for the boundedness are given in terms of Zygmund-type integral inequalities on $${\varphi(x, r)}$$ . No monotonicity type condition is imposed on $${\varphi(x, r)}$$ . In case $${\varphi}$$ has quasi- monotone properties, as a consequence of the main results, the conditions of the boundedness are also given in terms of the Matuszeska-Orlicz indices of the function $${\varphi}$$ . The proofs are based on pointwise estimates of the modulars defining the vanishing spaces

Maximal, potential and singular operators in the local “complementary” variable exponent Morrey type spaces

We consider local “complementary” generalized Morrey spaces ∁M{x0}p(⋅),ω(Ω) in which the p-means of function are controlled over Ω∖B(x0,r) instead of B(x0,r), where Ω⊂Rn is a bounded open set, p(x) is a variable exponent, and no monotonicity type condition is imposed onto the function ω(r) defining the “complementary” Morrey-type norm. In the case where ω is a power function, we reveal the relation of these spaces to weighted Lebesgue spaces. In the general case we prove the boundedness of the Hardy–Littlewood maximal operator and Calderon–Zygmund singular operators with standard kernel, in such spaces. We also prove a Sobolev type ∁M{x0}p(⋅),ω(Ω)→∁M{x0}q(⋅),ω(Ω)-theorem for the potential operators Iα(⋅), also of variable order. In all the cases the conditions for the boundedness are given it terms of Zygmund-type integral inequalities on ω(r), which do not assume any assumption on monotonicity of ω(r).

Boundedness of the maximal, potential and singular operators in the generalized variable exponent Morrey spaces

We consider generalized Morrey spaces ${\mathcal M}^{p(\cdot),\omega}(\Omega)$ with variable exponent $p(x)$ and a general function $\omega (x,r)$ defining the Morrey-type norm. In case of bounded sets $\Omega \subset {\mathsf R}^n$ we prove the boundedness of the Hardy-Littlewood maximal operator and Calderon-Zygmund singular operators with standard kernel, in such spaces. We also prove a Sobolev-Adams type ${\mathcal M}^{p(\cdot),\omega} (\Omega)\rightarrow {\mathcal M}^{q(\cdot),\omega} (\Omega)$-theorem for the potential operators $I^{\alpha(\cdot)}$, also of variable order. The conditions for the boundedness are given it terms of Zygmund-type integral inequalities on $\omega(x,r)$, which do not assume any assumption on monotonicity of $\omega(x,r)$ in $r$.

Estimates of commutators for multilinear operators on Herz-type spaces

In this paper, the boundedness of mulitilinear commutator [\( \vec b \), T] on Herz-type space is considered, where T is a standard Calderon-Zygmund singular operator and \( \vec b \) ∈ (BMO(ℝn))m.