The author studies the commutators generated by a suitable function a(x) on ℝⁿ and the oscillatory singular integral with rough kernel Ω(x)|x|ⁿ and polynomial phase, where Ω is homogeneous of degree zero on ℝⁿ, and a(x) is a BMO function or a Lipschitz function. Some mapping properties of these higher order commutators on $L^p\left(ℝⁿ\right)$, which are essential improvements of some well known results, are given.

We consider Littlewood-Paley functions associated with a non-isotropic dilation group on ${ℝ}^n$. We prove that certain Littlewood-Paley functions defined by kernels with no regularity concerning smoothness are bounded on weighted $L^p$ spaces, $1<p<\infty$, with weights of the Muckenhoupt class. This, in particular, generalizes a result of N. Rivière (1971).

The $L^p$ boundedness(1 < p < ∞) of Littlewood-Paley’s g-function, Lusin’s S function, Littlewood-Paley’s $g{*}_{\lambda }$-functions, and the Marcinkiewicz function is well known. In a sense, one can regard the Marcinkiewicz function as a variant of Littlewood-Paley’s g-function. In this note, we treat counterparts ${\mu }_S^{\varrho }$ and ${\mu }_{\lambda }^{*,\varrho }$ to S and $g{*}_{\lambda }$. The definition of ${\mu }_S^{\varrho }\left(f\right)$ is as follows: ${\mu }_S^{\varrho }\left(f\right)\left(x\right)=({ʃ}_{|y-x|<t}|1/t^{\varrho }{ʃ}_{\left|z\right|\le t}\Omega \left(z\right)/{\left(\right|z|}^{n-\varrho }{\left)f(y-z)dz\right|}^2\left(dydt\right)/\left(t^{n+1}\right){)}^{1/2}$, where Ω(x) is a homogeneous function of degree 0 and Lipschitz continuous of order β (0 < β ≤ 1) on the unit sphere $S^{n-1}$, and ${ʃ}_{S^{n-1}}\Omega \left(x^{\text{'}}\right)d\sigma \left(x^{\text{'}}\right)=0$. We show that...

In this paper we study the mapping properties of the one-sided fractional integrals in the Calderón-Hardy spaces ${\mathscr{H}}_{q,\alpha }^{p,+}\left(\omega \right)$ for $0<p\le 1$, $0<\alpha <\infty$ and $1<q<\infty$. Specifically, we show that, for suitable values of $p,q,\gamma ,\alpha$ and $s$, if $\omega \in A_s^{+}$ (Sawyer’s classes of weights) then the one-sided fractional integral $I_{\gamma }^{+}$ can be extended to a bounded operator from ${\mathscr{H}}_{q,\alpha }^{p,+}\left(\omega \right)$ to ${\mathscr{H}}_{q,\alpha +\gamma }^{p,+}\left(\omega \right)$. The result is a consequence of the pointwise inequality $$N_{q,\alpha +\gamma }^{+}\left(I_{\gamma }^{+}F;x\right)\le C_{\alpha ,\gamma }{N}_{q,\alpha }^{+}\left(F;x\right),$$ where $N_{q,\alpha }^{+}(F;x)$ denotes the Calderón maximal function.

We obtain the boundedness of Calderón-Zygmund singular integral operators $T$ of non-convolution type on Hardy spaces $H^p\left(𝒳\right)$ for $1/(1+ϵ)<p\le 1$, where $𝒳$ is a space of homogeneous type in the sense of Coifman and Weiss (1971), and $ϵ$ is the regularity exponent of the kernel of the singular integral operator $T$. Our approach relies on the discrete Littlewood-Paley-Stein theory and discrete Calderón’s identity. The crucial feature of our proof is to avoid atomic decomposition and molecular theory in contrast to what was...

Let w be in the Muckenhoupt $A_{\infty }$ weight class. We show that the Riesz transforms are bounded on the weighted Carleson measure space $CM{O}_w^p$, the dual of the weighted Hardy space $H_w^p$, 0 < p ≤ 1.

The aim of this paper is to study singular integrals T generated by holomorphic kernels defined on a natural neighbourhood of the set $z{\zeta }^{-1}:z,\zeta \in ,z\ne \zeta$, where is a star-shaped Lipschitz curve, $=exp\left(iz\right):z=x+iA\left(x\right),A^{\text{'}}\in L^{\infty }[-\pi ,\pi ],A(-\pi )=A\left(\pi \right)$. Under suitable conditions on F and z, the operators are given by (1) $TF\left(z\right)=p.v.{\int }_{(}z{\eta }^{-1})F\left(\eta \right)(d\eta /\eta ).$ We identify a class of kernels of the stated type that give rise to bounded operators on $L^2(,|d\left|\right)$. We establish also transference results relating the boundedness of kernels on closed Lipschitz curves to corresponding results on periodic, unbounded curves.

We introduce a new type of variable exponent function spaces $M{\dot{K}}_{q,p(·)}^{\alpha (·),\lambda }\left({ℝ}^n\right)$ of Morrey-Herz type where the two main indices are variable exponents, and give some propositions of the introduced spaces. Under the assumption that the exponents $\alpha$ and $p$ are subject to the log-decay continuity both at the origin and at infinity, we prove the boundedness of a wide class of sublinear operators satisfying a proper size condition which include maximal, potential and Calderón-Zygmund operators and their commutators of BMO...

Let $(X,d,\mu )$ be a metric measure space endowed with a distance $d$ and a nonnegative Borel doubling measure $\mu$. Let $L$ be a non-negative self-adjoint operator of order $m$ on $L^2\left(X\right)$. Assume that the semigroup ${\mathrm{e}}^{-tL}$ generated by $L$ satisfies the Davies-Gaffney estimate of order $m$ and $L$ satisfies the Plancherel type estimate. Let $H_L^p\left(X\right)$ be the Hardy space associated with $L.$ We show the boundedness of Stein’s square function ${𝒢}_{\delta }\left(L\right)$ arising from Bochner-Riesz means associated to $L$ from Hardy spaces $H_L^p\left(X\right)$ to $L^p\left(X\right)$, and also study the boundedness...

Let s ∈ ℝ, p ∈ (0,1] and q ∈ [p,∞). It is proved that a sublinear operator T uniquely extends to a bounded sublinear operator from the Triebel-Lizorkin space ${Ḟ}_{p,q}^s\left(ℝⁿ\right)$ to a quasi-Banach space ℬ if and only if sup${\left|\right|T\left(a\right)\left|\right|}_{ℬ}$: a is an infinitely differentiable (p,q,s)-atom of ${Ḟ}_{p,q}^s\left(ℝⁿ\right)$ < ∞, where the (p,q,s)-atom of ${Ḟ}_{p,q}^s\left(ℝⁿ\right)$ is as defined by Han, Paluszyński and Weiss.

Some boundedness results are established for sublinear operators on the homogeneous Herz spaces. As applications, some new theorems about the boundedness on homogeneous Herz spaces for commutators of singular integral operators are obtained.

Sufficient conditions for the boundedness of the Hausdorff operators in the Hardy spaces $H^p$, 0 < p < 1, on the real line are proved. Two related negative results are also given.

In this paper, the boundedness properties for some Toeplitz type operators associated to the Riesz potential and general integral operators from Lebesgue spaces to Orlicz spaces are proved. The general integral operators include singular integral operator with general kernel, Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.