Let $\Omega \in L^s\left({\mathrm{S}}^{n-1}\right)$ for $s\ge 1$ be a homogeneous function of degree zero and $b$ a BMO function. The commutator generated by the Marcinkiewicz integral ${\mu }_{\Omega }$ and $b$ is defined by $${\displaystyle [b,{\mu }_{\Omega }]\left(f\right)\left(x\right)=({\int }_0^{\infty }{\left|{\int }_{|x-y|\le t}\frac{\Omega (x-y)}{{|x-y|}^{n-1}}[b\left(x\right)-b\left(y\right)]f\left(y\right)\mathrm{d}y{|}^2\frac{\mathrm{d}t}{t^3}\right)}^{1/2}.}$$ In this paper, the author proves the $(L^{p(·)}\left({ℝ}^n\right),L^{p(·)}\left({ℝ}^n\right))$-boundedness of the Marcinkiewicz integral operator ${\mu }_{\Omega }$ and its commutator $[b,{\mu }_{\Omega }]$ when $p(·)$ satisfies some conditions. Moreover, the author obtains the corresponding result about ${\mu }_{\Omega }$ and $[b,{\mu }_{\Omega }]$ on Herz spaces with variable exponent.

A classical theorem of Coifman, Rochberg, and Weiss on commutators of singular integrals is extended to the case of generalized Lp spaces with variable exponent.

In this paper, the boundedness of a large class of sublinear commutator operators $T_b$ generated by a Calderón-Zygmund type operator on a generalized weighted Morrey spaces $M_{p,\varphi }\left(w\right)$ with the weight function $w$ belonging to Muckenhoupt’s class $A_p$ is studied. When $1<p<\infty$ and $b\in \mathrm{BMO}$, sufficient conditions on the pair $({\varphi }_1,{\varphi }_2)$ which ensure the boundedness of the operator $T_b$ from $M_{p,{\varphi }_1}\left(w\right)$ to $M_{p,{\varphi }_2}\left(w\right)$ are found. In all cases the conditions for the boundedness of $T_b$ are given in terms of Zygmund-type integral inequalities on $({\varphi }_1,{\varphi }_2)$, which do not require...

A sufficient condition for boundedness on Herz-type spaces of the commutator generated by a Lipschitz function and a weighted Hardy operator is obtained.

We investigate weighted norm inequalities for the commutator of a fractional integral operator and multiplication by a function. In particular, we show that, for $\mu ,\lambda \in A_{p,q}$ and α/n + 1/q = 1/p, the norm $\left|\right|[b,I_{\alpha }]:L^p\left({\mu }^p\right)\to L^q\left({\lambda }^q\right)\left|\right|$ is equivalent to the norm of b in the weighted BMO space BMO(ν), where $\nu =\mu {\lambda }^{-1}$. This work extends some of the results on this topic existing in the literature, and continues a line of investigation which was initiated by Bloom in 1985 and was recently developed further by the first author, Lacey, and Wick.

By analyzing the connection between complex Hadamard matrices and spectral sets, we prove the direction "spectral ⇒ tile" of the Spectral Set Conjecture, for all sets A of size |A| ≤ 5, in any finite Abelian group. This result is then extended to the infinite grid Zd for any dimension d, and finally to Rd.

In this paper, we obtain some strong and weak type continuity properties for the maximal operator associated with the commutator of the Bochner-Riesz operator on Hardy spaces, Hardy type spaces and weak Hardy type spaces.

We study the boundedness in $L^p{(}^n)$ of the projections onto spaces of functions with spectrum contained in horizontal strips. We obtain some results concerning convergence along nonisotropic regions of harmonic extensions of functions in $L^p{(}^n)$ with spectrum included in these horizontal strips.

We show that $L^2$-bounded singular integrals in metric spaces with respect to general measures and kernels converge weakly. This implies a kind of average convergence almost everywhere. For measures with zero density we prove the almost everywhere existence of principal values.