We give a new Calderón-Zygmund decomposition for Sobolev spaces on a doubling Riemannian manifold. Our hypotheses are weaker than those of the already known decomposition which used classical Poincaré inequalities.

We establish new estimates for the Laplacian, the div-curl system, and more general Hodge systems in arbitrary dimension $n$, with data in $L^1$. We also present related results concerning differential forms with coefficients in the limiting Sobolev space $W^{1,n}$.

We consider operators of the form $\left(\Omega f\right)\left(y\right)={ʃ}_{-\infty }^{\infty }\Omega (y,u)f\left(u\right)du$ with Ω(y,u) = K(y,u)h(y-u), where K is a Calderón-Zygmund kernel and $h\in L^{\infty }$ (see (0.1) and (0.2)). We give necessary and sufficient conditions for such operators to map the Besov space ${Ḃ}_1^{0,1}$ (= B) into itself. In particular, all operators with $h\left(y\right)=e^{{i\left|y\right|}^a}$, a > 0, a ≠ 1, map B into itself.

We study the $L^p$ mapping properties of a family of strongly singular oscillatory integral operators on ℝⁿ which are non-homogeneous in the sense that their kernels have isotropic oscillations but non-isotropic singularities.

We revisit the properties of Bessel-Riesz operators and present a different proof of the boundedness of these operators on generalized Morrey spaces. We also obtain an estimate for the norm of these operators on generalized Morrey spaces in terms of the norm of their kernels on an associated Morrey space. As a consequence of our results, we reprove the boundedness of fractional integral operators on generalized Morrey spaces, especially of exponent $1$, and obtain a new estimate for their norm.

We give norm inequalities for some classical operators in amalgam spaces and in some subspaces of Morrey space.

We obtain the factorization theorem for Hardy space via the variable exponent Lebesgue spaces. As an application, it is proved that if the commutator of Coifman, Rochberg and Weiss $[b,T]$ is bounded on the variable exponent Lebesgue spaces, then $b$ is a bounded mean oscillation (BMO) function.