Narrow Results

We define distributions of anisotropic order on manifolds, and establish their immediate properties. The central result is the Schwartz kernel theorem for such distributions, allowing the representation of continuous operators from ${\mathrm{\text{C}}}_c^l(X)$ to ${({\mathrm{\text{C}}}_c^m(Y))}^{\prime }$ by kernels, which we prove to be distributions of order $l$ in $x$, but higher, although still finite, order in $y$. Our main motivation for introducing these distributions is to obtain the new result that H-distributions (Antonić and Mitrović), a recently introduced generalization of H-measures are, in fact, distributions of order 0 (i.e. Radon measures) in $x\in R^d$, and of finite order in $\xi \in {\mathrm{\text{S}}}^{d-1}$. This allows us to obtain some more precise results on H-distributions, hopefully allowing for further applications to partial differential equations.

The purpose of this paper is to investigate the mapping properties of the strongly singular convolution operators on general weighted modulation spaces for 0 < p ≤ ∞, 0 < q ≤ ∞ and s ∈ ℝ. Our results show that modulation spaces are good substitutions for Lebesgue spaces.

The purpose of this paper is to investigate some properties of the Wiener amalgam spaces. As applications, we study the boundedness properties of some singular convolution operators on such spaces. From our results, we will see that, besides modulation spaces, Wiener amalgam spaces are another good substitutions for Lebesgue spaces.