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We study the boundedness of rough Fourier integral and pseudodifferential operators, defined by general rough H\"ormander class amplitudes, on Banach and quasi-Banach $L^p$ spaces. Thereafter we apply the aforementioned boundedness in order to improve on some of the existing boundedness results for H\"ormander class bilinear pseudodifferential operators and certain classes of bilinear (as well as multilinear) Fourier integral operators. For these classes of amplitudes, the boundedness of the aforementioned Fourier integral operators turn out to be sharp. Furthermore we also obtain results for rough multilinear operators.

We consider the Moutard transformation which is a two-dimensional version of the well-known Darboux transformation. We give an algebraic interpretation of the Moutard transformation as a conjugation in an appropriate ring and the corresponding version of the algebro-geometric formalism for two-dimensional Schroedinger operators. An application to some problems of the spectral theory of two-dimensional Schroedinger operators and to the $(2+1)$-dimensional Novikov--Veselov equation is sketched.

The paper concerns algebras of almost periodic pseudodifferential operators on $\mathbb R^d$ with symbols in H\"ormander classes. We study three representations of such algebras, one of which was introduced by Coburn, Moyer and Singer and the other two inspired by results in probability theory by Gladyshev. Two of the representations are shown to be unitarily equivalent for nonpositive order. We apply the results to spectral theory for almost periodic pseudodifferential operators acting on $L^2$ and on the Besicovitch Hilbert space of almost periodic functions.

We prove that if the Hardy-Littlewood maximal operator is bounded on a separable Banach function space $X(\mathbb{R}^n)$ and on its associate space $X'(\mathbb{R}^n)$ and a maximally modulated Calder\'on-Zygmund singular integral operator $T^\Phi$ is of weak type $(r,r)$ for all $r\in(1,\infty)$, then $T^\Phi$ extends to a bounded operator on $X(\mathbb{R}^n)$. This theorem implies the boundedness of the maximally modulated Hilbert transform on variable Lebesgue spaces $L^{p(\cdot)}(\mathbb{R})$ under natural assumptions on the variable exponent $p:\mathbb{R}\to(1,\infty)$. Applications of the above result to the boundedness and compactness of pseudodifferential operators with $L^\infty(\mathbb{R},V(\mathbb{R}))$-symbols on variable Lebesgue spaces $L^{p(\cdot)}(\mathbb{R})$ are considered. Here the Banach algebra $L^\infty(\mathbb{R},V(\mathbb{R}))$ consists of all bounded measurable $V(\mathbb{R})$-valued functions on $\mathbb{R}$ where $V(\mathbb{R})$ is the Banach algebra of all functions of bounded total variation.

We prove that if the Hardy-Littlewood maximal operator is bounded on a separable Banach function space $X(\mathbb{R}^n)$ and on its associate space $X'(\mathbb{R}^n)$ and a maximally modulated Calder\'on-Zygmund singular integral operator $T^\Phi$ is of weak type $(r,r)$ for all $r\in(1,\infty)$, then $T^\Phi$ extends to a bounded operator on $X(\mathbb{R}^n)$. This theorem implies the boundedness of the maximally modulated Hilbert transform on variable Lebesgue spaces $L^{p(\cdot)}(\mathbb{R})$ under natural assumptions on the variable exponent $p:\mathbb{R}\to(1,\infty)$. Applications of the above result to the boundedness and compactness of pseudodifferential operators with $L^\infty(\mathbb{R},V(\mathbb{R}))$-symbols on variable Lebesgue spaces $L^{p(\cdot)}(\mathbb{R})$ are considered. Here the Banach algebra $L^\infty(\mathbb{R},V(\mathbb{R}))$ consists of all bounded measurable $V(\mathbb{R})$-valued functions on $\mathbb{R}$ where $V(\mathbb{R})$ is the Banach algebra of all functions of bounded total variation.

We study the structure and asymptotic behavior of the resolvent of elliptic cone pseudodifferential operators acting on weighted Sobolev spaces over a compact manifold with boundary. We obtain an asymptotic expansion of the resolvent as the spectral parameter tends to infinity, and use it to derive corresponding heat trace and zeta function expansions as well as an analytic index formula.