It is known that on a Hilbert space, the sum of a real scalar-type operator and a commuting well-bounded operator is well-bounded. The corresponding property has been shown to be fail on $L^p$ spaces, for $1 \lt p \neq 2 \lt \infty$. We show that it does hold however on every Banach space X such that $X$ or $X*% is a Grothendieck space. This class notably includes $L^1$ and $C(K)$ spaces.

The connection between holomorphic and monogenic functions in sectors is used to construct an analytic functional calculus for several sectorial operators acting in a Banach space. The results are applied to the $H^\infty$-functional calculus for the differentiation operators on a Lipschitz surface.

We consider generalized Calderón-Zygmund operators whose kernel takes values in the space of all continuous linear operators between two Banach spaces. In the spirit of the $T(1)$ theorem of David and Journé we prove boundedness results for such operators on vector-valued Riesz potential spaces. This improves and generalizes a result by Hytönen and Weis.

If $A,B$ are sectorial operators on a Hilbert space with the same domain and range, and if $\parallel Ax \parallel \approx \parallel Bx \parallel$ and $\parallel A^{-1}x\parallel \approx \parallel B^{-1}x \approx$, then it is a result of Auscher, McIntosh and Nahmod that if $A$ has an $H^\infty$-calculus then so does $B$. On an arbitrary Banach space this is true with the additional hypothesis on B that it is almost R-sectorial as was shown by the author, Kunstmann and Weis in a recent preprint. We give an alternative approach to this result.

The fundamental solution of the heat equation on $\mathboldR^n$ is known as the heat kernel which is also the transition density of a Brownian motion. Similar statements hold when $\mathboldR^n$ is replaced by a Lie group. We briefly demonstrate how the results on $\mathboldR^n$ concerning the heat kernel and Brownian motion may be easily transferred to compact Lie groups using the wrapping map of Dooley and Wildberger.

We describe Salem’s proof of the Rademacher-Menshov Theorem, which shows that one constant works for all orthogonal expansions in all $L^2$-spaces. By changing the emphasis in Salem’s proof we produce a lower bound for sums of vectors coming from bi-orthogonal sets of vectors in a Hilbert space. This inequality is applied to sums of columns of an invertible matrix and to Lebesgue constants.

We consider commutator estimates in non-commutative (operator) $L^p$-spaces associated with general semi-finite von Neumann algebra. We discuss the difficulties which appear when one considers commutators with an unbounded operator in non-commutative $L^p$-spaces with $p \neq \infty$. We explain those difficulties using the example of the classical differentiation operator.

In the Euclidean context, tent spaces, introduced by Coifman, Meyer and Stein, admit an atomic decomposition. We generalize this decomposition to the case of spaces of homogeneous type.