The essence of an approach to the boundedness of singular integral operators based on several parameterized classes of new conditions one of which includes, in particular, Hörmander condition and its known variations is exposed. Most of the attention is paid to the comparison with known results in the same settings. The feature of dependence from some parameters being integer is revealed. As some of applications, the existence of functional calculus and variants of Littlewood-Paley-type decompositions in its terms without any requirements of smoothness or absolute value bounds of the kernels of the corresponding holomorphic semigroups is shown.

A key result in describing the asymptotic behaviour of bounded solutions of differential equations is the classical result of Bohl-Bohr: If $phi : \mathbb{R \rightarrow C$ is almost periodic and $P\phi(t) = \int_t^0 \phi(s)ds$ is bounded then $P\phi$ is almost periodic too. In this paper we reveal a new property of almost periodic functions: If $\psi(t) = t^N \phi(t)$ where $phi$ is almost periodic and $P\psi(t)/(1 + |t|)^N is bounded then $P\phi$ is bounded and hence almost periodic. As a consequence of this result and a theorem of Kadets, we obtain results on the almost periodicity of the primitive of Banach space valued almost periodic functions. This allows us to resolve the asymptotic behaviour of unbounded P solutions of differential equations of the form \sum_{j=0}^m b_ju^{(j)}(t) = t^N \phi(t). The results are new even for scalar valued functions. The techniques include the use of reduced Beurling spectra and ergodicity for functions of polynomial growth.

Let $\Omega$ be a subset of a space of homogeneous type. Let $A$ be the infinitesimal generator of a positive semigroup with Gaussian kernel bounds on $L^2(\Omega)$. We then show Gaussian heat kernel bounds for operators of the type $bA$ where $b$ is a bounded, complex valued function.

Let $X$ be a space of homogeneous type and $T$ a singular integral operator which is bounded on $L^2(X)$. We give a sufficient condition on the kernel of $T$ so that the maximal truncated operator $T_*$, which is defined by $T_*f(x) = sup_{\epsilon\lt0} |T_\epsilonf(x)|$, to be of weak type (1, 1). Our condition is weaker than the usual Hörmander type condition. Applications include the dominated convergence theorem of holomorphic functional calculi of linear elliptic operators on irregular domains.

Suppose that $M$ is a compact Riemannian manifold with boundary and $u$ is an $L^2$-normalized Dirichlet eigenfunction with eigenvalue $\lambda$. Let $\psi$ be its normal derivative at the boundary. Scaling considerations lead one to expect that the $L^2$ norm of will grow as $\lamdba^(1/2)$ as $\lamda \rightarrow \infty$. We sketch proofs of an upper bound of the form $\Vert \psi |Vert_2^2 \leq C\lambda$ for any Riemannian manifold, and a lower bound $c\lamda \leq \Vert \psi \Vert_2^2$ provided that $M$ has no trapped geodesics (see the main Theorem for a precise statement). Here $c$ and $C$ are positive constants that depend on $M$, but not on $\lamda$. Full details will appear in [3].

Given a system $A = (A_1, . . . , A_n)$ of linear operators whose real linear combinations have spectra contained in a fixed sector in $\mathbbC$ and satisfy resolvent bounds there, functions $f(A)$ of the system $A$ of operators can be formed for monogenic functions f having decay at zero and infinity in a corresponding sector in $\mathbb^n+1$. The paper discusses how the functional calculus $f \mapsto f(A)$ might be extended to a larger class of monogenic functions and its relationship with a functional calculus for analytic functions in a sector of $\mathbbC_n$.

Given a number $0 \lt \omega \leq \frac{\pi}{2}$, an $\omega$-accretive operator is a sectorial operator $A$ on Hilbert space whose numerical range lies in the closed sector of all $z \in \mathbbC$ such that $\vert Arg(z)\vert \leq \omega$. It is easy to check that any such operator admits bounded imaginary powers, with $\Vert A^{it} \Vert \leq e^{\omega\vert t \vert$ for any $t \in \mathbbR$. We show that conversely, $A$ is similar to an $\omega$-accretive operator if $\Vert A^{it} \Vert \leq e^{\omega\vert t \vert$ for any $t \in \mathbbR$

In this note we prove estimates of Jacobian determinants of $Du$ on strongly Lipschitz domains $\Omega$ in $\mathbbR^2$. The theorem consists of two parts: one is an estimate in terms of the $BMO_r(\Omega)$ norm for $u$ in the Sobolev space $W^{1,2}(\Omega,\mathbbR^@)$ with boundary zero, and another is an estimate in terms of the $BMO_z(\omega)$ norm for $u$ in $W^{1,2}(\Omega,\mathbbR^2)$ with no boundary conditions.

We combine the Cantor-Lebesgue Theorem and Uniform Boundedness Principle to prove a divergence result for Cesàro and Bochner-Riesz means of spherical harmonic expansions.

The paper studies two types of results on inducing monogenic functions in $\mathboldR_1^n$. One is based on McIntosh’s formula and the other is along the line of Fueter’s Theorem. Applications are summarized and a new application on monogenic sinc function interpolation is introduced.

We review the large time behaviour of the semigroup kernel associated with a homogeneous operator on a Lie group with polynomial growth. We consider complex second-order operators and two classes of higher order operators.

We survey the various local and global variants of $T(1)$ and $T(b)$ theorems which have appeared in their literature, and outline their proofs based on a local wavelet coefficient perspective. This survey is based on recent work in [4] with Pascal Auscher, Steve Hofmann, Camil Muscalu, and Christoph Thiele.

Let $K = \{A,B} \subset M^{Nxn$ with rank$(A − B) \gt 1$ and $\Omega \subset \mathbbR^n$ be a bounded arcwise connected Lipschitz domain. We show that there is a direct estimate of the size of the $\epsilon$-neighborhood $K_\epsilon$ of $K$ such that $K_\epsilon = \barB_\epsilon(A) \bigcup \barB_\epsilon(B)$ separates gradient Young measures, that is, if $(u_j) \subset W^{1,1}(\Omega, \mathbbR^N)$ is bounded and $\int_\Omega$ dist$(Du_j, K_\epsilon)dx \rightarrow 0$ as $j \rightarrow \infty$, then up to a subsequence, either $\int_Omega dist(DU_j, \barB_epsilon(A))dx \rightarrow 0$ or $\int_Omega dist(Du_j, \barB_epsilon(B))dx \rightarrow 0$.