We address aseismic fault slip and the onset of seismicity resulting from depletion-induced or injection-induced stresses in reservoirs with pre-existing vertical or inclined faults. Building on classic results, we discuss semi-analytical modelling techniques for fault slip including dislocation theory, Cauchy-type singular integral equations and the use of Chebyshev polynomials for their solution and an eigenvalue-based stability analysis. We consider slip patch development during depletion for faults with zero, constant static and slip-weakening friction, and our results confirm earlier findings based on numerical simulation, in particular the aseismic growth of two slip patches that may subsequently merge and/or become unstable resulting in nucleation of seismic slip. New findings include improved approximate expressions for the induced seismic moment per unit strike length and a description of the effect of coupling between the slip patches which affects both forward simulation and eigenvalue computation for high values of the ratio of fault throw to reservoir height. Our implementation based on analytical inversion and semi-analytical integration with Chebyshev polynomials is more efficient and more robust than our numerical integration approach. It is not yet well suited for Monte Carlo simulation, which typically requires sub-second simulation times, but with some further development that option seems to be within reach. Moreover, our results offer a possibility for embedded fault modelling in large-scale numerical simulation tools.

Several results in the existing literature establish Euclidean density theorems of the following strong type. These results claim that every set of positive upper Banach density in the Euclidean space of an appropriate dimension contains isometric copies of all sufficiently large elements of a prescribed family of finite point configurations. So far, all results of this type discussed linear isotropic dilates of a fixed point configuration. In this paper, we initiate the study of analogous density theorems for families of point configurations generated by anisotropic dilations, i.e., families with power-type dependence on a single parameter interpreted as their size. More specifically, we prove nonisotropic power-type generalizations of a result by Bourgain on vertices of a simplex, a result by Lyall and Magyar on vertices of a rectangular box, and a result on distance trees, which is a particular case of the treatise of distance graphs by Lyall and Magyar. Another source of motivation for this paper is providing additional evidence for the versatility of the approach stemming from the work of Cook, Magyar, and Pramanik and its modification used recently by Durcik and the present author. Finally, yet another purpose of this paper is to single out anisotropic multilinear singular integral operators associated with the above combinatorial problems, as they are interesting on their own.

We study the Muskat problem describing the vertical motion of two immiscible fluids in a two-dimensional homogeneous porous medium in an Lp-setting with p ∈ (1, ∞). The Sobolev space $W_p^s(\mathbb R)$ with s = 1+1/p is a critical space for this problem. We prove, for each s ∈ (1+1/p, 2) that the Rayleigh–Taylor condition identifies an open subset of $W_p^s(\mathbb R)$ within which the Muskat problem is of parabolic type. This enables us to establish the local well-posedness of the problem in all these subcritical spaces together with a parabolic smoothing property.

A new characterization of $\text {CMO}(\mathbb R^n)$ is established replying upon local mean oscillations. Some characterizations of iterated compact commutators on weighted Lebesgue spaces are given, which are new even in the unweighted setting for the first order commutators.

Let $p(\cdot ):\mathbb{R}^{n}\rightarrow (0,\infty )$ be a variable exponent function satisfying the globally log-Hölder continuous condition. In this paper, we obtain the boundedness of paraproduct operators $\unicode[STIX]{x1D70B}_{b}$ on variable Hardy spaces $H^{p(\cdot )}(\mathbb{R}^{n})$, where $b\in \text{BMO}(\mathbb{R}^{n})$. As an application, we show that non-convolution type Calderón–Zygmund operators $T$ are bounded on $H^{p(\cdot )}(\mathbb{R}^{n})$ if and only if $T^{\ast }1=0$, where $\frac{n}{n+\unicode[STIX]{x1D716}}<\text{ess inf}_{x\in \mathbb{R}^{n}}p\leqslant \text{ess sup}_{x\in \mathbb{R}^{n}}p\leqslant 1$ and $\unicode[STIX]{x1D716}$ is the regular exponent of kernel of $T$. Our approach relies on the discrete version of Calderón’s reproducing formula, discrete Littlewood–Paley–Stein theory, almost orthogonal estimates, and variable exponents analysis techniques. These results still hold for variable Hardy space on spaces of homogeneous type by using our methods.

In this paper we study the Hilbert transformations over ${{L}^{2}}(\mathbb{R})$ and ${{L}^{2}}(\mathbb{T})$ from the viewpoint of symmetry. For a linear operator over ${{L}^{2}}(\mathbb{R})$ commutative with the $ax\,+\,b$ group, we show that the operator is of the form $\lambda I+\eta H$, where $I$ and $H$ are the identity operator and Hilbert transformation, respectively, and $\lambda ,\eta $ are complex numbers. In the related literature this result was proved by first invoking the boundedness result of the operator using some machinery. In our setting the boundedness is a consequence of the boundedness of the Hilbert transformation. The methodology that we use is the Gelfand–Naimark representation of the $ax\,+\,b$ group. Furthermore, we prove a similar result on the unit circle. Although there does not exist a group like the $ax\,+\,b$ group on the unit circle, we construct a semigroup that plays the same symmetry role for the Hilbert transformations over the circle ${{L}^{2}}(\mathbb{T})$.

In this paper we characterize the compactness of the commutator $\left[ b,\,T \right]$ for the singular integral operator on the Morrey spaces ${{L}^{p,\lambda }}\left( {{\mathbb{R}}^{n}} \right)$. More precisely, we prove that if $b\,\in \,\text{VMO}\left( {{\mathbb{R}}^{n}} \right)$, the $\text{BMO}\left( {{\mathbb{R}}^{n}} \right)$-closure of $C_{c}^{\infty }\left( {{\mathbb{R}}^{n}} \right)$, then $\left[ b,\,T \right]$ is a compact operator on the Morrey spaces ${{L}^{p,\lambda }}\left( {{\mathbb{R}}^{n}} \right)$ for $1\,<\,p\,<\,\infty $ and $0\,<\,\lambda \,<\,n$. Conversely, if $b\,\in \,\text{BMO}\left( {{\mathbb{R}}^{n}} \right)$ and $\left[ b,\,T \right]$ is a compact operator on the ${{L}^{p,\lambda }}\left( {{\mathbb{R}}^{n}} \right)$ for some $p\,\left( 1\,<\,p\,<\,\infty \right)$, then $b\,\in \,\text{VMO}\left( {{\mathbb{R}}^{n}} \right)$. Moreover, the boundedness of a rough singular integral operator $T$ and its commutator $\left[ b,\,T \right]$ on ${{L}^{p,\lambda }}\left( {{\mathbb{R}}^{n}} \right)$ are also given. We obtain a sufficient condition for a subset in Morrey space to be a strongly pre-compact set, which has interest in its own right.

${{H}^{p}}$ estimate for the multilinear operators which are finite sums of pointwise products of singular integrals and fractional integrals is given. An application to Sobolev space and some examples are also given.

The authors establish the boundedness on the Herz spaces and the weak Herz spaces for a large class of rough singular integral operators and their corresponding fractional versions. Applications are given to Fefferman's rough singular integral operators, their fractional versions, their commutators with BMO() functions and Ricci-Stein oscillatory singular integral operators. Some new results are obtained.

Let $b(t)$ be an ${{L}^{\infty }}$ function on $\mathbf{R}$, $\Omega ({y}')$ be an ${{H}^{1}}$ function on the unit sphere satisfying the mean zero property (1) and ${{Q}_{m}}(t)$ be a real polynomial on $\mathbf{R}$ of degree $m$ satisfying ${{Q}_{m}}(0)\,=\,0$. We prove that the singular integral operator

$${{T}_{Qm,}}b\left( f \right)\left( x \right)=p.v.\int\limits_{\mathbf{R}}^{n}{b\left( \left| y \right| \right)}\Omega \left( y \right){{\left| y \right|}^{-n}}f\left( x-{{Q}_{m}}\left( \left| y \right| \right){y}' \right)\,\,dy$$

is bounded in ${{L}^{p}}({{\mathbf{R}}^{n}})$ for $1<p<\infty $, and the bound is independent of the coefficients of ${{Q}_{m}}(t)$.