For $b\in BMO(\mathbb{R}^n)$ and $k\in\mathbb{N}$, the $k$-th order maximal commutator of the singular integral operator $T$ with rough variable kernels is defined by $$ T^{\ast}_{b,k}f(x) = \sup_{\varepsilon > 0} \biggl| \int_{|x-y| > \varepsilon} \frac{\Omega(x,x-y)}{|x-y|^n} (b(x)-b(y))^{k} f(y) dy \biggl|. $$ In this paper the authors prove that the $k$-th order maximal commutator $T^{\ast}_{b,k}$ is a bounded operator on $L^2(\mathbb{R}^n)$ if $\Omega$ satisfies the same conditions given by Calderón and Zygmund. Moreover, the $L^2$-boundedness of the $k$-th order commutator of the rough maximal operator $M_\Omega$ with variable kernel, which is defined by $$ M_{\Omega;b,k}f(x) = \sup_{r > 0} \dfrac{1}{r^{n}} \int_{|x-y| < r} |\Omega(x,x-y)| |b(x)-b(y)|^{k} |f(y)| dy, $$ is also given here. These results obtained in this paper are substantial improvement and extension of some known results.

The aim of this paper is to present the construction, out of the Kohn-Rossi complex, of a new hypoelliptic operator $Q_L$ on almost CR manifolds equipped with a real structure. The operator acts on all $(p,q)$-forms, but when restricted to $(p,0)$-forms and $(p,n)$-forms it is a sum of squares up to sign factor and lower order terms. Therefore, only a finite type condition condition is needed to have hypoellipticity on those forms. However, outside these forms $Q_L$ may fail to be hypoelliptic, as it is shown in the example of the Heisenberg group $\mathbb{H}^{5}$.

Motivated by an old paper of Wells [J. London Math. Soc. {\bf 2} (1970), 549-556] we define the space $X\otimes Y$, where $X$ and $Y$ are "homogeneous" Banach spaces of analytic functions on the unit disk $\mathbb{D}$, by the requirement that $f$ can be represented as $f=\sum_{j=0}^\infty g_n * h_n$, with $g_n\in X$, $h_n\in Y$ and $\sum_{n=1}^\infty \|g_n\|_X \|h_n\|_Y < \infty$. We show that this construction is closely related to coefficient multipliers. For example, we prove the formula $((X\otimes Y),Z)=(X,(Y,Z))$, where $(U,V)$ denotes the space of multipliers from $U$ to $V$, and as a special case $(X\otimes Y)^*=(X,Y^*)$, where $U^*=(U,H^\infty)$. We determine $H^1\otimes X$ for a class of spaces that contains $H^p$ and $\ell^p$ $(1\le p\le 2)$, and use this together with the above formulas to give quick proofs of some important results on multipliers due to Hardy and Littlewood, Zygmund and Stein, and others.

In this work, we concentrate our interest and efforts on general variational (or optimization) problems which do not have solutions necessarily, but which do have approximate solutions (or solutions within $\varepsilon > 0$). We shall see how to recover all the (exact) minimizers of the relaxed version of the original problem (by closed-convexification of the objective function) in terms of the $\varepsilon $-minimizers of the original problem. Applications to two approximation problems in a Hilbert space setting will be shown.

In this paper we study the local behavior of a solution to second order elliptic operators with sharp singular coefficients in lower order terms. One of the main results is the bound on the vanishing order of the solution, which is a quantitative estimate of the strong unique continuation property. Our proof relies on Carleman estimates with carefully chosen phases. A key strategy in the proof is to derive doubling inequalities via three-sphere inequalities. Our method can also be applied to certain elliptic systems with similar singular coefficients.

We define discrete and continuous Menger-type curvatures. The discrete curvature scales the volume of a $(d+1)$-simplex in a real separable Hilbert space $H$, whereas the continuous curvature integrates the square of the discrete one according to products of a given measure (or its restriction to balls). The essence of this paper is to establish an upper bound on the continuous Menger-type curvature of an Ahlfors regular measure $\mu$ on $H$ in terms of the Jones-type flatness of $\mu$ (which adds up scaled errors of approximations of $\mu$ by $d$-planes at different scales and locations). As a consequence of this result we obtain that uniformly rectifiable measures satisfy a Carleson-type estimate in terms of the Menger-type curvature. Our strategy combines discrete and integral multiscale inequalities for the polar sine with the "geometric multipoles" construction, which is a multiway analog of the well-known method of fast multipoles.

As a step in developing a non-commutative Calderón-Zygmund theory, J. Parcet (J. Funct. Anal. {\bf 256} (2009), no. 2, 509-593) established a new pseudo-localisation principle for classical singular integrals, showing that $Tf$ has small $L^2$ norm outside a set which only depends on $f\in L^2$ but not on the arbitrary normalised Calderón-Zygmund operator $T$. Parcet also asked if a similar result holds true in $L^p$ for $p\in(1,\infty)$. This is answered in the affirmative in the present paper. The proof, which is based on martingale techniques, even somewhat improves on the original $L^2$ result.

A new proof of both analytic and $C^{\infty} $ hypoellipticity of Kohn's operator is given using FBI techniques introduced by J. Sjöstrand. The same proof allows us to obtain both kind of hypoellipticity at the same time.

We study submetries between Alexandrov spaces and show how some of the usual features of Riemannian submersions fail due to the lack of smoothness.

In this paper the center of a Leavitt path algebra is computed for a wide range of situations. A basis as a $K$-vector space is found for $Z(L(E))$ when $L(E)$ enjoys some finiteness condition such as being artinian, semisimple, noetherian and locally noetherian. The main result of the paper states that a simple Leavitt path algebra $L(E)$ is central (i.e. the center reduces to the base field $K$) when $L(E)$ is unital and has zero center otherwise. Finally, this result is extended, under some mild conditions, to the case of exchange Leavitt path algebras.

We study multi-parameter Carnot-Carathéodory balls, generalizing results due to Nagel, Stein and Wainger in the single parameter setting. The main technical result is seen as a uniform version of the theorem of Frobenius. In addition, we study maximal functions associated to certain multi-parameter families of Carnot-Carathéodory balls.