Analysis of Partial Differential Equations (APDE)http://hdl.handle.net/20.500.11824/12019-03-16T07:25:29Z2019-03-16T07:25:29ZBloom type inequality for bi-parameter singular integrals: efficient proof and iterated commutatorsLi K.Martikainen H.Vuorinen E.http://hdl.handle.net/20.500.11824/9502019-03-16T02:00:10Z2019-03-14T00:00:00ZBloom type inequality for bi-parameter singular integrals: efficient proof and iterated commutators
Li K.; Martikainen H.; Vuorinen E.
Utilising some recent ideas from our bilinear bi-parameter theory, we give an efficient proof of a two-weight Bloom type inequality
for iterated commutators of linear bi-parameter singular integrals. We prove that if $T$ is a bi-parameter singular integral
satisfying the assumptions of the bi-parameter representation theorem, then
$$
\| [b_k,\cdots[b_2, [b_1, T]]\cdots]\|_{L^p(\mu) \to L^p(\lambda)} \lesssim_{[\mu]_{A_p}, [\lambda]_{A_p}} \prod_{i=1}^k\|b_i\|_{{\rm{bmo}}(\nu^{\theta_i})} ,
$$
where $p \in (1,\infty)$, $\theta_i \in [0,1]$, $\sum_{i=1}^k\theta_i=1$, $\mu, \lambda \in A_p$, $\nu := \mu^{1/p}\lambda^{-1/p}$. Here
$A_p$ stands for the bi-parameter weights in $\mathbb R^n \times \mathbb R^m$ and ${\rm{bmo}}(\nu)$ is a suitable weighted little BMO space.
We also simplify the proof of the known first order case.
2019-03-14T00:00:00ZBoundary Triples for the Dirac Operator with Coulomb-Type Spherically Symmetric PerturbationsCassano B.Pizzichillo F.http://hdl.handle.net/20.500.11824/9472019-03-13T02:00:10Z2019-02-01T00:00:00ZBoundary Triples for the Dirac Operator with Coulomb-Type Spherically Symmetric Perturbations
Cassano B.; Pizzichillo F.
We determine explicitly a boundary triple for the Dirac operator $H:=-i\alpha\cdot \nabla + m\beta + \mathbb V(x)$ in $\mathbb R^3$, for $m\in\mathbb R$ and $\mathbb V(x)= |x|^{-1} ( \nu \mathbb{I}_4 +\mu \beta -i \lambda \alpha\cdot{x}/|x|\,\beta)$, with $\nu,\mu,\lambda \in \mathbb R$. Consequently we determine all the self-adjoint realizations of $H$ in terms of the behaviour of the functions of their domain in the origin. When $\sup_{x} |x| |\mathbb V(x)| \leq 1$, we discuss the problem of selecting the distinguished extension requiring that its domain is included in the domain of the appropriate quadratic form.
2019-02-01T00:00:00ZSparse bounds for maximal rough singular integrals via the Fourier transformDi Plinio F.Hytönen T.Li K.http://hdl.handle.net/20.500.11824/9462019-03-13T02:00:11Z2019-03-12T00:00:00ZSparse bounds for maximal rough singular integrals via the Fourier transform
Di Plinio F.; Hytönen T.; Li K.
We prove a quantified sparse bound for the maximal truncations of convolution-type singular integrals with suitable Fourier decay of the kernel. Our result extends the sparse domination principle by Conde-Alonso, Culiuc, Ou and the first author to the maximally truncated case, and covers the rough homogeneous singular integrals $T_\Omega$ on $\mathbb R^d$ with bounded angular part $\Omega$ having vanishing integral on the sphere. Among several consequences, we obtain new quantitative weighted norm inequalities for the maximal truncation of $T_\Omega$, extending a result by Roncal, Tapiola and the second author.
A convex-body valued version of the sparse bound is also deduced and employed towards novel matrix-weighted norm inequalities for
the maximal truncations of $T_\Omega$. Our result is quantitative, but even the qualitative statement is new, and the present approach via sparse domination is the only one currently known for the matrix weighted bounds of this class of operators.
2019-03-12T00:00:00ZThe excluded volume of two-dimensional convex bodies: shape reconstruction and non-uniquenessTaylor J. M.http://hdl.handle.net/20.500.11824/9412019-03-07T02:00:09Z2019-02-05T00:00:00ZThe excluded volume of two-dimensional convex bodies: shape reconstruction and non-uniqueness
Taylor J. M.
In the Onsager model of one-component hard-particle systems, the entire phase behaviour is dictated by a function of relative orientation, which represents the amount of space excluded to one particle by another at this relative orientation. We term this function the excluded volume function. Within the context of two-dimensional convex bodies, we investigate this excluded volume function for one-component systems addressing two related questions. Firstly, given a body can we find the excluded volume function? Secondly, can we reconstruct a body from its excluded volume function? The former is readily answered via an explicit Fourier series representation, in terms of the support function. However we show the latter question is ill-posed in the sense that solutions are not unique for a large class of bodies. This degeneracy is well characterised however, with two bodies admitting the same excluded volume function if and only if the Fourier coefficients of their support functions differ only in phase. Despite the non-uniqueness issue, we then propose and analyse a method for reconstructing a convex body given its excluded volume function, by means of a discretisation procedure where convex bodies are approximated by zonotopes with a fixed number of sides. It is shown that the algorithm will always asymptotically produce a best least-squares approximation of the trial function, within the space of excluded volume functions of centrally symmetric bodies. In particular, if a solution exists, it can be found. Results from a numerical implementation are presented, showing that with only desktop computing power, good approximations to solutions can be readily found.
2019-02-05T00:00:00Z