Harmonic Analysis
http://hdl.handle.net/20.500.11824/3
Wed, 24 Apr 2019 07:54:02 GMT2019-04-24T07:54:02ZBloom type upper bounds in the product BMO setting
http://hdl.handle.net/20.500.11824/965
Bloom type upper bounds in the product BMO setting
Li K.; Martikainen H.; Vuorinen E.
We prove some Bloom type estimates in the product BMO setting. More specifically,
for a bounded singular integral $T_n$ in $\mathbb R^n$ and a bounded singular integral $T_m$ in $\mathbb R^m$ we prove that
$$
\| [T_n^1, [b, T_m^2]] \|_{L^p(\mu) \to L^p(\lambda)} \lesssim_{[\mu]_{A_p}, [\lambda]_{A_p}} \|b\|_{{\rm{BMO}}_{\rm{prod}}(\nu)},
$$
where $p \in (1,\infty)$, $\mu, \lambda \in A_p$ and $\nu := \mu^{1/p}\lambda^{-1/p}$ is the Bloom weight. Here $T_n^1$ is $T_n$ acting on the first variable,
$T_m^2$ is $T_m$ acting on the second variable, $A_p$ stands for the bi-parameter weights of $\mathbb R^n \times \mathbb R^m$ and
${\rm{BMO}}_{\rm{prod}}(\nu)$ is a weighted product BMO space.
Mon, 08 Apr 2019 00:00:00 GMThttp://hdl.handle.net/20.500.11824/9652019-04-08T00:00:00ZBloom type inequality for bi-parameter singular integrals: efficient proof and iterated commutators
http://hdl.handle.net/20.500.11824/950
Bloom 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.
Thu, 14 Mar 2019 00:00:00 GMThttp://hdl.handle.net/20.500.11824/9502019-03-14T00:00:00ZSparse bounds for maximal rough singular integrals via the Fourier transform
http://hdl.handle.net/20.500.11824/946
Sparse 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.
Tue, 12 Mar 2019 00:00:00 GMThttp://hdl.handle.net/20.500.11824/9462019-03-12T00:00:00ZUnique determination of the electric potential in the presence of a fixed magnetic potential in the plane
http://hdl.handle.net/20.500.11824/935
Unique determination of the electric potential in the presence of a fixed magnetic potential in the plane
Caro P.; Rogers K.
For electric and magnetic potentials with compact support, we consider the magnetic Schrödinger equation with fixed positive energy. Under a mild additional regularity hypothesis, and with fixed magnetic potential, we show that the scattering solutions uniquely determine the electric potential. For this we develop the method of Bukhgeim for the purely electric Schrödinger equation.
Sat, 01 Dec 2018 00:00:00 GMThttp://hdl.handle.net/20.500.11824/9352018-12-01T00:00:00Z