Harmonic Analysis
http://hdl.handle.net/20.500.11824/3
2021-07-24T02:23:37ZRESTRICTED TESTING FOR POSITIVE OPERATORS
http://hdl.handle.net/20.500.11824/1288
RESTRICTED TESTING FOR POSITIVE OPERATORS
Hytönen, T.; Li, K.; Sawyer, E.
We prove that for certain positive operators T, such as the Hardy-Littlewood maximal function and fractional integrals, there is a constant D>1, depending only on the dimension n, such that the two weight norm inequality
\begin{equation*} \int_{\mathbb{R}^{n}}T\left( f\sigma \right)
^{2}d\omega \leq C\int_{\mathbb{ R}^{n}}f^{2}d\sigma \end{equation*}
holds for all $f\geq 0$ if and only if the (fractional) $A_2$ condition holds, and the restricted testing condition
$\int_{Q}T\left( 1_{Q}\sigma
\right) ^{2}d\omega \leq C\left\ | Q\right\ |_{\sigma } $
holds for all cubes $Q$ satisfying $\left\ | 2Q\right\ |_{\sigma }\leq D\left\ | Q\right\ |_{\sigma }$. If T is linear, we require as well that the dual restricted testing condition
$\begin{equation*} \int_{Q}T^{\ast }\left(
1_{Q}\omega \right) ^{2}d\sigma \leq C\left\ | Q\right\ |_{\omega }
\end{equation*}
holds for all cubes Q satisfying $\left\ | 2Q\right\ |_{\omega }\leq D\left\ | Q\right\ |_{\omega }$.
2020-01-01T00:00:00ZExtensions of the John-Nirenberg theorem and applications
http://hdl.handle.net/20.500.11824/1243
Extensions of the John-Nirenberg theorem and applications
Canto, J.; Pérez, C.
The John–Nirenberg theorem states that functions of bounded mean oscillation are
exponentially integrable. In this article we give two extensions of this theorem. The first one
relates the dyadic maximal function to the sharp maximal function of Fefferman–Stein, while
the second one concerns local weighted mean oscillations, generalizing a result of Muckenhoupt
and Wheeden. Applications to the context of generalized Poincaré type inequalities and to the
context of the $C_p$ class of weights are given. Extensions to the case of polynomial BMO type
spaces are also given.
2021-01-01T00:00:00ZConvergence over fractals for the Schrödinger equation
http://hdl.handle.net/20.500.11824/1240
Convergence over fractals for the Schrödinger equation
Luca, R.; Ponce-Vanegas, F.
We consider a fractal refinement of the Carleson problem for the Schrödinger equation, that is to identify the
minimal regularity needed by the solutions to converge pointwise to their initial data almost everywhere with respect to the $\alpha$-Hausdorff measure ($\alpha$-a.e.). We extend to the fractal setting ($\alpha < n$) a recent counterexample of Bourgain \cite{Bourgain2016}, which is sharp in the Lebesque measure setting ($\alpha = n$). In doing so we recover the necessary condition from \cite{zbMATH07036806} for pointwise convergence~$\alpha$-a.e. and we extend it to the range $n/2<\alpha \leq (3n+1)/4$.
2021-01-01T00:00:00ZMultilinear operator-valued calderón-zygmund theory
http://hdl.handle.net/20.500.11824/1213
Multilinear operator-valued calderón-zygmund theory
Di Plinio, F.; Li, K.; Martikainen, H.; Vuorinen, E.
We develop a general theory of multilinear singular integrals with operator- valued kernels, acting on tuples of UMD Banach spaces. This, in particular, involves investigating multilinear variants of the R-boundedness condition naturally arising in operator-valued theory. We proceed by establishing a suitable representation of mul- tilinear, operator-valued singular integrals in terms of operator-valued dyadic shifts and paraproducts, and studying the boundedness of these model operators via dyadic- probabilistic Banach space-valued analysis. In the bilinear case, we obtain a T(1)-type theorem without any additional assumptions on the Banach spaces other than the nec- essary UMD. Higher degrees of multilinearity are tackled via a new formulation of the Rademacher maximal function (RMF) condition. In addition to the natural UMD lat- tice cases, our RMF condition covers suitable tuples of non-commutative Lp-spaces. We employ our operator-valued theory to obtain new multilinear, multi-parameter, operator- valued theorems in the natural setting of UMD spaces with property α.
2020-01-01T00:00:00Z