Harmonic Analysis
http://hdl.handle.net/20.500.11824/3
2021-04-06T14:30:42ZExtensions 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.
2020-01-01T00:00:00ZEnd-point estimates, extrapolation for multilinear muckenhoupt classes, and applications
http://hdl.handle.net/20.500.11824/1210
End-point estimates, extrapolation for multilinear muckenhoupt classes, and applications
Li, K.; Martell, J.M.; Martikainen, H.; Ombrosi, S.; Vuorinen, E.
In this paper we present the results announced in the recent work by the first, second, and fourth authors of the current paper concerning Rubio de Francia extrapolation for the so-called multilinear Muckenhoupt classes. Here we consider the situations where some of the exponents of the Lebesgue spaces ap- pearing in the hypotheses and/or in the conclusion can be possibly infinity. The scheme we follow is similar, but, in doing so, we need to develop a one-variable end-point off-diagonal extrapolation result. This complements the correspond- ing “finite” case obtained by Duoandikoetxea, which was one of the main tools in the aforementioned paper. The second goal of this paper is to present some applications. For example, we obtain the full range of mixed-norm estimates for tensor products of bilinear Caldero ́n-Zygmund operators with a proof based on extrapolation and on some estimates with weights in some mixed-norm classes. The same occurs with the multilinear Caldero ́n-Zygmund operators, the bilin- ear Hilbert transform, and the corresponding commutators with BMO functions. Extrapolation along with the already established weighted norm inequalities eas- ily give scalar and vector-valued inequalities with multilinear weights and these include the end-point cases.
2019-01-01T00:00:00Z