Harmonic Analysishttp://hdl.handle.net/20.500.11824/32019-08-16T00:56:01Z2019-08-16T00:56:01Z$A_1$ theory of weights for rough homogeneous singular integrals and commutatorsPérez C.Rivera-Ríos I.Roncal L.http://hdl.handle.net/20.500.11824/10052019-08-13T01:00:12Z2019-01-01T00:00:00Z$A_1$ theory of weights for rough homogeneous singular integrals and commutators
Pérez C.; Rivera-Ríos I.; Roncal L.
Quantitative $A_1-A_\infty$ estimates for rough homogeneous singular integrals $T_{\Omega}$ and commutators of $\BMO$ symbols and $T_{\Omega}$ are obtained. In particular the following estimates are proved:
\[
\|T_\Omega \|_{L^p(w)}\le c_{n,p}\|\Omega\|_{L^\infty} [w]_{A_1}^{\frac{1}{p}}\,[w]_{A_{\infty}}^{1+\frac{1}{p'}}\|f\|_{L^p(w)}
\]
and
\[
\| [b,T_{\Omega}]f\| _{L^{p}(w)}\leq c_{n,p}\|b\|_{\BMO}\|\Omega\|_{L^{\infty}} [w]_{A_1}^{\frac{1}{p}}[w]_{A_{\infty}}^{2+\frac{1}{p'}}\|f\|_{L^{p}\left(w\right)},
\]
for $1<p<\infty$ and $1/p+1/p'=1$.
2019-01-01T00:00:00ZOn the absolute divergence of Fourier series in the infinite dimensional torusFernández E.Roncal L.http://hdl.handle.net/20.500.11824/10042019-08-13T01:00:10Z2019-03-22T00:00:00ZOn the absolute divergence of Fourier series in the infinite dimensional torus
Fernández E.; Roncal L.
In this note we present some simple counterexamples, based on quadratic forms in infinitely many variables, showing that the implication
$f\in C^{(\infty}(\mathbb{T}^\omega)\Longrightarrow\sum_{\bar{p}\in\mathbb{Z}^\infty}|\widehat{f}(\bar{p})|<\infty$ is false. There are functions of the class $C^{(\infty}(\mathbb{T}^\omega)$ (depending on an infinite number of variables) whose Fourier series diverges absolutely. This fact establishes a significant difference to what happens in the finite dimensional case.
2019-03-22T00:00:00ZImproved fractional Poincaré type inequalities in John domainsCejas E.Drelichman I.Martínez-Perales J.http://hdl.handle.net/20.500.11824/10002019-08-06T01:00:10Z2019-01-01T00:00:00ZImproved fractional Poincaré type inequalities in John domains
Cejas E.; Drelichman I.; Martínez-Perales J.
We obtain improved fractional Poincaré inequalities in John domains of a metric space $(X, d)$ endowed with a doubling measure $\mu$ under some mild regularity conditions on the measure $\mu$. We also give sufficient conditions on a bounded domain to support fractional Poincaré type inequalities in this setting.
2019-01-01T00:00:00ZOn extension problem, trace hardy and Hardy’s inequalities for some fractional LaplaciansBoggarapu P.Roncal L.Thangavelu S.http://hdl.handle.net/20.500.11824/9732019-05-03T01:00:08Z2019-09-01T00:00:00ZOn extension problem, trace hardy and Hardy’s inequalities for some fractional Laplacians
Boggarapu P.; Roncal L.; Thangavelu S.
We obtain generalised trace Hardy inequalities for fractional powers of general operators given by sums of squares of vector fields. Such inequalities are derived by means of particular solutions of an extended equation associated to the above-mentioned operators. As a consequence, Hardy inequalities are also deduced. Particular cases include Laplacians on stratified groups, Euclidean motion groups and special Hermite operators. Fairly explicit expressions for the constants are provided. Moreover, we show several characterisations of the solutions of the extension problems associated to operators with discrete spectrum, namely Laplacians on compact Lie groups, Hermite and special Hermite operators.
2019-09-01T00:00:00Z