Rubio de Francia's extrapolation theory: estimates for the distribution function

Let $T$ be an arbitrary operator bounded from $L^{p_0}(w)$ into $L^{p_0, \infty}(w)$ for every weight $w$ in the Muckenhoupt class $A_{p_0}$. It is proved in this article that the distribution function of $Tf$ with respect to any weight $u$ can be essentially majorized by the distribution function of $Mf$ with respect to $u$ (plus an integral term easy to control). As a consequence, well-known extrapolation results, including results in a multilinear setting, can be obtained with very simple proofs. New applications in extrapolation for two-weight problems and estimates on rearrangement invariant spaces are established too.Comment: 29 page

A multiplier theorem using the Schechter's method of interpolation

AbstractLet m be a measurable bounded function and let us assume that there exists a bounded functions S so that m(ξ)S(ξ)it−1 is a Fourier multiplier on Lp uniformly in t∈R. Then, using the analytic interpolation theorem of Stein, one can show that necessarily m is a Lp multiplier. The purpose of this work is to show that under the above conditions, it holds that, for every k∈N, m(logS)k∈Mp. The technique is based on the Schechter's interpolation method

New Extrapolation Estimates

AbstractGiven a sublinear operator T satisfying that ‖TχA‖Lp(ν)⩽Cp−1‖χA‖Lp(μ), for every measurable set A and every 1<p⩽p0, with C independent of A and p, we show that supr>0∫∞1/rλνTf(y)dy1+log+r≲∫M|f(x)|(1+log+|f(x)|)dμ(x). This estimate allows us to improve Yano's extrapolation theorem and also to obtain that for every f∈LlogL(μ), r→∞∫∞1/rλνTf(y)dylogr≲‖f‖1. Other types of extrapolation results are also given

Modified Gravity at Astrophysical Scales

Using a perturbative approach we solve stellar structure equations for low-density (solar-type) stars whose interior is described with a polytropic equation of state in scenarios involving a subset of modified gravity theories. Rather than focusing on particular theories, we consider a model-independent approach in which deviations from General Relativity are effectively described by a single parameter $\xi$. We find that for length scales below those set by stellar General Relativistic radii the modifications introduced by modified gravity can affect the computed values of masses and radii. As a consequence, the stellar luminosity is also affected. We discuss possible further implications for higher density stars and observability of the effects before described.Comment: 12 pages, 7figures, matches published versio

A unified approach for commutators theorems in interpolation theory, a survey

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II. Letter from Professor D e C arro to Dr. P earson , on the vaccine inoculation

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