We study weighted Poincaré and Poincaré-Sobolev type inequalities with an explicit analysis on the dependence on the Ap constants of the involved weights. We obtain inequalities of the form 1991 Mathematics Subject Classification. Primary: 42B25. Secondary: 42B20.
t. We present a general approach for proving the optimality
of the exponents on weighted estimates. We show that if an operator T
satisfies a bound like
kT kLp(w) ≤ c [w]
β
Ap
w ∈ Ap,
then the optimal lower bound for β is closely related to the asymptotic
behaviour of the unweighted L
p
norm kT kLp(Rn) as p goes to 1 and +∞,
which is related to Yano’s classical extrapolation theorem. By combining
these results with the known weighted inequalities, we derive the
sharpness of the exponents, without building any specific example, for
a wide class of operators including maximal-type, Calder´on–Zygmund
and fractional operators. In particular, we obtain a lower bound for the
best possible exponent for Bochner-Riesz multipliers. We also present
a new result concerning a continuum family of maximal operators on
the scale of logarithmic Orlicz functions. Further, our method allows to
consider in a unified way maximal operators defined over very general
Muckenhoupt bases.Ministerio de Ciencia e InnovaciónJunta de Andalucí
Abstract. In this note we study the behavior of the size of Furstenberg sets with respect to the size of the set of directions defining it. For any pair α, β ∈ (0, 1], we will say that a set E ⊂ R 2 is an F αβ -set if there is a subset L of the unit circle of Hausdorff dimension at least β and, for each direction e in L, there is a line segment ℓe in the direction of e such that the Hausdorff dimension of the set E ∩ ℓe is equal or greater than α. The problem is considered in the wider scenario of generalized Hausdorff measures, giving estimates on the appropriate dimension functions for each class of Furstenberg sets. As a corollary of our main results, we obtain that dim(E) ≥ max α + β 2 ; 2α + β − 1 for any E ∈ F αβ . In particular we are able to extend previously known results to the "endpoint" α = 0 case.
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