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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 $$ |T|_{L^{p}(w)}le c, [w]^{beta}_{A_p} qquad w in A_{p}, $$ then the optimal lower bound for $beta$ is closely related to the asymptotic behaviour of the unweighted $L^p$ norm $|T|_{L^p(mathbb{R}^n)}$ as $p$ goes to 1 and $+infty$, which is related to Yanos 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, Calderon--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.
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In this note we prove the estimate $M^{sharp}_{0,s}(Tf)(x) le c,M_gamma f(x)$ for general fractional type operators $T$, where $M^{sharp}_{0,s}$ is the local sharp maximal function and $M_gamma$ the fractional maximal function, as well as a local version of this estimate. This allows us to express the local weighted control of $Tf$ by $M_gamma f$. Similar estimates hold for $T$ replaced by fractional type operators with kernels satisfying H{o}rmander-type conditions or integral operators with homogeneous kernels, and $M_gamma $ replaced by an appropriate maximal function $M_T$. We also prove two-weight, $L^p_v$-$L^q_w$ estimates for the fractional type operators described above for $1<p< q<infty$ and a range of $q$. The local nature of the estimates leads to results involving generalized Orlicz-Campanato and Orlicz-Morrey spaces.
Let $ Tf =sum_{ I} varepsilon_I langle f,h_{I^+}rangle h_{I^-}$. Here, $ lvert varepsilon _Irvert=1 $, and $ h_J$ is the Haar function defined on dyadic interval $ J$. We show that, for instance, begin{equation*} lVert T rVert _{L ^{2} (w) to L ^{2} (w)} lesssim [w] _{A_2 ^{+}} . end{equation*} Above, we use the one sided $ A_2$ characteristic for the weight $ w$. This is an instance of a one sided $A_2$ conjecture. Our proof of this fact is difficult, as the very quick known proofs of the $A_2$ theorem do not seem to apply in the one sided setting.
Let $mathsf M_{mathsf S}$ denote the strong maximal operator on $mathbb R^n$ and let $w$ be a non-negative, locally integrable function. For $alphain(0,1)$ we define the weighted sharp Tauberian constant $mathsf C_{mathsf S}$ associated with $mathsf M_{mathsf S}$ by $$ mathsf C_{mathsf S} (alpha):= sup_{substack {Esubset mathbb R^n 0<w(E)<+infty}}frac{1}{w(E)}w({xinmathbb R^n:, mathsf M_{mathsf S}(mathbf{1}_E)(x)>alpha}). $$ We show that $lim_{alphato 1^-} mathsf C_{mathsf S} (alpha)=1$ if and only if $win A_infty ^*$, that is if and only if $w$ is a strong Muckenhoupt weight. This is quantified by the estimate $mathsf C_{mathsf S}(alpha)-1lesssim_{n} (1-alpha)^{(cn [w]_{A_infty ^*})^{-1}}$ as $alphato 1^-$, where $c>0$ is a numerical constant; this estimate is sharp in the sense that the exponent $1/(cn[w]_{A_infty ^*})$ can not be improved in terms of $[w]_{A_infty ^*}$. As corollaries, we obtain a sharp reverse Holder inequality for strong Muckenhoupt weights in $mathbb R^n$ as well as a quantitative imbedding of $A_infty^*$ into $A_{p}^*$. We also consider the strong maximal operator on $mathbb R^n$ associated with the weight $w$ and denoted by $mathsf M_{mathsf S} ^w$. In this case the corresponding sharp Tauberian constant $mathsf C_{mathsf S} ^w$ is defined by $$ mathsf C_{mathsf S} ^w alpha) := sup_{substack {Esubset mathbb R^n 0<w(E)<+infty}}frac{1}{w(E)}w({xinmathbb R^n:, mathsf M_{mathsf S} ^w (mathbf{1}_E)(x)>alpha}).$$ We show that there exists some constant $c_{w,n}>0$ depending only on $w$ and the dimension $n$ such that $mathsf C_{mathsf S} ^w (alpha)-1 lesssim_{w,n} (1-alpha)^{c_{w,n}}$ as $alphato 1^-$ whenever $win A_infty ^*$ is a strong Muckenhoupt weight.
Using modern techniques of dyadic harmonic analysis, we are able to prove sharp estimates for the Bergman projection and Berezin transform and more general operators in weighted Bergman spaces on the unit ball in $mathbb{C}^n$. The estimates are in terms of the Bekolle-Bonami constant of the weight.
Let $U_1, ldots, U_n$ be a collection of commuting measure preserving transformations on a probability space $(Omega, Sigma, mu)$. Associated with these measure preserving transformations is the ergodic strong maximal operator $mathsf M ^ast _{mathsf S}$ given by [ mathsf M ^ast _{mathsf S} f(omega) := sup_{0 in R subset mathbb{R}^n}frac{1}{#(R cap mathbb{Z}^n)}sum_{(j_1, ldots, j_n) in Rcap mathbb{Z}^n}big|f(U_1^{j_1}cdots U_n^{j_n}omega)big|, ] where the supremum is taken over all open rectangles in $mathbb{R}^n$ containing the origin whose sides are parallel to the coordinate axes. For $0 < alpha < 1$ we define the sharp Tauberian constant of $mathsf M ^ast _{mathsf S}$ with respect to $alpha$ by [ mathsf C ^ast _{mathsf S} (alpha) := sup_{substack{E subset Omega mu(E) > 0}}frac{1}{mu(E)}mu({omega in Omega : mathsf M ^ast _{mathsf S} chi_E (omega) > alpha}). ] Motivated by previous work of A. A. Solyanik and the authors regarding Solyanik estimates for the geometric strong maximal operator in harmonic analysis, we show that the Solyanik estimate [ lim_{alpha rightarrow 1}mathsf C ^ast _{mathsf S}(alpha) = 1 ] holds, and that in particular we have [mathsf C ^ast _{mathsf S}(alpha) - 1 lesssim_n (1 - frac{1}{alpha})^{1/n}] provided that $alpha$ is sufficiently close to $1$. Solyanik estimates for centered and uncentered ergodic Hardy-Littlewood maximal operators associated with $U_1, ldots, U_n$ are shown to hold as well. Further directions for research in the field of ergodic Solyanik estimates are also discussed.
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