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We show some non-standard Poincaré type estimates in the biparametric setting with appropriate weights. We will derive these results using variants from classical estimates exploiting the interplay between maximal functions and fractional integrals. We also provide a sharper result by using extrapolation techniques.

Given a family $${\mathcal {Z}}=\{\Vert \cdot \Vert _{Z_Q}\}$$ Z = { ‖ · ‖ Z Q } of norms or quasi-norms with uniformly bounded triangle inequality constants, where each Q is a cube in $${\mathbb {R}}^n$$ R n , we provide an abstract estimate of the form $$\begin{aligned} \Vert f-f_{Q,\mu }\Vert _{Z_Q}\le c(\mu )\psi ({\mathcal {Z}})\Vert f\Vert _{\mathrm {BMO}(\mathrm {d}\mu )} \end{aligned}$$ ‖ f - f Q , μ ‖ Z Q ≤ c ( μ ) ψ ( Z ) ‖ f ‖ BMO ( d μ ) for every function $$f\in \mathrm {BMO}(\mathrm {d}\mu )$$ f ∈ BMO ( d μ ) , where $$\mu $$ μ is a doubling measure in $${\mathbb {R}}^n$$ R n and $$c(\mu )$$ c ( μ ) and $$\psi ({\mathcal {Z}})$$ ψ ( Z ) are positive constants depending on $$\mu $$ μ and $${\mathcal {Z}}$$ Z , respectively. That abstract scheme allows us to recover the sharp estimate $$\begin{aligned} \Vert f-f_{Q,\mu }\Vert _{L^p \left( Q,\frac{\mathrm {d}\mu (x)}{\mu (Q)}\right) }\le c(\mu )p\Vert f\Vert _{\mathrm {BMO}(\mathrm {d}\mu )}, \qquad p\ge 1 \end{aligned}$$ ‖ f - f Q , μ ‖ L p Q , d μ ( x ) μ ( Q ) ≤ c ( μ ) p ‖ f ‖ BMO ( d μ ) , p ≥ 1 for every cube Q and every $$f\in \mathrm {BMO}(\mathrm {d}\mu )$$ f ∈ BMO ( d μ ) , which is known to be equivalent to the John–Nirenberg inequality, and also enables us to obtain quantitative counterparts when $$L^p$$ L p is replaced by suitable strong and weak Orlicz spaces and $$L^{p(\cdot )}$$ L p ( · ) spaces. Besides the aforementioned results we also generalize [(Ombrosi in Isr J Math 238:571-591, 2020), Theorem 1.2] to the setting of doubling measures and obtain a new characterization of Muckenhoupt’s $$A_\infty $$ A ∞ weights.

We study maximal operators related to bases on the infinite-dimensional torus $\mathbb{T}^\omega$. {For the normalized Haar measure $dx$ on $\mathbb{T}^\omega$ it is known that $M^{\mathcal{R}_0}$, the maximal operator associated with the dyadic basis $\mathcal{R}_0$, is of weak type $(1,1)$, but $M^{\mathcal{R}}$, the operator associated with the natural general basis $\mathcal{R}$, is not. We extend the latter result to all $q \in [1,\infty)$. Then we find a wide class of intermediate bases $\mathcal{R}_0 \subset \mathcal{R}' \subset \mathcal{R}$, for which maximal functions have controlled, but sometimes very peculiar behavior.} Precisely, for given $q_0 \in [1, \infty)$ we construct $\mathcal{R}'$ such that $M^{\mathcal{R}'}$ is of restricted weak type $(q,q)$ if and only if $q$ belongs to a predetermined range of the form $(q_0, \infty]$ or $[q_0, \infty]$. Finally, we study the weighted setting, considering the Muckenhoupt $A_p^\mathcal{R}(\mathbb{T}^\omega)$ and reverse H\"older $\mathrm{RH}_r^\mathcal{R}(\mathbb{T}^\omega)$ classes of weights associated with $\mathcal{R}$. For each $p \in (1, \infty)$ and each $w \in A_p^\mathcal{R}(\mathbb{T}^\omega)$ we obtain that $M^{\mathcal{R}}$ is not bounded on $L^q(w)$ in the whole range $q \in [1,\infty)$. Since we are able to show that \[ \bigcup_{p \in (1, \infty)}A_p^\mathcal{R}(\mathbb{T}^\omega) = \bigcup_{r \in (1, \infty)} \mathrm{RH}_r^\mathcal{R}(\mathbb{T}^\omega), \] the unboundedness result applies also to all reverse H\"older weights.

In this paper we give a geometric condition which ensures that $(q,p)$-Poincaré-Sobolev inequalities are implied from generalized $(1,1)$-Poincaré inequalities related to $L^1$ norms in the context of product spaces. The concept of eccentricity plays a central role in the paper. We provide several $(1,1)$-Poincaré type inequalities adapted to different geometries and then show that our selfimproving method can be applied to obtain special interesting Poincaré-Sobolev estimates. Among other results, we prove that for each rectangle $R$ of the form $R=I_1\times I_2 \subset \mathbb{R}^{n}$ where $I_1\subset \mathbb{R}^{n_1}$ and $I_2\subset \mathbb{R}^{n_2}$ are cubes with sides parallel to the coordinate axes, we have that % \begin{equation*} \left( \frac{1}{w(R)}\int_{ R } |f -f_{R}|^{p_{δ,w}^*} \,wdx\right)^{\frac{1}{p_{δ,w}^*}} \leq c\,(1-δ)^{\frac1p}\,[w]_{A_{1,\mathfrak{R}}}^{\frac1p}\, \Big(a_1(R)+a_2(R)\Big), \end{equation*} % where $δ\in (0,1)$, $w \in A_{1,\mathfrak{R}}$, $\frac{1}{p} -\frac{1}{ p_{δ,w}^* }= \fracδ{n} \, \frac{1}{1+\log [w]_{A_{1,\mathfrak{R}}}}$ and $a_i(R)$ are bilinear analog of the fractional Sobolev seminorms $[u]_{W^{δ,p}(Q)}$ (See Theorem 2.18). This is a biparameter weighted version of the celebrated fractional Poincaré-Sobolev estimates with the gain $(1-δ)^{\frac1p}$ due to Bourgain-Brezis-Minorescu., In the first version of the paper there was an issue with the last two inequalities on page 5, since the factor ${\delta}^{1/p}$ on the right-hand side has to be omitted in the general case. This is not relevant to our contribution in the paper, but in order to correct this issue and not propagate the imprecision, we decided to remove the factor ${\delta}^{1/p}$ from every incorrect appearance

We study minimal integrability conditions via Luxemburg-type expressions with respect to generalized oscillations that imply the membership of a given function f to the space BMO. Our method is simple, sharp and flexible enough to be adapted to several different settings, like spaces of homogeneous type, non doubling measures on R n and also BMO spaces defined over more general bases than the basis of cubes.

We present sharp quantitative weighted norm inequalities for the Hardy- Littlewood maximal function in the context of locally compact abelian groups, obtaining an improved version of the so-called Buckley's theorem. On the way, we prove a precise reverse Hölder inequality for Muckenhoupt A∞ weights and provide a valid version of the "open property" for Muckenhoupt Ap weights. Fil: Paternostro, Victoria. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina Fil: Rela, Ezequiel. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina

We provide quantitative weighted estimates for the $$L^p(w)$$ norm of a maximal operator associated to cube skeletons in $${\mathbb {R}}^n$$ . The method of proof differs from the usual in the area of weighted inequalities since there are no covering arguments suitable for the geometry of skeletons. We use instead a combinatorial strategy that allows to obtain, after a linearization and discretization, $$L^p$$ bounds for the maximal operator from an estimate related to intersections between skeletons and k-planes.

In this note we generalize the definition of Fujii-Wilson condition providing quantitative characterizations of some interesting classes of weights, such as $A_\infty$, $A_\infty^{weak}$ and $C_p$, in terms of BMO type spaces suited to them. We will provide as well some self improvement properties for some of those generalized BMO spaces and some quantitative estimates for Bloom's BMO type spaces., Comment: 15 pages

We present dimension-free reverse Hölder inequalities for strong $A^{\ast}_p$ weights, $1 \le p < \infty$. We also provide a proof for the full range of local integrability of $A^{\ast}_1$ weights. The common ingredient is a multidimensional version of Riesz’s “rising sun” lemma. Our results are valid for any nonnegative Radon measure with no atoms. For $p = \infty$, we also provide a reverse Ho ̈lder inequality for certain product measures. As a corollary we derive mixed $A^{\ast}_p − A^{\ast}_{\infty}$ weighted estimates., The third author is partially supported by grants UBACyT 20020130100403BA, and PIP (CONICET) 11220110101018.

In this article we present a new proof of a sharp Reverse Holder Inequality for A ∞ weights. Then we derive two applications: a precise open property of Muckenhoupt classes and, as a consequence of this last result, we obtain a simple proof of a sharp weighted bound for the Hardy–Littlewood maximal function involving A ∞ constants: ‖ M ‖ L p ( w ) ⩽ c ( 1 p − 1 [ w ] A p [ σ ] A ∞ ) 1 / p , where 1 p ∞ , σ = w 1 1 − p and c is a dimensional constant. Our approach allows us to extend the result to the context of spaces of homogeneous type and prove a weak Reverse Holder Inequality which is still sufficient to prove the open property for A p classes and the L p boundedness of the maximal function. In this latter case, the constant c appearing in the norm inequality for the maximal function depends only on the doubling constant of the measure μ and the geometric constant κ of the quasimetric.

A quantitative two weight theorem for the Hardy-Littlewood maximal operator is proved, improving the known ones. As a consequence, a new proof of the main results in papers by Hyt¨onen and the first author and Hyt¨onen, the first author and Rela is obtained which avoids the use of the sharp quantitative reverse Holder inequality for A∞ proved in those papers. Our results are valid within the context of spaces of homogeneous type without imposing the non-empty annuli condition. Fil: Pérez Moreno, Carlos. Universidad de Sevilla; España Fil: Rela, Ezequiel. Universidad de Sevilla; España. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina

In this survey we collect and discuss some recent results on the so-called “Furstenberg set problem”, which in its classical form concerns the estimates of the Hausdorff dimension (dimH) of the sets in the Fα-class: for a given α ∈ (0, 1], a set E ⊆ ℝ2 is in the Fα-class if for each e ∈ 𝕊 there exists a unit line segment l e in the direction of e such that dimH(l ∩ E) ≥ α. For α=1, this problem is essentially equivalent to the “Kakeya needle problem”. Define γ(α)= inf{dimH(E):E ∈F α}. The best-known results on γ(α) are the following inequalities

In this expository article we collect and discuss some recent results on different consequences of a Sharp Reverse Holder Inequality for \( A\infty \) weights. For two given operators T and S, we study \( L^{p}(w) \) bounds of Coifman– Fefferman type: $$ \parallel T\;f \parallel_{L^p}(w)\;\leq \; c_{n,w,p}\parallel S\;f \parallel _{L^p}(w),$$ that can be understood as a way to control T by S.

The following subexponential estimate for commutators is proved |[|\{x\in Q: |[b,T]f(x)|>tM^2f(x)\}|\leq c\,e^{-\sqrt{��\, t\|b\|_{BMO}}}\, |Q|, \qquad t>0.\] where $c$ and $��$ are absolute constants, $T$ is a Calder��n--Zygmund operator, $M$ is the Hardy Littlewood maximal function and $f$ is any function supported on the cube $Q$. It is also obtained \[|\{x\in Q: |f(x)-m_f(Q)|>tM_{1/4;Q}^#(f)(x) \}|\le c\, e^{-��\,t}|Q|,\qquad t>0,\] where $m_f(Q)$ is the median value of $f$ on the cube $Q$ and $M_{1/4;Q}^#$ is Str��mberg's local sharp maximal function. As a consequence it is derived Karagulyan's estimate \[|\{x\in Q: |Tf(x)|> tMf(x)\}|\le c\, e^{-c\, t}\,|Q|\qquad t>0,\] improving Buckley's theorem. A completely different approach is used based on a combination of "Lerner's formula" with some special weighted estimates of Coifman-Fefferman obtained via Rubio de Francia's algorithm. The method is flexible enough to derive similar estimates for other operators such as multilinear Calder��n--Zygmund operators, dyadic and continuous square functions and vector valued extensions of both maximal functions and Calder��n--Zygmund operators. On each case, $M$ will be replaced by a suitable maximal operator., To appear in Mathematische Annalen

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 $\alpha,\beta\in(0,1]$, we will say that a set $E\subset \R^2$ is an $F_{\alpha\beta}$-set if there is a subset $L$ of the unit circle of Hausdorff dimension at least $\beta$ and, for each direction $e$ in $L$, there is a line segment $\ell_e$ in the direction of $e$ such that the Hausdorff dimension of the set $E\cap\ell_e$ is equal or greater than $\alpha$. 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 $\dim(E)\ge\max\left\{\alpha+\frac{\beta}{2} ; 2\alpha+\beta -1\right\}$ for any $E\in F_{\alpha\beta}$. In particular we are able to extend previously known results to the ``endpoint'' $\alpha=0$ case., Comment: 13 pages

For $\alpha$ in $(0,1]$, a subset $E$ of $\RR$ is called Furstenberg set of type $\alpha$ or $F_\alpha$-set if for each direction $e$ in the unit circle there is a line segment $\ell_e$ in the direction of $e$ such that the Hausdorff dimension of the set $E\cap\ell_e$ is greater or equal than $\alpha$. In this paper we show that if $\alpha > 0$, there exists a set $E\in F_\alpha$ such that $\HH{g}(E)=0$ for $g(x)=x^{1/2+3/2\alpha}\log^{-\theta}(\frac{1}{x})$, $\theta>\frac{1+3\alpha}{2}$, which improves on the the previously known bound, that $H^{\beta}(E) = 0$ for $\beta>1/2+3/2\alpha$. Further, by refining the argument in a subtle way, we are able to obtain a sharp dimension estimate for a whole class of zero-dimensional Furstenberg type sets. Namely, for $\h_\gamma(x)=\log^{-\gamma}(\frac{1}{x})$, $\gamma>0$, we construct a set $E_\gamma\in F_{\h_\gamma}$ of Hausdorff dimension not greater than 1/2. Since in a previous work we showed that 1/2 is a lower bound for the Hausdorff dimension of any $E\in F_{\h_\gamma}$, with the present construction, the value 1/2 is sharp for the whole class of Furstenberg sets associated to the zero dimensional functions $\h_\gamma$., Comment: Final version

In this paper we prove some lower bounds on the Hausdorff dimension of sets of Furstenberg type. Moreover, we extend these results to sets of generalized Furstenberg type, associated to doubling dimension functions. With some additional growth conditions on the dimension function, we obtain a lower bound on the dimension of "zero dimensional" Furstenberg sets., 16 pages, 3 figures

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 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., Comment: Revised and corrected version. To appear in Math. Res. Lett