In this article we study a special class of non-doubling metric measure spaces for which there is a significant difference between the incidence of weak and restricted weak type $(p,p)$ inequalities for the centered and non-centered Hardy--Littlewood maximal operators, $M^c$ and $M$. As a corollary we extend the result obtained in [2].
In this article we characterize all possible cases that may occur in the relations between the sets of $p$ for which weak type $(p,p)$ and strong type $(p,p)$ inequalities for the Hardy--Littlewood maximal operators, both centered and non-centered, hold in the context of general metric measure spaces.
In this article we investigate a special class of non-doubling metric measure spaces in order to describe the possible configurations of $P_{k,\rm s}^{\rm c}$, $P_{k,\rm s}$, $P_{k,\rm w}^{\rm c}$ and $P_{k,\rm w}$, the sets of all $p \in [1, \infty]$ for which the weak and strong type $(p,p)$ inequalities hold for the centered and non-centered modified Hardy--Littlewood maximal operators, $M^{\rm c}_k$ and $M_k$, $k \geq 1$. For any fixed $k$ we describe the necessary conditions that $P_{k,\rm s}^{\rm c}$, $P_{k,\rm s}$, $P_{k,\rm w}^{\rm c}$ and $P_{k,\rm w}$ must satisfy in general and illu…
We characterize two-weight inequalities for certain maximal truncations of the Hilbert transform in terms of testing conditions on simpler functions. For 1<p<2 and two positive Borel measures u, v on R, we assume that u is doubling, and we consider maximal truncations T_# of the Hilbert transform. The norm estimate || T(f u) ||_{L^p(v)} < C || f ||_{L^p(u)} is characterized in terms of an A_p condition on the weights and two testing conditions. The first is the norm condition above, but the function f varies over bounded functions supported on a cube. The second is a dual weak-type condition, …
For the maximal operator $ M $ on $ \mathbb R ^{d}$, and $ 1< p , ρ< \infty $, there is a finite constant $ D = D _{p, ρ}$ so that this holds. For all weights $ w, σ$ on $ \mathbb R ^{d}$, the operator $ M (σ\cdot )$ is bounded from $ L ^{p} (σ) \to L ^{p} (w)$ if and only the pair of weights $ (w, σ)$ satisfy the two weight $ A _{p}$ condition, and this testing inequality holds: \begin{equation*} \int _{Q} M (σ\mathbf 1_{Q} ) ^{p} \; d w \lesssim σ( Q), \end{equation*} for all cubes $ Q$ for which there is a cube $ P \supset Q$ satisfying $ σ(P) < D σ(Q)$, and $ \ell P = ρ\ell Q$. This was re…
In this paper we develop a kind of A_p theory for Calderon-Zygmund operators in a non-homogeneous setting. Let μbe a Borel measure on \R^d which may be non doubling. The only condition that μmust satisfy is μ(B(x,r))\leq Cr^n for all x\in\R^d, r>0 and for some fixed n with 0<n\leq d. We introduce a maximal operator N, which coincides with the maximal Hardy-Littlewood operator if μ(B(x,r))\approx r^n for x\in\supp(μ), and we show that all n-dimensional Calderon-Zygmund operators are bounded on L^p(w dμ) if and only if N is bounded on L^p(w dμ), for a fixed p\in(1,\infty). Also, we prove that th…
We obtain improved fractional Poincaré inequalities in John domains of a metric space $(X, d)$ endowed with a doubling measure $μ$ under some mild regularity conditions on the measure $μ$. We also give sufficient conditions on a bounded domain to support fractional Poincaré type inequalities in this setting.
This paper studies the following weighted, fractional Bernstein inequality for spherical polynomials on $\sph$: \begin{equation}\label{4-1-TD-ab} \|(-Δ_0)^{r/2} f\|_{p,w}\leq C_w n^{r} \|f\|_{p,w}, \ \ \forall f\in Π_n^d, \end{equation} where $Π_n^d$ denotes the space of all spherical polynomials of degree at most $n$ on $\sph$, and $(-Δ_0)^{r/2}$ is the fractional Laplacian-Beltrami operator on $\sph$. A new class of doubling weights with conditions weaker than the $A_p$ is introduced, and used to fully characterize those doubling weights $w$ on $\sph$ for which the weighted Bernstein inequal…
Restricted testing conditions were considered recently. For the maximal operator, Li and Sawyer first obtained parental testing condition. Later, they showed that it suffices to restrict testing to doubling cubes. Chen and Lacey gave a similar restricted testing condition. In our paper, we discuss a version of the latter in the multilinear setting.
We consider L^p two weight inequalities for maximal truncations of dyadic Calderon-Zygmund operators. In the case of one weight being doubling, a characterization is given, and for the general case, sufficient conditions are given, including standard and non-standard testing conditions. The arguments of this paper parallel the authors prior work arXiv:0805.4711, but in this paper are much easier.
We answer a special case of a question of T. Hytonen regarding the two weight norm inequality for the maximal function M in the affirmative, namely that there is a constant D > 1, depending only on dimension n, such that the two weight norm inequality for M holds if and only if the A_2 condition holds, and the testing condition holds for all doubling cubes.
We present ten different characterizations of functions satisfying a weak reverse Hölder inequality on an open subset of a metric measure space with a doubling measure. Among others, we describe these functions as a class of weak $A_\infty$ weights, which is a generalization of Muckenhoupt weights that allows for nondoubling weights. Although our main results are modeled after conditions that hold true for Muckenhoupt weights, we also discuss two conditions for Muckenhoupt $A_\infty$ weights that fail to hold for weak $A_\infty$ weights.
We prove fractional Sobolev-Poincaré inequalities, capacitary versions of fractional Poincaré inequalities, and pointwise and localized fractional Hardy inequalities in a metric space equipped with a doubling measure. Our results generalize and extend earlier work where such inequalities have been considered in the Euclidean spaces or in the non-fractional setting in metric spaces. The results concerning pointwise and localized variants of fractional Hardy inequalities are new even in the Euclidean case.
We give a quantitative characterization of the pairs of weights $(w,v)$ for which the dyadic version of the one-sided Hardy-Littlewood maximal operator satisfies a restricted weak $(p,p)$ type inequality, for $1\leq p<\infty$. More precisely, given any measurable set $E_0$ the estimate \[w(\{x\in \mathbb{R}^n: M^{+,d}(\mathcal{X}_{E_0})(x)>t\})\leq \frac{C[(w,v)]_{A_p^{+,d}(\mathcal{R})}^p}{t^p}v(E_0)\] holds if and only if the pair $(w,v)$ belongs to $A_p^{+,d}(\mathcal{R})$, that is \[\frac{|E|}{|Q|}\leq [(w,v)]_{A_p^{+,d}(\mathcal{R})}\left(\frac{v(E)}{w(Q)}\right)^{1/p}\] for every dyadic …
Associated to the class of restricted-weak type weights for the Hardy operator, we find a new class of Lorentz spaces for which the normability property holds. This result is analogous to the characterization given by Sawyer for the classical Lorentz spaces. We also show that these new spaces are very natural to study the existence of equivalent norms described in terms of the maximal function
We prove generalizations of the Poincare and logarithmic Sobolev inequalities corresponding to the case of fractional derivatives in measure spaces with only a minimal amount of geometric structure. The class of such spaces includes (but is not limited to) spaces of homogeneous type with doubling measures. Several examples and applications are given, including Poincare inequalities for graph Laplacians, Fractional Poincare inequalities of Mouhot, Russ, and Sire [16], and implications for recent work of the author and R. M. Strain on the Boltzmann collision operator [10, 11, 9].
We prove necessary conditions on pairs of measures $(μ,ν)$ for a singular integral operator $T$ to satisfy weak $(p,p)$ inequalities, $1\leq p<\infty$, provided the kernel of $T$ satisfies a weak non-degeneracy condition first introduced by Stein, and the measure $μ$ satisfies a weak doubling condition related to the non-degeneracy of the kernel. We also show similar results for pairs of measures $(μ,σ)$ for the operator $T_σf = T(f\,dσ)$, which has come to play an important role in the study of weighted norm inequalities. Our major tool is a careful analysis of the strong type inequalities fo…
Let ${\cal P}_n^c$ denote the set of all algebraic polynomials of degree at most $n$ with complex coefficients. Let $$D^+ := \{z \in \mathbb{C}: |z| \leq 1, \, \, \Im(z) \geq 0\}$$ be the closed upper half-disk of the complex plane. For integers $0 \leq k \leq n$ let ${\mathcal F}_{n,k}^c$ be the set of all polynomials $P \in {\mathcal P}_n^c$ having at least $n-k$ zeros in $D^+$. Let $$\|f\|_A := \sup_{z \in A}{|f(z)|}$$ for complex-valued functions defined on $A \subset {\Bbb C}$. We prove that there are absolute constants $c_1 > 0$ and $c_2 > 0$ such that $$c_1 \left(\frac{n}{k+1}\right)^{1…
We define $(α_n)$ -regular sets in uniformly perfect metric spaces. This definition is quasisymmetrically invariant and the construction resembles generalized dyadic cubes in metric spaces. For these sets we then determine the necessary and sufficient conditions to be fat (or thin). In addition we discuss restrictions of doubling measures to these sets, and in particular give a sufficient condition to retain at least some of the restricted measures doubling on the set. Our main result generalizes and extends analogous results that were previously known to hold in the real-line.
We prove an endpoint version of the uniform Sobolev inequalities in Kenig-Ruiz-Sogge [8]. It was known that strong type inequalities no longer hold at the endpoints; however, we show that restricted weak type inequalities hold there, which imply the earlier classical result by real interpolation. The key ingredient in our proof is a type of interpolation first introduced by Bourgain [2]. We also prove restricted weak type Stein-Tomas restriction inequalities on some parts of the boundary of a pentagon, which completely characterizes the range of exponents for which the inequalities hold.
Let $X$ be a metric space equipped with a doubling measure. We consider weights $w(x)=\operatorname{dist}(x,E)^{-α}$, where $E$ is a closed set in $X$ and $α\in\mathbb R$. We establish sharp conditions, based on the Assouad (co)dimension of $E$, for the inclusion of $w$ in Muckenhoupt's $A_p$ classes of weights, $1\le p<\infty$. With the help of general $A_p$-weighted embedding results, we then prove (global) Hardy-Sobolev inequalities and also fractional versions of such inequalities in the setting of metric spaces.
In this note we describe some recent advances in the area of maximal function inequalities. We also study the behaviour of the centered Hardy-Littlewood maximal operator associated to certain families of doubling, radial decreasing measures, and acting on radial functions. In fact, we precisely determine when the weak type $(1,1)$ bounds are uniform in the dimension.
In this paper, the multilinear fractional strong maximal operator $\mathcal{M}_{\mathcal{R},α}$ associated with rectangles and corresponding multiple weights $A_{(\vec{p},q),\mathcal{R}}$ are introduced. Under the dyadic reverse doubling condition, a necessary and sufficient condition for two-weight inequalities is given. As consequences, we first obtain a necessary and sufficient condition for one-weight inequalities. Then, we give a new proof for the weighted estimates of multilinear fractional maximal operator $\mathcal{M}_α$ associated with cubes and multilinear fractional integral operato…
This paper continues the investigation begun in arXiv:1906.05602 of extending the T1 theorem of David and Journé, and optimal cancellation conditions, to more general weight pairs. The main additional tool developed here is a two weight restricted weak type inequality. Assuming σ and ω are locally finite positive Borel measures, with one of them an A infinity weight, we show that the two weight restricted weak type inequality for an α-fractional Calderón-Zygmund singular integral T holds if and (provided T is elliptic) only if the classical fractional Muckenhoupt condition holds. In the case α…
The purpose of this work is to describe an abstract theory of Hardy-Sobolev spaces on doubling Riemannian manifolds via an atomic decomposition. We study the real interpolation of these spaces with Sobolev spaces and finally give applications to Riesz inequalities.
We link the problem of estimating the lower Hausdorff dimension of PDE or Fourier constrained measures with Harnack's inequalities for the heat equation. Our approach provides new estimates in the case of Fourier constraints.
Here we show existence of numerous subsets of Euclidean and metric spaces that, despite having empty interior, still support Poincaré inequalities. Most importantly, our methods do not depend on any rectilinear or self-similar structure of the underlying space. We instead employ the notion of uniform domain of Martio and Sarvas. Our condition relies on the measure density of such subsets, as well as the regularity and relative separation of their boundary components. In doing so, our results hold true for metric spaces equipped with doubling measures and Poincaré inequalities in general, and f…
For $ 1\le k <n$, we prove that for functions $F,G$ on $ {\Bbb R}^{n}$, any $k$-dimensional affine subspace $H \subset {\Bbb R}^{n}$, and $p,q,r \ge 2$ with $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=1$, one has the estimate $$ {||(F*G)|_H||}_{L^{r}(H)} \leq {||F||}_{Λ^H_{2, p}({\Bbb R}^{n})} \cdot {||G||}_{Λ^H_{2, q}({\Bbb R}^{n})},$$ where the mixed norms on the right are defined by $$ {||F||}_{Λ^H_{2,p}({\Bbb R}^{n})}={(\int_{H^*} {(\int {|\hat{F}|}^2 dH_ξ^{\perp})}^{\frac{p}{2}} dξ)}^{\frac{1}{p}},$$ with $dH_ξ^{\perp}$ the $(n-k)$-dimensional Lebesgue measure on the affine subspace $H_ξ^{\perp}…
Let $X$ be a metric space with a doubling measure. Let $L$ be a nonnegative self-adjoint operator acting on $L^2(X)$, hence $L$ generates an analytic semigroup $e^{-tL}$. Assume that the kernels $p_t(x,y)$ of $e^{-tL}$ satisfy Gaussian upper bounds and Hölder's continuity in $x$ but we do not require the semigroup to satisfy the preservation condition $e^{-tL}1 = 1$. In this article we aim to establish the exponential-square integrability of a function whose square function associated to an operator $L$ is bounded, and the proof is new even for the Laplace operator on the Euclidean spaces ${\m…
The Assouad and quasi-Assouad dimensions of a metric space provide information about the extreme local geometric nature of the set. The Assouad dimension of a set has a measure theoretic analogue, which is also known as the upper regularity dimension. One reason for the interest in this notion is that a measure has finite Assouad dimension if and only if it is doubling. Motivated by recent progress on both the Assouad dimension of measures that satisfy a strong separation condition and the quasi-Assouad dimension of metric spaces, we introduce the notion of the quasi-Assouad dimension of a mea…
We study the boundedness of Riesz transforms in $L^p$ for $p>2$ on a doubling metric measure space endowed with a gradient operator and an injective, $ω$-accretive operator $L$ satisfying Davies-Gaffney estimates. If $L$ is non-negative self-adjoint, we show that under a reverse Hölder inequality, the Riesz transform is always bounded on $L^p$ for $p$ in some interval $[2,2+\varepsilon)$, and that $L^p$ gradient estimates for the semigroup imply boundedness of the Riesz transform in $L^q$ for $q \in [2,p)$. This improves results of \cite{ACDH} and \cite{AC}, where the stronger assumption of a …
In this paper, the authors characterize, in terms of pointwise inequalities, the classical Besov spaces $\dot B^s_{p,\,q}$ and Triebel-Lizorkin spaces $\dot F^s_{p,\,q}$ for all $s\in(0,\,1)$ and $p,\,q\in(n/(n+s),\,\infty],$ both in ${\mathbb R}^n$ and in the metric measure spaces enjoying the doubling and reverse doubling properties. Applying this characterization, the authors prove that quasiconformal mappings preserve $\dot F^s_{n/s,\,q}$ on $\rn$ for all $s\in(0,\,1)$ and $q\in(n/(n+s),\,\infty]$. A metric measure space version of the above morphism property is also established.
Restricted non-linear approximation is a type of N-term approximation where a measure $ν$ on the index set (rather than the counting measure) is used to control the number of terms in the approximation. We show that embeddings for restricted non-linear approximation spaces in terms of weighted Lorentz sequence spaces are equivalent to Jackson and Bernstein type inequalities, and also to the upper and lower Temlyakov property. As applications we obtain results for wavelet bases in Triebel-Lizorkin spaces by showing the Temlyakow property in this setting. Moreover, new interpolation results for …
A theorem of Lusin states that every Borel function on $R$ is equal almost everywhere to the derivative of a continuous function. This result was later generalized to $R^n$ in works of Alberti and Moonens-Pfeffer. In this note, we prove direct analogs of these results on a large class of metric measure spaces, those with doubling measures and Poincaré inequalities, which admit a form of differentiation by a famous theorem of Cheeger.
In this paper a reduction and equivalence theorems for the boundedness of the composition of a quasilinear operator $T$ with the Hardy and Copson operators in weighted Lebesgue spaces are proved. New equivalence theorems are obtained for the operator $T$ to be bounded in weighted Lebesgue spaces restricted to the cones of monotone functions, which allow to change the cone of non-decreasing functions to the cone of non-increasing functions and vice versa not changing the operator $T$. New characterizations of the weighted Hardy-type inequalities on the cones of monotone functions are given. The…
In this paper we characterize the validity of the Hardy-type inequality \begin{equation*} \left\|\left\|\int_s^{\infty}h(z)dz\right\|_{p,u,(0,t)}\right\|_{q,w,\infty}\leq c \,\|h\|_{1,v,\infty} \end{equation*} where $0<p< \infty$, $0<q\leq +\infty$, $u$, $w$ and $v$ are weight functions on $(0,\infty)$. It is pointed out that this characterization can be used to obtain new characterizations for the boundedness between weighted Lebesgue spaces for Hardy-type operators restricted to the cone of monotone functions and for the generalized Stieltjes operator.
