arXiv subjects
Publications and source records attributed to Stefanie Petermichl.
We prove the matrix $A_2$ conjecture for the dyadic square function, that is, a norm estimate of the matrix weighted square function, where the focus is on the sharp linear dependence on the matrix $A_2$ constant in the estimate. Moreover, we give a mixed estimate in terms of $A_2$ and $A_{\infty}$ constants. Key is a sparse domination of a process inspired by the integrated form of the matrix--weighted square function.
We present a new proof of the dimensionless $L^p$ boundedness of the Riesz vector on manifolds with bounded geometry. Our proof has the significant advantage that it allows for a much stronger conclusion, namely that of a new dimensionless weighted $L^p$ estimate with optimal exponent. Other than previous arguments, only a small part of our proof is based on special auxiliary functions, the core of the argument is a weak type estimate and a sparse decomposition of the stochastic process by X.D. Li, whose projection is the Riesz vector.
We characterize the boundedness of the commutators $[b, T]$ with biparameter Journé operators $T$ in the two-weight, Bloom-type setting, and express the norms of these commutators in terms of a weighted little $bmo$ norm of the symbol $b$. Specifically, if $μ$ and $λ$ are biparameter $A_p$ weights, $ν:= μ^{1/p}λ^{-1/p}$ is the Bloom weight, and $b$ is in $bmo(ν)$, then we prove a lower bound and testing condition $\|b\|_{bmo(ν)} \lesssim \sup \| [b, R_k^1 R_l^2]: L^p(μ) \rightarrow L^p(λ)\|$, where $R_k^1$ and $R_l^2$ are Riesz transforms acting in each variable. Further, we prove that for such symbols $b$ and any biparameter Journé operators $T$ the commutator $[b, T]:L^p(μ) \rightarrow L^p(λ)$ is bounded. Previous results in the Bloom setting do not include the biparameter case and are restricted to Calderón-Zygmund operators. Even in the unweighted, $p=2$ case, the upper bound fills a gap that remained open in the multiparameter literature for iterated commutators with Journé operators. As a by-product we also obtain a much simplified proof for a one-weight bound for Journé operators originally due to R. Fefferman.
We introduce the so called convex body valued sparse operators, which generalize the notion of sparse operators to the case of spaces of vector valued functions. We prove that Calderón--Zygmund operators as well as Haar shifts and paraproducts can be dominated by such operators. By estimating sparse operators we obtain weighted estimates with matrix weights. We get two weight $A_2$-$A_\infty$ estimates, that in the one weight case give us the estimate $$ \|T\|_{L^2(W)\to L^2 (W)} \le C [W]_{A_2}^{1/2} [W]_{A_\infty} \le C[W]_{A_2}^{3/2} $$ where $T$ is either a Calderon--Zygmund operator (with modulus of continuity satisfying the Dini condition), or a Haar shift or a paraproduct.
We show that the centered discrete Hilbert transform on integers applied to a function can be written as the conditional expectation of a transform of stochastic integrals, where the stochastic processes considered have jump components. The stochastic representation of the function and that of its Hilbert transform are under differential subordination and orthogonality relation with respect to the sharp bracket of quadratic covariation. This illustrates the Cauchy Riemann relations of analytic functions in this setting. This result is inspired by the seminal work of Gundy and Varopoulos on stochastic representation of the Hilbert transform in the continuous setting.
We give several sharp estimates for a class of combinations of second order Riesz transforms on Lie groups ${G}={G}_{x} \times {G}_{y}$ that are multiply connected, composed of a discrete abelian component ${G}_{x}$ and a connected component ${G}_{y}$ endowed with a biinvariant measure. These estimates include new sharp $L^p$ estimates via Choi type constants, depending upon the multipliers of the operator. They also include weak-type, logarithmic and exponential estimates. We give an optimal $L^q \to L^p$ estimate as well. It was shown recently by Arcozzi, Domelevo and Petermichl that such second order Riesz transforms applied to a function may be written as conditional expectation of a simple transformation of a stochastic integral associated with the function. The proofs of our theorems combine this stochastic integral representation with a number of deep estimates for pairs of martingales under strong differential subordination by Choi, Banuelos and Osekowski. When two continuous directions are available, sharpness is shown via the laminates technique. We show that sharpness is preserved in the discrete case using Lax-Richtmyer theorem.
We show that the norm of the vector of Riesz transforms as operator in the weighted Lebesgue space L^2(w) is bounded by a constant multiple of the first power of the Poisson-A_2 characteristic of w. The bound is free of dimension. Our argument requires an extension of Wittwer's linear estimate for martingale transforms to the vector valued setting with scalar weights, for which we indicate a proof. Extensions to L^p(w) for 1 1, the Poisson-A_2 class is properly included in the classical A_2 class.
We prove optimal ${L}^2$ bounds for a pair of Hilbert space valued differentially subordinate martingales under a change of law. The change of law is given by a process called a weight and sharpness in this context refers to the optimal growth with respect to the characteristic of the weight. The pair of martingales are adapted, uniformly integrable, and cadlag. Differential subordination is in the sense of Burkholder, defined through the use of the square bracket. In the scalar dyadic setting with underlying Lebesgue measure, this was proved by Wittwer, where homogeneity was heavily used. Recent progress by Thiele-Treil-Volberg and Lacey, independently, resloved the so-called non-homogenous case of discrete in time filtrations with two completely different proofs. The general case for continuous-in-time filtrations remained open and is adressed here. As a by-product, we give the needed explicit expression of a Bellman function of four variables for the weighted estimate of subordinate martingales with jumps.
We prove that the $A_\infty$ property of parabolic measure for operators in certain time-varying domains is equivalent to a Carleson measure property of bounded solutions. Kircheim, Kenig, Pipher, and T. Toro established this criterion on bounded solutions in the elliptic case, improving an earlier result of Dindos, Kenig and Pipher for solutions with data in BMO. The extension to the parabolic setting requires an approach to the key estimate that primarily exploits the maximum principle. For various classes of parabolic operators, this criterion makes it easier to establish the solvability of the Dirichlet problem with data in $L^p$ for some $p$ (see results of Rivera-Noriega), and also to quantify these results in several aspects.
We dominate non-integral singular operators by adapted sparse operators and derive optimal norm estimates in weighted spaces. Our assumptions on the operators are minimal and our result applies to an array of situations, whose prototype are Riesz transforms / multipliers or paraproducts associated with a second order elliptic operator. It also applies to such operators whose unweighted continuity is restricted to Lebesgue spaces with certain ranges of exponents $(p_0,q_0)$ where $1\le p_0<2<q_0\le \infty$. The norm estimates obtained are powers $α$ of the characteristic used by Auscher and Martell. The critical exponent in this case is $\mathfrak{p}=1+\frac{p_0}{q'_0}$. We prove $α=\frac1{p-p_0}$ when $p_0<p\le \mathfrak{p}$ and $α= \frac{q_0-1}{q_0-p}$ when $\mathfrak{p}\le p<q_0$. In particular, we are able to obtain the sharp $A_2$ estimates for non-integral singular operators which do not fit into the class of Calderón-Zygmund operators. These results are new even in the Euclidean space and are the first ones for operators whose kernel does not satisfy any regularity estimate.
Let $W$ denote a matrix $A_2$ weight. In this paper, we implement a scalar argument using the square function to deduce square-function type results for vector-valued functions in $L^2(\mathbb{R},\mathbb{C}^d)$. These results are then used to study the boundedness of the Hilbert transform and Haar multipliers on $L^2(\mathbb{R},\mathbb{C}^d)$. Our proof shortens the original argument by Treil and Volberg and improves the dependence on the $A_2$ characteristic. In particular, we prove that the Hilbert transform and Haar multipliers map $L^2(\mathbb{R},W,\mathbb{C}^d)$ to itself with dependence on on the $A_2$ characteristic at most $[W]_{A_2}^{\frac{3}{2}} \log [W]_{A_2}$.
We study a class of combinations of second order Riesz transforms on Lie groups that are multiply connected, composed of a discrete abelian component and a compact connected component. We prove sharp $L^{p}$ estimates for these operators, therefore generalising previous results. We construct stochastic integrals with jump components adapted to functions defined on our semi-discrete set. We show that these second order Riesz transforms applied to a function may be written as conditional expectation of a simple transformation of a stochastic integral associated with the function. The analysis shows that Ito integrals for the discrete component must be written in an augmented discrete tangent plane of dimension twice larger than expected, and in a suitably chosen discrete coordinate system. Those artifacts are related to the difficulties that arise due to the discrete component, where derivatives of functions are no longer local. Previous representations of Riesz transforms through stochastic integrals in this direction do not consider discrete components and jump processes.
We consider iterated commutators of multiplication by a symbol function and tensor products of Hilbert or Riesz transforms. We establish mixed BMO classes of symbols that characterize boundedness of these objects in $L^p$. Little BMO and product BMO, big Hankel operators and iterated commutators are the base cases of our results. We use operator theoretical methods and existing profound results on iterated commutators for the Hilbert transform case, while the general result in several variables is obtained through the construction of a Journé operator that models the behavior of the multiple Hilbert transform. Upper estimates for commutators with paraproduct free Journé operators as well as weak factorisation results are proven.
We show that multipliers of second order Riesz transforms on products of discrete abelian groups enjoy the $L^{p} $ estimate $p^{\ast} -1$, where $p^{\ast} = \max \{ p,q \}$ and $p$ and $q$ are conjugate exponents. This estimate is sharp if one considers all multipliers of the form $\sum_i σ_{i} R_{i} R^{\ast}_{i}$ with $| σ_{i} | \leqslant 1$ and infinite groups. In the real valued case, we obtain better sharp estimates for some specific multipliers, such as $\sum_{i} σ_{i} R_{i} R^{\ast}_{i}$ with $0 \leqslant σ_{i} \leqslant 1$. These are the first known precise $L^{p} $ estimates for discrete Calderón-Zygmund operators.
We give a criterion on collections of Calderon-Zygmund operators to classify product BMO by means of iterated commutators.
We study the possible analogous of the Div-Curl Lemma in classical harmonic analysis and partial differential equations, but from the point of view of the multi-parameter setting. In this context we see two possible Div-Curl lemmas that arise. Extensions to differential forms are also given.
As a corollary to our main theorem we give a new proof of the result that the norm of the Hilbert transform on L^2(w) has norm bounded by a the A_2 characteristic of a weight to the first power, a theorem of one of us. This new proof begins as the prior proofs do, by passing to Haar shifts. Then, we apply a deep two-weight T1 theorem of Nazarov-Treil-Volberg, to reduce the matter to checking a certain carleson measure condition. This condition is checked with a corona decomposition of the weight. Prior proofs of this type have used Bellman functions, while this proof is flexible enough to address all Haar shifts at the same time.
We establish new $p$-estimates for the norm of the generalized Beurling--Ahlfors transform $\mathcal{S}$ acting on form-valued functions. Namely, we prove that $\norm{\mathcal{S}}_{L^p(\R^n;Λ)\to L^p(\R^n;Λ)}\leq n(p^{*}-1)$ where $p^*=\max\{p, p/(p-1)\},$ thus extending the recent Nazarov--Volberg estimates to higher dimensions. The even-dimensional case has important implications for quasiconformal mappings. Some promising prospects for further improvement are discussed at the end.