arXiv subjects
Publications and source records attributed to Błażej Wróbel.
This is a survey article about recent developments in dimension-free estimates for maximal functions corresponding to the Hardy--Littlewood averaging operators associated with convex symmetric bodies in $\mathbb R^d$ and $\mathbb Z^d$.
In this paper we prove a Marcinkiewicz-type multiplier result for the spherical Fourier transform on products of rank one noncompact symmetric spaces.
Dimension-free bounds will be provided in maximal and $r$-variational inequalities on $\ell^p(\mathbb Z^d)$ corresponding to the discrete Hardy-Littlewood averaging operators defined over the cubes in $\mathbb Z^d$. We will also construct an example of a symmetric convex body in $\mathbb Z^d$ for which maximal dimension-free bounds fail on $\ell^p(\mathbb Z^d)$ for all $p\in(1, \infty)$. Finally, some applications in ergodic theory will be discussed.
We prove an $L^p$ spectral multiplier theorem for functions of the $K$-invariant sublaplacian $L$ acting on the space of functions of fixed $K$-type on the group $SL(2,\mathbb{R}).$ As an application we compute the joint $L^p(SL(2,\mathbb{R}))$ spectrum of $L$ and the derivative along $K$.
We study dimension-free $L^p$ inequalities for $r$-variations of the Hardy--Littlewood averaging operators defined over symmetric convex bodies in $\mathbb R^d$.
An explicit Bellman function is used to prove a bilinear embedding theorem for operators associated with general multi-dimensional orthogonal expansions on product spaces. This is then applied to obtain $L^p,$ $1<p<\infty,$ boundedness of appropriate vectorial Riesz transforms, in particular in the case of Jacobi polynomials. Our estimates for the $L^p$ norms of these Riesz transforms are both dimension-free and linear in $\max(p,p/(p-1)).$ The approach we present allows us to avoid the use of both differential forms and general spectral multipliers.
We propose a framework for bilinear multiplier operators defined via the (bivariate) spectral theorem. Under this framework we prove Coifman-Meyer type multiplier theorems and fractional Leibniz rules. Our theory applies to bilinear multipliers associated with the discrete Laplacian on $\mathbb{Z}^d,$ general bi-radial bilinear Dunkl multipliers, and to bilinear multipliers associated with the Jacobi expansions.
We characterise, for each $p$ in $[1,\infty) \setminus \{2\}$, the class of $L^p$ spherical multipliers on homogeneous trees in terms of $L^p$ Fourier multipliers on the torus.
We prove the strong continuity of spectral multiplier operators associated with dilations of certain functions on the general Hardy space $H^1_L$ introduced by Hofmann, Lu, Mitrea, Mitrea, Yan. Our results include the heat and Poisson semigroups as well as the group of imaginary powers.
We prove that the existence of a Mihlin-Hörmander functional calculus for an operator $L$ implies the boundedness on $L^p$ of both the maximal operators and the continuous square functions build on spectral multipliers of $L.$ The considered multiplier functions are finitely smooth and satisfy an integral condition at infinity. In particular multipliers of compact support are admitted.
We obtain a general Marcinkiewicz-type multiplier theorem for mixed systems of strongly commuting operators $L=(L_1,...,L_d);$ where some of the operators in $L$ have only a holomorphic functional calculus, while others have additionally a Marcinkiewicz-type functional calculus. Moreover, we prove that specific Laplace transform type multipliers of the pair $(\mathcal{L},A)$ are of certain weak type $(1,1).$ Here $\mathcal{L}$ is the Ornstein-Uhlenbeck operator while $A$ is a non-negative operator having Gaussian bounds for its heat kernel. Our results include the Riesz transforms $A(\mathcal{L}+A)^{-1}$ and $\mathcal{L}(\mathcal{L}+A)^{-1}.$
This thesis is devoted to the study of multivariate (joint) spectral multipliers for systems of strongly commuting non-negative self-adjoint operators, $L=(L_1,\ldots,L_d),$ on $L^2(X,ν),$ where $(X,ν)$ is a measure space. By strong commutativity we mean that the operators $L_r,$ $r=1,\ldots,d,$ admit a joint spectral resolution $E(λ).$ In that case, for a bounded function $m\colon [0,\infty)^d\to \mathbb{C},$ the multiplier operator $m(L)$ is defined on $L^2(X,ν)$ by $$m(L)=\int_{[0,\infty)^d}m(λ)dE(λ).$$ By spectral theory, $m(L)$ is then bounded on $L^2(X,ν).$ The purpose of the dissertation is to investigate under which assumptions on the multiplier function $m$ it is possible to extend $m(L)$ to a bounded operator on $L^p(X,ν),$ $1<p<\infty.$ The crucial assumption we make is the $L^p(X,ν),$ $1\leq p\leq \infty,$ contractivity of the heat semigroups corresponding to the operators $L_r,$ $r=1,\ldots,d.$ Under this assumption we generalize the results of [S. Meda, Proc. Amer. Math. Soc. 1990] to systems of strongly commuting operators. As an application we derive various multivariate multiplier theorems for particular systems of operators acting on separate variables. These include e.g. Ornstein-Uhlenbeck, Hermite, Laguerre, Bessel, Jacobi, and Dunkl operators. In some particular cases, we obtain presumably sharp results. Additionally, we demonstrate how a (bounded) holomorphic functional calculus for a pair of commuting operators, is useful in the study of dimension free boundedness of various Riesz transforms.
By using an $H^{\infty}$ joint functional calculus for strongly commuting operators, we derive a scheme to deduce the $L^p$ boundedness of certain $d$-dimensional Riesz transforms from the $L^p$ boundedness of appropriate one-dimensional Riesz transforms. Moreover, the $L^p$ bounds we obtain are independent of the dimension. The scheme is applied to Riesz transforms connected with orthogonal expansions and discrete Riesz transforms on products of groups with polynomial growth.
Let H(f)(x)=\int_{(0,infty)^d} f(v) E_{x}(v) dν(v), be the multivariable Hankel transform, where E_{x}(v)=\prod_{k=1}^d (x_k v_k)^{-a_k+1/2} J_{a_k-1/2}(x_k v_k), dν(v)=v^a dv, a=(a_1,...,a_d). We give sufficient conditions on a bounded continuous function m(v) which guarantee that the operator H(m Hf) is bounded on L^p(dν) and of weak-type (1,1), or bounded on the Hardy space H^1((0,infty)^d, dν) in the sense of Coifman-Weiss.