arXiv subjects
Publications and source records attributed to Paweł Plewa.
We prove sharp both-sided estimates for the $W$-invariant trigonometric Dunkl heat kernel in rank one case, including root systems $A_1$ and $BC_1$. Consequently, we obtain sharp bounds for the non-compact Jacobi heat kernel in the full range of the type parameters.
The main goal of the paper is to prove convergence in norm and pointwise almost everywhere on $L^p$, $p\in (1,\infty)$, for certain multiparameter polynomial ergodic averages in the spirit of Dunford and Zygmund for continuous flows. We will pay special attention to quantitative aspects of pointwise convergence phenomena from the point of view of uniform oscillation estimates for multiparameter polynomial Radon averaging operators. In the proof of our main result we develop flexible Fourier methods that exhibit and handle the so-called "parameters-gluing'' phenomenon, an obstruction that arises in studying oscillation and variation inequalities for multiparameter polynomial Radon operators. We will also discuss connections of our main result with a multiparameter variant of the Bellow-Furstenberg problem.
We give a unified and optimized proof of the sharp bounds for the Jacobi heat kernel, which were obtained gradually in several papers in recent years. We lay particular emphasis on tracing and estimating all constants appearing throughout the entire reasoning. This allows us to quantitatively control the multiplicative constants in the Jacobi heat kernel bounds in terms of the parameters involved. Consequently, analogous control extends to a number of interrelated heat kernels. In particular, we obtain quantitative control in terms of the associated dimension for the spherical heat kernel and for all other heat kernels on compact rank one symmetric spaces.
Let $G=N\rtimes \mathbb{R}$, where $N$ is a Carnot group and $\mathbb{R}$ acts on $N$ via automorphic dilations. Homogeneous left-invariant sub-Laplacians on $N$ and $\mathbb{R}$ can be lifted to $G$, and their sum is a left-invariant sub-Laplacian $Δ$ on $G$. We prove that the first-order Riesz transforms $X Δ^{-1/2}$ are bounded on $L^p(G)$ for all $p\in(1,\infty)$, where $X$ is any horizontal left-invariant vector field on $G$. This extends a previous result by Vallarino and the first-named author, who obtained the bound for $p\in(1,2]$. The proof makes use of an operator-valued spectral multiplier theorem, recently proved by the authors, and hinges on estimates for products of modified Bessel functions and their derivatives.
Let $L_ν= -\partial_x^2-(ν-1)x^{-1} \partial_x$ be the Bessel operator on the half-line $X_ν= [0,\infty)$ with measure $x^{ν-1} \,\mathrm{d} x$. In this work we study singular integral operators associated with the Laplacian $Δ_ν= -\partial_u^2 + e^{2u} L_ν$ on the product $G_ν$ of $X_ν$ and the real line with measure $\mathrm{d} u$. For any $ν\geq 1$, the Laplacian $Δ_ν$ is left-invariant with respect to a noncommutative hypergroup structure on $G_ν$, which can be thought of as a fractional-dimension counterpart to $ax+b$ groups. In particular, equipped with the Riemannian distance associated with $Δ_ν$, the metric-measure space $G_ν$ has exponential volume growth. We prove a sharp $L^p$ spectral multiplier theorem of Mihlin--Hörmander type for $Δ_ν$, as well as the $L^p$-boundedness for $p \in (1,\infty)$ of the associated first-order Riesz transforms. To this purpose, we develop a Calderón--Zygmund theory à la Hebisch--Steger adapted to the nondoubling structure of $G_ν$, and establish large-time gradient heat kernel estimates for $Δ_ν$. In addition, the Riesz transform bounds for $p > 2$ hinge on an operator-valued spectral multiplier theorem, which we prove in greater generality and may be of independent interest.
Let $G$ be the semidirect product $N \rtimes \mathbb{R}$, where $N$ is a stratified Lie group and $\mathbb{R}$ acts on $N$ via automorphic dilations. Homogeneous left-invariant sub-Laplacians on $N$ and $\mathbb{R}$ can be lifted to $G$, and their sum $Δ$ is a left-invariant sub-Laplacian on $G$. In previous joint work of Ottazzi, Vallarino and the first-named author, a spectral multiplier theorem of Mihlin--Hörmander type was proved for $Δ$, showing that an operator of the form $F(Δ)$ is of weak type $(1,1)$ and bounded on $L^p(G)$ for all $p \in (1,\infty)$ provided $F$ satisfies a scale-invariant smoothness condition of order $s > (Q+1)/2$, where $Q$ is the homogeneous dimension of $N$. Here we show that, if $N$ is a group of Heisenberg type, or more generally a direct product of Métivier and abelian groups, then the smoothness condition can be pushed down to the sharp threshold $s>(d+1)/2$, where $d$ is the topological dimension of $N$. The proof is based on lifting to $G$ weighted Plancherel estimates on $N$ and exploits a relation between the functional calculi for $Δ$ and analogous operators on semidirect extensions of Bessel--Kingman hypergroups.
Let $W$ be a finite reflection group associated with root system $R$ in $\mathbb R^d$. Let $C_+$ denote a positive Weyl chamber distinguished by a choice of $R_+$, a set of positive roots. We define and investigate Hardy and BMO spaces on $C_+$ in the framework of boundary conditions given by a homomorphism $η\in\mathrm{Hom}(W,\hat{\mathbb{Z}}_2)$ which attaches the $\pm$ signs to the facets of $C_+$. Specialized to orthogonal root systems, atomic decompositions in $H^1_η$ and $h^1_η$ are obtained and the duality problem is also treated.
Dependencies of the optimal constants in strong and weak type bounds will be studied between maximal functions corresponding to the Hardy--Littlewood averaging operators over convex symmetric bodies acting on $\mathbb R^d$ and $\mathbb Z^d$. Firstly, we show, in the full range of $p\in[1,\infty]$, that these optimal constants in $L^p(\mathbb R^d)$ are always not larger than their discrete analogues in $\ell^p(\mathbb Z^d)$; and we also show that the equality holds for the cubes in the case of $p=1$. This in particular implies that the best constant in the weak type $(1,1)$ inequality for the discrete Hardy--Littlewood maximal function associated with centered cubes in $\mathbb Z^d$ grows to infinity as $d\to\infty$, and if $d=1$ it is equal to the largest root of the quadratic equation $12C^2-22C+5=0$. Secondly, we prove dimension-free estimates for the $\ell^p(\mathbb Z^d)$ norms, $p\in(1,\infty]$, of the discrete Hardy--Littlewood maximal operators with the restricted range of scales $t\geq C_q d$ corresponding to $q$-balls, $q\in[2,\infty)$. Finally, we extend the latter result on $\ell^2(\mathbb Z^d)$ for the maximal operators restricted to dyadic scales $2^n\ge C_q d^{1/q}$.
We consider a nonnegative self-adjoint operator $L$ on $L^2(X)$, where $X\subseteq \mathbb{R}^d$. Under certain assumptions, we prove atomic characterizations of the Hardy space $$H^1(L) = ł\{f\in L^1(X) \ : \ \ {\|}\sup_{t>0} \ |\exp(-tL)f \ | \ {\|}_{L^1(X)}<\infty\ \}.$$ We state simple conditions, such that $H^1(L)$ is characterized by atoms being either the classical atoms on $X\subseteq \mathbb{R}^d$ or local atoms of the form $|Q|^{-1}χ_Q$, where $Q\subseteq X$ is a cube (or cuboid). One of our main motivation is to study multidimensional operators related to orthogonal expansions. We prove that if two operators $L_1, L_2$ satisfy the assumptions of our theorem, then the sum $L_1 + L_2$ also does. As a consequence, we give atomic characterizations for multidimensional Bessel, Laguerre, and Schrödinger operators. As a by-product, under the same assumptions, we characterize $H^1(L)$ also by the maximal operator related to the subordinate semigroup $\exp(-tL^ν)$, where $ν\in(0,1)$.
Hardy's inequality on $H^p$ spaces, $p\in(0,1]$, in the context of orthogonal expansions is investigated for general basis on a subset of $\mathbb{R}^d$ with Lebesgue measure. The obtained result is applied to various Hermite, Laguerre, and Jacobi expansions. For that purpose some delicate estimates of the higher order derivatives for the underlying functions and of the associated heat kernels are proved. Moreover, sharpness of studied Hardy's inequalities is justified by a construction of an explicit counterexample, which is adjusted to all considered settings.
Four Jacobi settings are considered in the context of Hardy's inequality: the trigonometric polynomials and functions, and the corresponding symmetrized systems. In the polynomial cases sharp Hardy's inequality is proved for the type parameters $α,β\in(-1,\infty)^d$, whereas in the function systems for $α,β\in[-1/2,\infty)^d$. The ranges of these parameters are the widest in which the corresponding orthonormal bases are composed of bounded functions. Moreover, the sharp $L^1$-analogues of Hardy's inequality are obtained with the same restrictions on the parameters $α$ and $β$.
Sharp multi-dimensional Hardy's inequality for the Laguerre functions of Hermite type is proved for the type parameter $\al\in[-1/2,\infty)^d$. As a consequence we obtain the corresponding result for the generalized Hermite expansions. In particular, it validate that the known version of Hardy's inequality for the Hermite functions is sharp.
A method of proving Hardy's type inequality for orthogonal expansions is presented in a rather general setting. Then sharp multi-dimensional Hardy's inequality associated with the Laguerre functions of convolution type is proved for type index $\al\in[-1/2,\infty)^d$. The case of the standard Laguerre functions is also investigated. Moreover, the sharp analogues of Hardy's type inequality involving $L^1$ norms in place of $H^1$ norms are obtained in both settings.
Hardy's inequality for Laguerre expansions of Hermite type with the index $\al\in(\{-1/2\}\cup[1/2,\infty))^d$ is proved in the multi-dimensional setting with the exponent $3d/4$. We also obtain the sharp analogue of Hardy's inequality with $L^1$ norm replacing $H^1$ norm at the expense of increasing the exponent by an arbitrarily small value.
Homogeneous Besov and Triebel-Lizorkin spaces associated with multi-dimensional Laguerre function expansions of Hermite type with index $α\in [-1/2,\infty)^d\backslash (-1/2,1/2)^d$, $d\geq 1$, are defined and investigated. To achieve expected goals Schwartz type spaces on $R^d_+$ are introduced and then tempered type distributions are constructed. Also, ideas from a recent paper of Bui and Duong on Besov and Triebel-Lizorkin spaces associated with Hermite functions expansions are used. This means, in particular, using molecular decomposition and an appropriate form of a Calderón reproducing formula.