Search

We introduce the centred and the uncentred triangular maximal operators $\mathcal T$ and $\mathcal U$, respectively, on any locally finite tree in which each vertex has at least three neighbours. We prove that both $\mathcal T$ and $\mathcal U$ are bounded on $L^p$ for every $p$ in $(1,\infty]$, that $\mathcal T$ is also bounded on $L^1(\mathfrak T)$, and that $\mathcal U$ is not of weak type $(1,1)$ on homogeneous trees. Our proof of the $L^p$ boundedness of $\mathcal U$ hinges on the geometric approach of A. C\'ordoba and R. Fefferman. We also establish $L^p$ bounds for some related maximal operators. Our results are in sharp contrast with the fact that the centred and the uncentred Hardy--Littlewood maximal operators (on balls) may be unbounded on $L^p$ for every $p<\infty$ even on some trees where the number of neighbours is uniformly bounded., Comment: 14 pages

In this paper we study the $L^p$ boundedness of the centred and the uncentred Hardy--Littlewood maximal operators on the class $\Upsilon_{a,b}$, $2\leq a\leq b$, of trees with $(a,b)$-bounded geometry. We find the sharp range of $p$, depending on $a$ and $b$, where the centred maximal operator is bounded on $L^p(\mathfrak T)$ for all $\mathfrak T$ in $\Upsilon_{a,b}$. We show that there exists a tree in $\Upsilon_{a,b}$ for which the uncentred maximal function is bounded on $L^p$ if and only if $p=\infty$. We also extend these results to graphs which are strictly roughly isometric, in the sense of Kanai, to trees in the class $\Upsilon_{a,b}$., Comment: 30 pages

In this paper we investigate the validity of first and second order $L^{p}$ estimates for the solutions of the Poisson equation depending on the geometry of the underlying manifold. We first present $L^{p}$ estimates of the gradient under the assumption that the Ricci tensor is lower bounded in a local integral sense and construct the first counterexample showing that they are false, in general, without curvature restrictions. Next, we obtain $L^p$ estimates for the second order Riesz transform (or, equivalently, the validity of $L^{p}$ Calder\'on--Zygmund inequalities) on the whole scale $1

In this paper we establish inclusions and noninclusions between various Hardy type spaces on noncompact Riemannian manifolds $M$ with Ricci curvature bounded from below, positive injectivity radius and spectral gap. Our first main result states that, if $\mathscr{L}$ is the positive Laplace-Beltrami operator on $M$, then the Riesz-Hardy space $H^1_\mathscr{R}(M)$ is the isomorphic image of the Goldberg type space $\mathfrak{h}^1(M)$ via the map $\mathscr{L}^{1/2} (\mathscr{I} + \mathscr{L})^{-1/2}$, a fact that is false in $\mathbb{R}^n$. Specifically, $H^1_\mathscr{R}(M)$ agrees with the Hardy type space $\mathfrak{X}^{1/2}(M)$ recently introduced by the the first three authors; as a consequence, we prove that $\mathfrak{h}^1(M)$ does not admit an atomic characterisation. Noninclusions are mostly proved in the special case where the manifold is a Damek-Ricci space $S$. Our second main result states that $H^1_\mathscr{R}(S)$, the heat Hardy space $H^1_\mathscr{H}(S)$ and the Poisson-Hardy space $H^1_\mathscr{P}(S)$ are mutually distinct spaces, a fact which is in sharp contrast to the Euclidean case, where these three spaces agree., Comment: 28 pages

We establish sharp-in-time kernel and dispersive estimates for the Schr\"odinger equation on non-compact Riemannian symmetric spaces of any rank. Due to the particular geometry at infinity and the Kunze-Stein phenomenon, these properties are more pronounced in large time and enable us to prove the global-in-time Strichartz inequality for a larger family of admissible couples than in the Euclidean case. Consequently, we obtain the global well-posedness for the corresponding semilinear equation with lower regularity data and some scattering properties for small powers which are known to fail in the Euclidean setting. The crucial kernel estimates are achieved by combining the stationary phase method based on a subtle barycentric decomposition, a subordination formula of the Schr\"odinger group to the wave propagator and an improved Hadamard parametrix., Comment: 19 pages, major revision with more details

We prove a radial maximal function characterisation of the local atomic Hardy space h^1(M) on a Riemannian manifold M with positive injectivity radius and Ricci curvature bounded from below. As a consequence, we show that an integrable function belongs to h^1(M) if and only if either its local heat maximal function or its local Poisson maximal function are integrable. A key ingredient is a decomposition of H\"older cut-offs in terms of an appropriate class of approximations of the identity, which we obtain on arbitrary Ahlfors-regular metric measure spaces and generalises a previous result of A. Uchiyama., Comment: 31 pages

We prove that if $\tau$ is a large positive number, then the atomic Goldberg-type space $\mathfrak{h}^1(N)$ and the space $\mathfrak{h}_{\mathcal R_\tau}^1(N)$ of all integrable functions on $N$ whose local Riesz transform $\mathcal R_\tau$ is integrable are the same space on any complete noncompact Riemannian manifold $N$ with Ricci curvature bounded from below and positive injectivity radius. We also relate $\mathfrak{h}^1(N)$ to a space of harmonic functions on the slice $N\times (0,\delta)$ for $\delta>0$ small enough., Comment: 51 pages

We introduce a decreasing one-parameter family $\mathfrak{X}^{\gamma}(M)$, $\gamma>0$, of Banach subspaces of the Hardy-Goldberg space $\mathfrak{h}^1(M)$ on certain nondoubling Riemannian manifolds with bounded geometry and we investigate their properties. In particular, we prove that $\mathfrak{X}^{1/2}(M)$ agrees with the space of all functions in $\mathfrak{h}^1(M)$ whose Riesz transform is in $L^1(M)$, and we obtain the surprising result that this space does not admit an atomic decomposition., Comment: 21 pages

In this paper we prove a Marcinkiewicz-type multiplier result for the spherical Fourier transform on products of rank one noncompact symmetric spaces.

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.

In this paper we define a space $\ghu{M}$ of Hardy--Goldberg type on a measured metric space satisfying some mild conditions. We prove that the dual of $\ghu{M}$ may be identified with $\gbmo{M}$, a space of functions with "local" bounded mean oscillation, and that if $p$ is in $(1,2)$, then $\lp{M}$ is a complex interpolation space between $\ghu{M}$ and $\ld{M}$. This extends previous results of Strichartz, Carbonaro, Mauceri and Meda, and Taylor. Applications to singular integral operators on Riemannian manifolds are given.

Suppose that (M,d,m) is an unbounded metric measure space, which possesses two geometric properties, called "isoperimetric property" and "approximate midpoint property", and that the measure m is locally doubling. The isoperimetric property implies that the volume of balls grows at least exponentially with the radius. Hence the measure m is not globally doubling. In this paper we define an atomic Hardy space H1(m), where atoms are supported only on "small balls", and a corresponding space BMO(m) of functions of bounded mean oscillation, where the control is only on the oscillation over small balls. We prove that BMO(m) is the dual of H1(m) and that an inequality of John-Nirenberg type on small balls holds for functions in BMO(m). Furthermore, we show that the Lp(m) spaces are intermediate spaces between H1(m) and BMO(m), and we develop a theory of singular integral operators acting on function spaces on M. Finally, we show that our theory is strong enough to give H1(m)-L1(m) and L1(m)-BMO(m) estimates for various interesting operators on Riemannian manifolds and symmetric spaces which are unbounded on L1(m) and on L\infty(m)., Comment: 32 pages

In this paper we prove sharp weak type 1 estimates for spherical Fourier multipliers on symmetric spaces of the noncompact type. This complements earlier results of J.-Ph. Anker and A.D. Ionescu.