In dimension n = 1, we obtain $L^{p_1}(\mathbb R) \times\dots\times L^{p_m}(\mathbb R)$ to $L^p(\mathbb R)$ boundedness for the multilinear spherical maximal function in the largest possible open set of indices and we provide counterexamples that indicate the optimality of our results.

For decreasing sequences $\{t_{n}\}_{n=1}^{\infty }$ converging to zero and initial data $f\in H^s(\mathbb {R}^N)$, $N\geq 2$, we consider the almost everywhere convergence problem for sequences of Schrödinger means ${\rm e}^{it_{n}\Delta }f$, which was proposed by Sjölin, and was open until recently. In this paper, we prove that if $\{t_n\}_{n=1}^{\infty }$ belongs to Lorentz space ${\ell }^{r,\infty }(\mathbb {N})$, then the a.e. convergence results hold for $s>\min \{\frac {r}{\frac {N+1}{N}r+1},\,\frac {N}{2(N+1)}\}$. Inspired by the work of Lucà-Rogers, we construct a counterexample to show that our a.e. convergence results are sharp (up to endpoints). Our results imply that when $0< r<\frac {N}{N+1}$, there is a gain over the a.e. convergence result from Du-Guth-Li and Du-Zhang, but not when $r\geq \frac {N}{N+1}$, even though we are in the discrete case. Our approach can also be applied to get the a.e. convergence results for the fractional Schrödinger means and nonelliptic Schrödinger means.

Let $M$ be a nondoubling parabolic manifold with ends. First, this paper investigates the boundedness of the maximal function associated with the heat semigroup ${\mathcal{M}}_{\unicode[STIX]{x1D6E5}}f(x):=\sup _{t>0}|e^{-t\unicode[STIX]{x1D6E5}}f(x)|$ where $\unicode[STIX]{x1D6E5}$ is the Laplace–Beltrami operator acting on $M$. Then, by combining the subordination formula with the previous result, we obtain the weak type $(1,1)$ and $L^{p}$ boundedness of the maximal function ${\mathcal{M}}_{\sqrt{L}}^{k}f(x):=\sup _{t>0}|(t\sqrt{L})^{k}e^{-t\sqrt{L}}f(x)|$ on $L^{p}(M)$ for $1<p\leq \infty$ where $k$ is a nonnegative integer and $L$ is a nonnegative self-adjoint operator satisfying a suitable heat kernel upper bound. An interesting thing about the results is the lack of both doubling condition of $M$ and the smoothness of the operators’ kernels.

Let M(u), H(u) be the maximal operator and Hilbert transform along the parabola (t, ut2). For U ⊂ (0, ∞) we consider Lp estimates for the maximal functions sup u∈U|M(u)f| and sup u∈U|H(u)f|, when 1 < p ≤ 2. The parabolas can be replaced by more general non-flat homogeneous curves.

Let $(X,d,{\it\mu})$ be a metric measure space satisfying the doubling, reverse doubling and noncollapsing conditions. Let $\mathscr{L}$ be a nonnegative self-adjoint operator on $L^{2}(X,d{\it\mu})$ satisfying a pointwise Gaussian upper bound estimate and Hölder continuity for its heat kernel. In this paper, we introduce the Hardy spaces $H_{\mathscr{L}}^{p}(X)$, $0<p\leq 1$, associated to $\mathscr{L}$ in terms of grand maximal functions and show that these spaces are equivalently characterised by radial and nontangential maximal functions.

Mixed norm inequalities for directional operators are closely related to the boundedness problems of several important operators in harmonic analysis. In this paper we prove weighted inequalities for some one-dimensional one-sided maximal functions. Then by applying these results, we establish mixed norm inequalities for directional maximal operators which are defined from these one-dimensional maximal functions. We also estimate the constants in these inequalities.

In this paper, we study the ${{L}^{p}}$ mapping properties of a class of singular integral operators with rough kernels belonging to certain block spaces. We prove that our operators are bounded on ${{L}^{p}}$ provided that their kernels satisfy a size condition much weaker than that for the classical Calderón–Zygmund singular integral operators. Moreover, we present an example showing that our size condition is optimal. As a consequence of our results, we substantially improve a previously known result on certain maximal functions.

We provide uniqueness results for holomorphic functions in the Nevanlinna class which bridge those previously obtained by Hayman and by Lyubarskii and Seip. In particular, we propose certain classes of hyperbolically separated sequences in the disc, in terms of the rate of non-tangential accumulation to the boundary (the outer limits of this spectrum of classes being, respectively, the sequences with a non-tangential cluster set of positive measure, and the sequences satisfying the Blaschke condition). For each of those classes, we give a critical condition of radial decrease on the modulus which will force a Nevanlinna class function to vanish identically.

AMS 2000 Mathematics subject classification: Primary 30D50; 30D55

Let $E$ be a UMD Banach space, and $L$ a positive self-adjoint operator in $\mathrm{L}^2$ of Laplace type, for which the imaginary powers $L^{-\ri t}$ form a $C_0$-group of exponential growth $0\leq\alpha\lt \pi$ on $\mathrm{L}^p(E)$, where $1\lt p\lt\lt \infty$. Suppose $G(z)$ is holomorphic inside and on the boundary of the sector $\{z:z\neq0,\ |\arg z|\leq\phi\}$, and $z^\kappa G(z)\rightarrow0$ uniformly as $z\rightarrow\infty$ for some $\kappa\gt0$ and $\phi\gt\alpha$. Then $G(tL)$ $(t \gt0)$ defines a bounded family of linear operators on $\mathrm{L}^p(E)$; and the maximal operator $f\mapsto\sup_{t \gt 0}\|G(tL)f\|_E$ is bounded on the domain of $\log L$. The proof uses transference methods. These hypotheses hold for the maximal solution operators for the heat, wave and Schrödinger equations, and for Cesàro sums.

AMS 2000 Mathematics subject classification: Primary 47D03; 42B25; 47D09