We consider the zero-resistivity limit for Hasegawa–Wakatani equations in a cylindrical domain when the initial data are Stepanov almost-periodic in the axial direction. First, we prove the existence of a solution to Hasegawa–Wakatani equations with zero resistivity; second, we obtain uniform a priori estimates with respect to resistivity. Such estimates can be obtained in the same way as for our previous results; therefore, the most important contribution of this paper is the proof of the existence of a local-in-time solution to Hasegawa–Wakatani equations with zero resistivity. We apply the theory of Bohr–Fourier series of Stepanov almost-periodic functions to such a proof.

Let $\mathcal{M}$ and G denote, respectively, the maximal operator and the geometric maximal operator associated with the dyadic lattice on $\mathbb{R}^d$ .

1. (i) We prove that for any 0 < p < ∞, any weight w on $\mathbb{R}^d$ and any measurable f on $\mathbb{R}^d$ , we have Fefferman–Stein-type estimate

   $$\begin{equation*}||G(f)||_{L^p(w)}\leq e^{1/p}||f||_{L^p(\mathcal{M}w)}.\end{equation*}$$
   For each p, the constant e1/p is the best possible.

2. (ii) We show that for any weight w on $\mathbb{R}^d$ and any measurable f on $\mathbb{R}^d$ ,

   $$\begin{equation*}\int_{\mathbb{R}^d} G(f)^{1/\mathcal{M}w}w\mbox{d}x\leq e\int_{\mathbb{R}^d} |f|^{1/w}w\mbox{d}x\end{equation*}$$
   and prove that the constant e is optimal.

Let $H=-\unicode[STIX]{x1D6E5}+V$ be a Schrödinger operator with some general signed potential $V$ . This paper is mainly devoted to establishing the $L^{q}$ -boundedness of the Riesz transform $\unicode[STIX]{x1D6FB}H^{-1/2}$ for $q>2$ . We mainly prove that under certain conditions on $V$ , the Riesz transform $\unicode[STIX]{x1D6FB}H^{-1/2}$ is bounded on $L^{q}$ for all $q\in [2,p_{0})$ with a given $2<p_{0}<n$ . As an application, the main result can be applied to the operator $H=-\unicode[STIX]{x1D6E5}+V_{+}-V_{-}$ , where $V_{+}$ belongs to the reverse Hölder class $B_{\unicode[STIX]{x1D703}}$ and $V_{-}\in L^{n/2,\infty }$ with a small norm. In particular, if $V_{-}=-\unicode[STIX]{x1D6FE}|x|^{-2}$ for some positive number $\unicode[STIX]{x1D6FE}$ , $\unicode[STIX]{x1D6FB}H^{-1/2}$ is bounded on $L^{q}$ for all $q\in [2,n/2)$ and $n>4$ .

The problem of pricing arithmetic Asian options is nontrivial, and has attracted much interest over the last two decades. This paper provides a method for calculating bounds on option prices and approximations to option deltas in a market where the underlying asset follows a geometric Lévy process. The core idea is to find a highly correlated, yet more tractable proxy to the event that the option finishes in-the-money. The paper provides a means for calculating the joint characteristic function of the underlying asset and proxy processes, and relies on Fourier methods to compute prices and deltas. Numerical studies show that the lower bound provides accurate approximations to prices and deltas, while the upper bound provides good though less accurate results.

We show that if a collection of lines in a vector space over a finite field has “dimension” at least $2(d-1)+\unicode[STIX]{x1D6FD}$ , then its union has “dimension” at least $d+\unicode[STIX]{x1D6FD}$ . This is the sharp estimate of its type when no structural assumptions are placed on the collection of lines. We also consider some refinements and extensions of the main result, including estimates for unions of $k$ -planes.

We prove a family of sharp bilinear space–time estimates for the half-wave propagator $\text{e}^{\text{i}t\sqrt{-\unicode[STIX]{x1D6E5}}}$ . As a consequence, for radially symmetric initial data, we establish sharp estimates of this kind for a range of exponents beyond the classical range.

We raise a question of whether the Riesz transform on $\mathbb{T}^{n}$ or $\mathbb{Z}^{n}$ is characterized by the ‘maximal semigroup symmetry’ that the transform satisfies. We prove that this is the case if and only if the dimension is one, two or a multiple of four. This generalizes a theorem of Edwards and Gaudry for the Hilbert transform on $\mathbb{T}$ and $\mathbb{Z}$ in the one-dimensional case, and extends a theorem of Stein for the Riesz transform on $\mathbb{R}^{n}$. Unlike the $\mathbb{R}^{n}$ case, we show that there exist infinitely many linearly independent multiplier operators that enjoy the same maximal semigroup symmetry as the Riesz transforms on $\mathbb{T}^{n}$ and $\mathbb{Z}^{n}$ if the dimension $n$ is greater than or equal to three and is not a multiple of four.

In this note we give a simple proof of the endpoint regularity for the uncentred Hardy–Littlewood maximal function on $\mathbb{R}$ . Our proof is based on identities for the local maximum points of the corresponding maximal functions, which are of interest in their own right.

The so-called triangular Hilbert transform is an elegant trilinear singular integral form which specializes to many well-studied objects of harmonic analysis. We investigate $L^{p}$ bounds for a dyadic model of this form in the particular case when one of the functions on which it acts is essentially one dimensional. This special case still implies dyadic analogues of boundedness of the Carleson maximal operator and of the uniform estimates for the one-dimensional bilinear Hilbert transform.

We consider the random functions $S_{N}(z):=\sum _{n=1}^{N}z(n)$, where $z(n)$ is the completely multiplicative random function generated by independent Steinhaus variables $z(p)$. It is shown that $\mathbb{E}|S_{N}|\gg \sqrt{N}(\log N)^{-0.05616}$ and that $(\mathbb{E}|S_{N}|^{q})^{1/q}\gg _{q}\sqrt{N}(\log N)^{-0.07672}$ for all $q>0$.

Let $\text{H}$ be a subgroup of some locally compact group $\text{G}$. Assume that $\text{H}$ is approximable by discrete subgroups and that $\text{G}$ admits neighborhood bases which are almost invariant under conjugation by finite subsets of $\text{H}$. Let $m:\text{G}\rightarrow \mathbb{C}$ be a bounded continuous symbol giving rise to an $L_{p}$-bounded Fourier multiplier (not necessarily completely bounded) on the group von Neumann algebra of $\text{G}$ for some $1\leqslant p\leqslant \infty$. Then, $m_{\mid _{\text{H}}}$ yields an $L_{p}$-bounded Fourier multiplier on the group von Neumann algebra of $\text{H}$ provided that the modular function ${\rm\Delta}_{\text{G}}$ is equal to 1 over $\text{H}$. This is a noncommutative form of de Leeuw’s restriction theorem for a large class of pairs $(\text{G},\text{H})$. Our assumptions on $\text{H}$ are quite natural, and they recover the classical result. The main difference with de Leeuw’s original proof is that we replace dilations of Gaussians by other approximations of the identity for which certain new estimates on almost-multiplicative maps are crucial. Compactification via lattice approximation and periodization theorems are also investigated.

In an extrapolation argument, we prove certain $L^{p}\,(1<p<\infty )$ estimates for nonisotropic Marcinkiewicz operators associated to surfaces under the integral kernels given by the elliptic sphere functions ${\rm\Omega}\in L(\log ^{+}L)^{{\it\alpha}}({\rm\Sigma})$ and the radial function $h\in {\mathcal{N}}_{{\it\beta}}(\mathbb{R}^{+})$. As applications, the corresponding results for parametric Marcinkiewicz integral operators related to area integrals and Littlewood–Paley $g_{{\it\lambda}}^{\ast }$-functions are given.

In this paper, we study Hardy spaces associated with non-negative self-adjoint operators and develop their vector-valued theory. The complex interpolation scales of vector-valued tent spaces and Hardy spaces are extended to the endpoint p=1. The holomorphic functional calculus of L is also shown to be bounded on the associated Hardy space H 1 L (X). These results, along with the atomic decomposition for the aforementioned space, rely on boundedness of certain integral operators on the tent space T 1(X).

We develop a strategy making extensive use of tent spaces to study parabolic equations with quadratic nonlinearities as for the Navier–Stokes system. We begin with a new proof of the well-known result of Koch and Tataru on the well-posedness of Navier–Stokes equations in $\mathbb{R}^{n}$ with small initial data in $\mathit{BMO}^{-1}(\mathbb{R}^{n})$ . We then study another model where neither pointwise kernel bounds nor self-adjointness are available.

We give a new direct proof of the local Tb theorem in the Euclidean setting and under the assumption of dual exponents. This theorem provides a flexible framework for proving the boundedness of a Calderón–Zygmund operator, supposing the existence of systems of local accretive functions. We assume that the integrability exponents on these systems of functions are of the form 1/p + 1/q ⩽ 1, the ‘dual case’ 1/p + 1/q = 1 being the most difficult one. Our proof is direct: it avoids a reduction to the perfect dyadic case unlike some previous approaches. The principal point of interest is in the use of random grids and the corresponding construction of the corona. We also use certain twisted martingale transform inequalities.

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.

We prove continuity in generalized parabolic Morrey spaces of sublinear operators generated by the parabolic Calderón—Zygmund operator and by the commutator of this operator with bounded mean oscillation (BMO) functions. As a consequence, we obtain a global -regularity result for the Cauchy—Dirichlet problem for linear uniformly parabolic equations with vanishing mean oscillation (VMO) coefficients.

The purpose of the paper is to introduce a novel “splitting” procedure which can be helpful in the derivation of explicit formulas for various Bellman functions. As an illustration, we study the action of the dyadic maximal operator on $L^{p}$. The associated Bellman function $\mathfrak{B}_{p}$, introduced by Nazarov and Treil, was found explicitly by Melas with the use of combinatorial properties of the maximal operator, and was later rediscovered by Slavin, Stokolos and Vasyunin with the use of the corresponding Monge–Ampère partial differential equation. Our new argument enables an alternative simple derivation of $\mathfrak{B}_{p}$.