Related papers: The Kakeya maximal operator on the variable Lebesg…

Given a space of homogeneous type we give sufficient conditions on a variable exponent {p(.)} so that the fractional maximal operator {M_{\eta}} maps {L^{p(.)}(X)} to {L^{q(.)}(X)}, where {1/p(.) - 1/q(.) = {\eta}}. In the endpoint case we…

In this paper, the main aim is to consider the boundedness of commutators of multilinear Calder\'{o}n-Zygmund operators with Lipschitz functions in the context of the variable exponent Lebesgue spaces. Furthermore, the variable versions of…

In this paper, we will introduce and study several types of Kakeya inequalities by the maximal functions in Hardy spaces in $\RR^n$,\,$(n\geq2)$, and we could obtain several inequalities associated with the Kakeya inequalities. We will show…

Two-weight criteria of various type for the Hardy-Littlewood maximal operator and singular integrals in variable exponent Lebesgue spaces defined on the real line are established.

We prove that if the Hardy-Littlewood maximal operator is bounded on a separable Banach function space $X(\mathbb{R}^n)$ and on its associate space $X'(\mathbb{R}^n)$ and a maximally modulated Calder\'on-Zygmund singular integral operator…

We study mapping properties of the centered Hardy--Littlewood maximal operator $\mathcal{M}$ acting on Lorentz spaces $L^{p,q}(\mathfrak{X})$ in the context of certain non-doubling metric measure spaces $\mathfrak{X}$. The special class of…

Given sparse collections of measurable sets $\mathcal S_k$, $k=1,2,\ldots ,N$, in a general measure space $(X,\mathfrak M,\mu)$, let $ \Lambda_{\mathcal S_k}$ be the sparse operator, corresponding to $\mathcal S_k$. We show that the maximal…

In this paper we study a variant of the uncentred Hardy--Littlewood maximal operator on Damek--Ricci spaces in which balls are replaced by suitable half balls. Perhaps surprisingly, such modified maximal operator has better boundedness…

We derive Maximal Kakeya estimates for functions over $\mathbb{Z}/N\mathbb{Z}$ proving the Maximal Kakeya conjecture for $\mathbb{Z}/N\mathbb{Z}$ for general $N$ as stated by Hickman and Wright [HW18]. The proof involves using polynomial…

In this article we investigate $L^p$ boundedness of the spherical maximal operator $\mathfrak{m}^\alpha$ of (complex) order $\alpha$ on the $n$-dimensional hyperbolic space $\mathbb{H}^n$, which was introduced and studied by El Kohen. We…

We introduce a notion of (finite order) lacunarity in higher dimensions for which we can bound the associated directional maximal operators in $L^p(\mathbb{R}^n)$, with $p>1$. In particular, we are able to treat the classes previously…

We study the Hardy-Littlewood maximal operator in the Musielak-Orlicz-Sobolev space $W^{1,\varphi}(\mathbb{R}^n)$. Under some natural assumptions on $\varphi$ we show that the maximal function is bounded and continuous in…

We give sufficient conditions on the exponent $p: \mathbb R^d\rightarrow [1,\infty)$ for the boundedness of the non-centered Gaussian maximal function on variable Lebesgue spaces $L^{p(\cdot)}(\mathbb R^d, \gamma_d)$, as well as of the new…

We establish the sharp growth order, up to epsilon losses, of the $L^2$-norm of the maximal directional averaging operator along a finite subset $V$ of a polynomial variety of arbitrary dimension $m$, in terms of cardinality. This is an…

Let $K\subset \mathbb R$ be a regular compact set and let $g(z)=g_{\overline{\mathbb C}\setminus K}(z,\infty)$ be the Green function for $\overline{\mathbb C}\setminus K$ with pole at infinity. For $\delta>0$, define $$ G(\delta):=\max\{…

We study maximal operators associated to singular averages along finite subsets $\Sigma$ of the Grassmannian $\mathrm{Gr}(d,n)$ of $d$-dimensional subspaces of $\mathbb R^n$. The well studied $d=1$ case corresponds to the the directional…

In this paper we derive the restricted weighted maximal operator, defined by ${\sup }_{k\in \mathbb{N}}\left(\left\vert \sigma _{k}F\right\vert/A^2_k\right)$ of Fej\'er means of Walsh-Fourier series and prove that the it is bounded from the…

In this article we obtain an "off-diagonal" version of the Fefferman-Stein vector-valued maximal inequality on weighted Lebesgue spaces with variable exponents. As an application of this result and the atomic decomposition developed in [12]…

The dimension of Kakeya sets can be bounded using sum-difference exponents $\SD(R;s)$ for various sets of rational slopes $R$ and output slope $s$; the arithmetic Kakeya conjecture, which implies the Kakeya conjecture in all dimensions,…

We prove $L^p\to L^q$ estimates for local maximal operators associated with dilates of codimension two spheres in Heisenberg groups; these are sharp up to two endpoints. The results can be applied to improve currently known bounds on sparse…