Related papers: $L^p$ estimates for angular maximal functions asso…

Let $A_tf(x)=\int f(x+ty)d\sigma(y)$ denote the spherical means in $\Bbb R^d$ ($d\sigma$ is surface measure on $S^{d-1}$, normalized to $1$). We prove sharp estimates for the maximal function $M_E f(x)=\sup_{t\in E}|A_tf(x)|$ where $E$ is a…

In this article we consider a modification of the Stein's spherical maximal operator of complex order $\alpha$ on ${\mathbb R^n}$: $$ {\mathfrak M}^\alpha_{[1,2]} f(x) =\sup\limits_{t\in [1,2]} \big| {1\over \Gamma(\alpha) } \int_{|y|\leq…

We study a parametrized family of strong maximal fractional operators. We prove their $L^p$ to $L^q$ boundedness for $1<p\le q<\infty$.

In this paper we obtain a new boundedness criterion for the maximal operator $M$ on variable exponent spaces $L^{p(\cdot)}$. It is formulated in terms of the variable exponent analogue of the well known weighted $A_{\infty}$ condition.

A convolution operator in $\mathbb{R}^d$ with kernel in $L_q$ acts from $L_p$ to $L_s$, where $1/p+1/q=1+1/s$. The main theorem states that if $1<q,p,s<\infty$, then there exists an $L_p$ function of unit norm on which the $s$-norm of the…

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…

We study the boundedness problem for maximal operators $\mathbb{M}$ associated to averages along families of finite type curves in the plane, defined by $$\mathbb{M}f(x) \, := \, \sup_{1 \leq t \leq 2} \left|\int_{\mathbb{C}} f(x-ty) \,…

In this paper we establish $L^p$ boundedness properties for maximal operators, Littlewood-Paley functions and variation operators involving Poisson semigroups and resolvent operators associated with nonsymmetric Ornstein-Uhlenbeck…

We investigate $L^p$ boundedness of the maximal function defined by the averaging operator $f\to \mathcal{A}_t^s f$ over the two-parameter family of tori $\mathbb{T}_t^{s}:=\{ ( (t+s\cos\theta)\cos\phi,\,(t+s\cos\theta)\sin\phi,\,…

In dimensions $n\ge 2$ we obtain $L^{p_1}(\mathbb R^n) \times\dots\times L^{p_m}(\mathbb R^n)$ to $L^p(\mathbb R^n)$ boundedness for the multilinear spherical maximal function in the largest possible open set of indices and we provide…

We prove the $L^p$ boundedness of a maximal operator associated with a dyadic frequency decomposition of a Fourier multiplier, under a weak regularity assumption.

For $m \geq 2$, let $(\mathbb{Z}_{m+1}^N, |\cdot|)$ denote the group equipped with the so-called $l^0$ metric, \[ |y| = \left| \big( y(1), \dots, y(N) \big) \right| := | \{1 \leq i \leq N : y(i) \neq 0 \} |,\] and define the…

Several estimates for singular integrals, maximal functions and the spherical summation operator are given in the spaces $L^p_{\text{rad}}L^2_{\text{ang}}(\mathbb{R}^n)$, $n\geq 2$.

In this article we prove a maximal $L^p$-regularity result for stochastic convolutions, which extends Krylov's basic mixed $L^p(L^q)$-inequality for the Laplace operator on ${\mathbb{R}}^d$ to large classes of elliptic operators, both on…

We study the boundedness problem for maximal operators $\M$ associated to smooth hypersurfaces $S$ in 3-dimensional Euclidean space. For $p>2,$ we prove that if no affine tangent plane to $S$ passes through the origin and $S$ is analytic,…

Many coupled evolution equations can be described via $2\times2$-block operator matrices of the form $\mathcal{A}=\begin{bmatrix} A & B \\ C & D \end{bmatrix}$ in a product space $X=X_1\times X_2$ with possibly unbounded entries. Here, the…

L. Diening \cite{D1} obtained the following dual property of the maximal operator $M$ on variable Lebesque spaces $L^{p(\cdot)}$: if $M$ is bounded on $L^{p(\cdot)}$, then $M$ is bounded on $L^{p'(\cdot)}$. We extend this result to weighted…

In this paper, we investigate $L^p$ bounds of maximal Fourier multiplier operators with dilation of fractional dimensions. For Fourier multipliers, we suggest a criterion related to dimensions of dilation sets which guarantees $L^p$ bounds…

We study the boundedness problem for maximal operators $\mathcal{M}$ associated to averages along families of hypersurfaces $S$ of finite type in $\mathbb{R}^n.$ In this paper, we prove that if $S$ is a finite type hypersurface which is of…

This is a revised version of the doctoral dissertation of the same title, written under the supervision of Professor Krzysztof Stempak in 2019. For general (possibly nondoubling) metric measure spaces various properties of the associated…