Related papers: Extension and averaging operators for finite field…

We derive sparse bounds for the bilinear spherical maximal function in any dimension $d\geq 1$. When $d\geq 2$, this immediately recovers the sharp $L^p\times L^q\to L^r$ bound of the operator and implies quantitative weighted norm…

We consider the Kato problem and extensions for degenerate elliptic operators of arbitrary order $2m$ ($m\geq 1$), whose coefficients are measurable, complex-valued and satisfy the G$\mathring{a}$rding inequality with respect to a…

We consider $\ell^r$ extensions of Calderon-Zygmund operators on weighted spaces $L^p(w)$ with $w$ an $A_p$ weight and $1 < p < \infty$. We give quantitative estimates of these operators' norm in terms of a given weight's $A_p$…

Let $L$ be a linear operator in $L^2(\mathbb{R}^n)$ which generates a semigroup $e^{-tL}$ whose kernels $p_t(x,y)$ satisfy the Gaussian upper bound. In this paper, we investigate several kinds of weighted norm inequalities for the conical…

Integration over curved manifolds with higher codimension and, separately, discrete variants of continuous operators, have been two important, yet separate themes in harmonic analysis, discrete geometry and analytic number theory research.…

In this article, we continue the study of $L^p$-boundedness of the maximal operator $\mathcal M_S$ associated to averages along isotropic dilates of a given, smooth hypersurface $S$ in 3-dimensional Euclidean space. We focus here on small…

We study both averaging and maximal averaging problems for Product $j$-varieties defined by $\Pi_j=\{x\in \mathbb F_q^d: \prod_{k=1}^d x_k=j\}$ for $j\in \mathbb F_q^*,$ where $\mathbb F_q^d$ denotes a $d$-dimensional vector space over the…

In this paper we consider logarithmic operators in two different contexts: the adapted to (continuous) Schr\"odinger operators and the classical discrete setting. The Schr\"odinger operator $\mathcal L_V$ on $\mathbb R^d$ is defined as…

Extending the methods developed in the author's previous paper and using adapted coordinate systems in two variables, an L^p boundedness theorem is proven for maximal operators over hypersurfaces in R^3 when p > 2. When the best possible p…

We prove $l^p$-improving estimates for the averaging operator along the discrete paraboloid in the sharp range of $p$ in all dimensions $n\ge 2$.

We prove that for a finite type curve in $\mathbb R^3$ the maximal operator generated by dilations is bounded on $L^p$ for sufficiently large $p$. We also show the endpoint $L^p \to L^{p}_{1/p}$ regularity result for the averaging operators…

Projection operators are important in Analysis, Optimization and Algorithm. It is well known that these operators are firmly nonexpansive. In this paper, we provide an exact result that sharpens this well-known result. We develop the theory…

We decompose the discrete bilinear spherical averaging operator into simpler operators in several ways. This leads to a wide array of extensions, such as to the simplex averaging operator, and applications, such as to operator bounds.

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,\,…

For a finite $E \subset \R^2$, $f:E \rightarrow \R$, and $p>2$, we produce a continuous $F:\R^2 \rightarrow \R$ depending linearly on $f$, taking the same values as $f$ on $E$, and with $L^{2,p}(\R^2)$ semi-norm minimal up to a factor…

A general classification of linear differential and finite-difference operators possessing a finite-dimensional invariant subspace with a polynomial basis is given. The main result is that any operator with the above property must have a…

We discuss how graph expansion is related to the behavior of $L^{p}$-functions on the covering tree. We show that the non-trivial eigenvalues of the adjacency operator on aa $(q+1)$-regular graph are bounded by $q^{1/p}+q^{(p-1)/p}$ - the…

In this paper it is proposed a very simple method for estimating the maximal operator in $L_1$. Using this method one can considerably improve the existing theorems on convergence almost-everywhere of eigenfunction expansions of an…

We prove $L^p\times L^q\rightarrow L^r$ bounds for certain lacunary bilinear maximal averaging operators with parameters satisfying the H\"older relation $1/p+1/q=1/r$. The boundedness region that we get contains at least the interior of…

We study a conical extension of averaged nonexpansive operators and the role it plays in convergence analysis of fixed point algorithms. Various properties of conically averaged operators are systematically investigated, in particular, the…