Related papers: Circular average relative to fractal measures

We investigate the $L^p$ mapping properties of maximal functions associated with analytic hypersurfaces in $\mathbb R^d$, with a particular emphasis on the role of transversality. Around points that are not transversal, we show that the…

We obtain sharp bounds for the number of n--cycles in a finite graph as a function of the number of edges, and prove that the complete graph is optimal in more ways than could be imagined. We prove sharp estimates on both the sum of k-th…

We prove essentially optimal $L^p(\mathbb{R})$-estimates for variational variants of the maximal Fourier multiplier operators considered by Bourgain in his work on pointwise convergence of polynomial ergodic averages. As a corollary of our…

We establish sharp $L^p$ integral mean estimates for $(\alpha,\beta)$-harmonic functions on the unit disk. Explicit bounds for the functions and their partial derivatives are obtained in terms of boundary data, by means of the associated…

We study approximation properties of linear sampling operators in the spaces $L_p$ for $1\le p<\infty$. By means of the Steklov averages, we introduce a new measure of smoothness that simultaneously contains information on the smoothness of…

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…

A cornerstone of geometric reconstruction, rotation averaging seeks the set of absolute rotations that optimally explains a set of measured relative orientations between them. In spite of being an integral part of bundle adjustment and…

We verify the conjecture of [CFKRS] for the mean square near the critical point of Dirichlet L-functions for a composite modulus q. We also prove a kind of reciprocity formula when the second moment for a prime modulus is twisted by a…

The aim of this paper is to establish the decay estimate for the fractional wave equation semigroup on H-type groups given by $e^{it\Delta^\alpha}$, $0<\alpha<1$. Combing the dispersive estimate and a standard duality argument, we also…

Surface area and mean width of a cylinder (the convex hull of two parallel disks) in R^3 are computed. It is more difficult to obtain analogous results for a cone (the convex hull of a disk D and a point p). Oblique formulas for mean width,…

We discuss the asymptotic behaviour for the best constant in L^p-L^q estimates for trigonometric polinomials and for an integral operator which is related to the solution of inhomogeneous Schrodinger equations. This gives us an opportunity…

We will explain how to compute the exact $L^p$ operator norm of a "quadratic perturbation" of the real part of the Ahlfors--Beurling operator. For the lower bound estimate we use a new approach of constructing a sequence of laminates…

We consider fluctuations of error terms $\Delta(x)$ appearing in the asymptotic formula for a summatory function of coefficients of the Dirichlet series. These are quantified via $\Omega$ and $\Omega_{\pm}$ estimates. We obtain $\Omega$…

Under the formalism of annealed averaging of the partition function, a type of random multifractal measures with their multipliers satisfying exponentially distributed is investigated in detail. Branching emerges in the curve of generalized…

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) \,…

We consider divergence form elliptic operators L = - div A(x)\nabla, defined in the half space R^{n+1}_+, n \geq 2, where the coefficient matrix A(x) is bounded, measurable, uniformly elliptic, t-independent, and not necessarily symmetric.…

In this paper, we determine the sharp \((p,q)\) range for \(L^p\)--\(L^q\) bounds of convolution operators \(f\mapsto \mu*f\) associated with fractal measures \(\mu\in \mathcal P_{\alpha,\beta}(\mathbb R^d)\), namely, compactly supported…

We study the $q$-analogue of the average of Montgomery's function $F(\alpha, T)$ over bounded intervals. Assuming the Generalized Riemann Hypothesis for Dirichlet $L$-functions, we obtain upper and lower bounds for this average over an…

We define fractional power of the Dunkl Laplacian, fractional modulus of smoothness and fractional $K$-functional in $L^p$-space with the Dunkl weight. As application, we prove direct and inverse theorems of approximation theory, and some…

As periodic orbit theory works badly on computing the observable averages of dynamical systems with intermittency, we propose a scheme to cooperate with cycle expansion and perturbation theory so that we can deal with intermittent systems…