Related papers: $L^p$ bound for the Hilbert transform along variab…

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…

In this paper, we develop some basic techniques towards the Riemann hypothesis for higher rank non-abelian zeta functions of an integral regular projective curve of genus $g$ over a finite field $\mathbb F_q$. As an application of the…

In this article, we characterize the range of $\alpha$ for which the helical maximal function is bounded from $L^p(|x|^\alpha)$ to itself for $3<p<\infty$. Our result is optimal for $4\leq p<\infty,$ except possibly at end-points.

We consider the problem of proving $L^p$ bounds for eigenfunctions of the Laplacian in the high frequency limit in the presence of nonpositive curvature and more generally, manifolds without conjugate points. In particular, we prove…

We show that if the dyadic Hilbert transform with values in a Banach space is $L^p$ bounded, then so is the Hilbert transform, with a linear relation of the bounds. This result is the counterpart of [arXiv:2212.00090] where the opposite…

Let $G$ be a two-step nilpotent Lie group, identified via the exponential map with the Lie-algebra $\mathfrak g=\mathfrak g_1\oplus\mathfrak g_2$, where $[\mathfrak g,\mathfrak g]\subset \mathfrak g_2$. We consider maximal functions…

The Hilbert matrix $\mathcal{H}_{n,m} = (n+m+ 1)^{-1}$ has been extensively studied in previous literature. In this paper we look at generalized Hilbert operators arising from measures on the interval $[0, 1]$, such that the Hilbert matrix…

The three-dimensional Hilbert transform takes scalar data on the boundary of a domain in R3 and produces the boundary value of the vector part of a quaternionic monogenic (hyperholomorphic) function of three real variables, for which the…

In this paper, we investigate a class of fractional Hardy type operators $\mathscr{H}_{\beta_{1},\cdots,\beta_{m}}$ defined on higher-dimensional product spaces…

We study the $L^{p},$ $1\leqslant p\leqslant \infty,$ boundedness for Riesz transforms of the form $V^{a}(-\frac{1}{2}\Delta+V)^{-a},$ where $a>0$ and $V$ is a non-negative potential. We prove that $V^{a}(-\frac{1}{2}\Delta+V)^{-a}$ is…

This paper gives necessary conditions and slightly stronger sufficient conditions for a holomorphic function to be the Segal-Bargmann transform of a function in L^p(R^d) with respect to a Gaussian measure. The proof relies on a family of…

We prove $L^p$-bounds on the Fourier transform of measures $\mu$ supported on two dimensional surfaces. Our method allows to consider surfaces whose Gauss curvature vanishes on a one-dimensional submanifold. Under a certain non-degeneracy…

We obtain necessary and sufficient conditions on weights for a wide class of integral transforms to be bounded between weighted $L^p-L^q$ spaces, with $1\leq p\leq q\leq \infty$. The kernels $K(x,y)$ of such transforms are only assumed to…

We establish the maximal regularity for nonautonomous Ornstein-Uhlenbeck operators in $L^p$-spaces with respect to a family of invariant measures, where $p\in (1,+\infty)$. This result follows from the maximal $L^p$-regularity for a class…

Building on the (Rank I) LGC-methodology introduced by the second author and on the novel perspective employed in the time-frequency discretization of the non-resonant bilinear Hilbert--Carleson operator, we develop a new, versatile method…

In the case (4/3)<p<4, and assuming a pair of locally finite positive Borel measures on the real line have no common point masses, we prove variants of two conjectures of T. Hyt\"onen and E. Vuorinen from 2018 on two weight testing theorems…

We compute the Hilbert coefficients of a graded module with pure resolution and discuss lower and upper bounds for these coefficients for arbitrary graded modules.

We establish the sharp growth rate, in terms of cardinality, of the $L^p$ norms of the maximal Hilbert transform $H_\Omega$ along finite subsets of a finite order lacunary set of directions $\Omega \subset \mathbb R^3$, answering a question…

We establish an L^2 \times L^2 to L^1 estimate for the bilinear Hilbert transform along a curve defined by a monomial. Our proof is closely related to multi-linear oscillatory integrals.

We consider the maximal regularity problem for non-autonomous evolution equations of the form $u(t) + A(t) u(t) = f(t)$ with initial data $u(0) = u\_0$ . Each operator $A(t)$ is associated with a sesquilinear form $a(t; *, *)$ on a Hilbert…