Related papers: Some one-sided estimates for oscillatory singular …

$(L_p, L_q)$ estimates are obtained for oscillatory potentials $(K^\alphaf)(x)=\int\limits_{R^n}\frac{\exp(i|y|)}{|y|^{n-\alpha}}f(x-y)dy$, $0<\alpha<n$, $n\geq 2$, whose symbol has a singularity on the unit sphere. These potentials are…

Let $T$ be a Calder\'on-Zygmund singular integral operator. In this paper, we will show some weighted boundedness properties of commutator $[b,T]$ on the weighted Morrey spaces $L^{p,\kappa}(w)$ under appropriate conditions on the weight…

We mostly survey results concerning the $L^2$ boundedness of oscillatory and Fourier integral operators. This article does not intend to give a broad overview; it mainly focusses on a few topics directly related to the work of the authors.

In this paper, we establish sufficient conditions for a singular integral $T$ to be bounded from certain Hardy spaces $H^p_L$ to Lebesgue spaces $L^p$, $0< p \le 1$, and for the commutator of $T$ and a BMO function to be weak-type bounded…

We consider partitions $p_{w}(n)$ of a positive integer $n$ arising from the generating functions \[ \sum_{n=1}^\infty p_{w}(n) z^n = \prod_{m \in \mathbb{N}} (1-z^m)^{-w(m)}, \] where the weights $w(m)$ are M\"{o}bius convolutions. We…

We prove some weighted $L_p$ estimates for generalized harmonic extensions in the half-space.

We study the approximation numbers of weighted composition operators $f\mapsto w\cdot(f\circ\varphi)$ on the Hardy space $H^2$ on the unit disc. For general classes of such operators, upper and lower bounds on their approximation numbers…

The aim of this paper is to obtain new Hardy inequalities with double singular weights - at an interior point and on the boundary of the domain. These inequalities give us the possibility to derive estimates from below of the first…

We consider harmonic functions in the unit ball of $\mathbb{R}^{n+1}$ that are unbounded near the boundary but can be estimated from above by some (rapidly increasing) radial weight $w$. Our main result gives some conditions on $w$ that…

Let $L$ be a closed, densely defined operator of type $ \omega $ on $ L^2(\mathbb{R}^n)$ with $0 \leq \omega < \pi/2 $. We assume that $ L $ possesses a bounded $ H_\infty $-functional calculus and that its heat kernel satisfies suitable…

We prove interior weighted Hessian estimates in Orlicz spaces for nondivergence type elliptic equations with a lower order term which involves a nonnegative potential satisfying a reverse H\"older type condition.

We study weakly stable hyperbolic boundary problems with highly oscillatory coefficients that are large, $O(1)$, compared to the small wavelength $\eps$ of oscillations. Such problems arise, for example, in the study of classical questions…

We give a quantitative characterization of the pairs of weights $(w,v)$ for which the dyadic version of the one-sided Hardy-Littlewood maximal operator satisfies a restricted weak $(p,p)$ type inequality, for $1\leq p<\infty$. More…

We describe an elementary method for bounding a one-dimensional oscillatory integral in terms of an associated non-oscillatory integral. The bounds obtained are efficient in an appropriate sense and behave well under perturbations of the…

In this work, we establish continuity properties of strongly singular integral operators for extreme values of $p$. Particularly, weighted $L^\infty$-$BMO$ boundedness is obtained, generalizing Miyachi's result to the context of Muckenhoupt…

Elliptic estimates in Hardy classes are proved on domains with minimally smooth boundary. The methodology is different from the original methods of Chang/Krantz/Stein.

Given a unilateral shift $B_w$ (determined by a bounded sequence $w$), a sequence $x \in \ell^2$ is "hypercyclic" for $w$ iff the forward iterates of $x$ under $B_w$ are dense in $\ell^2$. We show that it is possible to make the set of $x…

We develop a weighted mixed-norm $L_q(L_p)$-estimates for solutions to fractional evolution equations of the form \[ \partial_t^\alpha w(t,x) = \phi(\Delta) w(t,x) + h(t,x), \quad w(0,\cdot) = w_0, \quad t > 0, \; x \in \mathbb{R}^d, \]…

In this paper we establish results on the existence of nontangential limits for weighted $\Cal A$-harmonic functions in the weighted Sobolev space $W_w^{1,q}(\Bbb B^n)$, for some $q>1$ and $w$ in the Muckenhoupt $A_q$ class, where $\Bbb…

We prove quantitative, one-weight, weak-type estimates for maximal operators, singular integrals, fractional maximal operators and fractional integral operators. We consider a kind of weak-type inequality that was first studied by…