Related papers: The (weak-$L^2$) Boundedness of The Quadratic Carl…

We consider a basic $d$-adic model for the scattering transform on the line. We prove $L^2$ bounds for this scattering transform and a weak $L^2$ bound for a Carleson type maximal operator. The latter implies boundedness of $d$-adic models…

For any Calder\'on-Zygmund operator $ T$, any weight $ w$, and $ \alpha >1$, the operator $ T$ is bounded as a map from $ L ^{1} (M _{ L \log\log L (\log\log\log L) ^{\alpha } } w )$ into weak-$L^1(w)$. The interest in questions of this…

We extend in two directions the notion of perturbations of Carleson type for the Dirichlet problem associated to an elliptic real second-order divergence-form (possibly degenerate, not necessarily symmetric) elliptic operator. First, in…

In this work we provide a criterion for the global weak (1,1) type of integral operators which are known to be locally uniformly of weak (1,1) type. As an application, we establish the global weak (1,1) type for a class of Fourier integral…

Based on the tile discretization elaborated by the author in "The Polynomial Carleson Operator", we develop a Calderon-Zygmund type decomposition of the Carleson operator. As a consequence, through a unitary method that makes no use of…

In this paper we introduce a new double phase Baouendi-Grushin type operator with variable coefficients. We give basic properties of the corresponding functions space and prove a compactness result. In the second part, using topological…

We use the chiral quark model to estimate the coefficients of the weak chiral lagrangian as obtained from the bosonization of the ten relevant operators of the $\Delta S = 1$ effective quark lagrangian. All contributions of order $N_c^2$ as…

In this study, $(1,1)-$weak type boundedness of square function $S_{\alpha,\psi}$ is obtained by using Nazarov-Treil and Volberg technique. Also using this result, the $(1,1)-$ weak type boundedness of $g^{*}_{\lambda,\psi}$ operator is…

In this paper, we investigate the weak-type regularity of the Bergman projection. The two domains we focus on are the polydisc and the Hartogs triangle. For the polydisc we provide a proof that the weak-type behavior is of "$L\log L$" type.…

Doubling metric measure spaces provide a natural framework for singular integral operators. In contrast, the study of maximally modulated singular integral operators, the so-called Carleson operators, has largely been limited to Euclidean…

\noindent The paper establishes weak convergence in $C[0,1]$ of normalized stochastic processes, generated by Toeplitz type quadratic functionals of a continuous time Gaussian stationary process, exhibiting long-range dependence. Both…

In this paper, we investigate the boundedness of bilinear Calder\'on-Zygmund operators $T$ from ${L^{p_1}\left(w_1\right)} \times {L^{p_2}\left(w_2\right)}$ to ${L^{p,\infty}\left(v_{\vec{w}}\right)}$ with the stopping time method, where $1…

We obtain a complete characterization of the weak-type $(1,1)$ for Haar shift operators in terms of generalized Haar systems adapted to a Borel measure $\mu$ in the operator-valued setting. The main technical tool in our method is a…

Let $T$ be a Fourier integral operator on $\R^n$ of order $-(n-1)/2$. It was shown by Seeger, Sogge, and Stein that $T$ mapped the Hardy space $H^1$ to $L^1$. In this note we show that $T$ is also of weak-type $(1,1)$. The main ideas are a…

We prove that the lacunary Carleson operator is bounded from $L \log L$ to $L^{1}$. This result is sharp. The proof is based on two newly introduced concepts: 1) the \emph{time-frequency regularization of a measurable set} and 2) the…

Let $P(\Delta)$ be a polynomial of the Laplace operator $\Delta=\sum_{j=1}^n\frac{\partial^2}{\partial x^2_j}$ on $\mathbb{R}^n$. We prove the existence of weak solutions of the equation $P(\Delta)u=f$ and the existence of a bounded right…

We prove $L^p$ bounds, $\frac{d^2 + 4d + 2}{(d+1)^2} < p < 2(d+1)$, for maximal linear modulations of singular integrals along paraboloids with frequencies in certain subspaces of $\mathbb{R}^{d+1}$, for $d \geq 2$. This generalizes…

We study the mixed local and nonlocal double phase parabolic equation \begin{align*} \partial_t u(x,t)-\mathrm{div}(a(x,t)|\nabla u|^{q-2}\nabla u) +\mathcal{L}u(x,t)=0 \end{align*} in $Q_T=\Omega\times(0,T)$, where $\mathcal{L}$ is the…

We provide elementary proofs that the 2-variation Carleson operator $V_2$ along with explicit bilinear multipliers adapted to $\{\xi_1 + \xi_2 = 0\}$ satisfy no $L^p$ estimates. Furthermore, we obtain $L^p \rightarrow L^p$ estimates when $2…

We prove a Carleman estimate for elliptic second order partial differential operators with Lipschitz continuous coefficients. The Carleman estimate is valid for any complex-valued function $u\in W^{2,2}$ with support in a punctured ball of…