We study the near $L^1$ behavior of the maximally quadratically modulated Hilbert transform \[ \mathcal{C}_2f(x) := \sup_{\lambda} \left|\operatorname{p.v.} \int_{\mathbb{R}}f(x-y)e^{2 \pi i \lambda y^2} \frac{\mathrm{d} y }{y} \right| \] and its lacunary counterpart obtained by restricting $\lambda$ to $2^{\mathbb{Z}}.$ We prove that if $\Phi$ is a Young function satisfying $ \Phi(t)=o\bigl(t\log_2t\bigr)$ then $\mathcal{C}_{2,\mathsf{lac}}$, and therefore $\mathcal{C}_2$ as well, does not satisfy a corresponding $\Phi$-modular estimate. In particular, neither the lacunary nor the full quadratic operator is of weak type $(1,1)$. In the positive direction, we establish an $L\log_1 L$ modular estimate for $\mathcal{C}_2$ and a $L (\log_2L)^2 \log_4L$ estimate for $\mathcal{C}_{2,\mathsf{lac}}.$
Let $d\geq 2$ and $T^*_{\Omega,b}$ be the commutator of the rough maximal singular integral on $\mathbb{R}^d$ defined by $$T^*_{\Omega,b}f(x)=\sup_{\varepsilon>0}\left|\int_{|x-y|>\varepsilon}\bigl(b(x)-b(y)\bigr)\frac{\Omega(x-y)}{|x-y|^d}f(y)\,dy\right|.$$ Assume that $\Omega\in L(\log L)^2(\mathbb{S}^{d-1})$ has mea...
We study regularity for the variable exponent quasilinear $\mathfrak{p}\left(x\right)$-Laplace type equation on domains in Heisenberg groups and Euclidean spaces. We establish borderline continuity estimates for the appropriate first order derivatives of solutions. More precisely, we show that for any weak solution $u...
We study the Stokes operator with no-slip boundary conditions on the spaces $\mathrm{L}_{\sigma,n}(\Omega)$ and ${\mathrm{L}^1(\Omega,\mathbb{C}^d)}/{\nabla \mathrm{W}^{1,1}(\Omega,\mathbb{C})}$, where $\Omega\subset\mathbb{R}^d$ is an arbitrary bounded $\mathrm{C}^{1,\alpha}$-domain. We show that the Stokes operator o...
Let $\mathcal{A}$ be the normalized class of analytic functions $f$ in the unit disc $\mathbb{D} := \{z \in \mathbb{C} : \vert{}z\vert{}<1\}$. For $\beta>1$, let $\mathcal{N}(\beta)$ denote the subclass of $\mathcal{A}$ satisfying $\text{Re}\{1 + z f''(z)/f'(z)\}<\beta$ for $z \in \mathbb{D}$. The main purpose of this...
Let $\lambda>-1/2$, $\lambda\neq0$, and let $\Delta_\lambda=-\frac{d^2}{dx^2}-\frac{2\lambda}{x}\frac{d}{dx}$ be the Bessel operator on $\mathbb R_+=(0,\infty)$ studied by Muckenhoupt and Stein (TAMS 1965). Andersen and Kerman (Studia Math. 1981) proved that for $1<p<\infty$, the Bessel Riesz transform $R_\lambda=\frac...
Let $\Omega\subset\mathbb{R}^d$, $d\ge2$, be a bounded connected domain with boundary of class $C^{1,\alpha}$, where $0<\alpha<1$. For the adjoint Neumann--Poincar\'e operator $K^*_{\partial\Omega}$, normalised so that its distinguished eigenvalue is $1/2$, let $\lambda_j^+(\Omega)$ denote the upper min--max values on...
Matthew J. Colbrook, Siavash Sadeghi· 0 citations
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