Related papers: Sharp $\ell^q(L^p)$ decoupling for paraboloids

An $L_{q}(L_{p})$-theory of divergence and non-divergence form parabolic equations is presented. The main coefficients are supposed to belong to the class $VMO_{x}$, which, in particular, contains all measurable functions depending only on…

Lipschitz decomposition is a useful tool in the design of efficient algorithms involving metric spaces. While many bounds are known for different families of finite metrics, the optimal parameters for $n$-point subsets of $\ell_p$, for $p >…

For domains in $\mathbb{R}^d$, $d\geq 2$, we prove universal upper and lower bounds on the product of the bottom of the spectrum for the Laplacian to the power $p>0$ and the supremum over all starting points of the $p$-moments of the exit…

In this paper, we establish an $\ell^2$ decoupling inequality for the hypersurface \[\Big\{(\xi_1,...,\xi_{n-1},\xi_1^m+...+\xi_{n-1}^m): (\xi_1,...,\xi_{n-1}) \in [0,1]^{n-1}\Big\}\]associated with the decomposition adapted to hypersufaces…

In this article, we establish an $\ell^2$ decoupling inequality for the surface $$F_4^2:=\Big\{(\xi_1,\xi_2,\xi_1^4+\xi_2^4): (\xi_1,\xi_2) \in [0,1]^2\Big\}$$ associated with the decomposition adapted to finite type geometry from our…

We prove the boundedness of the maximal operator and Hilbert transform along certain variable parabolas in $L^p$ for $p>p_0$ with some $p_0\in (1, 2)$. Connections with the Hilbert transform along vector fields and the polynomial Carleson's…

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 show weighted non-autonomous $L^q(L^p)$ maximal regularity for families of complex second-order systems in divergence form under a mixed regularity condition in space and time. To be more precise, we let $p,q \in (1,\infty)$ and we…

We show that, for every $1 \leq p < +\infty$ and for every Borel probability measure $\mathbb{P}$ over $\mathbb{R}$, every element of $L^{p}(\mathbb{R}, \mathscr{B}_{\mathbb{R}}, \mathbb{P})$ is the $L^{p}$-limit of some sequence of bounded…

We prove the $W^{1,2}_{p}$-solvability of second order parabolic equations in nondivergence form in the whole space for $p\in (1,\infty)$. The leading coefficients are assumed to be measurable in one spatial direction and have vanishing…

In this paper we prove higher regularity for 2m-th order parabolic equations with general boundary conditions. This is a kind of maximal L_p-L_q regularity with differentiability, i.e. the main theorem is isomorphism between the solution…

An $L_{p}$-theory of divergence and non-divergence form elliptic and parabolic equations is presented. The main coefficients are supposed to belong to the class $VMO_{x}$, which, in particular, contains all functions independent of $x$.…

We study the homogeneous elliptic systems of order $2\ell$ with real constant coefficients on Lipschitz domains in $R^n$, $n\ge 4$. For any fixed $p>2$, we show that a reverse H\"older condition with exponent $p$ is necessary and sufficient…

We prove the Lp,q-solvability of parabolic equations in divergence form with full lower-order terms. The coefficients and non-homogeneous terms belong to mixed Lebesgue spaces with the lowest integrability conditions. In particular, the…

We give a short linear--algebraic proof of the inequality \[ \|x\|_1\,\|x\|_\infty \le \frac{1+\sqrt{p}}{2}\,\|x\|_2^2, \] valid for every \(x\in\mathbb{R}^p\). This inequality relates three fundamental norms on finite-dimensional spaces…

We introduce the notion of decoupling for operators, and prove an equivalence between classical $\ell^qL^p$ decoupling for functions and $\ell^q\mathcal{S}^p$ decoupling for operators on bounded sets in $\mathbb{R}^{2d}$. We also show that…

We obtain $L^p(L^q)$ maximal regularity estimates for time dependent second order elliptic operators in divergence form with rough dependencies in the spatial variables.

In this article, we aim to study decoupling inequality for a specific degenerate hypersurface in $\mathbb{R}^4$. Inspired by the work of Bourgain--Demeter and Li--Zheng, we consider the hypersurface…

We establish the $L^p(\mathbb{R}^3)$ boundedness of the helical maximal function for the sharp range $p>3$. Our results improve the previous known bounds for $p>4$. The key ingredient is a new microlocal smoothing estimate for averages…

We prove two types of results. First we develop the decoupling theory for hypersurfaces with nonzero Gaussian curvature, which extends our earlier work from \cite{BD3}. As a consequence of this we obtain sharp (up to $\epsilon$ losses)…