Related papers: On $L_p$-estimates for elliptic and parabolic equa…

We derive a convergent expansion of the generalized hypergeometric function ${}_{p-1}F_p$ in terms of the Bessel functions ${}_{0}F_1$ that holds uniformly with respect to the argument in any horizontal strip of the complex plane. We…

We provide sparse estimates for gradients of solutions to divergence form elliptic partial differential equations in terms of the source data. We give a general result of Meyers (or Gehring) type, a result for linear equations with VMO…

We establish a refined $L_p$-estimate ($p\geq 2$) for the stochastic heat equation on angular domains in $\mathbb{R}^2$ with mixed weights based on both, the distance to the boundary and the distance to the vertex. This way we can capture…

The main purpose of this paper is to capture the asymptotic behavior for solutions to a class of nonlinear elliptic and parabolic equations with the anisotropic weights consisting of two power-type weights of different dimensions near the…

We establish the unique solvability of solutions in Sobolev spaces to linear parabolic equations in a more general form than those in the literature. A distinguishing feature of our equations is the inclusion of a half-order time derivative…

In this paper we study the existence and partial regularity of weak solutions to an elliptic-parabolic system that models the single-phase miscible displacement of one incompressible fluid by another in a porous media. The system is…

We consider the weak-type inequality for Littlewood-Paley square functions on A_p weighted Lebesgue spaces. Of interest is the sharp in the A_p characteristic estimate. The case of 1<p<2 is subcritical, and the sharp power of 1/p is…

We study both divergence and non-divergence form parabolic and elliptic equations in the half space $\{x_d>0\}$ whose coefficients are the product of $x_d^\alpha$ and uniformly nondegenerate bounded measurable matrix-valued functions, where…

In this paper we establish sharp weighted bounds (Buckley type theorems) for one{sided maximal and fractional integral operators in terms of one{sided $A_p$ characteristics. Appropriate sharp bounds for strong maximal functions, multiple…

In this paper, we prove sharp estimates and existence results for anisotropic nonlinear elliptic problems with lower order terms depending on the gradient. Our prototype is: $ \left\{ \begin{array}{ll} -\mathcal Q_{p}u =[H(Du)]^{q}+f(x)…

In this paper, we study parabolic equations in divergence form with coefficients that are singular degenerate as some Muckenhoupt weight functions in one spatial variable. Under certain conditions, weighted reverse H\"{o}lder's inequalities…

We give an overview of the generalized Calder\'on-Zygmund theory for "non-integral" singular operators, that is, operators without kernels bounds but appropriate off-diagonal estimates. This theory is powerful enough to obtain weighted…

We prove sharp weak and strong type weighted estimates for a class of dyadic operators that includes majorants of both standard singular integrals and square functions. Our main new result is the optimal bound…

This paper studies the expected $L_p$-discrepancy ($2 \leq p < \infty$) for stratified sampling schemes under importance sampling. We introduce a parametric family of equivolume partitions $\Omega_{\theta,\sim}$ and leverage recent exact…

We prove a local version of Fefferman-Stein inequality for the local sharp maximal function, and a local version of John-Nirenberg inequality for locally BMO functions, in the framework of locally homogeneous spaces, in the sense of…

We prove $L^p$-parabolic a-priori estimates for $\partial_t u + \sum_{i,j=1}^d c_{ij}(t)\partial_{x_i x_j}^2 u = f $ on $R^{d+1}$ when the coefficients $c_{ij}$ are locally bounded functions on $R$. We slightly generalize the usual…

We study elliptic and parabolic systems in divergence form with degenerate or singular coefficients. Under the conormal boundary condition on the flat boundary, we establish boundary Schauder type estimates when the coefficients have…

A representation of the sharp coefficient in a pointwise estimate for the gradient of the generalized Poisson integral of a function $f$ on ${\mathbb R}^n$ is obtained under the assumption that $f$ belongs to $L^p$. The explicit value of…

We show $L^p$ estimates for square roots of second order complex elliptic systems $L$ in divergence form on open sets in $\mathbb{R}^d$ subject to mixed boundary conditions. The underlying set is supposed to be locally uniform near the…

We discuss $L_p$-estimates for finite difference schemes approximating parabolic, possibly degenerate, SPDEs, with initial conditions from $W^m_p$ and free terms taking values in $W^m_p.$ Consequences of these estimates include an…