Related papers: Boundary Smoothness conditions for functions in $R…

The following result was announced in the earlier version(s) of this paper: On weakly compactly generated Banach spaces which admit a Lipschitz, C^{p} smooth bump function, one can uniformly approximate uniformly continuous, bounded,…

Numerical solutions of differential equations are usually not smooth functions. However, they should resemble the smoothness of the corresponding real solutions in one way or another. In two of our recent papers, a kind of spacial…

Let $M$ be the maximal operator associated to a smooth curve in $\mathbb R^3$ which has nonvanishing curvature and torsion. We prove that $M$ is bounded on $L^p$ if and only if $p>3$.

We consider second order elliptic operators with real, nonsymmetric coefficient functions which are subject to mixed boundary conditions. The aim of this paper is to provide uniform resolvent estimates for the realizations of these…

We consider nonhomogeneous fractional $p$-Laplace equations defined on a bounded nonsmooth domain which goes beyond the Lipschitz category. Under a sufficient flatness assumption on the domain in the sense of Reifenberg, we establish…

Let $p$ be a real number greater than one and let $X$ be a locally compact, noncompact metric measure space that satisfies certain conditions. The $p$-Royden and $p$-harmonic boundaries of $X$ are constructed by using the $p$-Royden algebra…

We discuss the smoothness and strict convexity of the solution of the $L_p$ Minkowski problem when $p<1$ and the given measure has a positive density function.

To verify the universal validity of the "two-sided" monotonicity condition introduced in [8], we will apply it to include more classical examples. The present paper selects the $L^{p}$ convergence case for this purpose. Furthermore, Theorem…

Various new sufficient conditions for representation of a function of several variables as an absolutely convergent Fourier integral are obtained in the paper. The results are given in terms of $L^p$ integrability of the function and its…

We prove $L^p$ boundedness results, $p > 2$, for local maximal averaging operators over a smooth 2D hypersurface $S$ with either a $C^1$ density function or a density function with a singularity that grows as $|(x,y)|^{-\beta}$ for $\beta <…

In this paper we consider the Laplace operator with Dirichlet boundary conditions on a smooth domain. We prove that it has a bounded $H^\infty$-calculus on weighted $L^p$-spaces for power weights which fall outside the classical class of…

The article considers the Lorentz space $L_{p,\tau}(\mathbb{T}^{m})$, $2\pi$ of periodic functions of many variables and $S_{p,\tau,\theta}^{0, \overline{b}}\mathbf{B}$, $S_{p, \tau, \theta}^{0, \overline{b}}B$ -- spaces of functions with…

The completeness on the boundary (in the sense of Picone) of certain systems related to the III and IV BVPs for the elasticity system is proved. The completeness is obtained in both $L^p$ ($1\leq 1<\infty$) and uniform norms.

The condition number of a smooth convex function, namely the ratio of its smoothness to strong convexity constants, is closely tied to fundamental properties of the function. In particular, the condition number of a quadratic convex…

We study the boundedness problem for maximal operators $\M$ associated to smooth hypersurfaces $S$ in 3-dimensional Euclidean space. For $p>2,$ we prove that if no affine tangent plane to $S$ passes through the origin and $S$ is analytic,…

We describe some sufficient conditions, under which smooth and compactly supported functions are or are not dense in the fractional Sobolev space $W^{s,p}(\Omega)$ for an open, bounded set $\Omega\subset\mathbb{R}^{d}$. The density property…

We prove almost sharp upper bounds for the $L^p$ norms of eigenfunctions of the full ring of invariant differential operators on a compact locally symmetric space, as well as their restrictions to maximal flat subspaces. Our proof combines…

We consider generic families $X_\param$ of smooth dynamical systems depending on parameters $\param\in P$ where $P$ is a 2-dimensional simply connected domain and assume that each $X_\param$ only has a finite number of restpoints and…

We consider the design of smoothings of the (coordinate-wise) max function in $\mathbb{R}^d$ in the infinity norm. The LogSumExp function $f(x)=\ln(\sum^d_i\exp(x_i))$ provides a classical smoothing, differing from the max function in value…

We study smoothness of generalized solutions of nonlocal elliptic problems in plane bounded domains with piecewise smooth boundary. The case where the support of nonlocal terms can intersect the boundary is considered. We find conditions…