Based on the theory of variable exponent spaces, we study the regularity of local minimizers for a class of functionals with variable growth and discontinuous coefficients. Under suitable assumptions, we obtain local Hölder continuity of minimizers.

In this paper we prove that a Gaussian white noise on the d-dimensional torus has paths in the Besov spaces $B_{p,\infty }^{-d/2}{(}^d)$ with p ∈ [1,∞). This result is shown to be optimal in several ways. We also show that Gaussian white noise on the d-dimensional torus has paths in the Fourier-Besov space $b{̂}_{p,\infty }^{-d/p}{(}^d)$. This is shown to be optimal as well.

In this paper, the authors introduce a kind of local Hardy spaces in Rn associated with the local Herz spaces. Then the authors investigate the regularity in these local Hardy spaces of some nonlinear quantities on superharmonic functions on R2. The main results of the authors extend the corresponding results of Evans and Müller in a recent paper.

We prove that the exit times of diffusion processes from a bounded open set Ω almost surely belong to the Besov space $B_{p,q}^{\alpha }\left(\Omega \right)$ provided that pα < 1 and 1 ≤ q < ∞.

In this paper we consider the regularity problem for the commutators $\left(\right[b,R_k]{)}_{1\le k\le n}$ where $b$ is a locally integrable function and $(R_j{)}_{1\le j\le n}$ are the Riesz transforms in the $n$-dimensional euclidean space ${ℝ}^n$. More precisely, we prove that these commutators $\left(\right[b,R_k]{)}_{1\le k\le n}$ are bounded from $L^p$ into the Besov space ${\dot{B}}_p^{s,p}$ for $1\&lt;p\&lt;+\infty$ and $0\&lt;s\&lt;1$ if and only if $b$ is in the $BMO$-Triebel-Lizorkin space ${\dot{F}}_{\infty }^{s,p}$. The reduction of our result to the case $p=2$ gives in particular that the commutators $\left(\right[b,R_k]{)}_{1\le k\le n}$ are bounded form $L^2$ into the Sobolev space ${\dot{H}}^s$ if and only if $b$ is in the $BMO$-Sobolev...

For s>0, we consider bounded linear operators from $D\left({ℝ}^n\right)$ into $D^{\text{'}}\left({ℝ}^n\right)$ whose kernels K satisfy the conditions $|{\partial }_x^{\gamma }K(x,y)|\le C_{\gamma }{|x-y|}^{-n+s-\left|\gamma \right|}$ for x≠y, |γ|≤ [s]+1, $|{\nabla }_y{\partial }_x^{\gamma }K(x,y)|\le C_{\gamma }{|x-y|}^{-n+s-\left|\gamma \right|-1}$ for |γ|=[s], x≠y. We establish a new criterion for the boundedness of these operators from $L^2\left({ℝ}^n\right)$ into the homogeneous Sobolev space ${Ḣ}^s\left({ℝ}^n\right)$. This is an extension of the well-known T(1) Theorem due to David and Journé. Our arguments make use of the function T(1) and the BMO-Sobolev space. We give some applications to the Besov and Triebel-Lizorkin spaces as well as some other potential...

In this paper the Dirichlet problem for a linear elliptic equation in an open, bounded subset of ${\mathbb{R}}^2$ is studied. Regularity properties of the solutions are proved, when the data are $L^1$-functions or Radon measures. In particular sharp assumptions which guarantee the continuity of solutions are given.