Related papers: Characterization of Sobolev spaces on the sphere

The purpose of this paper is to characterize all the entire solutions of the homogeneous Helmholtz equation (solutions in $\mathbb{R}^d$) arising from the Fourier extension operator of distributions in Sobolev spaces of the sphere…

We establish characterization of $H^1$ Sobolev spaces by certain square functions, improving previous results.

In this paper, the authors characterize Sobolev spaces $W^{\alpha,p}({\mathbb R}^n)$ with the smoothness order $\alpha\in(0,2]$ and $p\in(\max\{1, \frac{2n}{2\alpha+n}\},\infty)$, via the Lusin area function and the Littlewood-Paley…

We present a new characterization of higher-order Sobolev spaces on the sphere. Building on the approach of Barcel\'o et al. (2020), we refine the square function they introduced for this purpose. In particular, we provide a detailed…

We prove a characterization of some $L^p$-Sobolev spaces involving the quadratic symmetrization of the Calder\'on commutator kernel, which is related to a square function with differences of difference quotients. An endpoint weak type…

In this paper we introduce a polynomial frame on the unit sphere $\sph$ of $\mathbb{R}^d$, for which every distribution has a wavelet-type decomposition. More importantly, we prove that many function spaces on the sphere $\sph$, such as…

In this paper, motivated by the analysis of the fractional Laplace equation on the unit disk in $\mathbb{R}^{2}$, we establish a characterization of the weighted Sobolev space $H_{\beta}^{s}(\Omega)$ in terms of the decay rate of…

Let $p\in(1,\infty)$ and $q\in[1,\infty)$. In this article, the authors characterize the Triebel-Lizorkin space ${F}^\alpha_{p,q}(\mathbb{R}^n)$ with smoothness order $\alpha\in(0,2)$ via the Lusin-area function and the…

In this work we investigate the Sobolev space $\mathrm{H}^{1}(\partial\Omega)$ on a strong Lipschitz boundary $\partial\Omega$, i.e., $\Omega$ is a strong Lipschitz domain. In most of the literature this space is defined via charts and…

We provide new characterizations of the $BMO$-Sobolev space $I_{\alpha}(BMO)$ for the range $0 < \alpha <2$. When $0 < \alpha <1$, our characterizations are in terms of square functions measuring multiscale approximation of constants, and…

Let $S^2$ be the 2-dimensional unit sphere and let $J_\alpha $ denote the nonlinear functional on the Sobolev space $H^{1,2}(S^2)$ defined by $$ J_\alpha(u) = \frac{\alpha}{4}\int_{S^2}|\nabla u|^2 d\omega + \int_{S^2} u d\omega -\ln…

We study the estimation of quadratic Sobolev-type integral functionals of an unknown density on the unit sphere. The functional is defined through fractional powers of the Laplace--Beltrami operator and provides a global measure of…

Let $\Omega\subset\mathbb{C}^n$ be a bounded domain with smooth boundary, whose Bergman projection $B$ maps the Sobolev space $H^{k_{1}}(\Omega)$ (continuously) into $H^{k_{2}}(\Omega)$. We establish two smoothing results: (i) the full…

In this paper we present a new characterization of Sobolev spaces on Euclidian spaces ($\mathbb{R}^n$). Our characterizing condition is obtained via a quadratic multiscale expression which exploits the particular symmetry properties of…

Fractional Sobolev spaces $\widehat{H}^s(\mathbb{R})$ have been playing important roles in analysis of many mathematical subjects. In this work, we re-consider fractional Sobolev spaces under the perspective of fractional operators and…

It is well-known that the embedding of the Sobolev space of weakly differentiable functions into H\"{o}lder spaces holds if the integrability exponent is higher than the space dimension. In this paper, the embedding of the Sobolev functions…

A new representation is proposed for functions in a Sobolev space with dominating mixed smoothness on an $N$-dimensional hyperrectangle. In particular, it is shown that these functions can be expressed in terms of their highest-order mixed…

Motivated by an open question going back to P.Malliavin and P.-A.Meyer (and closely related to the foundational work of S.Watanabe) on whether Malliavin-Watanabe-Sobolev regularity admits a characterization in terms of a holomorphic Laplace…

This paper considers filtered polynomial approximations on the unit sphere $\mathbb{S}^d\subset \mathbb{R}^{d+1}$, obtained by truncating smoothly the Fourier series of an integrable function $f$ with the help of a "filter" $h$, which is a…

In this paper we study reproducing kernel Hilbert spaces of arbitrary smoothness on the sphere $\mathbb{S}^d \subset \mathbb{R}^{d+1}$. The reproducing kernel is given by an integral representation using the truncated power function…