The purpose of this paper is to study the smoothing properties (in $L^p$ Sobolev spaces) of operators of the form $f\mapsto ψ(x) \int f(γ_t(x)) K(t)\: dt$, where $γ_t(x)$ is a $C^\infty$ function defined on a neighborhood of the origin in $(t,x)\in\mathbb{R}^N\times \mathbb{R}^n$, satisfying $γ_0(x)\equiv x$, $ψ$ is a $C^\infty$ cut-off function supported on a small neighborhood of $0\in \mathbb{R}^n$, and $K$ is a "multi-parameter fractional kernel" supported on a small neighborhood of $0\in \mathbb{R}^N$. When $K$ is a Calderón-Zygmund kernel these operators were studied by Christ, Nagel, Stein, and Wainger, and when $K$ is a multi-parameter singular kernel they were studied by the author and Stein. In both of these situations, conditions on $γ$ were given under which the above operator is bounded on $L^p$ ($1<p<\infty$). Under these same conditions, we introduce non-isotropic $L^p$ Sobolev spaces associated to $γ$. Furthermore, when $K$ is a fractional kernel which is smoothing of an order which is close to $0$ (i.e., very close to a singular kernel) we prove mapping properties of the above operators on these non-isotropic Sobolev spaces. As a corollary, under the conditions introduced on $γ$ by Christ, Nagel, Stein, and Wainger, we prove optimal smoothing properties in isotropic $L^p$ Sobolev spaces for the above operator when $K$ is a fractional kernel which is smoothing of very low order.
In the paper [G1] the author proved $L^p$ Sobolev regularity results for averaging operators over hypersurfaces and connected them to associated Newton polyhedra. In this paper, we use rather different resolution of singularities techniques to prove $L^p$ Sobolev regularity results for a class of averaging operators over surfaces which can be of any codimension.
We extend the theorems of [G1] on $L^p$ to $L^p_s$ Sobolev improvement for translation invariant Radon and fractional singular Radon transforms over hypersurfaces, proving $L^p$ to $L^q_s$ boundedness results for such operators. Here $q \geq p$ but $s$ can be positive, negative, or zero. For many such operators we will have a triangle $Z \subset (0,1) \times (0,1) \times {\mathbb R} $ such that one has $L^p$ to $L^q_{s}$ boundedness for $({1 \over p}, {1 \over q}, s)$ beneath $Z$, and in the case of Radon transforms one does not have $L^p$ to $L^q_{s}$ boundedness for $({1 \over p}, {1 \over q…
In this paper, we discuss the trace operator for homogeneous fractional Sobolev spaces over infinite strip-like domains. We determine intrinsic seminorms on the trace space that allow for a bounded right inverse. The intrinsic seminorm includes two features previously used to describe the trace of homogeneous Sobolev spaces, a relation between the two disconnected components of the trace and the screened Sobolev seminorm. However, unlike its homogeneous Sobolev space equivalent, fractional Sobolev spaces require a far screened Sobolev seminorm that captures the non-local properties of fraction…
We obtain improved fractional Poincaré and Sobolev Poincaré inequalities including powers of the distance to the boundary in John, $s$-John domains and Hölder-$α$ domains, and discuss their optimality.
This paper may be viewed as a companion paper to [G1]. In that paper, $L^2$ Sobolev estimates derived from a Newton polyhedron-based resolution of singularities method are combined with interpolation arguments to prove $L^p$ to $L^q_s$ estimates, some sharp up to endpoints, for translation invariant Radon transforms over hypersurfaces and related operators. Here $q \geq p$ and $s$ can be positive, negative, or zero. In this paper, we instead use $L^2$ Sobolev estimates derived from the resolution of singularities methods of [G2] and combine with analogous interpolation arguments, again resulti…
We prove a sharp $L^p$-Sobolev regularity results for a class of generalized Radon transforms for families of curves in a three dimensional manifold, with folding canonical relations. The proof relies on decoupling inequalities by Wolff and by Bourgain-Demeter for plate decompositions of thin neighborhoods of cones.
Our main purpose is to establish Gagliardo-Nirenberg type inequalities using fractional homogeneous Sobolev spaces, and homogeneous Besov spaces. In particular, we extend some of the results obtained by the authors in [1, 2, 3, 7, 16, 21].
We prove a certain improved fractional Sobolev-Poincaré inequality on John domains; the proof is based on the equivalence of the corresponding weak and strong type inequalities. We also give necessary conditions for the validity of an improved fractional Sobolev-Poincaré inequality, in particular, we show that a domain having a finite measure and satisfying this inequality, and a `separation property', is a John domain.
We establish a mixed norm estimate for the Radon transform in the plane when the set of directions has fractional dimension. This estimate is used to prove a result about an exceptional set of directions connected with projections of planar sets. That leads to a conjecture analogous to a well-known conjecture of Furstenberg.
We characterize the real interpolation space between a weighted $L^p$ space and a weighted Sobolev space in arbitrary bounded domains in $\mathbb{R}^n$, with weights that are positive powers of the distance to the boundary.
We study the fractional Laplacian and the homogeneous Sobolev spaces on R^d , by considering two definitions that are both considered classical. We compare these different definitions, and show how they are related by providing an explicit correspondence between these two spaces, and show that they admit the same representation. Along the way we also prove some properties of the fractional Laplacian.
We define a parametric Radon transform $R$ that assigns to a Sobolev function on the cylinder $\mathbb{S}\times \mathbb{R}$ in $\mathbb{R}^3$ its mean values along sets $E_ζ$ formed by the intersections of planes through the origin and the cylinder. We show that $R$ is a continuous operator, prove an inversion formula, provide a support theorem, as well as a characterization of its null space. We conclude by presenting a formula for the dual transform $R^*$. We show that $R$ and its dual $R^*$ are related to the right-sided and left-sided Chebyshev fractional integrals. Using this relationship…
$L^p$ to $L^p_β$ boundedness theorems are proven for translation invariant averaging operators over hypersurfaces in Euclidean space. The operators can either be Radon transforms or averaging operators with multiparameter fractional integral kernel. In many cases, the amount $β> 0 $ of smoothing proven is optimal up to endpoints, and in such situations this amount of smoothing can be computed explicitly through the use of appropriate Newton polyhedra.
Taking inspiration from a recent paper by Bergounioux, Leaci, Nardi and Tomarelli we study the Riemann-Liouville fractional Sobolev space $W^{s, p}_{RL, a+}(I)$, for $I = (a, b)$ for some $a, b \in \mathbb{R}, a < b$, $s \in (0, 1)$ and $p \in [1, \infty]$; that is, the space of functions $u \in L^{p}(I)$ such that the left Riemann-Liouville $(1 - s)$-fractional integral $I_{a+}^{1 - s}[u]$ belongs to $W^{1, p}(I)$. We prove that the space of functions of bounded variation and the fractional Sobolev space, $BV(I)$ and $W^{s, 1}(I)$, continuously embed into $W^{s, 1}_{RL, a+}(I)$. In addition, …
We consider discrete analogues of fractional Radon transforms involving integration over paraboloids defined by positive definite quadratic forms. We prove that such discrete operators extend to bounded operators from $\ell^p$ to $\ell^q$ for a certain family of kernels. The method involves an intricate spectral decomposition according to major and minor arcs, motivated by ideas from the circle method of Hardy and Littlewood. Techniques from harmonic analysis, in particular Fourier transform methods and oscillatory integrals, as well as the number theoretic structure of quadratic forms, expone…
We characterize the fractional Sobolev inequality with fractional isocapacitary and isoperimetric inequalities. We give a sufficient condition and examples so that the fractional capacity of the closure of an open set is bounded above by the fractional perimeter of its interior.
As our main result we prove a variant of the fractional Hardy-Sobolev-Maz'ya inequality for half spaces. This result contains a complete answer to a recent open question by Musina and Nazarov. In the proof we apply a new version of the fractional Hardy-Sobolev inequality that we establish also for more general unbounded John domains than half spaces.
In this paper, we prove the Brezis-Gallouet-Wainger type inequality involving the BMO norm, the fractional Sobolev norm, and the logarithmic norm of $\mathcal{\dot{C}}^η$, for $η\in(0,1)$.
Let (X, \r{ho},μ) be a space of homogeneous type, a variable exponent satisfying the globally log-Holder continuous condition. In this article, the author introduce the variable fractional Sobolev spaces on X via Hajłasz gradient. Using various maximal functions, several characterizations of this space are established.
This paper presents a self-contained new theory of weak fractional differential calculus and fractional Sobolev spaces in one-dimension. The crux of this new theory is the introduction of a weak fractional derivative notion which is a natural generalization of integer order weak derivatives; it also helps to unify multiple existing fractional derivative definitions and characterize what functions are fractionally differentiable. Various calculus rules including a fundamental theorem of calculus, product and chain rules, and integration by parts formulas are established for weak fractional deri…
We prove existence and uniqueness of mild solutions to Sobolev type fractional nonlocal dynamic equations in Banach spaces. The Sobolev nonlocal condition is considered in terms of a Riemann-Liouville fractional derivative. A Lagrange optimal control problem is considered, and existence of a multi-integral solution obtained. Main tools include fractional calculus, semigroup theory, fractional power of operators, a singular version of Gronwall's inequality, and Leray-Schauder fixed point theorem. An example illustrating the theory is given.
We prove fractional Hardy--Sobolev--Maz'ya inequality for balls and a half-space, partially answering the open problem posed by Frank and Seiringer [arXiv:0906.1561v1 [math.FA], 2009] We note that for half-spaces this inequality has been recently obtained by Sloane [arXiv:1004.4828v1 [math.FA], 2010].
In this note we establish the boundedness properties of local maximal operators $M_G$ on the fractional Sobolev spaces $W^{s,p}(G)$ whenever $G$ is an open set in $\mathbb{R}^n$, $0<s<1$ and $1<p<\infty$. As an application, we characterize the fractional $(s,p)$-Hardy inequality on a bounded open set $G$ by a Maz'ya-type testing condition localized to Whitney cubes.
We prove generalizations of the Poincare and logarithmic Sobolev inequalities corresponding to the case of fractional derivatives in measure spaces with only a minimal amount of geometric structure. The class of such spaces includes (but is not limited to) spaces of homogeneous type with doubling measures. Several examples and applications are given, including Poincare inequalities for graph Laplacians, Fractional Poincare inequalities of Mouhot, Russ, and Sire [16], and implications for recent work of the author and R. M. Strain on the Boltzmann collision operator [10, 11, 9].
We first prove the equivalence of two definitions of Riemann-Liouville fractional integral on time scales, then by the concept of fractional derivative of Riemann-Liouville on time scales, we introduce fractional Sobolev spaces, characterize them, define weak fractional derivatives, and show that they coincide with the Riemann-Liouville ones on time scales. Next, we prove equivalence of some norms in the introduced spaces and derive their completeness, reflexivity, separability and some imbeddings. Finally, as an application, by constructing an appropriate variational setting, using the mounta…
We characterize one-sided weighted Sobolev spaces $W^{1,p}(\mathbb{R},ω)$, where $ω$ is a one-sided Sawyer weight, in terms of a.e.~and weighted $L^p$ limits as $α\to1^-$ of Marchaud fractional derivatives of order $α$. Similar results for weighted Sobolev spaces $W^{2,p}(\mathbb{R}^n,ν)$, where $ν$ is an $A_p$-Muckenhoupt weight, are proved in terms of limits as $s\to1^-$ of fractional Laplacians $(-Δ)^s$. These are Bourgain--Brezis--Mironescu-type characterizations for weighted Sobolev spaces. We also complement their work by studying a.e.~and weighted $L^p$ limits as $α,s\to0^+$.
The purpose of this article is twofold. The first is to strengthen fractional Sobolev type inequalities in Besov spaces via the classical Lorentz space. In doing so, we show that the Sobolev inequality in Besov spaces is equivalent to the fractional Hardy inequality and the iso-capacitary type inequality. Secondly, we will strengthen fractional Sobolev type inequalities in Besov spaces via capacitary Lorentz spaces associated with Besov capacities. For this purpose, we first study the embedding of the associated capacitary Lorentz space to the classical Lorentz space. Then, the embedding of th…
This book provides a gentle introduction to fractional Sobolev spaces, which play a central role in the calculus of variations, partial differential equations, and harmonic analysis. The first part deals with fractional Sobolev spaces of one variable. It covers the definition, standard properties, extensions, embeddings, Hardy inequalities, and interpolation inequalities. The second part deals with fractional Sobolev spaces of several variables. The author studies completeness, density, homogeneous fractional Sobolev spaces, embeddings, necessary and sufficient conditions for extensions, Gagli…
In the cases where there is no Sobolev-type or Gagliardo-Nirenberg-type fractional estimate involving $\lvert u\rvert_{W^{s,p}}$, we establish alternative estimates where the strong $L^p$ norms are replaced by Lorentz norms.
Galerkin Approximation of the Fractional Sobolev Constant
Simion Stoilow Institute of Mathematics of the Romanian Academy（罗马尼亚科学院西奥多·斯托伊洛瓦数学研究所）；National University of Science and Technology Politehnica Bucharest（布加勒斯特理工大学）；Academy of Romanian Scientists（罗马尼亚科学家学院）
