We consider the Sobolev (Bessel potential) spaces H^ell(R^d, C), and their standard norms || ||_ell (with ell integer or noninteger). We are interested in the unknown sharp constant K_{ell m n d} in the inequality || f g ||_{ell} \leqs K_{ell m n d} || f ||_{m} || g ||_n (f in H^m(R^d, C), g in H^n(R^d, C); 0 <= ell <= m <= n, m + n - ell > d/2); we derive upper and lower bounds K^{+}_{ell m n d}, K^{-}_{ell m n d} for this constant. As examples, we give a table of these bounds for d=1, d=3 and many values of (ell, m, n); here the ratio K^{-}_{ell m n d}/K^{+}_{ell m n d} ranges between 0.75 and 1 (being often near 0.90, or larger), a fact indicating that the bounds are close to the sharp constant. Finally, we discuss the asymptotic behavior of the upper and lower bounds for K_{ell, b ell, c ell, d} when 1 <= b <= c and ell -> + Infinity. As an example, from this analysis we obtain the ell -> + Infinity limiting behavior of the sharp constant K_{ell, 2 ell, 2 ell, d}; a second example concerns the ell -> + Infinity limit for K_{ell, 2 ell, 3 ell, d}. The present work generalizes our previous paper [16], entirely devoted to the constant K_{ell m n d} in the special case ell = m = n; many results given therein can be recovered here for this special case.
Following Sarason's classification of the densely defined multiplication operators over the Hardy space, we classify the densely defined multipliers over the Sobolev space, $W^{1,2}[0,1]$. In this paper we find that the collection of such multipliers for the Sobolev space is exactly the Sobolev space itself. This sharpens a result of Shields concerning bounded multipliers. The densely defined multiplication operators over the subspace $W_0 = \{f \in W^{1,2}[0,1] : f(0)=f(1)=0 \}$ are also classified. In this case the densely defined multiplication operators can be written as a ratio of functio…
For a pointwise multiplier $φ$ of the Hardy-Sobolev space $H^2_β$ on the open unit ball $\bn$ in $\cn$, we study spectral properties of the multiplication operator $M_φ: H^2_β\to H^2_β$. In particular, we compute the spectrum and essential spectrum of $M_φ$ and develop the Fredholm theory for these operators.
We study the operator $\mathcal{A}$ of multiplication by an independent variable in a matrix Sobolev space $W^2(M)$. In the cases of finite measures on $[a,b]$ with $(2\times 2)$ and $(3\times 3)$ real continuous matrix weights of full rank it is shown that the operator $\mathcal{A}$ is symmetrizable. Namely, there exist two symmetric operators $\mathcal{B}$ and $\mathcal{C}$ in a larger space such that $\mathcal{A} f = \mathcal{C} \mathcal{B}^{-1} f$, $f\in D(\mathcal{A})$. As a corollary, we obtain some new orthogonality conditions for the associated Sobolev orthogonal polynomials. These con…
We show that every vector lattice homomorphism $T$ between Sobolev spaces can be represented by a composition and a multiplication, that is, $T$ is of the form $Tu(x)=u(h(x))g(x)$ for quasi every/almost every $x$ and all $u$.
We investigate some multiplication properties of Kato-Sobolev spaces by adapting the techniques used in the study of Beurling algebras by Coifman and Meyer. We develop an analytic functional calculus for Kato-Sobolev algebras based on an integral representation formula belonging A. P. Calderón. Also we study the Schatten-von Neumann properties of pseudo-differential operators with symbols in the Kato-Sobolev spaces.
In this paper we obtain some practical criteria to bound the multiplication operator in Sobolev spaces with respect to measures in curves. As a consequence of these results, we characterize the weighted Sobolev spaces with bounded multiplication operator, for a large class of weights. To have bounded multiplication operator has important consequences in Approximation Theory: it implies the uniform bound of the zeros of the corresponding Sobolev orthogonal polynomials, and this fact allows to obtain the asymptotic behavior of Sobolev orthogonal polynomials. We also obtain some non-trivial resul…
For n > d/2, the Sobolev (Bessel potential) space H^n(R^d, C) is known to be a Banach algebra with its standard norm || ||_n and the pointwise product; so, there is a best constant K_{n d} such that || f g ||_{n} <= K_{n d} || f ||_{n} || g ||_{n} for all f, g in this space. In this paper we derive upper and lower bounds for these constants, for any dimension d and any (possibly noninteger) n > d/2. Our analysis also includes the limit cases n -> (d/2) and n -> + Infinity, for which asymptotic formulas are presented. Both in these limit cases and for intermediate values of n, the lower bounds …
We investigate some multiplication properties of Kato-Sobolev spaces by adapting the techniques used in the study of Beurling algebras by Coifman and Meyer. Also we develop an analytic functional calculus for Kato-Sobolev algebras based on an integral representation formula belonging A. P. Calderón.
The question is addressed of when a Sobolev type space, built upon a general rearrangement-invariant norm, on an $n$-dimensional domain, is a Banach algebra under pointwise multiplication of functions. A sharp balance condition among the order of the Sobolev space, the strength of the norm, and the (ir)regularity of the domain is provided for the relevant Sobolev space to be a Banach algebra. The regularity of the domain is described in terms of its isoperimetric function. Related results on the boundedness of the multiplication operator into lower-order Sobolev type spaces are also establishe…
Under certain restrictions we describe the set of all pointwise multipliers in case of Sobolev and Besov spaces of dominating mixed smoothness. In addition we shall give necessary and sufficient conditions for the case that these spaces form algebras with respect to pointwise multiplication.
Banach Hardy-Sobolev Spaces on the Upper Half-plane and Operator Theory
College of Mathematics and Informatics, South China Agricultural University（华南农业大学数学与信息学院）；Faculty of Innovation Engineering, Macau University of Science and Technology（澳门科技大学创新工程学院）
We derive sharp Sobolev inequalities for Sobolev spaces on metric spaces. In particular, we obtain new sharp Sobolev embeddings and Faber-Krahn estimates for Hörmander vector fields.
Using Gutzmer's formula, due to Lassalle, we characterise the image of Sobolev spaces under the Segal-Bargmann transform on compact Riemannian symmetric spaces.
The inclusion relations between the $L^p$-Sobolev spaces and the modulation spaces is determined explicitly. As an application, mapping properties of unimodular Fourier multiplier $e^{i|D|^α}$ between $L^p$-Sobolev spaces and modulation spaces are discussed.
Holomorphic removability is of importance in dynanical systems theory. P.W.Jones found that Sobolev removability can serve as a substitute. He proved a strong strong result for Hölder domains. The present work gives some results on the this and related matters.
The kernel of composition operator $C_T$ on Orlicz-Sobolev space is obtained. Using the kernel, a necessary and a sufficient condition for injectivity of composition operator $C_T$ has been established. Composition operators on Orlicz-Sobolev space with finite ascent as well as infinite ascent have been characterized.
Sobolev wavefront sets and $2$-microlocal spaces play a key role in describing and analyzing the singularities of distributions in microlocal analysis and solutions of partial differential equations. Employing the continuous shearlet transform to Sobolev spaces, in this paper we characterize the microlocal Sobolev wavefront sets, the $2$-microlocal spaces, and local Hölder spaces of distributions/functions. We then establish the connections among Sobolev wavefront sets, $2$-microlocal spaces, and local Hölder spaces through the continuous shearlet transform.
We consider a homogeneous fractional Sobolev space obtained by completion of the space of smooth test functions, with respect to a Sobolev--Slobodecki\uı norm. We compare it to the fractional Sobolev space obtained by the $K-$method in real interpolation theory. We show that the two spaces do not always coincide and give some sufficient conditions on the open sets for this to happen. We also highlight some unnatural behaviors of the interpolation space. The treatment is as self-contained as possible.
We apply the metrical approach to Sobolev spaces, which arise in various evolution PDEs. Functions from those spaces are defined on an interval and take values in a family of Banach spaces. In this case we adapt the definition of Newtonian spaces. For a monotone family, we show the existence of weak derivative, obtain an isomorphism to the standard Sobolev space, and provide some scalar characteristics.
In this paper we prove structural and topological characterizations of the screened Sobolev spaces with screening functions bounded below and above by positive constants. We generalize a method of interpolation to the case of seminormed spaces. This method, which we call the truncated method, generates the screened Sobolev subfamily and a more general screened Besov scale. We then prove that the screened Besov spaces are equivalent to the sum of a Lebesgue space and a homogeneous Sobolev space and provide a Littlewood-Paley frequency space characterization.
The continouity and compactness of embedding operators in in Sobolev-Lions type spaces are derived. By applying this result separability properties of degenerate anisotropic differential operator equations, well-posedeness and Strichartz type estimates for solution of corresponding parabolic problem are established
We obtain some nonlocal characterizations for a class of variable exponent Sobolev spaces arising in nonlinear elasticity theory and in the theory of electrorheological fluids. We also get a singular limit formula extending Nguyen results to the anisotropic case.
The present paper is devoted to a theory of profile decomposition for bounded sequences in \emph{homogeneous} Sobolev spaces, and it enables us to analyze the lack of compactness of bounded sequences. For every bounded sequence in homogeneous Sobolev spaces, the sequence is asymptotically decomposed into the sum of profiles with dilations and translations and a double suffixed residual term. One gets an energy decomposition in the homogeneous Sobolev norm. The residual term becomes arbitrarily small in the critical Lebesgue or Sobolev spaces of lower order, and then, the results of decompositi…
In this paper, we obtain an isometry between the Fock-Sobolev space and the Gauss-Sobolev space. As an application, we use multipliers on the Gauss-Sobolev space to characterize the boundedness of an integral operator on the Fock-Sobolev space.
We study the boundary traces of Newton-Sobolev, Hajlasz-Sobolev, and BV (bounded variation) functions. Assuming less regularity of the domain than is usually done in the literature, we show that all of these function classes achieve the same "boundary values", which in particular implies that the trace spaces coincide provided that they exist. Many of our results seem to be new even in Euclidean spaces but we work in a more general complete metric space equipped with a doubling measure and supporting a Poincare inequality.
We obtain symmetrization inequalities in the context of Fractional Hajlasz-Sobolev spaces in the setting of rearrangement invariant spaces and prove that for a large class of measures our symmetrization inequalities are equivalent to the lower bound of the measure.
We generalize in this article the classical Sobolev's and Sobolev's trace inequalities on the Grand Lebesgue Spaces under monomial weight instead the classical Lebesgue or grand Lebesgue Spaces. We will distinguish the classical Sobolev's inequality and the so-called trace Sobolev's inequality.
In this article, the authors establish a new characterization of the Musielak--Orlicz--Sobolev space on $\mathbb{R}^n$, which includes the classical Orlicz--Sobolev space, the weighted Sobolev space and the variable exponent Sobolev space as special cases, in terms of sharp ball averaging functions. Even in a special case, namely, the variable exponent Sobolev space, the obtained result in this article improves the corresponding result obtained by P. Hästö and A. M. Ribeiro [Commun. Contemp. Math. 19 (2017), 1650022, 13 pp] via weakening the assumption $f\in L^1(\mathbb R^n)$ into $f\in L_{\ma…
We introduce Herz-Sobolev spaces, which unify and generalize the classical Sobolev spaces. We will give a proof of the Sobolev-type embedding for these function spaces. All these results generalize the classical results on Sobolev spaces. Some remarks on Caffarelli--Kohn--Nirenberg inequality are given.
We define Euler-Hilbert-Sobolev spaces and obtain embedding results on homogeneous groups using Euler operators, which are homogeneous differential operators of order zero. Sharp remainder terms of $L^{p}$ and weighted Sobolev type and Sobolev-Rellich inequalities on homogeneous groups are given. Most inequalities are obtained with best constants. As consequences, we obtain analogues of the generalised classical Sobolev type and Sobolev-Rellich inequalities. We also discuss applications of logarithmic Hardy inequalities to Sobolev-Lorentz-Zygmund spaces. The obtained results are new already in…
We extend in this article the classical Sobolev inequalities for the module of continuity for the functions belonging to the integer order Sobolev's space on the Sobolev-Bilateral Grand Lebesgue spaces. As a consequence, we deduce the fractional Orlicz-Sobolev imbedding theorems and investigate the rectangle module of continuity of non-Gaussian multiparameter random fields.
Our goal in this article is to construct HK-Sobolev spaces on $\R^\infty$ which contains Sobolev spaces as dense embedding. We discuss that the sequence of weak solution of Sobolev spaces are convergence strongly in HK-Sobolev space. Also, we obtain that the Sobolev space through Bessel Potential is densely contained in HK-Sobolev spaces. Finally we find sufficient condition for the solvability of the divergence equation $\nabla.F= f,$ for $f$ is an element of the subspace $K{S^p}[\R_I^n]$ and $n \in \N$, in the SoboHK-Sobolev space $WS^{k,p}[\R_I^n] $ with the help of Fourier transformation.
It is known that for Banach valued functions there are several approaches to define a Sobolev class. We compare the usual definition via weak derivatives with the Reshetnyak-Sobolev space and with the Newtonian space; in particular, we provide sufficient conditions when all three agree. As well we revise the difference quotient criterion and the property of Lipschitz mapping to preserve Sobolev space when it acting as a superposition operator.
We investigate two types of characterizations for anisotropic Sobolev and BV spaces. In particular, we establish anisotropic versions of the Bourgain-Brezis-Mironescu formula, including the magnetic case both for Sobolev and BV functions.
In this paper we consider a generalized conditional-type Holder- inequality and investigate some classic properties of multiplication conditional expectation type operators on Orlicz-spaces.
We determine multiplication and convolution topological algebras for classes of $ω$-ultradifferentiable functions of Beurling type. Hypocontinuity and discontinuity of the multiplication and convolution mappings are also investigated.
Muthukumar and Ponnusamy \cite{MP-Tp-spaces} studied the multiplication operators on $\mathbb{T}_p$ spaces. In this article, we mainly consider multiplication operators between $\mathbb{T}_p$ and $\mathbb{T}_q$ ($p\neq q$). In particular, we characterize bounded and compact multiplication operators from $\mathbb{T}_{p}$ to $\mathbb{T}_{q}$. For $p\neq q$, we prove that there are no invertible multiplication operators from $\mathbb{T}_{p}$ to $\mathbb{T}_{q}$ and also there are no isometric multiplication operators from $\mathbb{T}_{p}$ to $\mathbb{T}_{q}$. Finally, we discuss about fixed point…
In this paper, we study the sharp Poincaré inequality and the Sobolev inequalities in the higher order Lorentz--Sobolev spaces in the hyperbolic spaces. These results generalize the ones obtained in \cite{Nguyen2020a} to the higher order derivatives and seem to be new in the context of the Lorentz--Sobolev spaces defined in the hyperbolic spaces.
In this paper, we study the matrix multiplication operators on Banach function spaces and discuss their applications in semigroups for solving the abstract Cauchy problem.
We study in this short preprint the theory of trigonometric approximation in the so-called Banach functional rearrangement invariant Sobolev-Grand Lebesgue Spaces.
In this paper, Lambert multipliers acting between Orlicz spaces are characterized based on conditional expectation operators. Also, we give necessary and sufficient conditions to *-multiplication operators to be closed range. Finally, the necessary condition under which a *-multiplication operator to be Fredholm operator is investigated.
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 establish characterizations on the Fourier transform of functions of fractional Sobolev spaces, thereby giving another equivalent definition.
We list multiplier and convolutor spaces of the spaces occurring in L. Schwartz' "Théorie des distributions". Furthermore we clarify whether the multiplications and convolutions are continuous or not.
Let $d$ be a metric on $R^n$ and let $C^{m,(d)}(R^n)$ be the space of $C^m$-function on $R^n$ whose partial derivatives of order $m$ belong to the space $Lip(R^n;d)$. We show that the homogeneous Sobolev space $L^{m+1}_p(R^n),p>n,$ can be represented as a union of $C^{m,(d)}(R^n)$-spaces where $d$ belongs to a family of metrics on $R^n$ with certain "nice" properties. This enables us in several important cases to give intrinsic characterizations of the restrictions of Sobolev spaces to arbitrary closed subsets of $R^n$. In particular, we generalize the classical Whitney extension theorem for t…
We deal with Orlicz-Sobolev embeddings in open subsets of $\mathbb{R}^n$. A necessary and sufficient condition is established for the existence of an optimal, i.e. largest possible, Orlicz-Sobolev space continuously embedded into a given Orlicz space. Moreover, the optimal Orlicz-Sobolev space is exhibited whenever it exists. Parallel questions are addressed for Orlicz-Sobolev embeddings into Orlicz spaces with respect to a Frostman measure, and, in particular, for trace embeddings on the boundary.
By a stronger compact boundary embedding theorem in Musielak-Orlicz-Sobolev space developed in the paper, variational method is employed to deal with the nonlinear elliptic equation with the nonlinear Neumann boundary condition in the framework of Musielak-Orlicz-Sobolev space.
In this paper, first we investigate closed range multiplication conditional type operators between two Lp-spaces. Then we characterize Fredholm ones when the underlying measure space is non-atomic. Finally we give some examples.
