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 functions in $W_0$ where the denominator is non-vanishing. This is proved using a contructive argument.
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…
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…
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…
While the symbol map for the collection of bounded Toeplitz operators is well studied, there has been little work on a symbol map for densely defined Toeplitz operators. In this work a family of candidate symbols, the Sarason Sub-Symbols, is introduced as a means of reproducing the symbol of a densely defined Toeplitz operator. This leads to a partial answer to a question posed by Donald Sarason in 2008. In the bounded case the Toeplitzness of an operator can be classified in terms of its Sarason Sub-Symbols. This justifies the investigation into the application of the Sarason Sub-Symbols on d…
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.
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.
We examine densely defined (but possibly unbounded) multiplication operators in Hilbert function spaces possessing a complete Nevanlinna-Pick (CNP) kernel. For such a densely defined operator $T$, the domains of $T$ and $T^*$ are reproducing kernel Hilbert spaces contractively contained in the ambient space. We study several aspects of these spaces, especially the domain of $T^*$, which can be viewed as analogs of the classical deBranges-Rovnyak spaces in the unit disk.
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 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 $u$.
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. 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.
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 a…
We discuss the construction of the full Sobolev (Hölder) scale for non-densely defined operators on a Banach space with rays of minimal growth. In particular, we give a construction for extrapolation- and Favard spaces of generators of (bi-continuous) semigroups, or which is essentially the same, Hille-Yosida operators on Saks spaces.
Recently it has been shown that any regular simple symmetric operator with deficiency indices (1,1) is unitarily equivalent to the operator of multiplication in a reproducing kernel Hilbert space of functions on the real line with the Kramer sampling property. This work has been motivated, in part, by potential applications to signal processing and mathematical physics. In this paper we exploit well-known results about de Branges-Rovnyak spaces and characteristic functions of symmetric operators to prove that any such a symmetric operator is in fact unitarily equivalent to multiplication by th…
In this paper we investigate the reproducing kernel Hilbert space where the polylogarithm appears as kernel functions. This investigation begins with the properties of functions in this space, and here a connection to the classical Hardy space is shown through the Bose-Einstein integral equation. Next we consider function theoretic operators over the polylogarithmic Hardy space, such as multiplication and Toeplitz operators. It is determined that there are only trivial densely defined multiplication operators (and therefore only trivial bounded multipliers) over this space, which makes this sp…
In this paper we consider unbounded weighted conditional type operators on the space Lp, we give some conditions under which they are densely defined and we obtain a dense subset of the domain. Also, we get that a WCT operator is continuous if and only of it is every where defined. A description of polar decomposition, spectrum and spectral radius in this context are provided. Finally, we investigate hyperexpansive WCT operators on the Hilbert space L2. As a consequence hyperexpansive multiplication operators are investigated.
Along the line of singular value estimates for commutators by Rochberg-Semmes, Lord-McDonald-Sukochev-Zanin and Fan-Lacey-Li, we establish the endpoint weak Schatten class estimate for commutators of Riesz transforms with multiplication operator $M_f$ on Heisenberg groups via homogeneous Sobolev norm of the symbol $f$. The new tool we exploit is the construction of a singular trace formula on Heisenberg groups, which, together with the use of double operator integrals, allows us to bypass the use of Fourier analysis and provides a solid foundation to investigate the singular values estimates f…
