Boundedness Of Operators Research Articles

On the $$R$$ R -boundedness of stochastic convolution operators

The \(R\)-boundedness of certain families of vector-valued stochastic convolution operators with scalar-valued square integrable kernels is the key ingredient in the recent proof of stochastic maximal \(L^p\)-regularity, \(2<p<\infty \), for certain classes of sectorial operators acting on spaces \(X=L^q(\mu )\), \(2\le q<\infty \). This paper presents a systematic study of \(R\)-boundedness of such families. Our main result generalises the afore-mentioned \(R\)-boundedness result to a larger class of Banach lattices \(X\) and relates it to the \(\ell ^{1}\)-boundedness of an associated class of deterministic convolution operators. We also establish an intimate relationship between the \(\ell ^{1}\)-boundedness of these operators and the boundedness of the \(X\)-valued maximal function. This analysis leads, quite surprisingly, to an example showing that \(R\)-boundedness of stochastic convolution operators fails in certain UMD Banach lattices with type \(2\).

Weighted boundedness of multilinear operators associated to singular integral operators with non-smooth kernels

In this paper, we establish the weighted sharp maximal function inequalities for a multilinear operator associated to a singular integral operator with non-smooth kernel. As an application, we obtain the boundedness of the operator on weighted Lebesgue and Morrey spaces. MSC:42B20, 42B25.

Weighted boundedness of a multilinear operator associated to a singular integral operator with general kernels

In this paper, we establish the weighted sharp maximal function inequalities for a multilinear operator associated to a singular integral operator with general kernels. As an application, we obtain the boundedness of the operator on weighted Lebesgue and Morrey spaces. MSC:42B20, 42B25.

Study of the boundedness of one-sided operators and the well-posedness of KdV type dispersion equations

Many important issues in Mathematics and Physics can often attribute to the boundedness of operators on some function spaces. The study for the boundedness of singular integral is one of the important components in Harmonic analysis. A variety of methods and techniques in Harmonic analysis have been widely used in the study of partial differential equations. By the space-time estimates for oscillatory type integral operators on Lebesgue spaces or Morrey type spaces and semigroup theory, one can obtain the well-posedness for the Cauchy problem of nonlinear dispersion equations in the low order Sobolev spaces. We give the definitions of one kind of one-sided oscillatory singular integral operators and study some classical boundedness of these operators. Inspired by the classical results for the characterizations via commutators, we introduce the idea of characterizations via commutators on Morrey spaces and obtain characterizations via commutators with symbol belonging to two kinds of function spaces on these spaces. The results obtained above offer some new tools to the study of partial differential equations. Finally, combining energy method and Number theory, we establish the well-posedness of the Cauchy problem for a class of KdV type dispersion equations.

The boundedness of multilinear operators on generalized Morrey spaces over the quasi-metric space of non-homogeneous type

In this paper we study the multilinear fractional integral operators, the multilinear Calderon-Zygmund operators and the multi-sublinear maximal operators defined on the quasi-metric space with non-doubling measure. We obtain the boundedness of these operators on the generalized Morrey spaces over the quasi-metric space of non-homogeneous type.

Boundedness criterion of Journéʼs class of singular integrals on multiparameter Hardy spaces

This article concerns nonconvolutional type operators (also known as Journéʼs type operators) associated with a multiparameter family of dilations given by (x1,x2,…,xm)→(δ1x1,δ2x2,…,δmxm) where x1∈Rn1,x2∈Rn2,…,xm∈Rnm and m⩾3. We are especially interested in the boundedness of such operators on the multiparameter Hardy spaces. This work is motivated by Pipherʼs result on the boundedness of these operators from the multiparameter Hp spaces to Lp spaces for 0<p⩽1, and Journéʼs counter-example which shows that the number of parameter plays a crucial role in boundedness of singular integral operators on multiparameter Hardy spaces. Journéʼs work shows that there is a sharp difference between the situations for two and three or more parameters. We establish in this paper the necessary and sufficient conditions under which the singular integral operators in Journéʼs class are bounded on multiparameter Hp spaces (0<p⩽1) with arbitrary number of parameters.

Regularizations of general singular integral operators

In the theory of singular integral operators significant effort is often required to rigorously define such an operator. This is due to the fact that the kernels of such operators are not locally integrable on the diagonal s=t , so the integral formally defining the operator T or its bilinear form \langle Tf, g \rangle is not well defined (the integrand in not in L^1 ) even for “nice” f and g . However, since the kernel only has singularities on the “diagonal” s=t , the bilinear form \langle Tf, g \rangle is well defined, say, for bounded compactly supported functions with separated supports. One of the standard ways to interpret the boundedness of a singular integral operators is to consider the regularized kernel [ K_\varepsilon(s, t) = K(s, t), m((s-t)/\varepsilon), ] where the cut-off function m is 0 in a neighborhood of the origin, so the integral operators T_\varepsilon with kernel K_\varepsilon are well defined (at least on a dense set). Then instead of asking about the boundedness of the operator T , which is not well defined, one can ask about uniform boundedness (in \varepsilon ) of the regularized operators T_\varepsilon . For the standard regularizations one usually considers truncated operators T_\varepsilon with m(s) =\mathbf{1}_{[1, \infty)} (|s|) , although smooth cut-off functions were also considered in the literature. The main result of the paper is that for a wide class of singular integral operators (including the classical Calderón–Zygmund operators in nonhomogeneous two weight settings), the so called restricted L^p boundedness, i.e., the uniform estimate [ |\langle Tf, g \rangle| \le C ,|f| p ,|g| {p'} ] for bounded compactly supported f and g with separated supports implies the uniform L^p -boundedness of regularized operators T_\varepsilon for any reasonable choice of smooth cut-off function m . For example, any m \in C^\infty(\mathbb{R}^N) }, m\equiv 0 in a neighborhood of 0 , and such that 1-m is compactly supported would work. If the kernel K satisfies some additional assumptions (which are satisfied for classical singular integral operators like the Hilbert transform, Cauchy transform, Ahlfors–Beurling transform, and generalized Riesz transforms), then the restricted L^p boundedness also implies the uniform L^p boundedness of the classical truncated operators T_\varepsilon ( m(s) =\mathbf{1}_{[1, \infty)} (|s|) ).

Boundedness of operators on Hardy spaces

The boundedness of a linear or sublinear operator from the Hardy space Hp(ℝd) to a quasi-Banach space is investigated.

The boundedness and the fredholm property of integral operators with anisotropically homogeneous kernels of compact type and variable coefficients

In the space Lp(ℝn), 1 < p < ∞, we study a new wide class of integral operators with anisotropically homogeneous kernels. We obtain sufficient conditions for the boundedness of operators from this class. We consider the Banach algebra generated by operators with anisotropically homogeneous kernels of compact type and multiplicatively slowly oscillating coefficients. We establish a relationship between this algebra and multidimensional convolution operators, and construct a symbolic calculus for it. We also obtain necessary and sufficient conditions for the Fredholm property of operators from this algebra.

Atomic Decomposition and Boundedness of Operators on Weighted Hardy Spaces

AbstractIn this article, we establish a new atomic decomposition for , where the decomposition converges in -norm rather than in the distribution sense. As applications of this decomposition, assuming that T is a linear operator bounded on and 0 &lt; p ≤ 1, we obtain (i) if T is uniformly bounded in -norm for all w-p-atoms, then T can be extended to be bounded from to ; (ii) if T is uniformly bounded in -norm for all w-p-atoms, then T can be extended to be bounded on ; (iii) if T is bounded on , then T can be extended to be bounded from to .

Global Quantization of Pseudo-Differential Operators on Compact Lie Groups, SU(2), 3-sphere, and Homogeneous Spaces

Global quantization of pseudo-differential operators on compact Lie groups is introduced relying on the representation theory of the group rather than on expressions in local coordinates. Operators on the 3-dimensional sphere and on group SU(2) are analysed in detail. A new class of globally defined symbols is introduced giving rise to the usual Hormander's classes of operators $\Psi^m(G)$, $\Psi^m(S^3)$ and $\Psi^m(SU(2))$. Properties of the new class and symbolic calculus are analysed. Properties of symbols as well as $L^2$-boundedness and Sobolev $L^2$--boundedness of operators in this global quantization are established on general compact Lie groups.

Integral operators on B σ -Morrey-Campanato spaces

We show the boundedness of the Hardy-Littlewood maximal operator, singular and fractional integral operators, and more general sublinear operators on Bσ-Morrey-Campanato spaces. These function spaces have been introduced recently to unify central Morrey spaces, λ-central mean oscillation spaces and usual Morrey-Campanato spaces. Using the Bσ-Morrey-Campanato spaces, we can study both local and global regularities of functions simultaneously, and unify a series of results on the boundedness of operators on several classical function spaces.

Bounded operators on the weighted spaces of holomorphic functions on the polydiscs

Assuming that S is the space of functions of regular variation and ω = (ω1, …, ω n ), ω j ∈ S, by A p (ω) we denote the class of all holomorphic functions defined on the polydisc U n such that where dm 2n (z) is the 2n-dimensional Lebesgue measure on U n . In this article, we characterize the multipliers of A p (ω) (0 < p ≤ 1) and study the boundedness of Toeplitz operators on this space. As an application, a theorem about the division by good-inner functions in A p (ω) is proved. We also consider the generalized little Hankel and Berezin-type operators on A p (ω) (and on L 1(ω)) and prove some theorems about the boundedness of these operators.

Molecules in coorbit spaces and boundedness of operators

We study the notion of molecules in coorbit spaces. The main result states that if an operator, originally defined on an appropriate space of test functions, maps atoms to molecules, then it can be extended to a bounded operator on coorbit spaces. For tim

On fractional maximal function and fractional integrals associated with the Dunkl operator on the real line

In this paper we obtain necessary and sufficient conditions on the parameters for the boundedness of the Dunkl-type fractional maximal operator M β , and the Dunkl-type fractional integral operator I β from the spaces L p , α ( R ) to the spaces L q , α ( R ) , 1 < p < q < ∞ , and from the spaces L 1 , α ( R ) to the weak spaces WL q , α ( R ) , 1 < q < ∞ . In the case p = 2 α + 2 β , we prove that the operator M β is bounded from the space L p , α ( R ) to the space L ∞ , α ( R ) , and the Dunkl-type modified fractional integral operator I ˜ β is bounded from the space L p , α ( R ) to the Dunkl-type BMO space BMO α ( R ) . By this results we get boundedness of the operators M β and I β from the Dunkl-type Besov spaces B p θ , α s ( R ) to the spaces B q θ , α s ( R ) , 1 < p < q < ∞ , 1 / p − 1 / q = β / ( 2 α + 2 ) , 1 ⩽ θ ⩽ ∞ and 0 < s < 1 .

Singular operators and Fourier multipliers in weighted Lebesgue spaces with variable index

Mikhlin’s ideas and results related to the theory of spaces Lρp(·) with nonstandard growth are developed. These spaces are called Lebesgue spaces with variable index; they are used in mechanics, the theory of differential equations, and variational problems. The boundedness of Fourier multipliers and singular operators on the spaces Lρp(·) are considered. All theorems are derived from an extrapolation theorem due to Rubio de Francia. The considerations essentially use theorems on the boundedness of operators and maximal Hardy-Littlewood functions on Lebesgue spaces with constant index.

On the $H^1$--$L^1$ boundedness of operators

We prove that if $q$ is in $(1,\infty )$, $Y$ is a Banach space, and $T$ is a linear operator defined on the space of finite linear combinations of $(1,q)$-atoms in $\mathbb {R}^n$ with the property that \[ \sup \left \{ \Vert Ta \Vert Y: \text {$a$ is a $(1,q)$-atom} \right \} < \infty , \] then $T$ admits a (unique) continuous extension to a bounded linear operator from $H^1({\mathbb {R}^n})$ to $Y$. We show that the same is true if we replace $(1,q)$-atoms by continuous $(1,\infty )$-atoms. This is known to be false for $(1,\infty )$-atoms.

The exact solution of a linear integral equation with weakly singular kernel

A space W 2 1 [ a , b ] , which is proved to be a reproducing kernel space with simple reproducing kernel, is defined. The expression of its reproducing kernel function is given. Subsequently, a class of linear Volterra integral equation (VIE) with weakly singular kernel is discussed in the new reproducing kernel space. The reproducing kernel method of linear operator equation A u = f , which request the image space of operator A is W 2 1 [ a , b ] and operator A is bounded, is improved. Namely, the request for the image space is weakened to be L 2 [ a , b ] , and the boundedness of operator A is also not required. As a result, the exact solution of the equation is obtained. The numerical experiments show the efficiency of our method.

Interpolation of operators of weak type (ϕ, ϕ)

Considering the measurable and nonnegative functions ϕ on the half-axis [0, ∞) such that ϕ(0) = 0 and ϕ(t) → ∞ as t → ∞, we study the operators of weak type (ϕ, ϕ) that map the classes of ϕ-Lebesgue integrable functions to the space of Lebesgue measurable real functions on ℝn. We prove interpolation theorems for the subadditive operators of weak type (ϕ0, ϕ0) bounded in L ∞(ℝn) and subadditive operators of weak types (ϕ0, ϕ0) and (ϕ1, ϕ1) in L ϕ(ℝ n ) under some assumptions on the nonnegative and increasing functions ϕ(x) on [0, ∞). We also obtain some interpolation theorems for the linear operators of weak type (ϕ0, ϕ0) bounded from L ∞(ℝn) to BMO(ℝ n). For the restrictions of these operators to the set of characteristic functions of Lebesgue measurable sets, we establish some estimates for rearrangements of moduli of their values; deriving a consequence, we obtain a theorem on the boundedness of operators in rearrangement-invariant spaces.

Sobolev spaces associated to the harmonic oscillator

We define the Hermite-Sobolev spaces naturally associated to the harmonic oscillatorH = −δ+|x|2. Structural properties, relations with the classical Sobolev spaces, boundedness of operators and almost everywhere convergence of solutions of the Schrodinger equation are also considered.